General Chemistry I

Chemistry is the study of matter and the changes it undergoes. Because it sits between physics on one side and biology on the other, chemistry is sometimes called the central science: it borrows the mathematics and energy principles of physics and uses them to explain the molecules that make biology possible. Matter is anything that has mass and takes up space, from the air in your lungs to the screen you are reading to the distant clouds of gas between the stars. Almost everything you will ever study in this course is an attempt to answer two deceptively simple questions about a sample of matter: what is it made of, and how does it change? To make sense of the enormous variety of matter, chemists first sort it into a few clear categories, because the category tells you which rules apply. A carpenter who cannot tell oak from pine will make poor furniture, and a chemist who cannot tell an element from a compound from a mixture will make poor predictions. This first lesson builds that classification vocabulary, and it also introduces the habit of thinking on two levels at once: the macroscopic level of what you can see, weigh, and pour, and the particulate level of the atoms and molecules that explain what you see. Learning to move fluidly between these two levels is, in a sense, the whole art of chemistry.

Matter, mass, and the particulate view

Everything made of matter is built from unimaginably small particles called atoms, and often from groups of atoms bonded together called molecules. A single copper penny contains on the order of 10²² atoms, a number so large that if you could count one atom per second it would take longer than the age of the universe many times over. You never see individual atoms with your eyes, yet their collective behavior produces every property you do observe. When water feels wet, boils at a certain temperature, or dissolves sugar, those macroscopic facts are the visible summary of countless invisible particles jostling, attracting, and rearranging. Throughout this course, whenever you meet a new observation, get in the habit of asking what the particles must be doing to produce it. That single question turns memorization into understanding.

Two words that sound alike but mean different things are mass and weight. Mass is the amount of matter in an object, and it does not change with location: a bag of flour has the same mass on Earth, on the Moon, or drifting in deep space. Weight is the force that gravity exerts on that mass, and it does change with location, because gravity is weaker on the Moon than on Earth. On a bathroom scale you technically read weight, but because everyday scales are calibrated for Earth's gravity, we casually treat the reading as mass. In careful scientific work the distinction matters, and we will keep the two ideas separate.

Pure substances versus mixtures

The first and most important fork in classifying matter is whether a sample is a pure substance or a mixture. A pure substance has a fixed composition that does not vary from sample to sample, and it has a definite, reproducible set of properties: a fixed melting point, a fixed boiling point, a fixed density. Pure water always freezes at 0 degrees Celsius and boils at 100 degrees Celsius at sea level, no matter where the water came from. Pure substances come in two kinds, elements and compounds.

An element cannot be broken down into simpler substances by chemical means. There are about 118 known elements, of which roughly 90 occur naturally on Earth and the rest have been made in laboratories and particle accelerators. Each element has its own one- or two-letter symbol, such as oxygen (O), hydrogen (H), carbon (C), iron (Fe), or gold (Au). Some symbols come from Latin names, which is why iron is Fe (from ferrum), sodium is Na (from natrium), and gold is Au (from aurum). An element is the simplest form of matter that still retains a chemical identity; every atom in a sample of pure gold is a gold atom, and no ordinary chemical process can turn it into anything simpler.

A compound is two or more elements chemically bonded in a fixed, whole-number ratio, like water (H₂O), table salt (NaCl), or carbon dioxide (CO₂). The phrase "fixed, whole-number ratio" is doing a lot of work here. Every water molecule anywhere in the universe contains exactly two hydrogen atoms for every one oxygen atom, never 2.1 or 1.9. This constancy is called the law of definite proportions: a given compound always contains the same elements in the same proportion by mass. Water is always about 11 percent hydrogen and 89 percent oxygen by mass, whether it comes from a glacier, a raincloud, or your kitchen tap.

The most striking fact about compounds is that a compound has properties completely different from the elements inside it. Sodium is a soft, silvery metal so reactive that it bursts into flame when dropped into water, and chlorine is a poisonous, choking green gas once used as a chemical weapon. Yet when these two dangerous elements bond chemically, they make sodium chloride, an edible white solid you sprinkle on food and could not live without. This dramatic transformation is a signature of chemical bonding, and it is worth pausing on: the properties of a compound are not the average of its elements, they are something entirely new. The reason, which later lessons develop fully, is that bonding rearranges the electrons of the atoms, and it is the electrons that govern nearly all chemical behavior.

A mixture is a physical blend of two or more substances that keep their own identities and can be present in any ratio. This last point is the sharp contrast with compounds: while water is locked at a 2-to-1 ratio, you can make salt water with a pinch of salt, a spoonful, or enough to saturate the solution, and it is a valid salt-water mixture at every ratio. Because the components of a mixture keep their own chemical identities and are not bonded to one another, their individual properties are retained. Iron filings mixed with sand still respond to a magnet, and salt dissolved in water still tastes salty and can be recovered unchanged.

Mixtures divide further by how uniform they are. A homogeneous mixture, also called a solution, is uniform throughout, so any sample you scoop out looks and behaves the same as any other. Salt water, clean air, brass, and 14-karat gold are all homogeneous mixtures; the components are so evenly and finely dispersed that no boundaries between them are visible even under a microscope. A heterogeneous mixture has visibly different regions, so different scoops can have different compositions. Sand in water, a bowl of cereal in milk, granite, and oil-and-vinegar salad dressing are heterogeneous: you can point to the distinct parts. A single glass of chocolate milk that has been stirred well is roughly homogeneous, while the same glass with syrup pooled at the bottom is heterogeneous, which shows that the classification depends on the actual state of the sample, not just its ingredients.

Separating mixtures by physical methods

Because the parts of a mixture are not chemically bonded, they can be pulled apart by physical methods that exploit a difference in some physical property. No new substances are made in any of these separations, which is exactly what tells you a mixture, not a compound, was involved. The main techniques, each keyed to a specific property, are worth knowing by name.

• Filtration separates by particle size, letting a liquid pass through a porous barrier while trapping a suspended solid. Pouring sandy water through filter paper leaves the sand behind and lets clear water through.
• Evaporation removes a volatile liquid by turning it to vapor, leaving a dissolved solid behind. Boiling away salt water in a shallow pan recovers the solid salt.
• Distillation separates by boiling point, vaporizing a liquid and then condensing the vapor back into a separate container. It can purify water from salt water and separate two liquids that boil at different temperatures.
• Magnetic separation exploits magnetism, using a magnet to pull iron or other magnetic material out of a nonmagnetic mixture such as sand.
• Chromatography separates by how strongly different components cling to a stationary material as a moving solvent carries them along, which is how the many pigments in a drop of ink can be spread into a rainbow of separate bands.

Notice that every one of these methods targets a physical difference, size, boiling point, magnetism, or affinity for a surface, and none of them breaks any chemical bonds. That is precisely why a mixture can be undone this way while a compound cannot: splitting water into hydrogen and oxygen requires breaking chemical bonds and cannot be done by filtering or boiling.

The states of matter

Matter usually appears in one of three familiar states (also called phases): solid, liquid, and gas. The difference between them comes down to two particulate factors, how tightly the particles are held together and how much they move. This is the essence of the kinetic molecular view of matter, which you will meet again in force when you study gases.

A solid holds a fixed shape and a fixed volume because its particles are locked in place, close together, held by strong attractions, and able only to vibrate in place rather than move past one another. That is why a steel bar keeps its shape and a diamond is hard. A liquid keeps a fixed volume but takes the shape of its container, because its particles are still close together and strongly attracting, but now have enough energy to slide past one another while staying in contact. Water poured from a bottle into a glass keeps the same volume but adopts the glass's shape. A gas has neither fixed shape nor fixed volume; its particles are far apart, moving fast, and barely attracting one another, so they fly apart to fill whatever space they are given. A small amount of gas released in a room quickly spreads to fill it. A useful mental picture is a crowd of people: solids are people packed shoulder to shoulder standing still, liquids are the same crowd milling around and shuffling past each other, and gases are a few people sprinting freely across an empty stadium.

State	Shape	Volume	Particle spacing	Particle motion
Solid	Fixed	Fixed	Very close, ordered	Vibrate in place
Liquid	Takes container's shape	Fixed	Close, disordered	Slide past one another
Gas	Fills container	Fills container	Far apart	Fast, mostly free flight

These three are the states you meet in everyday life, but they are not the only ones. A fourth state, plasma, forms when a gas becomes so hot that electrons are stripped from atoms, leaving a glowing mixture of charged particles; it is actually the most common state of ordinary matter in the universe, because stars are made of it. Neon signs and lightning are small terrestrial examples. For this course, though, solid, liquid, and gas cover almost everything.

Changes of state

Adding or removing heat drives changes between the states, because heat changes how fast the particles move and therefore whether their attractions can hold them in place. Warming a solid until its particles gain enough energy to break free of their fixed positions is melting (the reverse, cooling a liquid until its particles lock into place, is freezing). Warming a liquid until particles at the surface, and eventually throughout, gain enough energy to escape into the gas phase is vaporization, which we call evaporation when it happens gradually at the surface and boiling when it happens rapidly throughout the liquid; the reverse, a gas cooling into a liquid, is condensation. Some substances skip the liquid state entirely and go straight from solid to gas, a change called sublimation, which is what solid carbon dioxide (dry ice) does as it produces its eerie fog, and the reverse, gas directly to solid, is deposition, which is how frost forms on a cold window. Crucially, none of these are chemical changes. In every case the substance is the same before and after, and only its physical arrangement has changed. Steam, liquid water, and ice are all H₂O; the atoms are identical, only the spacing and motion of the molecules differ.

Properties: physical versus chemical

To describe and identify matter, chemists catalog its properties, and every property falls into one of two families. A physical property can be observed or measured without changing the substance's chemical identity. Color, odor, density, hardness, melting point, boiling point, electrical conductivity, and state at room temperature are all physical properties. You can measure the density of a gold ring without turning the gold into anything else, so density is physical. A chemical property describes how a substance reacts to form new substances. Flammability (the tendency to burn), the tendency of iron to rust, the reactivity of an acid with a metal, and whether a substance is toxic or corrosive are chemical properties. The telltale sign of a chemical property is that you often cannot observe it without actually running the reaction and thereby changing the substance. You cannot know that gasoline is flammable by looking at it; you learn it by igniting a sample, which destroys that sample in the process.

Properties also split by whether they depend on the amount of matter present. An extensive property, such as mass, volume, or total energy, scales with the size of the sample: a bucket of water has more mass and more volume than a cup of water. An intensive property, such as density, temperature, color, or melting point, does not depend on sample size: a drop of water and a swimming pool of water have the same density and the same boiling point. Intensive properties are especially valuable for identifying substances, because they let you test a small sample and draw a conclusion about any amount of the material. This idea returns in the next lesson when density takes center stage.

Changes: physical versus chemical

The same physical-versus-chemical split applies to changes as well as properties. In a physical change, the substance is the same before and after; only its form or appearance changes. Melting ice gives liquid water, still H₂O. Dissolving sugar in tea gives sweet water, but the sugar molecules are unchanged and could, in principle, be recovered by evaporating the water. Tearing paper, boiling water, crushing a can, and mixing sand with gravel are all physical changes, because in each case no new substance appears. Changes of state and the dissolving of one substance in another are the most common physical changes you will classify.

In a chemical change, also called a chemical reaction, the atoms are rearranged into new substances with new properties, because chemical bonds are broken and new ones are formed. Burning wood produces ash, carbon dioxide, and water vapor, none of which can easily be turned back into wood. Iron rusting, food digesting, a battery discharging, and bread baking are all chemical changes. Because chemistry is fundamentally the study of chemical change, learning to recognize when one has happened is a core skill. Several observable clues, while none is foolproof alone, together give strong evidence that a chemical change has occurred:

• A color change that is not simply mixing colors, as when shiny iron turns to reddish-brown rust.
• A gas produced, seen as bubbling or fizzing, when no boiling is involved, as when an antacid tablet fizzes in water.
• A precipitate, a new solid, forming when two clear liquids are mixed.
• Heat or light released or absorbed, as in the heat and glow of a flame or the chill of certain reactions.
• An odor produced where there was none, signaling new molecules.

The reason no single clue is decisive is that some physical changes mimic these signs: boiling water produces bubbles of gas (water vapor) without any chemical change, and mixing two colored solutions can change the color by simple blending. This is why chemists look for a combination of evidence, and ultimately confirm a chemical change by testing whether the products truly are new substances with new properties. A precipitate that will not redissolve when the water evaporates, or a gas that reignites a glowing splint, points firmly to new chemistry.

Worked example. Classify each of the following as a physical or chemical change, and justify it. (1) An ice sculpture melts in the sun. (2) A silver spoon tarnishes to a dark coating over months. (3) Salt is dissolved in water and then the water is boiled away, recovering the salt. (4) A firework explodes with light and a bang. The answers: (1) physical, because liquid water is still H₂O and could be refrozen; (2) chemical, because silver reacts with sulfur compounds in the air to form a new substance, silver sulfide, on the surface; (3) physical overall, because dissolving disperses the unchanged salt and evaporation returns it intact; (4) chemical, because the explosive compounds are transformed into hot gases and new solids, releasing energy as light and sound. Working through cases like these trains the instinct you will use constantly.

Common misconceptions previewed

Two confusions trip up nearly every beginner, so it helps to name them early. First, students often assume that if something disappears, a chemical change has occurred; the classic case is dissolving salt or sugar, which looks like a vanishing act but is purely physical, since the solute is merely dispersed and fully recoverable. Second, students assume that the bubbles of boiling water are some new gas being generated chemically, when in fact they are just water vapor, the same H₂O in gaseous form. Keeping the particulate picture in mind, asking whether the actual molecules have been rebuilt into different molecules, resolves both.

Recap

In this lesson you learned that chemistry studies matter and its changes, and that reasoning well means shuttling between the macroscopic and particulate levels. You sorted matter into pure substances (elements and compounds) and mixtures (homogeneous and heterogeneous), and you saw that mixtures can be separated by physical methods precisely because their components are not chemically bonded. You reviewed the three common states of matter and the changes between them, all of which are physical, and you learned to distinguish physical from chemical properties and changes, along with the useful extensive-versus-intensive split. This classification vocabulary is the foundation for everything ahead. A balanced chemical equation, the goal of Module 5, is nothing more than a precise bookkeeping of a chemical change, and every quantitative tool you build in between exists to describe matter and track how it transforms.

Science runs on measurement, and measurement runs on agreed units. A number without a unit is almost meaningless in chemistry: the bare number 5 could be 5 grams, 5 liters, 5 kelvin, or 5 moles, and each tells a wildly different story. Reporting a measurement means reporting three things together, a number, a unit, and (as the next lesson shows) an honest indication of precision. This lesson establishes the units chemists agree to share, the elegant prefix system that scales those units across many orders of magnitude, the two temperature scales you must be able to convert between, and one of the most useful derived quantities in the whole subject, density. Mastering these now is not busywork; measurement is the language every later calculation is written in, and errors here propagate into every result downstream.

The SI system and its base units

Chemists worldwide use the SI system (from the French Systeme International d'Unites, the modern form of the metric system). Its power comes from being coherent and universal: scientists in every country report the same quantity in the same unit, so results can be compared and combined without confusion. SI is built on a small set of base units, each for one fundamental kind of quantity, and everything else is derived from them.

Quantity	Base unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K
Amount of substance	mole	mol
Electric current	ampere	A
Luminous intensity	candela	cd

For general chemistry, five of these dominate: the meter (m) for length, the kilogram (kg) for mass, the second (s) for time, the kelvin (K) for temperature, and the mole (mol) for amount of substance, which is the star of Module 4. Notice a quirk worth remembering: the base unit of mass is the kilogram, not the gram, even though the kilogram carries a prefix. That is a historical accident, and in practice everyday chemistry leans heavily on the gram (g) for mass and the liter (L) for volume, because these are conveniently sized for lab-scale work. The liter itself is a derived unit; it equals one cubic decimeter, a cube ten centimeters on each side.

Some quantities you use constantly are derived units, built by combining base units. Volume is length cubed, so its SI unit is the cubic meter, though the liter and milliliter are far more practical in the lab. Speed is length divided by time (meters per second). Density, the subject later in this lesson, is mass divided by volume. Energy, which appears in the thermochemistry lesson, has the derived unit called the joule. The habit of tracking which base units make up a derived unit will pay off enormously when you use dimensional analysis in the next lesson.

Prefixes scale the unit by powers of ten

The great strength of the metric system is that prefixes multiply a base unit by a power of ten, which makes very large and very small quantities manageable without switching to a whole new unit. This is a genuine advantage over customary units, where you must remember that 12 inches make a foot, 3 feet make a yard, and 1760 yards make a mile, each an arbitrary conversion. In SI, everything is a factor of ten, and the prefix names are consistent across all quantities: a kilometer is 1000 meters exactly as a kilogram is 1000 grams.

Prefix	Symbol	Meaning	Example
giga	G	1,000,000,000 (109)	1 GB is a billion bytes
mega	M	1,000,000 (106)	1 MW is a million watts
kilo	k	1000 (103)	1 kg is 1000 g
(base)	--	1 (100)	1 m, 1 g, 1 L
deci	d	0.1 (10-1)	1 dm is 0.1 m
centi	c	0.01 (10-2)	1 cm is 0.01 m
milli	m	0.001 (10-3)	1 mL is 0.001 L
micro	µ	0.000001 (10-6)	1 µm is a millionth of a meter
nano	n	0.000000001 (10-9)	1 nm spans a few atoms

From this table, 1 kg is 1000 g, 1 cm is 0.01 m, and 1 mL is 0.001 L. Moving up the ladder makes the number smaller (2000 g is 2 kg); moving down makes it larger (2 L is 2000 mL). A handy fact that comes up constantly in the lab: 1 mL is exactly the same volume as 1 cubic centimeter (cm³), because a milliliter is defined as one thousandth of a liter and a liter is a cube 10 cm on a side. This equivalence lets you compare the densities of solids (often reported in g/cm³) and liquids (often reported in g/mL) directly, since the two volume units are identical. The atomic-scale distances chemists care about, such as the roughly 0.1 to 0.3 nanometer size of an atom or the length of a chemical bond, are naturally expressed in nanometers or picometers (10^-12 m), which is exactly why those tiny prefixes matter to us.

Worked example. A sample of DNA is measured to be 340 nm long. Express this in meters and in millimeters. In meters, 340 nm times 10^-9 m per nm equals 3.4 times 10^-7 m. In millimeters, since 1 mm is 10^-3 m, that is 3.4 times 10^-7 m divided by 10^-3 m per mm, giving 3.4 times 10^-4 mm. The point is that the same physical length can wear many different unit-costumes; the prefix simply chooses a convenient size of number.

Temperature and its scales

Temperature is a measure of the average kinetic energy of the particles in a sample, how vigorously they are moving. Chemists routinely use two scales, and it is essential to know when each applies. The Celsius scale is convenient for everyday temperatures and is defined around water: water freezes at 0 degrees Celsius and boils at 100 degrees Celsius at ordinary atmospheric pressure, a tidy 100-degree span. The Kelvin scale uses degrees of the same size as Celsius, but starts its zero at the coldest temperature that can possibly exist. To convert between them you simply shift by 273.15:

K = degrees C + 273.15 and, rearranged, degrees C = K - 273.15

So water freezes at 0 degrees Celsius, which is 273.15 K, and boils at 100 degrees Celsius, which is 373.15 K. A key point students often miss: because the degree sizes are identical and the scales differ only by an offset, a temperature change of, say, 10 degrees Celsius is exactly the same as a change of 10 K. It is only absolute readings that differ by 273.15.

Why bother with Kelvin at all? Because the Kelvin scale has no negative values: 0 K is absolute zero, about -273.15 degrees Celsius, the temperature at which particle motion reaches its theoretical minimum and can go no lower. This makes Kelvin the required scale whenever temperature appears in a physical law that involves proportionality. Doubling a Kelvin temperature genuinely doubles the associated thermal energy, whereas doubling a Celsius temperature (from 10 to 20 degrees, say) has no such clean meaning because the Celsius zero is arbitrary. This is precisely why the gas laws in Module 6 insist on kelvin: a gas at 200 K really does exert twice the pressure, all else equal, as the same gas at 100 K, but no such statement holds for Celsius. Get in the habit now of converting to kelvin before doing any gas-law arithmetic.

For reference, a third scale, Fahrenheit, is still used in the United States for weather and cooking, with water freezing at 32 degrees F and boiling at 212 degrees F. It is not an SI scale and rarely appears in chemistry, but the conversion, degrees F equals 1.8 times degrees C plus 32, is worth recognizing.

Density: tying mass to volume

Density is one of the most useful derived quantities in chemistry. It is defined as mass divided by volume, and it answers the intuitive question, how much stuff is packed into a given space? A brick and a sponge of the same size have very different masses because they have very different densities. Density is usually reported in grams per milliliter (g/mL) for liquids or grams per cubic centimeter (g/cm³) for solids, and because 1 mL equals 1 cm³, those two units are numerically interchangeable. For gases, which are far less dense, grams per liter is more practical.

density = mass ÷ volume, often written d = m / V

The formula rearranges two ways, and you will use all three forms constantly: to find density, divide mass by volume; to find mass, multiply density by volume (m = d times V); to find volume, divide mass by density (V = m / d). Learning to move fluidly among these three is a small but high-value skill.

Density is the property that decides whether an object floats or sinks in a fluid. An object less dense than the fluid floats; one more dense sinks. Ice (density about 0.92 g/cm³) floats on liquid water (1.00 g/cm³), which is why icebergs float and, more profoundly, why lakes freeze from the top down rather than the bottom up, allowing fish to survive winter. Oil floats on water because it is less dense; a lead sinker plunges because lead is far denser.

Substance	Density (g/cm3 or g/mL)
Air (at room conditions)	about 0.0012
Ethanol	0.789
Ice	0.92
Water (liquid)	1.00
Aluminum	2.70
Iron	7.87
Lead	11.3
Gold	19.3

Worked example 1. A metal block has a mass of 54.0 g and a volume of 20.0 cm³. Its density is 54.0 g divided by 20.0 cm³, which equals 2.70 g/cm³. Comparing to the table, that value matches aluminum, so the block is very likely aluminum. Because density is a fixed, intensive property of a pure substance, it serves as a fingerprint for identifying an unknown material, a technique with a famous pedigree, discussed below.

Worked example 2. Density can also work backward to find a mass. If a liquid has a density of 0.789 g/mL (this is ethanol), then 50.0 mL of it has a mass of 50.0 mL times 0.789 g/mL, which equals 39.5 g. Notice how the milliliters in the volume cancel against the milliliters in the density, leaving grams. This cancellation of units is a preview of the dimensional analysis you will master in the next lesson, and it is a reliable check that you set the calculation up correctly.

Worked example 3. A student needs 100.0 g of mercury for an experiment and wants to know what volume that is, given mercury's density of 13.6 g/mL. Rearranging, volume equals mass divided by density, so V equals 100.0 g divided by 13.6 g/mL, which is 7.35 mL. Only a small volume of mercury is needed because it is very dense. This is the kind of everyday calculation that lets a chemist measure out a substance by volume when weighing is inconvenient.

Worked example 4: an irregular solid. How do you find the volume of an oddly shaped object like a small statue, which has no simple formula? By water displacement. Suppose a metal figurine is dropped into a graduated cylinder holding 25.0 mL of water, and the water rises to 32.4 mL. The figurine displaced 32.4 minus 25.0, or 7.4 mL, so its volume is 7.4 cm³. If the figurine's mass is 71 g, its density is 71 g divided by 7.4 cm³, about 9.6 g/cm³, suggesting it is not solid gold (which would be 19.3) but perhaps a copper alloy. This displacement trick is the modern echo of a very old story.

Why density is an intensive property

Because density is a ratio of two measured quantities that both scale with sample size, the size dependence cancels: doubling the amount of a substance doubles both its mass and its volume, leaving the ratio unchanged. A single drop of pure water and an entire swimming pool of pure water both have a density near 1.00 g/mL. Properties that are independent of sample size in this way are called intensive (density, temperature, color, melting point), while properties that scale with the amount of matter, such as mass and volume themselves, are called extensive. The distinction is not a mere vocabulary exercise; it has real practical weight. Intensive properties can identify a substance regardless of how much you have, which is exactly why a jeweler can test a small shaving of metal and draw a conclusion about the whole bar, and why a chemist can measure the density of a milliliter of an unknown liquid and know something true about a barrel of it.

The famous story of Archimedes

The link between density and identification is captured in the legend of Archimedes, the ancient Greek thinker, who was asked to determine whether a king's crown was pure gold or had been adulterated with cheaper silver, without damaging it. Realizing in his bath that a submerged object displaces its own volume of water, he could measure the crown's volume by displacement, combine it with its known mass, and compute its density. Because gold is much denser than silver, a crown of the correct mass but too large a volume would betray the fraud through a density lower than pure gold's. Whether or not he truly ran naked through the streets shouting "Eureka!", the principle is sound and is the same one used in Worked example 4 above. Density, an intensive property computed from two extensive ones, remains a workhorse for identifying materials to this day.

Common misconceptions

Three errors recur. First, students conflate mass with weight; remember that mass (in kilograms or grams) measures the quantity of matter and is the same everywhere, while weight is the gravitational force on that mass and varies with location. Second, students try to convert Celsius to Kelvin by multiplying; the scales are offset, not scaled, so you add 273.15 rather than multiply. Third, students assume heavier objects always sink; floating depends on density, not mass, which is why a massive steel ship floats (its average density, counting the air-filled hull, is less than water) while a small steel bolt sinks.

Recap

You now know the SI base units that anchor chemical measurement, especially the meter, kilogram, second, kelvin, and mole, and you can scale any of them across many orders of magnitude using metric prefixes from giga down to nano. You can convert between Celsius and Kelvin by shifting 273.15, and you understand why physical laws demand the absolute Kelvin scale. Finally, you can compute density in all three of its rearranged forms, use it to identify substances and predict floating, find the volume of irregular solids by displacement, and explain why density is an intensive property. The unit fluency built here feeds directly into the next lesson, where significant figures and dimensional analysis turn these measurements into reliable calculations.

Every measurement carries uncertainty, because every physical instrument has a limit to how finely it can read. A digital balance might read to the nearest 0.01 g and no further; a graduated cylinder might let you estimate to the nearest 0.1 mL. Significant figures, often shortened to sig figs, are the standard way chemists honestly report how precise a value is. The idea is simple but consequential: the significant figures in a measurement are all the digits known with certainty plus one final digit that is estimated. Reporting more digits than your instrument justifies is a false claim of precision, a kind of lie about your data, and reporting fewer throws away real information you actually possess. A ruler marked in whole millimeters lets you confidently read the millimeter marks and estimate one more digit, a tenth of a millimeter, by judging where the object falls between marks. It does not let you claim a hundredth of a millimeter, and writing ten decimal places would be pretending your cheap ruler is a laboratory-grade instrument. This lesson teaches you to count significant figures, to carry them correctly through calculations, and to wield dimensional analysis, the single most useful problem-solving technique in the entire course.

Precision, accuracy, and where uncertainty comes from

Before the rules, it helps to see why they exist. Two different measuring tools reading the same object can honestly give different numbers of digits. Suppose a pencil is measured as 15 cm on a meterstick marked only in centimeters, but as 15.2 cm on a ruler marked in millimeters, and 15.23 cm on a digital caliper. All three are correct reports; each conveys the precision of the tool used. The last written digit in each is the estimated one, uncertain by a bit, and everything before it is certain. Significant figures are the bookkeeping system that keeps this honesty intact as numbers flow through arithmetic, so that a computed answer never claims more certainty than the measurements that produced it. Keep in mind the distinction, developed more fully at the end of this lesson, between precision (how finely and reproducibly you can measure) and accuracy (how close your measurement is to the true value); significant figures track precision, not accuracy.

Counting significant figures: the rules

A short set of rules covers every case. Learn them until they are automatic, because you will apply them in nearly every numerical problem.

• Every nonzero digit is significant. The number 24.7 has three sig figs; 8391 has four.
• Zeros between nonzero digits (captive zeros) are always significant. The number 1005 has four sig figs; 40.06 has four.
• Leading zeros are never significant. They only mark the position of the decimal point. The number 0.0034 has two sig figs (the 3 and 4); 0.00080 has two sig figs (the 8 and the trailing 0).
• Trailing zeros are significant only when a decimal point is present. So 2.50 has three sig figs and 0.4200 has four, because the trailing zeros after a decimal point are meaningful. But a number like 250 without a decimal point is ambiguous: it could be two or three sig figs. Treat such a number as two sig figs unless told otherwise, or resolve the ambiguity by writing 250. (with a trailing decimal point) for three, or by using scientific notation.
• Exact numbers have unlimited significant figures. Counted quantities (12 eggs, 30 students) and defined conversions (exactly 100 cm per 1 m, exactly 1000 g per 1 kg) are perfectly precise by definition. They never limit the precision of a calculated result.

Number	Significant figures	Reason
48.2	3	All nonzero digits count
3005	4	Captive zeros count
0.00612	3	Leading zeros do not count
7.100	4	Trailing zeros after a decimal count
2500	2 (ambiguous)	No decimal point, so trailing zeros are unclear
2.500 × 103	4	Scientific notation removes ambiguity
60 (as in 60 seconds per minute)	unlimited	Exact defined value

Why scientific notation removes ambiguity

The trailing-zero problem disappears entirely in scientific notation, in which a number is written as a coefficient between 1 and 10 multiplied by a power of ten, and every digit shown in the coefficient is significant. Writing 2.5 times 10² says clearly "two sig figs," while 2.50 times 10² says "three sig figs," and there is no ambiguity because you write only the digits you mean. This clarity is one reason chemists reach for scientific notation constantly, but it is not the only reason. The subject is full of enormous and tiny numbers: Avogadro's number is 6.022 times 10²³, and the mass of a single hydrogen atom is about 1.67 times 10^-24 g. Writing these out with all their zeros would be error-prone and unreadable. Scientific notation also makes multiplying and dividing such numbers easy, since you multiply the coefficients and add the exponents. To convert a number into scientific notation, move the decimal point until one nonzero digit sits to its left, counting the moves: moving left gives a positive exponent (93,000,000 becomes 9.3 times 10⁷), and moving right gives a negative exponent (0.00042 becomes 4.2 times 10^-4).

Significant figures in multiplication and division

When you multiply or divide measurements, the answer keeps as many significant figures as the measurement with the fewest significant figures. The reasoning is that a result cannot be more precise than the least precise number that went into it; one weak link limits the whole chain, exactly as a chain is only as strong as its weakest link.

Example. Compute 4.56 times 1.4. The calculator shows 6.384, but 1.4 has only two sig figs while 4.56 has three, so the answer is limited to two sig figs and is reported as 6.4. Another example: 2001 divided by 3.2. The calculator gives 625.3125, but 3.2 has two sig figs, so the answer is 630 (or, to make the two sig figs unambiguous, 6.3 times 10²).

Significant figures in addition and subtraction

Addition and subtraction follow a different rule: the answer keeps as many decimal places as the value with the fewest decimal places. Here it is the position of the last certain digit, not the total count of significant figures, that governs, because when you line up the decimal points the uncertainty lives in a particular column.

Example. Compute 12.11 plus 0.3. The calculator shows 12.41, but 0.3 is known only to the tenths place (one decimal place), so the sum is rounded to the tenths place and reported as 12.4. Another example: 100.0 minus 0.444. The calculator gives 99.556, but 100.0 has only one decimal place, so the answer is 99.6. Notice that 100.0 has four sig figs while the answer 99.6 has three; the sig-fig count can change in addition and subtraction because the rule is about decimal places, not sig-fig count. This trips up many students, so keep the two rules mentally separate: multiply and divide by counting sig figs, add and subtract by counting decimal places.

Rounding correctly

To round a number to the desired precision, look at the first digit you are dropping. If it is less than 5, round down (leave the last kept digit unchanged); if it is greater than 5, round up (increase the last kept digit by one). If the dropped part is exactly 5 with nothing after, a common convention (round-half-to-even, or the "banker's rule") rounds to the nearest even digit to avoid a systematic upward bias, though for this course simply rounding 5 up is acceptable. More important than the tie-breaking convention is a habit that prevents real errors: round only at the very end of a multi-step problem. If you round at each intermediate step, small rounding errors accumulate and can shift the final digit. Carry one or two extra "guard" digits through the middle of a calculation, and round exactly once, at the finish, to the correct number of significant figures.

Dimensional analysis: the factor-label method

Dimensional analysis, also called the factor-label method or unit analysis, is the single most useful problem-solving tool in general chemistry, and it will carry you all the way through stoichiometry. The idea is to convert units by multiplying by fractions that are equal to one, called conversion factors, arranged so that the unwanted units cancel like common factors in algebra. Because every conversion factor equals one, multiplying by it changes the units of a quantity without changing the quantity itself.

A conversion factor is built from any equality between two units. Since 1000 g equals 1 kg, you can write it as either of two fractions, (1000 g / 1 kg) or (1 kg / 1000 g), both equal to one. You choose whichever orientation makes the unwanted unit cancel. To convert 3.50 kg to grams, you want kilograms to cancel, so you multiply by the factor with kilograms in the denominator:

3.50 kg × (1000 g ÷ 1 kg) = 3500 g

The kilograms in the numerator cancel the kilograms in the denominator, leaving grams. If you had accidentally used the upside-down factor, you would get units of kg²/g, which are meaningless, and that nonsense is your signal that the setup is wrong. This self-checking property is the great virtue of the method.

You can chain several conversion factors together in a single line, which is where the technique really shines. To convert a speed of 90.0 km/h into meters per second, you convert kilometers to meters and hours to seconds in one continuous product:

90.0 km/h × (1000 m ÷ 1 km) × (1 h ÷ 3600 s) = 25.0 m/s

Kilometers cancel against kilometers, hours cancel against hours, and you are left with exactly meters per second, the units the question asked for. The arithmetic is 90.0 times 1000 divided by 3600, which equals 25.0.

Worked example: a multi-step conversion. A recipe calls for 2.0 pounds of flour, and you want the mass in grams, knowing that 1 pound equals 453.6 g. Then 2.0 lb times (453.6 g / 1 lb) equals 910 g (limited to two sig figs by the 2.0). A harder one: convert 55 miles per hour to meters per second, given 1 mile equals 1609 m. Set it up as 55 mi/h times (1609 m / 1 mi) times (1 h / 3600 s), which is 55 times 1609 divided by 3600, or about 25 m/s. As always, watch the units cancel to leave m/s, and confirm the answer is sensible.

The discipline the method demands is worth spelling out. Always write the units, not just the numbers. Set each conversion factor so the unit you want to eliminate cancels. Cancel units explicitly on paper. Finally, check that the units you are left with match what the question asked for. If the units come out right, the arithmetic almost always follows, and if they come out wrong, you have caught a mistake before it cost you any points. This method scales all the way up to the multi-step mole conversions of Module 4 and the mass-to-mass stoichiometry of Module 5, where a single problem might chain grams to moles to moles to grams. Mastering the habit now pays dividends for the entire rest of the course.

Accuracy versus precision

Significant figures track precision, but precision is only half of good measurement. Accuracy is how close a measurement is to the true value; precision is how close repeated measurements are to one another. The two are independent. A dartboard analogy is standard and useful: darts clustered tightly but far from the bullseye are precise but not accurate; darts scattered all around but averaging on the bullseye are accurate on average but not precise; darts clustered on the bullseye are both. In the lab, a miscalibrated balance can be highly precise, reading 5.001, 5.002, and 5.001 g on three tries, yet inaccurate if the true mass is 4.900 g, because every reading is off by the same amount. Precision shows up as tight, reproducible significant figures; accuracy requires calibrating your instrument against a known standard. Confusing the two is one of the most common errors in interpreting data. When you later see a result reported as a mean plus or minus a standard deviation, you are seeing precision quantified, whereas a comparison to an accepted reference value speaks to accuracy.

Common misconceptions

Three errors are especially frequent. First, students think more decimal places always means a better answer; in fact, writing digits your instrument cannot support overstates precision and is incorrect. Second, students round after every step to keep numbers tidy; this lets errors accumulate and can change the final digit, so keep guard digits and round once at the end. Third, students let exact numbers limit their sig figs; counted and defined values have infinite significant figures and never constrain a result.

Recap

You can now count significant figures using the rules for nonzero digits, captive zeros, leading zeros, trailing zeros, and exact numbers, and you understand why scientific notation removes ambiguity. You can carry sig figs correctly through calculations, counting sig figs for multiplication and division but decimal places for addition and subtraction, and you know to round only at the end. Most importantly, you can perform dimensional analysis, chaining conversion factors so units cancel to deliver the units the problem requires. Finally, you can distinguish accuracy from precision. These quantitative habits underlie every calculation in the chapters ahead, from atomic masses to molar masses to gas laws.

Everything you have classified and measured so far, every element, compound, and mixture, is ultimately built from atoms. The word comes from the Greek atomos, meaning "uncuttable," a name coined by ancient philosophers who guessed that matter could not be divided forever. They were right that matter is grainy rather than infinitely smooth, but wrong that the atom is the end of the story: the atom itself is built from still smaller pieces, and it is those pieces, and especially their arrangement, that decide every chemical property you will ever study. This lesson opens up the atom. You will learn the three subatomic particles and where they live, how a simple count of protons defines an element's very identity, why the same element can come in slightly different masses (isotopes), and how the single number printed under each symbol on the periodic table, the atomic mass, is actually a carefully weighted population average rather than the mass of any real atom. Every one of these ideas is load-bearing for the rest of the course, because bonding, the periodic table, and the mole all rest on the architecture of the atom.

The three subatomic particles

Every atom, whether of gold or of hydrogen, shares the same basic architecture. At the center sits a tiny, extraordinarily dense nucleus containing positively charged protons and electrically neutral neutrons. Surrounding the nucleus in a diffuse cloud are negatively charged electrons, which have almost no mass. These three particles differ in the two properties that matter most for chemistry, charge and mass.

Particle	Symbol	Charge	Relative mass (amu)	Actual mass (kg)	Location
Proton	p+	+1	about 1	1.673 × 10-27	Nucleus
Neutron	n0	0	about 1	1.675 × 10-27	Nucleus
Electron	e-	-1	about 1/1836	9.109 × 10-31	Electron cloud

Read the mass column carefully, because it holds a fact that shapes the whole subject. A proton and a neutron have almost exactly the same mass, and each is roughly 1836 times heavier than an electron. The electron is so light that for the purpose of an atom's total mass we can essentially ignore it; virtually all of an atom's mass comes from its protons and neutrons in the nucleus. The charges, by contrast, come in tidy whole-number units: the proton carries exactly +1 and the electron exactly -1, equal in size and opposite in sign. This exact charge balance is why a complete atom, with equal numbers of protons and electrons, is electrically neutral overall.

The atom is mostly empty space

The most startling fact about an atom is how empty it is. The nucleus is fantastically small compared with the atom as a whole. A common and reliable analogy: if an atom were scaled up to the size of a large sports stadium, the nucleus would be about the size of a marble or a pea placed at the very center, with the electrons occupying the vast reaches out to the seats. In numbers, a typical atom is about 10^-10 m across (0.1 nanometer), while its nucleus is only about 10^-15 m across, so the atom is roughly 100,000 times wider than its nucleus. Yet that minuscule nucleus holds more than 99.9 percent of the atom's mass. The solidity of a table or a wall is an illusion created almost entirely by electrical repulsion between the outer electrons of atoms, not by any actual filling of space with matter. When you press your hand on a desk, essentially no particles touch; the electron clouds simply repel one another.

Atomic number defines the element

Of the three particles, one count reigns supreme. The atomic number (Z) is the number of protons in the nucleus, and it alone defines which element an atom is. Every carbon atom, without a single exception anywhere in the universe, has 6 protons. Add a seventh proton and you no longer have carbon; you have nitrogen, a completely different element with different chemistry. Remove one and you have boron. This is why the periodic table is ordered by atomic number, marching 1, 2, 3, 4 from hydrogen (1 proton) to helium (2 protons) to lithium (3 protons) and onward. The atomic number is the element's fingerprint.

In a neutral atom the number of electrons equals the number of protons, so the +1 charges of the protons are exactly canceled by the -1 charges of the electrons and the atom carries no net charge. A neutral carbon atom therefore has 6 protons and 6 electrons. Keep this default in mind: unless a charge is written, assume an atom is neutral and set electrons equal to protons.

Mass number and counting neutrons

The mass number (A) is the total count of the two heavy particles, protons plus neutrons, in the nucleus. Because it counts only whole particles, the mass number is always a whole number, unlike the atomic mass we will meet shortly. Since the mass number is protons plus neutrons and the atomic number is protons alone, a single subtraction recovers the neutron count:

number of neutrons = A - Z

You will do this subtraction many times, so make it automatic. Chemists write an atom's identity with two numbers stacked to the left of its symbol: the mass number A on top and the atomic number Z on the bottom, as in ²³₁₁Na for sodium-23. From that single symbol you can read everything: Z = 11 means 11 protons and (in a neutral atom) 11 electrons, and A = 23 means 23 - 11 = 12 neutrons. An equivalent and very common shorthand simply names the element and appends the mass number with a hyphen, as in "sodium-23" or "carbon-14," in which case you look up Z on the periodic table.

Worked example 1. How many protons, neutrons, and electrons are in a neutral atom of ⁵⁶₂₆Fe (iron-56)? The bottom number is the atomic number, Z = 26, so there are 26 protons, and because the atom is neutral there are also 26 electrons. The top number is the mass number, A = 56, so the neutron count is A - Z = 56 - 26 = 30 neutrons. Iron-56 is the most common isotope of iron and, not coincidentally, one of the most tightly bound nuclei in nature.

Worked example 2. An atom is described as having 17 protons and 20 neutrons. Identify it and give its mass number. The element is set by the proton count: 17 protons means atomic number 17, which is chlorine. The mass number is protons plus neutrons, 17 + 20 = 37, so this is chlorine-37, written ³⁷₁₇Cl.

Isotopes: same element, different mass

Here is a subtlety the ancient philosophers never imagined: atoms of a single element are not all identical. Isotopes are atoms of the same element (the same number of protons, and therefore the same chemical identity) that differ in their number of neutrons, and therefore in mass. The proton count is fixed, but the neutron count can vary, and each different neutron count is a different isotope.

Carbon is the classic example. It has three naturally significant isotopes, all with 6 protons but different numbers of neutrons:

• Carbon-12: 6 protons, 6 neutrons (mass number 12). This is by far the most abundant, making up about 98.9 percent of all carbon, and it is stable.
• Carbon-13: 6 protons, 7 neutrons (mass number 13). It makes up about 1.1 percent of carbon and is also stable.
• Carbon-14: 6 protons, 8 neutrons (mass number 14). It is exceedingly rare (about one atom in a trillion) and, crucially, radioactive, decaying slowly over thousands of years.

Because chemistry is governed almost entirely by electrons, and all three carbon isotopes have 6 electrons arranged identically, they behave essentially the same in chemical reactions. A carbon-14 atom bonds and reacts just like a carbon-12 atom. The extra neutrons change only the mass and, in some cases, the nuclear stability. Carbon-14's radioactivity is the basis of radiocarbon dating: living things take in carbon-14 from the atmosphere at a steady rate while alive, and once they die the carbon-14 slowly decays with a half-life of about 5,730 years, so measuring how much remains reveals how long ago the organism died. Carbon-12 and carbon-13, being stable, do not decay and last indefinitely.

Hydrogen offers another famous case, with isotopes so distinct they earned their own names: ordinary hydrogen (¹H, one proton and no neutrons, called protium), deuterium (²H, one proton and one neutron), and tritium (³H, one proton and two neutrons, which is radioactive). Deuterium combined with oxygen makes "heavy water," which is chemically water but noticeably denser. Isotopes are not a rare curiosity; most elements are naturally a mixture of two or more.

Atomic mass is a weighted average of isotopes

Now the isotopes force a question. If chlorine comes as a mix of chlorine-35 and chlorine-37, what single mass should the periodic table print for chlorine? The answer is the atomic mass (sometimes called atomic weight): the average mass of an element's atoms, weighted by how common each isotope is in nature. It is emphatically not a simple average of the isotope masses, because a rare isotope should count for less than a common one. The recipe is to multiply each isotope's mass by its fractional abundance (its percentage written as a decimal) and add the results:

atomic mass = (mass₁ × fraction₁) + (mass₂ × fraction₂) + ...

Worked example 3. Chlorine is about 75.8 percent chlorine-35 (mass 34.97 amu) and 24.2 percent chlorine-37 (mass 36.97 amu). Convert the percentages to decimal fractions, 0.758 and 0.242, and compute the weighted sum:

(0.758 × 34.97) + (0.242 × 36.97) = 26.51 + 8.95 = 35.45 amu

That is precisely why the periodic table lists chlorine as about 35.45 amu, even though no single chlorine atom actually has that mass. Every real chlorine atom is either mass 35 or mass 37; the tabulated 35.45 is a population average, closer to 35 than to 37 because chlorine-35 is the more common isotope. A useful sanity check falls out of this reasoning: a weighted average must always land between the lightest and heaviest isotope masses, and closer to whichever isotope is more abundant. If your computed atomic mass falls outside that range, you have made an arithmetic error.

Worked example 4. Copper has two stable isotopes: copper-63 (mass 62.93 amu, 69.2 percent abundant) and copper-65 (mass 64.93 amu, 30.8 percent abundant). Predict copper's atomic mass. Compute (0.692 × 62.93) + (0.308 × 64.93) = 43.55 + 20.00 = 63.55 amu. The periodic table indeed lists copper as about 63.55 amu, and note that the answer sits between 63 and 65, leaning toward 63 because copper-63 is more abundant, exactly as the sanity check predicts.

Worked example 5 (working backward). Sometimes you know the atomic mass and one isotope and must find an abundance. Boron has an atomic mass of 10.81 amu and two isotopes, boron-10 (10.01 amu) and boron-11 (11.01 amu). What fraction is boron-11? Let x be the fraction of boron-11, so (1 - x) is the fraction of boron-10. Then 10.01(1 - x) + 11.01x = 10.81. Expanding gives 10.01 - 10.01x + 11.01x = 10.81, so 10.01 + 1.00x = 10.81, which means 1.00x = 0.80, so x = 0.80. Boron is about 80 percent boron-11 and 20 percent boron-10, which matches the measured natural abundances closely.

The atomic mass unit and why it is chosen so cleverly

Mass on the atomic scale is measured in atomic mass units (amu), also written u and often called daltons (Da). The unit is defined so that one atom of carbon-12 weighs exactly 12 amu, which fixes 1 amu as one-twelfth the mass of a carbon-12 atom (about 1.66 × 10^-24 g). This choice is not arbitrary bookkeeping; it is one of the most elegant design decisions in all of chemistry. Because the amu is pegged to carbon-12, the average mass of an element's atoms in amu turns out to be numerically identical to the mass of one mole of that element in grams. Chlorine's atomic mass is 35.45 amu per atom and also 35.45 grams per mole. This numerical bridge between the invisible single-atom scale and the weighable gram scale is the foundation of the mole concept you will build in Module 4, and it is what lets a chemist weigh out a sample on an ordinary balance and immediately know how many atoms it contains.

Ions: atoms with a charge

While the proton count is locked for a given element, the electron count is not. If a neutral atom gains or loses one or more electrons, it becomes an ion, an atom carrying a net electric charge. Losing electrons leaves the protons in the majority, so the atom becomes a positively charged cation (for example, a sodium atom that loses one electron becomes Na⁺, with 11 protons but only 10 electrons). Gaining electrons tips the balance negative, producing an anion (a chlorine atom that gains one electron becomes Cl^-, with 17 protons and 18 electrons).

The essential point is that forming an ion changes the charge but never the element, because the protons in the nucleus are untouched. A sodium atom and a sodium ion, Na⁺, are both sodium; they simply differ in electron count and therefore in charge. Similarly, isotopes and ions are independent ideas: an isotope varies neutrons and changes mass, while an ion varies electrons and changes charge, and an atom can do both at once. This distinction becomes essential the moment we reach ionic bonding in Module 3, where the transfer of electrons to form cations and anions is the whole story.

Common misconceptions

Three errors recur with atomic structure. First, students think the atomic mass on the table is the mass of one atom; in truth it is a weighted average over all natural isotopes, so no single chlorine atom weighs 35.45 amu. Second, students expect isotopes to react differently in chemistry; because reactivity depends on electrons and isotopes share the same electron arrangement, they react essentially identically, differing only in mass and nuclear stability. Third, students think changing an atom's electrons changes what element it is; only the proton count sets the element, so gaining or losing electrons yields an ion, not a new element.

Recap

An atom consists of a dense central nucleus of protons and neutrons surrounded by a vast, mostly empty cloud of nearly massless electrons. The number of protons, the atomic number Z, defines the element and (in a neutral atom) equals the number of electrons. The mass number A counts protons plus neutrons, so neutrons equal A minus Z. Isotopes are atoms of one element with different neutron counts and therefore different masses; they behave the same chemically. The atomic mass on the periodic table is the abundance-weighted average of an element's isotope masses, expressed in atomic mass units defined by carbon-12, a choice that makes atomic mass in amu equal molar mass in grams per mole. Finally, gaining or losing electrons turns an atom into a charged ion without changing its elemental identity. With the atom's structure in hand, the next lesson asks how those electrons are actually arranged, which is what controls all of chemistry.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 2 "Atoms, Molecules, and Ions." https://openstax.org/books/chemistry-2e/pages/2-introduction
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 2 "Atoms, Molecules, and Ions."
3. National Institute of Standards and Technology (NIST), "Atomic Weights and Isotopic Compositions." https://www.nist.gov/pml/atomic-weights-and-isotopic-compositions-relative-atomic-masses
4. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 2.
5. IUPAC, "Atomic weights of the elements" (Commission on Isotopic Abundances and Atomic Weights). https://iupac.org/what-we-do/periodic-table-of-elements/

In the last lesson you saw that an atom is a dense nucleus wrapped in a cloud of electrons. Now comes the payoff, because an atom's entire chemistry, every bond it forms, every reaction it undergoes, is decided almost entirely by those electrons and, above all, by how they are arranged. Two atoms with the same electron arrangement in their outer region behave alike, which is the deep reason the periodic table works. This lesson develops the modern picture of electron arrangement: electrons do not swarm the nucleus at random but occupy a strict, layered structure of energy levels, subshells, and orbitals, filling that structure according to a small set of rules. From an atom's electron configuration you can read off its most important chemical property, its number of outer (valence) electrons, and you can understand why atoms bond at all. Getting comfortable here makes the periodic trends of the next lesson and the bonding of Module 3 feel almost inevitable rather than arbitrary.

Energy levels and quantization

Electrons occupy discrete energy levels, often called shells, numbered n = 1, 2, 3, and so on, counting outward from the nucleus. The number n is called the principal quantum number. This is one of the deepest surprises of the atomic world: an electron cannot have just any energy it likes, only certain specific, allowed energies, like a person who can stand on one step of a staircase or another but never float in between. This restriction is called quantization, and the whole framework built on it is quantum mechanics. Lower-numbered shells lie closer to the nucleus and are lower in energy, so they fill first, just as water fills the lowest parts of a container before rising.

The evidence for quantization is beautifully visible. When atoms are heated or given electrical energy, their electrons jump to higher shells and then fall back, emitting light of very specific colors as they drop. Each element produces its own unique set of sharp colored lines, its emission spectrum, rather than a continuous rainbow. Neon glows red-orange, sodium street lamps glow yellow, and hydrogen produces a distinctive line pattern. These sharp lines are the fingerprints of quantized energy levels: an electron falling from one specific level to another releases one specific packet of energy, seen as one specific color. If electrons could have any energy, the light would be a smooth continuum instead of discrete lines.

Each shell can hold only a limited number of electrons. The pattern is that shell n holds up to 2n² electrons: the first shell holds up to 2, the second up to 8, the third up to 18, and the fourth up to 32. For the main-group elements this course emphasizes, the outer shell behaves as though it seeks 8 electrons, an idea we will formalize as the octet rule at the end of the lesson.

Subshells: s, p, d, and f

Within each shell, electrons live in subshells labeled s, p, d, and f, each with a slightly different energy and a characteristic shape. The number of subshells available grows with the shell number: shell 1 has only an s subshell; shell 2 has s and p; shell 3 has s, p, and d; shell 4 has s, p, d, and f. The letters are historical (they come from spectroscopists' terms sharp, principal, diffuse, and fundamental), but their capacities follow a clean pattern that you should memorize:

Subshell	Number of orbitals	Maximum electrons	Shape
s	1	2	Spherical
p	3	6	Dumbbell (along x, y, z axes)
d	5	10	Mostly cloverleaf
f	7	14	Complex, multi-lobed

Orbitals and the Pauli exclusion principle

An orbital is a region of space where an electron is most likely to be found, essentially a three-dimensional map of probability rather than a fixed path. This is a genuine break from the older cartoon of electrons circling the nucleus like planets. We cannot say exactly where an electron is at a given instant; we can only describe the region it is likely to occupy, and that region is the orbital. An s orbital is a fuzzy sphere centered on the nucleus; the three p orbitals are dumbbell-shaped and point along the three perpendicular axes.

A fundamental rule limits how orbitals fill: by the Pauli exclusion principle, each orbital holds at most 2 electrons, and those two must have opposite spins (an intrinsic property of electrons, pictured as spinning clockwise or counterclockwise, and drawn as an up-arrow and a down-arrow). This is why the capacities double the orbital counts: the s subshell has 1 orbital and so holds 2 electrons; the p subshell has 3 orbitals and so holds 6; the d has 5 orbitals holding 10; the f has 7 orbitals holding 14.

Three rules for filling: Aufbau, Pauli, and Hund

Electrons fill this structure according to three rules working together. The first is the Aufbau principle (from the German Aufbau, "building up"): electrons occupy the lowest-energy subshells first, then fill upward. The order of increasing energy for the subshells we use most is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on. Notice the surprise that 4s fills before 3d; this ordering, which we revisit in the deep dive, is why the periodic table has the shape it does. The second rule is the Pauli exclusion principle just described, capping each orbital at two opposite-spin electrons.

The third is Hund's rule: when several orbitals of a subshell have the same energy, such as the three p orbitals, electrons spread out one to each orbital before any orbital receives a second electron, and these single electrons keep parallel (same-direction) spins. The intuition is simple: electrons are negatively charged and repel one another, so they take separate "rooms" before being forced to double up in the same room. Nitrogen, with three 2p electrons, therefore places one electron in each of the three 2p orbitals, leaving all three unpaired, rather than crowding two into one orbital.

Writing electron configurations

An electron configuration is a compact notation listing which subshells are occupied and how many electrons each holds, written in order of increasing energy. Each term shows the shell number, the subshell letter, and a superscript giving the electron count in that subshell. Here are several built by applying the three rules:

• Hydrogen (1 electron): 1s¹
• Helium (2 electrons): 1s² (the first shell is now full)
• Lithium (3 electrons): 1s² 2s¹ (the third electron starts the second shell)
• Carbon (6 electrons): 1s² 2s² 2p²
• Oxygen (8 electrons): 1s² 2s² 2p⁴
• Neon (10 electrons): 1s² 2s² 2p⁶ (the second shell is now full)
• Sodium (11 electrons): 1s² 2s² 2p⁶ 3s¹

A quick check that never fails: the superscripts must add up to the total number of electrons in the atom. For sodium, 2 + 2 + 6 + 1 = 11, matching sodium's 11 electrons. If your superscripts do not sum to the electron count, you have made an error somewhere and should recount.

Worked example 1. Write the electron configuration of phosphorus (atomic number 15). Fill in energy order: 1s holds 2, 2s holds 2, 2p holds 6 (that is 10 so far), 3s holds 2 (12 so far), and the remaining 3 electrons go into 3p. The configuration is 1s² 2s² 2p⁶ 3s² 3p³. Check: 2 + 2 + 6 + 2 + 3 = 15. By Hund's rule, those three 3p electrons occupy the three separate 3p orbitals singly, so phosphorus has three unpaired electrons.

Noble-gas shorthand. Longer configurations get tedious, so chemists abbreviate the inner (core) electrons using the previous noble gas in brackets. Sodium, 1s² 2s² 2p⁶ 3s¹, has the same first ten electrons as neon, so it is written [Ne] 3s¹. Phosphorus becomes [Ne] 3s² 3p³. This shorthand does more than save ink: it puts the chemically important outer electrons front and center, which is exactly what we care about next.

Valence electrons drive all chemistry

The electrons in the outermost shell are the valence electrons, and they are the ones that form bonds and dictate chemical behavior. The inner electrons, called core electrons, are held tightly by the nucleus and shielded deep inside the atom, so they rarely participate in reactions; they are essentially spectators. When sodium reacts, it is the single 3s electron that acts, not the ten core electrons.

For the main-group elements, there is a wonderfully simple shortcut: the number of valence electrons matches the group number on the periodic table, using the traditional labeling of the tall columns as Groups 1 through 8 (equivalently, the newer 1 through 18 labeling where the main-group columns are 1, 2, and then 13 through 18). Group 1 elements have 1 valence electron, Group 2 have 2, and Groups 13 through 18 have 3, 4, 5, 6, 7, and 8 respectively. Sodium is in Group 1 and has 1 valence electron (its 3s¹); oxygen is in Group 16 and has 6 valence electrons (its 2s² 2p⁴, which sums to 6). This is the mechanical reason elements in the same column behave alike: they share the same valence configuration, differing only in which shell that configuration sits in. Lithium, sodium, and potassium are all soft, reactive metals precisely because each has a lone s¹ valence electron ready to be lost.

Worked example 2. How many valence electrons does sulfur have, and how does that predict its behavior? Sulfur (atomic number 16) has the configuration [Ne] 3s² 3p⁴. The outermost shell is n = 3, which holds 2 + 4 = 6 electrons, so sulfur has 6 valence electrons, consistent with its position in Group 16. Being just two electrons short of a full outer octet, sulfur tends to gain or share two electrons, which is exactly why it commonly forms the sulfide ion S^2- and compounds like hydrogen sulfide, H₂S.

The octet rule

Atoms are especially stable when their outer shell is full, which for most main-group elements means having 8 valence electrons (an octet). This is the octet rule, and it is the single most useful idea for predicting how atoms bond. It explains at a glance why the noble gases in Group 18 (helium, neon, argon, and the rest) are so unreactive that several form essentially no compounds: their valence shells are already completely full, so they have no drive to gain, lose, or share electrons. Helium is content with 2 (its only shell, the first, is full at 2), while the others are content with 8.

Every atom that is not a noble gas is, in a sense, "trying" to reach the ultra-stable full-shell arrangement of the nearest noble gas, and it does so by gaining, losing, or sharing electrons. Sodium (one electron past neon's configuration) readily loses that one electron to become Na⁺, which now has neon's arrangement. Chlorine (one electron short of argon's configuration) readily gains one to become Cl^- with argon's arrangement. This drive toward a full octet is the thread that ties this lesson directly to the ionic and covalent bonding of Module 3. Do note that the octet rule is a powerful guideline rather than an unbreakable law: hydrogen and helium are stable with just 2 electrons, and a few elements such as boron and phosphorus can be stable with fewer or more than 8. Treat it as a strong default that predicts the vast majority of everyday chemistry.

Common misconceptions

Three confusions are worth flagging. First, students picture electrons orbiting the nucleus in neat circles like planets; in reality electrons occupy orbitals, fuzzy three-dimensional regions of probability, and we can describe only where an electron is likely to be, not a precise path. Second, students assume every atom obeys the octet rule perfectly; it is a strong guideline with real exceptions (hydrogen and helium want 2, and boron and phosphorus can deviate). Third, students think all of an atom's electrons take part in bonding; only the valence electrons do, while the core electrons stay locked inside as spectators.

Recap

Electrons occupy quantized energy levels (shells) that fill from the inside out, and each shell contains subshells (s, p, d, f) made of orbitals, with every orbital holding at most two opposite-spin electrons by the Pauli exclusion principle. Electrons fill lowest energy first (Aufbau) and spread out singly within equal-energy orbitals before pairing (Hund). An electron configuration records this arrangement, and its superscripts must sum to the electron count. The outermost, valence electrons drive all chemistry, and for main-group elements their number equals the group number. Atoms gain, lose, or share electrons to reach a full, stable octet, the guiding principle that the next module turns into a full theory of bonding. The very shape of the periodic table, as the next lesson reveals, is a direct picture of this electron-filling order.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 6 "Electronic Structure and Periodic Properties of Elements." https://openstax.org/books/chemistry-2e/pages/6-introduction
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 7 "Atomic Structure and Periodicity."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapters 6-7.
4. Averill, B., and Eldredge, P. General Chemistry: Principles, Patterns, and Applications (LibreTexts), Chapter 6 "The Structure of Atoms." https://chem.libretexts.org
5. National Institute of Standards and Technology (NIST), Atomic Spectra Database. https://www.nist.gov/pml/atomic-spectra-database

The previous two lessons built the atom from the inside out, ending with electron configurations and the drive toward a full octet. The periodic table is where all of that structure becomes visible and usable at a glance. It arranges the roughly 118 known elements by increasing atomic number in a grid whose very shape is a direct picture of electron configuration, and that arrangement is not a convenience but a profound organizing principle: elements with similar electron arrangements land in the same column and behave alike. Once you understand why the table has the shape it does, you can predict an element's size, how tightly it holds its electrons, and how hard it pulls on shared electrons, all from its position alone, without memorizing a single data point. This lesson explains the table's structure and then develops the three great periodic trends, atomic radius, ionization energy, and electronegativity, which together let you reason about nearly all of descriptive chemistry.

How the table is built: groups and periods

A vertical column is a group (or family). Elements in a group share the same number of valence electrons and therefore show similar chemical behavior; this is the single most important fact about the table. Sodium and potassium sit in the same group (Group 1) because each has one valence electron, and so both are soft, violently reactive metals. A horizontal row is a period. Moving across a period from left to right, a particular shell is being filled one electron at a time, from the first electron of that shell on the left to a full shell on the right. Because a shell fills, then a new one begins and fills again, the pattern of properties repeats at regular intervals, which is exactly why the arrangement is called periodic. The table is essentially the electron-filling order of the last lesson, wrapped into rows so that atoms with matching valence configurations stack into columns.

This is why the table has its distinctive block structure. The two leftmost columns are the s-block (two columns wide, matching the s subshell's capacity of 2), the six columns on the right are the p-block (six wide, matching the p subshell's 6), and the broad center is the d-block or transition metals (ten wide, matching the d subshell's 10). The shape of the periodic table is literally a map of subshell capacities.

Regions and families of the table

Metals occupy the left and center of the table. They are shiny, conduct heat and electricity well, are malleable (can be hammered into sheets) and ductile (can be drawn into wire), and tend to lose electrons to form cations. Most elements are metals. Nonmetals sit in the upper right; they are typically dull, poor conductors, brittle if solid, and tend to gain or share electrons. Dividing the two is a diagonal staircase of metalloids (such as boron, silicon, germanium, and arsenic) whose intermediate, in-between properties make them invaluable as semiconductors, the foundation of every computer chip.

Several named families are worth knowing by name and character:

• Group 1, the alkali metals (lithium, sodium, potassium, and so on): soft, silvery, and violently reactive, each with a single valence electron they lose eagerly. They react vigorously with water.
• Group 2, the alkaline earth metals (beryllium, magnesium, calcium, and so on): reactive but noticeably less so than the alkali metals, each with two valence electrons.
• Group 17, the halogens (fluorine, chlorine, bromine, iodine): reactive nonmetals with seven valence electrons, one short of an octet, so they readily gain an electron and form salts (the name means "salt-former").
• Group 18, the noble gases (helium, neon, argon, and so on): nearly inert, thanks to their full valence shells, which is why they resist forming compounds at all.
• The transition metals, the large central d-block (iron, copper, zinc, gold, and many more): typical metals, many of which can form ions of more than one charge, as you saw in nomenclature previews.

Effective nuclear charge: the engine behind every trend

Before stating the trends, it pays to name their common cause. An outer electron does not feel the full positive pull of every proton in the nucleus, because the inner (core) electrons partly block, or shield, that pull. The net attraction an outer electron actually experiences is called the effective nuclear charge. Going across a period, protons are added to the nucleus while the new electrons enter the same shell and shield each other poorly, so the effective nuclear charge felt by the outer electrons rises steadily. Going down a group, each step adds a whole new shell of electrons that shields the outer electrons from the nucleus, and the outer electrons also sit farther away, so the effective nuclear charge felt by the valence electrons stays comparatively low. Almost every periodic trend is a consequence of how effective nuclear charge changes as you move around the table, so keep this one idea in mind and the three trends below will feel like a single story told three ways.

Trend 1: Atomic radius

Atomic radius is a measure of the size of an atom, essentially how far the outer electron cloud extends from the nucleus. Two clear trends emerge. Going down a group, atoms get larger, because each new period adds an entirely new electron shell that sits farther out from the nucleus. Potassium is bigger than sodium, which is bigger than lithium, because each has one more occupied shell than the last.

Going across a period from left to right, atoms get smaller, a trend that surprises many students who reason that adding electrons should make an atom bigger. The resolution is effective nuclear charge. Across a period the added electrons all go into the same shell, adding little shielding, while the number of protons steadily grows. The rising effective nuclear charge pulls that outer shell in more and more tightly, so the atom contracts even as electrons are added. The net result is that the largest atoms sit in the bottom left of the table (cesium and francium) and the smallest in the top right (fluorine, setting aside the noble gases, which are measured differently).

Worked example 1. Which is larger, a magnesium atom or a chlorine atom? Both are in period 3, so neither has more shells than the other, and we compare across the period. Magnesium (Group 2) lies to the left of chlorine (Group 17). Because atomic radius decreases from left to right, magnesium, being farther left, is the larger atom. Chlorine's greater nuclear charge pulls its third shell in tighter, making it smaller.

Worked example 2. Rank lithium, sodium, and potassium by size. All three are in Group 1, so we compare down the group. Each step down adds a shell, so potassium (period 4) is largest, sodium (period 3) is in the middle, and lithium (period 2) is smallest: K > Na > Li.

Trend 2: Ionization energy

Ionization energy is the energy needed to remove an electron from an atom (specifically the outermost, most loosely held electron, which is called the first ionization energy). It runs essentially opposite to atomic size, and for the same reason: an electron held close to the nucleus, feeling a strong effective nuclear charge, is hard to pull away, while a distant, well-shielded electron leaves easily. So ionization energy increases across a period, because rising effective nuclear charge grips the outer electrons ever more tightly, and it decreases down a group, because the outermost electrons sit farther out and are better shielded, so they depart more easily.

This trend makes physical sense of chemical behavior. Metals on the left of the table have low ionization energies, which is exactly why they readily give up electrons to form positive ions, the hallmark of metallic reactivity. Nonmetals on the upper right have high ionization energies and cling to their electrons. The noble gases have the highest ionization energies of all, reflecting the extreme stability of their full shells.

Worked example 3. Which has the higher first ionization energy, sodium or chlorine? Both are in period 3, and chlorine lies to the right of sodium. Ionization energy rises across a period, so chlorine holds its electrons more tightly and has the higher ionization energy. This fits their chemistry: sodium easily loses an electron to become Na⁺, while chlorine resists losing electrons and instead tends to gain one.

Trend 3: Electronegativity

Electronegativity measures how strongly an atom pulls on the shared electrons within a chemical bond. It is not about losing or gaining electrons outright, but about tug-of-war over a shared pair. Like ionization energy, and again for the same effective-nuclear-charge reason, electronegativity increases across a period and decreases down a group. Fluorine, in the upper right (setting aside the noble gases, which usually do not bond), is the most electronegative element, assigned a value of about 4.0 on the widely used Pauling scale. Oxygen and nitrogen are close behind. At the opposite corner, cesium and francium at the bottom left are the least electronegative, near 0.7. Electronegativity is the single most useful number in the whole table for predicting bond type, because, as the next module shows, the difference in electronegativity between two bonded atoms determines whether their bond is nonpolar covalent, polar covalent, or ionic.

Putting the three trends together

The beauty of these trends is that they are not three facts to memorize but one fact seen from three angles. All three trace back to how strongly the nucleus grips the outer electrons, that is, to effective nuclear charge. Toward the top right of the table the grip is strong, so atoms there are small, hold their electrons tightly (high ionization energy), and pull hard on shared electrons (high electronegativity). Toward the bottom left the grip is weak, giving large atoms that give up electrons readily (low ionization energy) and pull only weakly on shared electrons (low electronegativity). If you simply remember the top-right corner as "small, hard to ionize, highly electronegative" and the bottom-left as its mirror opposite, you can rank any two main-group elements on all three properties without looking up a single number.

Worked example 4. Consider oxygen and sulfur, both in Group 16. Predict their relative size and electronegativity. Going down the group from oxygen (period 2) to sulfur (period 3) adds a shell, so sulfur is the larger atom. Electronegativity decreases down a group, so oxygen is the more electronegative of the two. Both conclusions follow directly from the top-right rule: oxygen sits higher and closer to the top-right corner, so it is smaller and more electronegative.

A worked ranking that ties it together

Worked example 5. Arrange the elements sodium (Na), aluminum (Al), and chlorine (Cl), all in period 3, in order of increasing electronegativity, and separately in order of increasing atomic radius. For electronegativity, the trend rises left to right across a period, so the order from lowest to highest is Na, Al, Cl. For atomic radius, the trend falls left to right, so the largest is the leftmost and the order from smallest to largest is Cl, Al, Na. Notice how the two orderings are exact reverses of each other, a direct reflection of the shared cause: the same rising nuclear grip that pulls the atom smaller also makes it hold and attract electrons more strongly.

Common misconceptions

Three misunderstandings are common with periodic trends. First, students expect atoms to grow larger across a period because electrons are being added; in fact they shrink, because the added electrons enter the same shell while the rising proton count pulls that shell inward, and radius grows only when you move down a group and add a whole new shell. Second, students think the noble gases are unreactive because their shells are somehow empty; the opposite is true, as their valence shells are completely full, the most stable arrangement of all. Third, students treat electronegativity and ionization energy as unrelated; both measure how tightly an atom holds electrons, so both rise toward the top right for the same underlying reason.

Recap

The periodic table orders the elements by atomic number, and its shape mirrors electron configuration: columns (groups) share valence-electron counts and thus chemical behavior, while rows (periods) fill a shell across from left to right. Metals dominate the left and center, nonmetals sit upper right, and metalloids form the dividing staircase, with named families like the alkali metals, halogens, and noble gases marking key columns. The three great trends, atomic radius, ionization energy, and electronegativity, all flow from a single cause, the effective nuclear charge felt by the outer electrons. Radius shrinks up and to the right, while ionization energy and electronegativity grow up and to the right. Mastering the "top-right corner is small, tightly holding, and strongly attracting" rule lets you predict and compare element properties on sight, and it feeds directly into the next module, where the electronegativity difference between two atoms decides the very type of bond they form.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 6.5 "Periodic Variations in Element Properties." https://openstax.org/books/chemistry-2e/pages/6-5-periodic-variations-in-element-properties
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 7 "Atomic Structure and Periodicity."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 7 "Periodic Properties of the Elements."
4. Royal Society of Chemistry, "Periodic Table." https://www.rsc.org/periodic-table
5. National Institute of Standards and Technology (NIST), "Atomic Reference Data for Electronic Structure Calculations." https://www.nist.gov/pml/atomic-reference-data-electronic-structure-calculations

You now know that atoms seek a full, stable outer shell, and that the periodic table lets you rank how strongly each atom holds and attracts electrons. This lesson uses those two ideas to answer one of the central questions of chemistry: why and how do atoms join together into compounds? A lone hydrogen atom or a lone chlorine atom is reactive and short-lived, yet H₂ and HCl are stable substances. The reason is energetic. Atoms bond because a bonded arrangement sits at lower energy than the separated atoms, and nature always tends toward lower energy, the way a ball rolls downhill. The octet rule is the guiding principle behind that descent: atoms gain, lose, or share electrons to reach eight electrons in the valence shell (or two, for the smallest atoms like hydrogen). There are two principal ways to accomplish this, ionic bonding and covalent bonding, and the entire distinction between them comes down to a single question, whether electrons are transferred outright or shared, which in turn is decided by the electronegativity difference between the two atoms. By the end you will be able to look at any pair of elements and predict what kind of bond they form.

Ionic bonding: complete electron transfer

When a metal meets a nonmetal, their electronegativities are very different, often dramatically so. The metal, sitting on the left of the table, has a low ionization energy and gives up one or more electrons easily; the nonmetal, on the upper right, has a high electronegativity and pulls those electrons away decisively. The metal thereby becomes a positively charged cation, and the nonmetal becomes a negatively charged anion. Opposite charges attract, and it is that electrostatic attraction between the resulting ions that constitutes the ionic bond.

The formation of sodium chloride is the archetype. A sodium atom (configuration [Ne] 3s¹) has one lonely valence electron; losing it leaves Na⁺ with the stable, full-shell configuration of neon. A chlorine atom (configuration [Ne] 3s² 3p⁵) is one electron short of a full octet; gaining that electron gives Cl^- the stable configuration of argon. So sodium hands its electron to chlorine, both achieve noble-gas configurations, and the oppositely charged Na⁺ and Cl^- lock together. Both atoms end up more stable than they were alone, which is why the process releases energy.

A crucial and often-missed point is that ionic compounds do not form individual molecules. There is no such thing as a single "NaCl molecule" floating in isolation in a salt crystal. Instead, the ions stack into a vast, repeating three-dimensional array called a crystal lattice, in which every Na⁺ is surrounded by six Cl^- neighbors and every Cl^- is surrounded by six Na⁺ neighbors, on and on through the whole crystal. The formula NaCl does not name a two-atom unit; it states the 1-to-1 ratio of ions in this enormous network.

This lattice structure explains the signature properties of ionic compounds. They are hard, brittle solids with high melting points (sodium chloride melts at 801 degrees Celsius) because melting requires breaking an immense number of strong electrostatic attractions all at once, which takes a great deal of energy. They are brittle because if you strike the crystal and force the layers to shift, ions of like charge suddenly line up next to one another, repel violently, and the crystal cleaves. And they conduct electricity only when melted or dissolved in water, never as a dry solid, because conduction requires mobile charges: in the rigid solid lattice the ions are locked in place, but melting or dissolving frees them to move and carry current. Solid table salt does not conduct, but molten salt or salt water does.

Covalent bonding: sharing electrons

When two nonmetals bond, the situation is different. Their electronegativities are similar and both hold their electrons tightly, so neither atom can simply strip electrons from the other. The solution nature finds is compromise: the atoms share a pair of electrons, and each atom counts the shared pair toward its own octet. A shared pair of electrons is a covalent bond.

Two hydrogen atoms each have one electron and need one more to fill their first shell; by sharing a single pair between them, both effectively have two electrons, and H₂ forms. Atoms can share more than one pair when needed. Oxygen atoms share two pairs, forming a double bond in O₂; nitrogen atoms share three pairs, forming a triple bond in N₂. The nitrogen triple bond is one of the strongest bonds in chemistry, which is a large part of why nitrogen gas is so remarkably unreactive and why it takes extreme conditions (as in the industrial Haber process) to make nitrogen react at all.

Covalent substances form discrete molecules, genuinely separate units like a single H₂O molecule or a single CO₂ molecule. This gives them properties that contrast sharply with ionic compounds. They tend to have much lower melting and boiling points, because while the covalent bonds within each molecule are strong, the attractions between separate molecules are comparatively weak, and it is those weaker between-molecule attractions that must be overcome to melt or boil the substance. Many molecular compounds are liquids or gases at room temperature (water, carbon dioxide, methane), whereas ionic compounds are always solids. Molecular compounds also usually do not conduct electricity, even when melted or dissolved, because they contain no free ions to carry charge.

Polar versus nonpolar covalent bonds

Sharing, it turns out, is not always equal, and this subtlety is enormously important. If the two bonded atoms have different electronegativities, the more electronegative atom pulls the shared electron pair closer to itself, spending more time with the electrons. That atom takes on a slight negative charge, written with the lowercase Greek delta as the partial charge, while the less electronegative atom is left slightly positive. This uneven sharing is a polar covalent bond, and the small opposing charges at its two ends are called partial charges; the bond is said to have a dipole. If instead the two atoms are identical, or so close in electronegativity that the pull is effectively even, the bond is nonpolar covalent.

The contrast is easy to see. The H-H bond in H₂ joins two identical atoms, so the electrons are shared perfectly evenly and the bond is purely nonpolar. The H-Cl bond in hydrogen chloride joins hydrogen (electronegativity about 2.1) to the much more electronegative chlorine (about 3.0), so chlorine hogs the shared pair, the chlorine end is partially negative, the hydrogen end is partially positive, and the bond is distinctly polar. These partial charges are the seed of molecular polarity, which the very next lesson develops into a full account of why water dissolves salt and oil does not.

Bonding is a continuous spectrum

The single most useful tool for predicting bond type is the electronegativity difference between the two atoms, which you can estimate from their positions on the table or read from a table of values. A common set of rough guidelines is:

Electronegativity difference	Bond type	What happens to the electrons	Example
Less than about 0.4	Nonpolar covalent	Shared essentially equally	H-H, C-H
About 0.4 to 1.7	Polar covalent	Shared unequally (partial charges)	H-O, H-Cl
Greater than about 1.7	Ionic	Transferred essentially completely	Na-Cl

The deep insight here is that bonding is not two (or three) separate boxes but a single continuous spectrum. At one extreme, two identical atoms share electrons perfectly evenly (nonpolar covalent). As the electronegativity difference grows, the sharing becomes more and more lopsided (polar covalent), the more electronegative atom claiming an ever-larger share of the electron density. At the far extreme, the difference is so large that one atom essentially seizes the electrons outright (ionic). Polar covalent bonds occupy the vast middle ground and can be thought of as part sharing, part transfer. There is no sharp line where sharing suddenly becomes transfer; the 1.7 cutoff is a useful convention, not a law of nature. Understanding this spectrum means you can predict a compound's bonding character from nothing more than the identities of its two elements and the electronegativity ranking you learned in Module 2.

Worked example 1. Classify the bonding in potassium fluoride, KF. Potassium is a Group 1 metal with a very low electronegativity (about 0.8), and fluorine is the most electronegative element (about 4.0). The difference is 4.0 - 0.8 = 3.2, far above 1.7, so the bond is ionic: potassium transfers its single valence electron to fluorine, forming K⁺ and F^- held in a crystal lattice.

Worked example 2. Classify the bonding in a carbon-oxygen bond, as found in carbon dioxide. Carbon has an electronegativity of about 2.5 and oxygen about 3.5, a difference of 1.0. That falls in the 0.4 to 1.7 range, so each C-O bond is polar covalent: the atoms share electrons, but oxygen pulls them closer, giving oxygen a partial negative charge. (Whether the whole CO₂ molecule is polar is a separate question about shape, answered in the next lesson.)

Worked example 3. Classify the bonding in a carbon-hydrogen bond. Carbon is about 2.5 and hydrogen about 2.1, a difference of only 0.4, right at the boundary and essentially nonpolar covalent. This tiny difference is precisely why hydrocarbons such as oil and gasoline are nonpolar and refuse to mix with polar water, a fact with huge practical consequences from cooking to oil spills.

Comparing ionic and covalent compounds at a glance

Because the two bonding types produce such different substances, it helps to see their contrasting properties side by side. The differences all trace back to the same root: ionic compounds are held together by strong electrostatic attractions extending through an entire lattice, while molecular compounds are discrete units held to each other only by weaker between-molecule forces.

Property	Ionic compounds	Molecular (covalent) compounds
Basic unit	Extended crystal lattice of ions	Discrete molecules
Melting/boiling point	High (often hundreds of degrees)	Low (many are liquids or gases)
State at room temperature	Always solid	Solid, liquid, or gas
Electrical conductivity	Conducts when melted or dissolved	Usually does not conduct
Made from	Metal + nonmetal	Nonmetal + nonmetal

A third type: metallic bonding

For completeness it is worth naming a third major kind of bonding, the one that holds pure metals and alloys together: metallic bonding. In a metal, the atoms release their valence electrons into a shared, mobile "sea" of electrons that flows freely among a lattice of positive metal ions. This simple picture elegantly explains the signature properties of metals. They conduct electricity and heat superbly because the mobile electrons carry charge and energy through the whole sample. They are malleable and ductile because the ion cores can slide past one another while the electron sea simply flows to accommodate them, without breaking rigid directional bonds the way a brittle ionic crystal would. And their shiny luster comes from those free electrons interacting with light. So while this lesson focuses on ionic and covalent bonding, remember that metals form a distinct and enormously important third category, and that all three types are ultimately just different solutions to the same problem of electrons seeking the lowest-energy arrangement.

Common misconceptions

Three errors are especially common. First, students imagine ionic compounds are made of discrete molecules, like little NaCl pairs; in reality they form extended crystal lattices, and the formula gives only the ratio of ions, not a molecular unit. Second, students think a bond must be either purely ionic or purely covalent; in truth bonding is a continuous spectrum, and most real bonds are polar covalent, somewhere in between. Third, students believe ionic compounds always conduct electricity; they conduct only when melted or dissolved, because only then are the ions free to move, while the solid holds its ions locked in place.

Recap

Atoms bond to reach lower energy, guided by the octet rule. When electronegativities differ greatly (a metal plus a nonmetal), electrons transfer and oppositely charged ions attract in an ionic bond, building a crystal lattice that makes the compound a hard, high-melting solid that conducts only when molten or dissolved. When electronegativities are similar (two nonmetals), atoms share electron pairs in covalent bonds, forming discrete molecules with lower melting points that generally do not conduct. Sharing can be equal (nonpolar covalent) or unequal (polar covalent, with partial charges), and the electronegativity difference places any bond along a smooth spectrum from nonpolar covalent through polar covalent to ionic. This spectrum, and especially the partial charges of polar bonds, sets up the next lesson, where the three-dimensional shape of a molecule determines whether those bond polarities add up to a polar molecule or cancel out.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 7 "Chemical Bonding and Molecular Geometry." https://openstax.org/books/chemistry-2e/pages/7-introduction
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 8 "Bonding: General Concepts."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 8 "Basic Concepts of Chemical Bonding."
4. Pauling, L. The Nature of the Chemical Bond, 3rd ed., Cornell University Press, 1960 (origin of the electronegativity scale).
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 8 "Ionic versus Covalent Bonding." https://chem.libretexts.org

In the last lesson you learned that nonmetals share electrons in covalent bonds to build molecules. This lesson gives you the tools to draw those molecules on paper and, more remarkably, to predict their three-dimensional shapes. A Lewis structure is a two-dimensional map of a molecule's valence electrons, in which bonding pairs are drawn as lines between atoms and lone pairs (nonbonding valence electrons that belong to a single atom) are drawn as pairs of dots. These humble dot-and-line diagrams are among the most powerful tools in all of chemistry, because from a correct Lewis structure you can determine a molecule's shape, and shape in turn governs an astonishing range of real properties: boiling point, whether a substance dissolves in water, its smell, its color, and its biological activity. The difference between a molecule that fits a receptor in your nose and one that does not, or between a drug that works and one that does not, is often a matter of shape. Learning to draw Lewis structures well, and then to read shapes off them using VSEPR theory, is therefore a genuinely core skill. We finish by connecting shape back to the bond polarity of the last lesson to decide whether a whole molecule is polar.

Drawing a Lewis structure, step by step

There is a reliable procedure that works for the small molecules of general chemistry. Follow these steps in order:

1. Count the total valence electrons contributed by all atoms. Add up the group-number valence electrons of every atom. For water, H₂O, each hydrogen brings 1 and oxygen brings 6, giving 2(1) + 6 = 8 electrons total. (For an ion, add one electron for each unit of negative charge and subtract one for each unit of positive charge.)
2. Choose a central atom. Pick the least electronegative atom, which is usually the one there is only one of. Never put hydrogen in the center, because hydrogen forms only one bond. For water, oxygen is central and the two hydrogens attach to it.
3. Connect the outer atoms to the center with single bonds, each bond being one shared pair (drawn as one line, using two electrons).
4. Distribute the remaining electrons as lone pairs, starting with the outer atoms, so that every atom reaches an octet. Hydrogen is the exception: it needs only 2 electrons and is already satisfied by its single bond.
5. If the central atom still lacks an octet, move one or more lone pairs from an outer atom into the bond to form a double or triple bond, sharing more pairs until the central atom is satisfied.

Worked example 1: water. With 8 total valence electrons, place oxygen in the center and draw two O-H single bonds (using 4 electrons). The remaining 4 electrons become two lone pairs on the oxygen. Now check: each hydrogen has 2 electrons (its bond), and oxygen has 8 (two bonds plus two lone pairs). All atoms are satisfied, and all 8 electrons are used. Water is H with a bent arrangement of two bonds and two lone pairs on oxygen.

Worked example 2: carbon dioxide. CO₂ has 4 (from carbon) + 2(6) (from two oxygens) = 16 valence electrons. Put carbon in the center with an oxygen on each side. If you use only single bonds and fill the oxygens' octets, carbon ends up with only 4 electrons, short of its octet. The fix is to convert a lone pair on each oxygen into a second bond, forming two double bonds: O=C=O. Now carbon has 8 electrons (two double bonds), each oxygen has 8 (one double bond plus two lone pairs), and all 16 electrons are accounted for. This is why CO₂ contains double bonds.

Worked example 3: ammonia. NH₃ has 5 (nitrogen) + 3(1) (three hydrogens) = 8 valence electrons. Nitrogen is central with three N-H single bonds (6 electrons used); the last 2 electrons form one lone pair on nitrogen. Nitrogen thus carries three bonds and one lone pair, a detail that will prove decisive for its shape.

VSEPR predicts the three-dimensional shape

A flat Lewis structure hides the fact that molecules are three-dimensional objects. VSEPR theory (Valence Shell Electron Pair Repulsion) turns the flat drawing into a shape, and its central idea could not be simpler: the groups of electrons around a central atom all carry negative charge and repel one another, so they spread out in space to get as far apart from each other as possible. Whatever arrangement maximizes the distance between the electron groups is the shape the molecule adopts.

The key counting rule is that each electron group is one region of electron density around the central atom. A single, double, or triple bond each counts as just one group, because a multiple bond still points in a single direction from the central atom. A lone pair also counts as one group. So you count the number of atoms bonded to the central atom plus the number of lone pairs on it. That total number of groups, and how many are lone pairs, fixes the geometry:

Bonding groups	Lone pairs	Shape	Angle	Example
2	0	Linear	180°	CO2
3	0	Trigonal planar	120°	BF3
4	0	Tetrahedral	109.5°	CH4
3	1	Trigonal pyramidal	about 107°	NH3
2	2	Bent	about 104.5°	H2O

Reading the table, notice that all of methane, ammonia, and water have four electron groups around the central atom, so all three are based on the tetrahedral arrangement. What differs is how many of those groups are lone pairs, which changes the shape of the atoms you actually see and shrinks the bond angle, as the next section explains.

Worked example 4: methane. In CH₄, carbon has four bonds to hydrogen and no lone pairs, so there are four electron groups. Four groups spread out as far as possible by pointing to the corners of a tetrahedron, giving bond angles of 109.5 degrees. Methane is tetrahedral. (A flat drawing with 90-degree angles is misleading; the real molecule is three-dimensional.)

Worked example 5: ammonia. From worked example 3, nitrogen has three bonds and one lone pair, again four electron groups, so the arrangement of groups is tetrahedral. But we only "see" the three hydrogen atoms and the nitrogen, and with one corner occupied by an invisible lone pair, the visible shape is a trigonal pyramid (like a tripod). Ammonia is trigonal pyramidal.

Lone pairs squeeze the bond angles

A subtle but observable effect explains the slightly odd angles in the table. A lone pair is held by only one nucleus, whereas a bonding pair is pinned tightly between two nuclei. As a result, a lone pair spreads out and occupies more angular space than a bonding pair, and it pushes the neighboring bonding pairs closer together, shrinking their angles below the ideal value. The evidence is a beautiful, steady progression across the three four-group molecules. Methane, with four bonds and zero lone pairs, shows the perfect tetrahedral angle of 109.5 degrees. Ammonia, with three bonds and one lone pair, is squeezed to about 107 degrees. Water, with two bonds and two lone pairs, is squeezed further to about 104.5 degrees. Each additional lone pair presses the bonds a little closer, and the measured angles confirm it. This is a direct, quantitative payoff of taking lone-pair repulsion seriously.

Shape decides molecular polarity

Here is where all of this pays off and connects back to the previous lesson. Recall that a polar covalent bond has a dipole, a small separation of charge. The critical realization is that a molecule can contain polar bonds and yet be nonpolar overall, if its shape is symmetric enough that the individual bond dipoles cancel out. You must consider both the bonds and the geometry; either one alone can mislead you.

Compare two molecules with the same two polar bonds arranged differently. Carbon dioxide has two polar C=O bonds, but the molecule is linear (O=C=O), so the two equal pulls point in exactly opposite directions and cancel perfectly, like two equally matched teams in a tug-of-war. The net result is zero: CO₂ is a nonpolar molecule despite its polar bonds. Water also has two polar O-H bonds, but its bent shape means the two pulls do not point oppositely; instead they point partly the same way and add up to a net dipole, with the oxygen end negative and the hydrogen side positive. Water is therefore a polar molecule. That polarity is the single most consequential fact in all of chemistry and biology: it is why water is an outstanding solvent for salts and sugars, why it has an unusually high boiling point for so small a molecule, and why it is the medium of life.

Worked example 6: is ammonia polar? Ammonia has three polar N-H bonds and a trigonal pyramidal shape. Because the shape is not symmetric (the lone pair sits on one side), the three bond dipoles do not cancel; they add to give a net dipole pointing through the nitrogen. Ammonia is polar, which is exactly why it dissolves so readily in water. Contrast this with a hypothetical flat, symmetric arrangement, which would cancel; it is the pyramidal shape, forced by the lone pair, that makes ammonia polar.

Worked example 7: the symmetric-cancellation test. Is carbon tetrachloride, CCl₄, polar? Each C-Cl bond is polar, because chlorine is more electronegative than carbon. But the molecule is tetrahedral and perfectly symmetric, with four identical chlorine atoms pulling equally toward the four corners of a tetrahedron. Those four pulls cancel exactly, just as four equally matched teams pulling on ropes tied to a central ring leave the ring motionless. So CCl₄ is nonpolar overall despite four polar bonds, which is why it behaves as a nonpolar solvent. The rule of thumb: when the central atom has no lone pairs and all the attached atoms are identical, the symmetry cancels the dipoles and the molecule is nonpolar.

When the octet rule bends: a brief note on resonance and exceptions

Lewis structures are powerful but not absolute, and a few common cases stretch the simple rules. Some molecules cannot be drawn as a single correct structure because the bonding is spread out; ozone, O₃, is the classic example, where a double bond could be drawn on either side and the real molecule is an average of both. This averaging is called resonance, and the true structure is a blend, or resonance hybrid, of the possible drawings, with the bonding electrons delocalized over the molecule rather than fixed in one spot. Separately, a handful of atoms legitimately violate the octet rule: boron is often stable with only 6 electrons (as in BF₃), and elements in the third period and beyond, such as sulfur and phosphorus, can hold more than 8 (as in SF₆, with 12 around sulfur). These are not failures of the theory so much as reminders that the octet rule is a strong default rather than an unbreakable law, a point worth keeping in mind as you draw structures.

Common misconceptions

Three misconceptions deserve attention. First, students assume any molecule with polar bonds must itself be polar; not so, because a symmetric shape can make the bond dipoles cancel, as in linear CO₂. Second, students count a double or triple bond as two or three electron groups in VSEPR; a multiple bond counts as a single group, since all its electrons point in one direction, and only the number of directions sets the shape. Third, students think lone pairs do not affect shape; lone pairs occupy space, repel bonding pairs, and both bend molecules and shrink their angles, which is exactly why water is bent rather than linear.

Recap

A Lewis structure maps a molecule's valence electrons as bonding lines and lone-pair dots, built by counting valence electrons, choosing a central atom, forming bonds, and completing octets (adding multiple bonds if needed). VSEPR theory then predicts shape from the simple principle that electron groups repel and spread as far apart as possible, with each bond or lone pair counting as one group: two groups give linear, three trigonal planar, and four tetrahedral, while lone pairs convert tetrahedral into trigonal pyramidal (ammonia) or bent (water) and squeeze the bond angles below 109.5 degrees. Finally, shape combined with bond polarity determines molecular polarity: symmetric molecules like CO₂ are nonpolar even with polar bonds, while bent water is polar. This link from electrons to shape to polarity underlies solubility, boiling points, and much of the chemistry of the modules ahead.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 7.3-7.6 (Lewis structures, molecular structure, and polarity). https://openstax.org/books/chemistry-2e/pages/7-3-lewis-symbols-and-structures
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 8 "Bonding: General Concepts."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 9 "Molecular Geometry and Bonding Theories."
4. Gillespie, R. J., and Hargittai, I. The VSEPR Model of Molecular Geometry, Dover Publications, 2012.
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 9 "Molecular Geometry and Covalent Bonding Models." https://chem.libretexts.org

Nomenclature is the systematic set of rules for naming chemical compounds so that any chemist, anywhere in the world, reads and writes the same name the same way. Its importance is hard to overstate. Without a shared naming system, chemistry would be chaos: a compound might have a dozen local names, and a formula written in one country might be misread in another, with dangerous consequences in medicine and industry. With nomenclature, a name becomes a precise set of instructions for building a formula, and a formula can be read back into a name, a reversible code. This lesson teaches you to name and to write formulas for the two great classes of binary compounds, ionic and molecular, plus the essential polyatomic ions that appear everywhere. The single most important habit to build first is a sorting reflex: before naming anything, decide which type of compound you have, because a metal combined with a nonmetal is ionic and follows one set of rules, while two nonmetals together form a molecular (covalent) compound and follow a completely different set. Mixing the two systems is the number-one source of naming errors, and deciding the type first prevents almost all of them.

Naming ionic compounds (metal plus nonmetal)

The rule for a simple ionic compound is short: name the cation (the metal) first, using its element name unchanged, then name the anion (the nonmetal) with its ending changed to -ide. So NaCl is sodium chloride, MgO is magnesium oxide, K₂S is potassium sulfide, and CaBr₂ is calcium bromide. Notice that no numbers or prefixes appear in these names; you do not say "sodium monochloride." The reason is that the charges of the ions are fixed, so the ratio is determined automatically and does not need to be stated.

To write a formula from a name, you balance the charges so the overall compound is electrically neutral (an ionic compound must have zero net charge). The quick tool for this is the criss-cross method: the numerical size of one ion's charge becomes the subscript of the other ion. Magnesium is Mg²⁺ and chloride is Cl^-. Criss-crossing, the 2 from magnesium's charge becomes the subscript on chlorine and the 1 from chlorine's charge becomes the (unwritten) subscript on magnesium, giving MgCl₂. Check that it is neutral: one Mg²⁺ contributes +2 and two Cl^- contribute -2, for a net of zero.

Worked example 1. Write the formula for aluminum oxide. Aluminum is Al³⁺ and oxide is O^2-. Criss-crossing the charges, the 3 goes to oxygen and the 2 goes to aluminum, giving Al₂O₃. Verify: two Al³⁺ give +6 and three O^2- give -6, a net of zero. Always reduce the subscripts to the smallest whole-number ratio; here 2 and 3 share no common factor, so Al₂O₃ is final.

Metals with more than one charge: Roman numerals

Many transition metals (and a few others) can form more than one possible ionic charge, so their names must specify which one is present. We do this with a Roman numeral in parentheses giving the charge on the metal ion. Iron, for instance, can be Fe²⁺ or Fe³⁺, so FeCl₂ is iron(II) chloride and FeCl₃ is iron(III) chloride. Similarly, copper can be Cu⁺ (copper(I)) or Cu²⁺ (copper(II)), and tin can be tin(II) or tin(IV). The Roman numeral is not optional for these metals; it is the only thing that tells you which compound is meant, and leaving it out makes the name ambiguous. A key point of confusion to avoid: the Roman numeral gives the charge on the metal, not the number of atoms present. Iron(III) means Fe³⁺; the actual number of atoms in the formula still comes from balancing charges.

Worked example 2. Write the formula for iron(III) oxide. The Roman numeral tells you iron is Fe³⁺, and oxide is O^2-. Criss-cross the 3 and the 2 to get Fe₂O₃ (the compound in rust). Working the other direction, if you are given the formula FeO and asked to name it, the single O^2- means the iron must be +2 to balance, so FeO is iron(II) oxide.

A handful of common metals essentially always show one charge, and by convention we skip the Roman numeral for them: the Group 1 metals are always +1, the Group 2 metals always +2, aluminum is always +3, and zinc is always +2 and silver always +1. So we write "zinc chloride," not "zinc(II) chloride," even though it is technically unambiguous either way.

Polyatomic ions

A polyatomic ion is a charged group of atoms covalently bonded together that travels through reactions as a single unit and carries an overall charge. These appear constantly in chemistry, so a core handful are worth memorizing cold, including both the formula and the charge:

Name	Formula	Charge
Ammonium	NH4+	+1
Hydroxide	OH-	-1
Nitrate	NO3-	-1
Carbonate	CO32-	-2
Sulfate	SO42-	-2
Phosphate	PO43-	-3

Most polyatomic ions are anions (ammonium is the important cationic exception), and many end in -ate. Compounds containing them are named just like other ionic compounds: cation first, then the polyatomic ion by its own name (its ending is not changed to -ide). So Na₂SO₄ is sodium sulfate, NH₄Cl is ammonium chloride, and KOH is potassium hydroxide.

A special rule of formula-writing applies when you need more than one of a polyatomic ion: enclose the whole ion in parentheses and place the subscript outside. Calcium nitrate is Ca(NO₃)₂, because Ca²⁺ needs two NO₃^- ions to balance its +2 charge, and the parentheses keep each intact nitrate unit together while the subscript 2 multiplies the whole thing. Writing it without parentheses would be wrong and ambiguous, because the reader could not tell whether the 2 applies to just the oxygen or to the entire nitrate group.

Worked example 3. Write the formula for iron(III) sulfate. Iron is Fe³⁺ (from the Roman numeral) and sulfate is SO₄^2-. Criss-crossing the 3 and 2, you need two iron ions and three sulfate ions: Fe₂(SO₄)₃. Verify neutrality: two Fe³⁺ give +6, three SO₄^2- give -6, net zero. The sulfate stays wrapped in parentheses because there is more than one of it.

Naming molecular compounds (two nonmetals)

When two nonmetals combine, the rules change completely. We do not balance charges, because the atoms share electrons rather than transferring them, and there are no fixed ionic charges to work from. Instead we use Greek prefixes to state explicitly how many of each atom is present. The common prefixes are mono (1), di (2), tri (3), tetra (4), penta (5), hexa (6), and on upward. The first element keeps its full name, and the second element takes the -ide ending; both may carry a prefix to show their count. Two small conventions: a leading "mono-" on the first element is always dropped (so CO is carbon monoxide, not monocarbon monoxide), and the trailing "a" or "o" of a prefix is often dropped before a vowel for smoother pronunciation (so it is "tetroxide," not "tetraoxide," and "monoxide," not "monooxide").

Examples make the pattern clear: CO is carbon monoxide, CO₂ is carbon dioxide, N₂O is dinitrogen monoxide (laughing gas), NO₂ is nitrogen dioxide, N₂O₄ is dinitrogen tetroxide, and SF₆ is sulfur hexafluoride. The prefixes are essential here precisely because two nonmetals can combine in many different ratios, and the prefix is the only thing distinguishing, say, carbon monoxide (a deadly gas) from carbon dioxide (the gas you exhale).

Worked example 4. Name P₂O₅. Both phosphorus and oxygen are nonmetals, so this is a molecular compound and we use prefixes. Two phosphorus atoms give "diphosphorus," and five oxygen atoms give "pentoxide" (dropping the a of penta before the o). The name is diphosphorus pentoxide.

The most common naming mistake

The single most frequent error, by a wide margin, is mixing the two systems, especially by using prefixes in an ionic name. Ionic names never use prefixes, because the fixed ionic charges already determine the ratio; CaCl₂ is simply calcium chloride, never "calcium dichloride." Molecular names always use prefixes, because two nonmetals can combine in several ratios and the prefix is the only way to tell them apart. The reliable defense is the sorting reflex from the start of this lesson: look at the compound, decide metal-plus-nonmetal (ionic, no prefixes, use Roman numerals for variable-charge metals) versus nonmetal-plus-nonmetal (molecular, use prefixes), and then commit fully to the matching rule set. Do that consistently and the great majority of naming mistakes simply never happen.

Naming acids

Acids form an important special category worth previewing, because they follow their own patterns tied to the anion they contain. An acid can be recognized by a formula that begins with hydrogen (in water, it releases H⁺ ions). Three patterns cover the common cases. An acid based on a simple -ide anion takes the form hydro...ic acid: HCl dissolved in water is hydrochloric acid, and H₂S is hydrosulfuric acid. An acid based on a polyatomic ion ending in -ate becomes an -ic acid: HNO₃ (from nitrate) is nitric acid and H₂SO₄ (from sulfate) is sulfuric acid. An acid based on a polyatomic ion ending in -ite becomes an -ous acid: HNO₂ (from nitrite) is nitrous acid. A helpful memory aid is "ate becomes ic, ite becomes ous." These acids are among the most-used chemicals in the world, so their names appear constantly.

Reading a formula back into a name

Nomenclature is meant to work in both directions, and translating a formula into a name is just as important as the reverse. Worked example 5. Name Cu₃(PO₄)₂. The metal copper can take more than one charge, so this is an ionic compound needing a Roman numeral, and it contains the polyatomic ion phosphate, PO₄^3-. To find copper's charge, balance the total: two phosphate ions carry a total of 2 × (-3) = -6, so the three copper ions must total +6, which means each copper is +2, that is copper(II). The name is copper(II) phosphate. Worked example 6. Name SO₃. Both sulfur and oxygen are nonmetals, so this is molecular and uses prefixes: one sulfur (no "mono" on the first element) and three oxygens give sulfur trioxide. Working formulas and names in both directions until it is automatic is the surest sign you have mastered nomenclature.

Common misconceptions

Three misconceptions recur. First, students use di- and tri- prefixes in ionic names; ionic names never take prefixes, since the fixed charges fix the ratio, so CaCl₂ is calcium chloride. Second, students read the Roman numeral as a count of atoms; it gives the charge on the metal ion (iron(III) means Fe³⁺), and the atom count comes from balancing charges. Third, students split polyatomic ions apart when writing formulas; a polyatomic ion stays together as a unit, wrapped in parentheses when more than one is needed, as in Ca(NO₃)₂.

Recap

Naming starts with sorting: metal plus nonmetal is ionic, nonmetal plus nonmetal is molecular. Ionic compounds are named cation first (unchanged) then anion with an -ide ending, with no prefixes, and a Roman numeral is added for metals of variable charge to state the metal's charge. Formulas follow from balancing charges to neutrality, often via the criss-cross method, and polyatomic ions are named by their own names and kept in parentheses when a formula needs more than one. Molecular compounds instead use Greek prefixes to count each atom, giving names like dinitrogen tetroxide. The overarching goal is a reversible code in which a name reconstructs a formula and a formula reconstructs a name. These naming skills are prerequisites for the equations and stoichiometry of Module 5, where you must read and write formulas fluently to balance reactions.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 2.6 "Molecular and Ionic Compounds" and Chapter 2.7 "Chemical Nomenclature." https://openstax.org/books/chemistry-2e/pages/2-7-chemical-nomenclature
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 2 "Atoms, Molecules, and Ions" (nomenclature section).
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 2.8 "Naming Inorganic Compounds."
4. International Union of Pure and Applied Chemistry (IUPAC), Nomenclature of Inorganic Chemistry (Recommendations 2005). https://iupac.org/what-we-do/nomenclature/
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 2.5-2.6 (ionic and molecular nomenclature). https://chem.libretexts.org

Atoms are unimaginably small and stupefyingly numerous. A single drop of water contains more molecules than there are stars in the observable universe, and there is no balance on Earth that can weigh one atom directly. Yet chemistry is a quantitative science: to run a reaction, a chemist needs to combine substances in exact numerical ratios of atoms and molecules. How can you count particles you cannot see and cannot handle one at a time? The answer is the mole (abbreviated mol), the single most important counting concept in all of chemistry and the bridge that connects the invisible atomic world to the visible, weighable world of grams. This lesson defines the mole and Avogadro's number, shows how to compute the molar mass of any substance from its formula, and teaches you to convert fluently among the three quantities you will use constantly: mass in grams, amount in moles, and number of particles. Everything in Module 5, all of stoichiometry, rests entirely on the mole reasoning you build here, so it is worth mastering completely.

The mole is a counting unit

The idea behind the mole is exactly like the idea behind a "dozen." A dozen always means 12 of something, whether eggs, roses, or doughnuts. In the same way, a mole always means one specific, fixed number of particles, no matter what the particles are. That number is Avogadro's number, approximately 6.022 × 10²³. So one mole of carbon atoms is 6.022 × 10²³ carbon atoms, one mole of water molecules is 6.022 × 10²³ water molecules, and one mole of electrons is 6.022 × 10²³ electrons.

This number is almost impossible to grasp intuitively, it is so large. If you had a mole of marbles, they would cover the entire surface of the Earth to a depth of many kilometers. A mole of seconds is vastly longer than the age of the universe. Why choose such a colossal number as the chemist's counting unit? Because atoms are so tiny that you need an astronomically large number of them to add up to any weighable amount of material. Avogadro's number is calibrated to be exactly the right size to bridge the two worlds, so that a mole of a substance is an amount you can actually hold in your hand and weigh on a laboratory balance.

Why 6.022 x 10^23?

Avogadro's number is not an arbitrary round figure; it is chosen with exquisite care so that one mole of a substance has a mass in grams numerically equal to that substance's atomic or molecular mass in atomic mass units (amu). This is the elegant payoff of the amu definition you met back in the atomic-structure lesson. One carbon-12 atom has a mass of exactly 12 amu, and, by the design of the mole, one mole of carbon-12 atoms has a mass of exactly 12 grams. One oxygen atom is about 16 amu, and one mole of oxygen atoms is about 16 grams. This single design choice is the beating heart of quantitative chemistry, because it means you can read an atomic mass off the periodic table, call it grams per mole, weigh out that many grams on an ordinary balance, and know with confidence that you are holding exactly one mole (that is, 6.022 × 10²³) of atoms. The mole turns the ability to weigh into the ability to count.

Molar mass

The molar mass of a substance is the mass of one mole of it, expressed in grams per mole (g/mol). For a pure element, the molar mass is simply the atomic mass read directly off the periodic table: carbon is 12.01 g/mol, iron is 55.85 g/mol, and so on. For a compound, you compute the molar mass by adding up the molar masses of every atom in the chemical formula, counting each atom the number of times it appears.

Worked example 1. Find the molar mass of water, H₂O. From the periodic table, hydrogen is about 1.008 g/mol and oxygen is about 16.00 g/mol. The formula has two hydrogens and one oxygen, so:

2 × 1.008 + 1 × 16.00 = 2.016 + 16.00 = 18.02 g/mol

So one mole of water, a staggering 6.022 × 10²³ molecules, weighs just 18.02 grams, which is only about a tablespoon. That vividly illustrates how tiny and numerous molecules are.

Worked example 2. Find the molar mass of carbon dioxide, CO₂. It contains one carbon and two oxygens: 12.01 + 2(16.00) = 12.01 + 32.00 = 44.01 g/mol.

Worked example 3 (with parentheses). Find the molar mass of calcium nitrate, Ca(NO₃)₂. The subscript 2 outside the parentheses multiplies everything inside, so the formula contains one calcium, two nitrogens, and six oxygens (2 × 3). Adding up: 1(40.08) + 2(14.01) + 6(16.00) = 40.08 + 28.02 + 96.00 = 164.10 g/mol. The most common error here is forgetting to multiply the atoms inside the parentheses by the outside subscript, so always distribute it carefully.

Converting among grams, moles, and particles

Molar mass and Avogadro's number are the two conversion factors that link the three quantities you care about most. Molar mass connects grams to moles, and Avogadro's number connects moles to particles. The core relationships are:

• moles = grams ÷ molar mass
• grams = moles × molar mass
• particles = moles × 6.022 × 10²³
• moles = particles ÷ 6.022 × 10²³

The single most important structural fact to absorb is that moles sit in the middle of every conversion. There is no direct one-step route from grams to number of particles; you must always pass through moles. Picture the mole as a central hub, a train station, with grams on one side (reached via molar mass) and number of particles on the other (reached via Avogadro's number). To travel from grams to particles you first ride into the mole station, then out the other side. Keeping this hub picture in mind prevents the common mistake of trying to multiply grams directly by Avogadro's number.

Worked example 4. How many molecules are in 36.0 g of water? First convert grams to moles by dividing by the molar mass: 36.0 g ÷ 18.02 g/mol = 2.00 mol. Then convert moles to molecules by multiplying by Avogadro's number: 2.00 mol × 6.022 × 10²³ molecules/mol = 1.20 × 10²⁴ molecules. Set problems like this up with the dimensional analysis you learned in Module 1, writing each factor so the unwanted unit cancels: grams cancel to leave moles, then moles cancel to leave molecules. When you trust the units to guide the setup, even long chains become routine.

Worked example 5 (going backward). A sample contains 3.011 × 10²³ atoms of carbon. What is its mass? First convert particles to moles by dividing by Avogadro's number: 3.011 × 10²³ atoms ÷ 6.022 × 10²³ atoms/mol = 0.500 mol. Then convert moles to grams by multiplying by carbon's molar mass: 0.500 mol × 12.01 g/mol = 6.01 g. Notice the path runs particles to moles to grams, passing through the mole hub, just in the reverse direction from the previous example.

Worked example 6 (a fuller chain). How many oxygen atoms are in 22.0 g of carbon dioxide? Convert grams to moles of CO₂: 22.0 g ÷ 44.01 g/mol = 0.500 mol CO₂. Convert to molecules: 0.500 mol × 6.022 × 10²³ = 3.011 × 10²³ molecules of CO₂. Finally, each CO₂ molecule contains 2 oxygen atoms, so multiply by 2 to get 6.022 × 10²³ oxygen atoms. This kind of multi-step reasoning, always routed through moles, is exactly the skill that makes stoichiometry possible.

Percent composition: what a compound is made of

Molar mass unlocks another useful quantity, the percent composition of a compound, which is the percentage by mass contributed by each element. You find it by dividing the total mass of each element in one mole of the compound by the compound's molar mass, then multiplying by 100. Worked example 7. What is the percent composition of water, H₂O (molar mass 18.02 g/mol)? The two hydrogens contribute 2(1.008) = 2.016 g, and the one oxygen contributes 16.00 g. So hydrogen is (2.016 / 18.02) × 100 = 11.19 percent and oxygen is (16.00 / 18.02) × 100 = 88.79 percent. Notice the two percentages add to essentially 100 percent, a built-in check. This is the quantitative version of the law of definite proportions from Module 1: water is always about 11 percent hydrogen and 89 percent oxygen by mass, no matter its source. Percent composition is used in industry to assess the purity of materials and the richness of ores, and to verify that a synthesized compound has the intended formula.

A sense of scale: how big is a mole, really?

It is worth dwelling on the sheer size of Avogadro's number, because grasping it makes the mole feel less like an abstract rule. If you had one mole of pennies and distributed them equally among all roughly 8 billion people on Earth, each person would receive about 75 trillion dollars' worth. One mole of ordinary drops of water would fill a cube of water many kilometers on each side. And yet, remarkably, one mole of water is just 18 grams, a mere tablespoon that fits in the palm of your hand. This staggering contrast, an astronomical count of molecules weighing only a few grams, is the whole reason the mole is indispensable: molecules are so tiny that only Avogadro's number of them adds up to a weighable, workable amount. Whenever the mole feels arbitrary, remember that it is the exact bridge between the unimaginably small (a single molecule) and the everyday (a spoonful you can measure on a balance).

Common misconceptions

Three misconceptions cause most mole errors. First, students think a mole is a unit of mass like the gram; a mole is a count, a fixed number of particles (6.022 × 10²³), just as a dozen is a count of 12. Second, students believe one mole of any substance weighs the same; a mole always contains the same number of particles, but its mass depends on the substance (a mole of water is 18 g, a mole of CO₂ is 44 g). Third, students try to convert grams straight to particles; you must go through moles, using molar mass to reach moles and Avogadro's number to reach particles, with the mole as the required stepping stone.

Recap

The mole is chemistry's counting unit: one mole is Avogadro's number, 6.022 × 10²³, of particles, chosen so that a substance's molar mass in grams per mole equals its atomic or molecular mass in amu. Molar mass is found by summing the atomic masses of all atoms in a formula, carefully distributing any subscript outside parentheses. The mole sits at the center of every conversion, linking grams (via molar mass) to particles (via Avogadro's number), so mass-to-particle problems always pass through moles. Handled with dimensional analysis, these conversions become mechanical. The mole is the indispensable foundation for the next module, where the coefficients of a balanced equation give mole ratios that let you predict exactly how much product a reaction will yield.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 3 "Composition of Substances and Solutions" (the mole and molar mass). https://openstax.org/books/chemistry-2e/pages/3-introduction
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 3 "Stoichiometry."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 3.4 "Avogadro's Number and the Mole."
4. National Institute of Standards and Technology (NIST), "The mole and the revised SI." https://www.nist.gov/si-redefinition/mole
5. International Bureau of Weights and Measures (BIPM), The International System of Units (SI Brochure), 9th ed., 2019 (definition of the mole). https://www.bipm.org/en/publications/si-brochure

You can now name compounds, write their formulas, and count particles with the mole. This lesson brings those skills together to describe the central event of chemistry itself: the chemical reaction. A chemical reaction rearranges atoms, breaking old bonds and forming new ones, to turn one set of substances into another with different properties. Burning, rusting, digesting, and photosynthesizing are all chemical reactions. We record a reaction in a compact symbolic form called a chemical equation, and the great task of this lesson is learning to balance those equations so they honor the fundamental law that atoms are never created or destroyed. Balancing is a skill you will use in every remaining topic of the course, because a correct balanced equation is the starting point for all of stoichiometry, so this lesson is worth practicing until it is second nature. We also survey the common types of reactions, so you can recognize patterns and even predict products.

The anatomy of a chemical equation

A chemical equation is a sentence written in the language of formulas. The reactants (the starting materials) go on the left, the products (the new substances formed) go on the right, and an arrow between them reads "yields" or "produces." For example, the equation for hydrogen burning in oxygen to form water is written 2 H₂ + O₂ yields 2 H₂O, with the arrow standing for "yields." A plus sign separates multiple reactants or multiple products.

Equations often include small italic letters in parentheses called state symbols, which tell you the physical phase of each substance: (s) for solid, (l) for liquid, (g) for gas, and (aq) for aqueous, meaning dissolved in water. So a fuller version might read 2 H₂(g) + O₂(g) yields 2 H₂O(l). The state symbols carry real information: whether a product forms as a solid (a precipitate) or a gas is often the visible sign that a reaction has occurred, and it matters for later work with solutions. A well-written, balanced equation is a precise, information-dense statement of exactly what a chemical change does.

The law of conservation of mass

The rule that governs every equation is the law of conservation of mass: in a chemical reaction, atoms are neither created nor destroyed, only rearranged. This principle was established by the French chemist Antoine Lavoisier in the late eighteenth century through meticulous weighing of reactants and products in sealed vessels; he showed that the total mass before and after a reaction is identical. Because matter is conserved, every atom that appears on the reactant side of an equation must reappear, in exactly the same total number, on the product side. If four hydrogen atoms go in, four must come out; they cannot vanish or multiply.

To make an equation obey this law, we adjust coefficients, the whole numbers written in front of formulas, until the count of every kind of atom is equal on both sides. A coefficient multiplies the entire formula that follows it: 2 H₂O means two water molecules, containing four hydrogen atoms and two oxygen atoms in all. The one thing you must never do is change the subscripts inside a formula, because a subscript is part of the substance's chemical identity. Changing H₂O to H₂O₂ does not balance an equation; it silently replaces water with hydrogen peroxide, a completely different and rather dangerous chemical. Balance only by placing coefficients in front of whole, correct formulas; never touch the subscripts.

Balancing step by step

Consider the combustion of propane, C₃H₈, the fuel in many gas grills. The unbalanced skeleton equation, which shows only what reacts and what forms but not yet in the right numbers, is:

C₃H₈ + O₂ → CO₂ + H₂O

Work through the atoms in a sensible order, saving the free element oxygen for last:

1. Carbon: there are 3 carbon atoms on the left in C₃H₈, but only 1 on the right in CO₂. Put a coefficient of 3 in front of CO₂. Now there are 3 carbon atoms on each side.
2. Hydrogen: there are 8 hydrogen atoms on the left in C₃H₈, but only 2 on the right in one H₂O. Put a 4 in front of H₂O, giving 4 × 2 = 8 hydrogen atoms. Now hydrogen balances at 8 on each side.
3. Oxygen: count the oxygen atoms now fixed on the right: 3 CO₂ contributes 3 × 2 = 6, and 4 H₂O contributes 4 × 1 = 4, for a total of 10 oxygen atoms. Put a 5 in front of O₂ on the left to supply 5 × 2 = 10. Oxygen now balances at 10 on each side.

The balanced equation is:

C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O

A final check confirms it: 3 carbon, 8 hydrogen, and 10 oxygen atoms on each side. The reason oxygen is saved for last is that it appears as a free element (O₂) whose coefficient can be tuned freely without disturbing any other element's balance.

Worked example 1: the Haber process. Balance N₂ + H₂ yields NH₃. Nitrogen appears as N₂ (2 atoms) on the left and 1 in each NH₃, so put a 2 in front of NH₃. That gives 2 × 3 = 6 hydrogen atoms on the right, so put a 3 in front of H₂ to supply 6 hydrogen atoms on the left. The balanced equation is N₂ + 3 H₂ yields 2 NH₃. Check: 2 nitrogen and 6 hydrogen on each side. This is the industrially vital reaction that fixes atmospheric nitrogen into ammonia for fertilizer, helping feed much of the world.

Worked example 2: clearing a fraction. Balance the combustion of ethane, C₂H₆ + O₂ yields CO₂ + H₂O. Carbon: 2 on the left, so put 2 CO₂. Hydrogen: 6 on the left, so put 3 H₂O. Oxygen on the right now totals 2(2) + 3(1) = 7 atoms, which would need 3.5 O₂, a fraction. To clear it, multiply the entire equation by 2: 2 C₂H₆ + 7 O₂ yields 4 CO₂ + 6 H₂O. Check: 4 carbon, 12 hydrogen, and 14 oxygen on each side. Doubling to remove a fraction is a standard and legitimate move.

Strategy and good habits

Balancing is part logic and part practice, but a few habits make it far easier and less error-prone. Balance the most complicated molecule first and save pure elements like O₂ or H₂ for last, because a lone element's coefficient can absorb whatever leftover count you need without disturbing anything else. Treat any polyatomic ion that survives the reaction unchanged (such as sulfate or nitrate appearing intact on both sides) as a single unit and balance it as one block rather than atom by atom, which saves considerable effort. When you finish, always recount every element on both sides to confirm the balance, and reduce the coefficients to the smallest whole-number ratio (dividing through by any common factor). If an odd count forces a fraction, double the entire equation to clear it, as in worked example 2.

Common reaction types

Reactions fall into recognizable families, and learning to spot a reaction's type helps you anticipate its products and check that your equation makes sense.

• Combination (synthesis): two or more reactants join into a single product, as in 2 H₂ + O₂ → 2 H₂O. The signature is many reactants becoming one product.
• Decomposition: a single compound breaks apart into two or more simpler substances, as in 2 H₂O₂ → 2 H₂O + O₂ (hydrogen peroxide breaking down). This is the reverse pattern of synthesis.
• Single replacement: one element displaces another from a compound, as in Zn + 2 HCl → ZnCl₂ + H₂, where zinc pushes hydrogen out of hydrochloric acid.
• Double replacement: two compounds swap partner ions, often producing a precipitate or a gas, as in AgNO₃ + NaCl → AgCl + NaNO₃, where solid silver chloride precipitates out.
• Combustion: a fuel, usually a hydrocarbon, burns in oxygen to produce carbon dioxide and water while releasing energy, as in the propane example above. Combustion is the reaction behind engines, furnaces, and campfires.

Worked example 3 (a double-replacement precipitation). Balance the reaction of lead(II) nitrate with potassium iodide, which forms a brilliant yellow precipitate: Pb(NO₃)₂ + KI → PbI₂ + KNO₃. Treat the nitrate as an intact block. Lead: 1 on each side, fine. Iodine: 2 in PbI₂ on the right, so put 2 KI on the left. That gives 2 potassium on the left, so put 2 KNO₃ on the right, which also makes 2 nitrate groups on each side. The balanced equation is Pb(NO₃)₂ + 2 KI → PbI₂ + 2 KNO₃. Notice how treating nitrate as a single unit, rather than balancing nitrogen and oxygen separately, made the work far easier.

Reactions in solution: molecular, ionic, and net ionic equations

Many reactions happen between dissolved ionic compounds, and chemists have three ways to write them, each showing a different level of detail. The molecular equation writes every substance as a complete formula, as in the lead iodide example above. The complete ionic equation shows every strong electrolyte that is dissolved (marked aq) as its separate ions, because that is how it truly exists in solution; only solids, liquids, and gases stay written as whole formulas. The net ionic equation then removes the ions that appear unchanged on both sides, called spectator ions, leaving only the species that actually take part in the chemical change. For the lead iodide reaction, potassium and nitrate are spectators (they float freely before and after), so the net ionic equation is simply Pb²⁺(aq) + 2 I^-(aq) → PbI₂(s). This stripped-down equation reveals the essence of the reaction: lead ions and iodide ions combine to form a solid. Net ionic equations are widely used because they focus attention on the real chemistry and ignore the ions that are just along for the ride.

Common misconceptions

Three misconceptions cause most balancing errors. First, students try to balance by changing subscripts; never change subscripts, because that changes the substances themselves (turning O₂ into O₃ swaps oxygen for ozone), and balancing is done only with coefficients in front of whole formulas. Second, students think balancing changes how much matter there is; balancing simply enforces conservation of mass, keeping every kind of atom equal on both sides, and coefficients bookkeep atoms rather than create or destroy them. Third, students believe an equation is wrong unless it uses the smallest coefficients; any set of coefficients that balances every atom is chemically valid, though by convention we reduce to the smallest whole-number ratio.

Recap

A chemical equation lists reactants on the left and products on the right, joined by a yields-arrow, with optional state symbols giving each substance's phase. The law of conservation of mass requires that every kind of atom appear in equal numbers on both sides, which we achieve by adjusting coefficients (whole numbers in front of formulas) and never by changing subscripts. A reliable strategy is to balance complex molecules first, save free elements for last, treat intact polyatomic ions as blocks, and double the equation to clear any fraction. Finally, reactions sort into families, synthesis, decomposition, single replacement, double replacement, and combustion, that help you recognize and predict chemistry. A correct balanced equation is the essential launchpad for the next lesson, stoichiometry, which turns those coefficients into quantitative predictions of how much reactant is consumed and how much product forms.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 4 "Stoichiometry of Chemical Reactions" (writing and balancing equations). https://openstax.org/books/chemistry-2e/pages/4-1-writing-and-balancing-chemical-equations
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapters 3 and 4 (chemical equations and reactions).
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 3.1 "Chemical Equations."
4. Encyclopaedia Britannica, "Antoine Lavoisier" and "Conservation of mass." https://www.britannica.com/biography/Antoine-Lavoisier
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 3 "Chemical Reactions." https://chem.libretexts.org

Everything in this course so far has been building toward this lesson. You can classify matter, structure atoms, predict bonding, name compounds, count with the mole, and balance equations. Stoichiometry puts all of it to work as the arithmetic of reactions: it uses a balanced equation to predict how much product will form from a given amount of reactant, or how much reactant is needed to make a target amount of product. This is the moment chemistry becomes genuinely predictive and industrially indispensable. A pharmaceutical company must know exactly how much of each ingredient to combine to make a batch of medicine; a rocket engineer must know precisely how much fuel and oxidizer to load. The central insight that makes all of this possible is simple and powerful: the coefficients of a balanced equation give the mole ratio in which substances react and form. This lesson develops mole ratios, mass-to-mass calculations, the crucial idea of the limiting reactant, and the measurement of a reaction's efficiency through percent yield.

The mole ratio: the heart of stoichiometry

Recall the balanced Haber-process equation, N₂ + 3 H₂ → 2 NH₃. Read at the molecular level, the coefficients say that 1 molecule of nitrogen reacts with 3 molecules of hydrogen to make 2 molecules of ammonia. But because a mole is just a fixed count of molecules, the very same coefficients also read at the mole level: 1 mole of N₂ reacts with 3 moles of H₂ to make 2 moles of NH₃. This is the profound convenience of the mole. The coefficients you found by balancing are directly the mole ratios of the reaction.

From those coefficients you can write mole ratios as conversion factors linking any two substances in the reaction, such as "3 mol H₂ per 1 mol N₂," "2 mol NH₃ per 1 mol N₂," or "2 mol NH₃ per 3 mol H₂." Each of these can be flipped as needed so that the unit you want to cancel sits in the denominator, exactly as in the dimensional analysis of Module 1. The one caution that cannot be repeated too often: the coefficients give ratios of moles (or molecules), never ratios of grams. This is precisely why the mole concept of the last module is indispensable here; you cannot do stoichiometry directly with masses.

Worked example 1 (mole to mole). How many moles of ammonia form from 2.0 mol of nitrogen, assuming plenty of hydrogen? Use the mole ratio from the balanced equation, 2 mol NH₃ per 1 mol N₂: 2.0 mol N₂ × (2 mol NH₃ / 1 mol N₂) = 4.0 mol NH₃. The nitrogen units cancel, leaving moles of ammonia.

Mass-to-mass calculations

Most real problems start and end in grams, because grams are what a balance actually measures. You cannot go straight from grams of one substance to grams of another, though, because the mole ratio only works in moles. The general path therefore has three steps, and, just as in the mole lesson, the mole sits in the middle of every one:

grams of A → moles of A → moles of B → grams of B

The first step (grams of A to moles of A) uses the molar mass of A. The middle step (moles of A to moles of B) uses the mole ratio from the balanced equation, the only step that involves the reaction itself. The final step (moles of B to grams of B) uses the molar mass of B. Memorize this three-step skeleton and mass-to-mass problems become mechanical.

Worked example 2 (mass to mass). For the combustion of propane, C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O, how many grams of CO₂ form from 22.0 g of propane? Use molar masses C₃H₈ = 44.10 g/mol and CO₂ = 44.01 g/mol.

1. Grams to moles of propane: 22.0 g ÷ 44.10 g/mol = 0.499 mol C₃H₈.
2. Mole ratio (3 mol CO₂ per 1 mol C₃H₈): 0.499 mol C₃H₈ × (3 mol CO₂ / 1 mol C₃H₈) = 1.50 mol CO₂.
3. Moles to grams of CO₂: 1.50 mol × 44.01 g/mol = 65.9 g CO₂.

So 22.0 g of propane produces about 65.9 g of carbon dioxide. A subtle but important observation: mass is conserved overall (the propane plus the oxygen consumed together weigh exactly as much as the CO₂ plus water produced), yet the mass of any single product does not equal the mass of the single reactant, because it is the mole ratio, not any mass ratio, that governs the conversion. Getting more grams of CO₂ (65.9) than grams of propane (22.0) is perfectly correct, because the oxygen also contributes mass to the carbon dioxide.

Limiting reactants

Real reactions rarely begin with a perfect stoichiometric ratio of ingredients; usually one reactant is present in excess and another runs short. The limiting reactant (or limiting reagent) is the one that runs out first, and it alone determines how much product can form. Once the limiting reactant is used up, the reaction stops, no matter how much of the other reactant remains. Whatever is left over of the other reactant is said to be in excess.

A kitchen analogy makes the idea concrete. Suppose a sandwich requires 2 slices of bread and 1 slice of cheese, which we could write as 2 bread + 1 cheese yields 1 sandwich. If you have 10 slices of bread but only 3 slices of cheese, how many sandwiches can you make? The bread could make 5 sandwiches, but the cheese can make only 3, so you make just 3 sandwiches. Cheese is the limiting ingredient, and you are left with 4 slices of bread in excess. The limiting reactant is not simply whichever you have less of in raw count; it is whichever runs out first once the recipe's ratio is taken into account.

The reliable method is to calculate how much product each reactant could make on its own, and the smaller answer wins; that reactant is limiting. Worked example 3. For N₂ + 3 H₂ → 2 NH₃, suppose you start with 4.0 mol N₂ and 9.0 mol H₂. From nitrogen: 4.0 mol N₂ × (2 mol NH₃ / 1 mol N₂) = 8.0 mol NH₃. From hydrogen: 9.0 mol H₂ × (2 mol NH₃ / 3 mol H₂) = 6.0 mol NH₃. Hydrogen yields less product, so hydrogen is the limiting reactant, only 6.0 mol of NH₃ can form, and nitrogen is in excess. Notice that hydrogen is limiting even though there are more moles of it than of nitrogen (9.0 versus 4.0); the mole ratio, not the raw amount, is what matters. The rule to internalize: whenever two starting amounts are given, always test for the limiting reactant before trusting any single-reactant calculation.

Worked example 4 (limiting reactant from grams). Consider 2 H₂ + O₂ → 2 H₂O with 4.0 g of H₂ (molar mass 2.02 g/mol) and 32.0 g of O₂ (molar mass 32.00 g/mol). Convert to moles: hydrogen is 4.0 / 2.02 = 1.98 mol, oxygen is 32.0 / 32.00 = 1.00 mol. From hydrogen: 1.98 mol H₂ × (2 mol H₂O / 2 mol H₂) = 1.98 mol water. From oxygen: 1.00 mol O₂ × (2 mol H₂O / 1 mol O₂) = 2.00 mol water. Hydrogen makes less, so hydrogen is limiting and about 1.98 mol of water forms. Always convert to moles first when the problem gives masses.

Theoretical yield and percent yield

The amount of product predicted by stoichiometry from the limiting reactant is called the theoretical yield: it is the maximum you could possibly obtain if everything went perfectly. In the real world, reactions rarely go perfectly. Some product is lost during transfer and purification, some reactant fails to react, and competing side reactions may consume material. The actual yield is the amount you really collect in the lab, and it is almost always less than the theoretical yield. The efficiency of a reaction is captured by the percent yield:

percent yield = (actual yield ÷ theoretical yield) × 100%

Worked example 5. If the theoretical yield of carbon dioxide from a reaction is 65.9 g but you actually isolate only 58.0 g, the percent yield is (58.0 g ÷ 65.9 g) × 100% = 88.0%. A percent yield of 88 percent means the reaction was fairly efficient, capturing most of the possible product. Chemists work hard to push percent yields higher, because in industry every percentage point represents saved money, saved raw material, and less waste. One important sanity check: a correct percent yield can never exceed 100 percent, because you cannot collect more product than the theoretical maximum. A calculated value above 100 percent signals an error, most often impure or still-wet product that weighs more than the pure substance should.

Stoichiometry in solution: titration

Stoichiometry is not confined to solids weighed on a balance; it extends naturally to reactions in solution, and this is where one of chemistry's most important techniques lives. Recall from the solutions work ahead that the moles of a dissolved reactant equal its molarity times its volume in liters (moles = M × V). That relationship lets you start a stoichiometry problem from a measured volume of solution. The premier application is titration, in which a solution of precisely known concentration is added drop by drop to a solution of unknown concentration until an indicator signals that the reaction is exactly complete (the equivalence point). At that instant, the mole ratio from the balanced equation, combined with the measured volumes, pins down the unknown concentration.

Worked example 6 (a titration). Suppose 25.0 mL of hydrochloric acid of unknown concentration is exactly neutralized by 30.0 mL of 0.100 M sodium hydroxide, following HCl + NaOH → NaCl + H₂O. First find the moles of NaOH used: 0.0300 L × 0.100 mol/L = 0.00300 mol NaOH. The mole ratio is 1 to 1, so 0.00300 mol of HCl reacted. That amount was contained in 25.0 mL, so the acid's concentration is 0.00300 mol ÷ 0.0250 L = 0.120 M. Titration like this is used everywhere from checking the acidity of a swimming pool or a wine, to certifying the purity of a medicine, to measuring pollutants in water, and it is a direct, practical payoff of mastering the humble mole ratio.

Common misconceptions

Three misconceptions trip up students most. First, they assume the limiting reactant is simply whichever one there is less of; you must account for the mole ratio, and a reactant present in larger amount can still be limiting if the equation demands even more of it, so always compare how much product each reactant would make. Second, they treat the coefficients as ratios of masses; coefficients give ratios of moles, so you must convert to moles, apply the ratio, then convert back. Third, they think a percent yield above 100 percent just means an unusually efficient reaction; it actually signals an error, since the theoretical yield is a true ceiling that a correct actual yield cannot exceed.

Recap

Stoichiometry uses the mole ratios given directly by a balanced equation's coefficients to relate the amounts of reactants and products. Mole-to-mole problems apply the ratio in one step, while mass-to-mass problems follow the three-step path grams of A to moles of A (via molar mass) to moles of B (via the mole ratio) to grams of B (via molar mass), always routing through moles. When two reactant amounts are given, the limiting reactant, the one that makes the least product, determines the theoretical yield, and the leftover reactant is in excess. The percent yield, actual over theoretical times 100 percent, measures real-world efficiency and can never exceed 100 percent. These techniques are the quantitative core of chemistry, and the final module extends them to gases, solutions, and the energy that accompanies reactions.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 4.3-4.4 (reaction stoichiometry, yields, limiting reactant). https://openstax.org/books/chemistry-2e/pages/4-3-reaction-stoichiometry
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 3 "Stoichiometry."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 3.6-3.7 (limiting reactants and theoretical yields).
4. Tro, N. J. Chemistry: A Molecular Approach, 5th ed., Pearson, 2020, Chapter 4 "Chemical Quantities and Aqueous Reactions."
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 3.4 "Mass Relationships in Chemical Equations." https://chem.libretexts.org

Of the three states of matter you met in Module 1, gases are by far the easiest to describe mathematically, and this lesson shows why and how. Gases behave in strikingly simple and predictable ways, far simpler than liquids or solids, for one underlying reason: their particles are so far apart that the particles themselves take up almost no space and barely attract one another. This near-independence of the particles means that all gases, whatever they are made of, obey very nearly the same numerical laws. Four measurable quantities completely describe a sample of gas, and the gas laws are just the relationships among them: pressure (P), the force the particles exert per unit area on the container walls; volume (V), the space the gas fills; temperature (T), which measures the average kinetic energy of the particles; and amount in moles (n). One rule overrides everything in this lesson and must never be forgotten: temperature must always be expressed in kelvin, never Celsius, because the gas laws are proportionalities that only make sense measured from an absolute zero point. This lesson builds from the simple gas laws to the powerful ideal gas law and the molar-volume shortcut.

Pressure, and why kelvin is mandatory

Before the laws, a word on two of the quantities. Pressure arises because gas particles are in constant motion and continually collide with the walls of their container; each tiny collision pushes on the wall, and the sum of countless collisions per unit area is the pressure. Common units include the atmosphere (atm), the kilopascal (kPa), and millimeters of mercury (mmHg or torr); this lesson uses atmospheres. Temperature measures the average kinetic energy of the particles, that is, how fast they are moving on average. The reason kelvin is required is that the gas laws involve direct proportions, and a proportion only works from a true zero. At 0 kelvin (absolute zero) particle motion is at its theoretical minimum, so a gas at 200 K genuinely has twice the thermal energy of the same gas at 100 K. The Celsius scale has an arbitrary zero (the freezing point of water), so "twice the Celsius temperature" is physically meaningless, and using Celsius in a gas law gives nonsense, especially near or below 0 degrees Celsius. Always convert to kelvin first, using K = degrees C + 273.15.

The simple gas laws

Each simple gas law holds two of the four quantities constant and describes how the remaining two relate. They are named for the scientists who discovered them.

• Boyle's law (constant temperature and amount): pressure and volume are inversely related. Squeeze a gas into half its volume and its pressure doubles, because the same number of particles now strike a smaller wall area more frequently. Mathematically, P₁V₁ = P₂V₂.
• Charles's law (constant pressure and amount): volume is directly proportional to temperature in kelvin. Heat a gas and it expands, because faster-moving particles push the walls outward until pressure is restored. Mathematically, V₁/T₁ = V₂/T₂.
• Gay-Lussac's law (constant volume and amount): pressure is directly proportional to temperature in kelvin. Heat a sealed rigid container and its internal pressure rises, which is exactly why aerosol cans carry a warning against heating or incinerating, since the rising pressure can burst the can. Mathematically, P₁/T₁ = P₂/T₂.

Worked example 1 (Boyle's law). A gas occupies 4.0 L at 1.0 atm. If it is compressed to 1.0 L at constant temperature, what is the new pressure? Using P₁V₁ = P₂V₂: (1.0 atm)(4.0 L) = P₂(1.0 L), so P₂ = 4.0 atm. Reducing the volume to one-fourth raised the pressure fourfold, as the inverse relationship predicts.

Worked example 2 (Charles's law). A balloon holds 2.0 L of gas at 300 K. If it is warmed to 600 K at constant pressure, what is its new volume? Using V₁/T₁ = V₂/T₂: 2.0 L / 300 K = V₂ / 600 K, so V₂ = 2.0 L × (600 / 300) = 4.0 L. Doubling the kelvin temperature doubled the volume. Note how essential kelvin is here; using 27 and 327 degrees Celsius would give a completely wrong ratio.

These three simple laws combine into the combined gas law, which handles a fixed amount of gas moving between two different sets of conditions, allowing pressure, volume, and temperature all to change at once:

P₁V₁ / T₁ = P₂V₂ / T₂

Worked example 3 (combined gas law). A gas occupies 5.0 L at 2.0 atm and 300 K. What volume will it occupy at 1.0 atm and 450 K? Rearrange to solve for V₂: V₂ = P₁V₁T₂ / (T₁P₂) = (2.0 × 5.0 × 450) / (300 × 1.0) = 4500 / 300 = 15 L. Lowering the pressure and raising the temperature both act to expand the gas, so a larger volume makes sense.

The ideal gas law

The most powerful relationship of all ties every one of the four quantities together in a single compact equation:

PV = nRT

Here R is the ideal gas constant, a fixed number of nature equal to 0.0821 L·atm/(mol·K) when pressure is in atmospheres and volume in liters. (In SI units R is 8.314 J/(mol·K), but the 0.0821 value is the one to use with atm and L.) Unlike the simpler laws, which compare two states of a gas, the ideal gas law describes a single state, so if you know any three of the four variables P, V, n, and T, you can solve for the fourth.

Worked example 4. What volume does 2.00 mol of an ideal gas occupy at 1.00 atm and 273 K? Rearrange PV = nRT to solve for V:

V = nRT / P = (2.00 mol × 0.0821 L·atm/(mol·K) × 273 K) ÷ 1.00 atm = 44.8 L

The units of R are deliberately chosen so that, with atmospheres, liters, moles, and kelvin, the moles, atmospheres, and kelvin all cancel and the answer comes out in liters, another reward for the careful dimensional analysis you practiced in Module 1. Worked example 5. How many moles of gas are in a 10.0 L container at 2.00 atm and 300 K? Solve for n: n = PV / RT = (2.00 × 10.0) / (0.0821 × 300) = 20.0 / 24.63 = 0.812 mol.

Standard conditions and molar volume

STP, standard temperature and pressure, is defined as 0 degrees Celsius (273.15 K, often rounded to 273 K) and 1 atm, a common reference point for comparing gases on equal footing. A remarkable consequence follows straight from the ideal gas law: at STP, one mole of any ideal gas occupies about 22.4 liters, regardless of what the gas is. This value is called the molar volume at STP, and it is a tremendously useful shortcut, letting you convert directly between moles of a gas and its volume at STP without solving the full PV = nRT equation each time (using 22.4 L per mole as a conversion factor).

The fact that one mole of light hydrogen and one mole of much heavier carbon dioxide occupy the very same 22.4 L at STP is a vivid demonstration of a deep principle: gas volume depends only on the number of particles present, not on their mass or chemical identity. This is Avogadro's law (equal volumes of gases at the same temperature and pressure contain equal numbers of molecules), the very principle that made gas-phase stoichiometry so clean in Module 5, where the coefficients of a balanced equation also give the ratio of gas volumes. Worked example 6. What volume does 0.500 mol of any gas occupy at STP? Simply multiply by the molar volume: 0.500 mol × 22.4 L/mol = 11.2 L, no full equation needed.

Dalton's law of partial pressures

Real gases are usually mixtures. The air you breathe is roughly 78 percent nitrogen, 21 percent oxygen, and 1 percent other gases. Dalton's law of partial pressures states that in a mixture of gases that do not react, each gas exerts the same pressure it would exert if it alone occupied the container, called its partial pressure, and the total pressure is simply the sum of all the partial pressures. In symbols, P_total = P₁ + P₂ + P₃ + and so on. This makes sense from the kinetic picture: because the particles of an ideal gas do not interact, each type contributes its collisions with the walls independently of the others. Worked example 7. A container holds oxygen at 0.60 atm and nitrogen at 1.40 atm. What is the total pressure? By Dalton's law, P_total = 0.60 + 1.40 = 2.00 atm. Dalton's law matters for scuba diving (where the partial pressure of oxygen must be kept in a safe range at depth), for the exchange of oxygen and carbon dioxide in your lungs, and for any gas collected over water, where the measured pressure includes the partial pressure of water vapor.

Connecting gases to stoichiometry

The ideal gas law and molar volume let you fold gases directly into the stoichiometry of Module 5, because they convert between a gas's volume and its moles. Worked example 8. How many liters of oxygen at STP are needed to completely burn 0.100 mol of propane, given C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O? First use the mole ratio: 0.100 mol C₃H₈ × (5 mol O₂ / 1 mol C₃H₈) = 0.500 mol O₂. Then convert moles of oxygen to a volume at STP with the molar volume: 0.500 mol × 22.4 L/mol = 11.2 L of oxygen. This kind of problem, moving from a chemical equation to a real gas volume, is exactly why gas laws appear alongside stoichiometry in general chemistry, and it is used constantly in engineering combustion and industrial reactions.

Common misconceptions

Three misconceptions are common with gases. First, students think Celsius temperatures are acceptable in gas-law calculations if used consistently; the laws depend on absolute temperature, so kelvin is mandatory, and Celsius gives nonsense near or below 0 degrees. Second, students assume heavier gases always occupy more volume; at the same temperature and pressure, equal moles of any gas occupy equal volume regardless of mass, so a mole of dense CO₂ and a mole of light H₂ both fill about 22.4 L at STP. Third, students believe real gases obey PV = nRT exactly; it is an idealization that works well at ordinary conditions but breaks down at high pressure and low temperature, where particle volume and attractions can no longer be ignored.

Recap

A gas is described by pressure, volume, temperature, and moles, and temperature must always be in kelvin. The simple gas laws each hold two variables fixed: Boyle's (P and V inversely related), Charles's (V proportional to T), and Gay-Lussac's (P proportional to T), which merge into the combined gas law P₁V₁/T₁ = P₂V₂/T₂ for a fixed amount changing conditions. The ideal gas law PV = nRT links all four variables in a single state, with R = 0.0821 L·atm/(mol·K). At STP (0 degrees C, 1 atm), one mole of any ideal gas fills about 22.4 L, a molar-volume shortcut rooted in Avogadro's law. These relationships let you predict how gases respond to changing conditions, a skill applied constantly from weather to engines to the chemistry of the atmosphere.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 9 "Gases." https://openstax.org/books/chemistry-2e/pages/9-introduction
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 5 "Gases."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 10 "Gases."
4. National Institute of Standards and Technology (NIST), "Fundamental Physical Constants: molar gas constant R." https://physics.nist.gov/cgi-bin/cuu/Value?r
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 10 "Gases." https://chem.libretexts.org

Most chemistry, and essentially all of the chemistry of life, happens not with pure substances but in solution. A solution is a homogeneous mixture of a solute (the substance being dissolved, usually present in the smaller amount) in a solvent (the substance doing the dissolving, usually present in the larger amount). When the solvent is water, by far the most important case in chemistry and biology, the mixture is called an aqueous solution and is labeled (aq), a state symbol you met in the equations lesson. Solutions surround us: the ocean, blood, sap, soft drinks, and the reagents in nearly every chemistry experiment are all solutions. Because reactions in solution are so common, chemists need a precise way to state how much solute a solution contains, and quantifying that concentration, especially through the unit called molarity, is the central skill of this lesson. We also examine why water dissolves so much, how to prepare a solution of a target concentration, and how to dilute a concentrated stock solution, all of which are everyday laboratory operations.

Why water dissolves so much: like dissolves like

Water is often called the universal solvent because it dissolves an enormous range of substances, and the reason traces directly back to the molecular polarity you studied in Module 3. Water is a bent, polar molecule with partially positive hydrogen ends and a partially negative oxygen end. When an ionic solid like table salt is placed in water, the partially negative oxygen ends of many water molecules cluster around each positive Na⁺ ion, while the partially positive hydrogen ends surround each negative Cl^- ion. This crowd of oriented water molecules tugs the ions out of the crystal lattice and keeps them dispersed and separated in solution, a process called dissolving, or for ionic solutes specifically, dissociation into free ions. It is those free, mobile ions that allow salt water to conduct electricity.

The guiding principle is the memorable rule "like dissolves like": polar solvents dissolve polar and ionic solutes, while nonpolar solvents dissolve nonpolar solutes. This is not merely a rhyme but a statement of real molecular physics, because a solvent can only pull a solute apart if the solvent-solute attractions are comparable to the attractions being broken. It explains why nonpolar oil refuses to mix with polar water: water molecules attract each other strongly and have nothing comparable to offer the nonpolar oil molecules, so the two separate into layers. The same principle explains why greasy (nonpolar) stains need a nonpolar solvent or a soap to remove them, and why polar substances like sugar and salt dissolve so readily in water.

Molarity: the chemist's concentration unit

Concentration expresses how much solute is present in a given amount of solution, the quantitative difference between weak tea and strong tea. Chemists use several concentration measures, but by far the most common in general chemistry is molarity (M), defined as the number of moles of solute per liter of solution:

molarity (M) = moles of solute ÷ liters of solution

Molarity is popular precisely because it is expressed in moles, which connects it directly to the balanced-equation stoichiometry of Module 5, letting you carry a reaction calculation straight into solution work. There is one detail that catches many students: the denominator is liters of solution, the total final volume after mixing, not liters of solvent. In practice you dissolve the solute in some solvent and then add more solvent up to the final volume mark, rather than adding a full liter of solvent to the solute. This matters because the solute itself takes up some space.

Worked example 1. Dissolve 0.50 mol of NaCl in enough water to make 2.0 L of solution. The molarity is moles divided by liters: 0.50 mol ÷ 2.0 L = 0.25 M. Worked example 2 (from a mass). If a problem gives a mass instead of moles, convert to moles first using molar mass. Find the concentration of a solution made from 58.44 g of NaCl (molar mass 58.44 g/mol) dissolved to make 1.0 L of solution. Since 58.44 g is exactly 1.00 mol of NaCl, the concentration is 1.00 mol ÷ 1.0 L = 1.0 M. This 1.0 M sodium chloride solution is a common laboratory benchmark.

Preparing a solution of known molarity

Very often you know the concentration and volume you want and need to figure out how much solute to weigh out. Rearranging the definition gives the moles of solute required: moles = molarity × liters. Worked example 3. To prepare 0.500 L of 0.200 M glucose solution, you need 0.200 mol/L × 0.500 L = 0.100 mol of glucose. You would then convert that to a mass using glucose's molar mass (about 180.16 g/mol): 0.100 mol × 180.16 g/mol = 18.0 g. The procedure in the lab is to weigh out 18.0 g of glucose, dissolve it in some water in a volumetric flask, and then add water up to the 0.500 L calibration mark. This exact calculation is performed countless times a day in laboratories preparing reagents, so it is worth being fluent in it.

Dilution

Laboratories usually keep reagents as concentrated stock solutions and dilute them down to the working concentration an experiment needs. Dilution is the process of adding more solvent to a solution, which spreads the same amount of solute through a larger volume and thereby lowers the concentration. The essential insight, and the thing that makes dilution calculations easy, is that during a dilution the number of moles of solute does not change at all; you are only adding solvent, not solute, so the amount of dissolved material stays fixed while the volume grows.

Because moles = M × V, and the moles of solute stay constant before and after dilution, the product M × V must be the same on both sides. This gives the dilution equation:

M₁V₁ = M₂V₂

where M₁ and V₁ are the concentration and volume before dilution and M₂ and V₂ are the values after. Because the volumes appear on both sides, they can be in any consistent unit (milliliters are fine, as long as both are milliliters).

Worked example 4. You have concentrated 12.0 M hydrochloric acid and need 500 mL of 3.0 M acid. How much of the concentrated acid should you start with? Solve for V₁: V₁ = M₂V₂ / M₁ = (3.0 M × 500 mL) ÷ 12.0 M = 125 mL. So you measure out 125 mL of the concentrated acid and add water up to a total volume of 500 mL. One safety note that is genuinely important and worth memorizing: always add acid to water, never water to acid. Mixing concentrated acid with water releases significant heat, and if you pour water onto acid, that heat can be generated in a small spot and cause the mixture to boil and spatter dangerously; adding acid slowly to a larger volume of water spreads the heat out safely. A common mnemonic is "do as you oughta, add acid to water."

Worked example 5. What is the concentration if you dilute 100 mL of 2.0 M solution to a total volume of 500 mL? Solve for M₂: M₂ = M₁V₁ / V₂ = (2.0 M × 100 mL) ÷ 500 mL = 0.40 M. Diluting fivefold (from 100 mL to 500 mL) lowered the concentration fivefold (from 2.0 M to 0.40 M), exactly as expected.

Solubility and saturation

A natural question is how much solute a given amount of solvent can dissolve. The solubility of a substance is the maximum amount that will dissolve in a given quantity of solvent at a specified temperature. A solution that holds less than this maximum is unsaturated and can dissolve more; a solution holding exactly the maximum is saturated; and, under special careful conditions, a solution can be coaxed into holding more than the normal maximum, becoming temporarily supersaturated and prone to dumping the excess solute suddenly if disturbed. For most solids dissolving in water, solubility rises with temperature, which is why hot water dissolves more sugar than cold water and why rock-candy crystals grow as a hot sugar solution cools and can no longer hold all its dissolved sugar. Gases behave oppositely: gas solubility falls as temperature rises, which is why a warm carbonated drink goes flat faster and why warm river water holds less dissolved oxygen for fish. Gas solubility also rises with pressure (Henry's law), which is exactly why a sealed soda bottle stays fizzy under pressure and releases a rush of bubbles the moment you open it and drop the pressure.

Electrolytes and why salt water conducts

Solutions differ sharply in whether they conduct electricity. A substance that dissociates into ions when it dissolves is an electrolyte, and its solution conducts electricity because those free ions carry charge. Ionic compounds like sodium chloride are strong electrolytes: essentially every formula unit splits into Na⁺ and Cl^- ions, so salt water conducts well. Many molecular compounds, like sugar, dissolve without forming ions at all; these are nonelectrolytes, and a sugar solution does not conduct even though the sugar is fully dissolved. This is the practical reason a solution's ability to conduct electricity can reveal whether the solute dissolved as ions or as neutral molecules, and it is why the fluids in your body, rich in dissolved ions called electrolytes, are essential for nerve signals and muscle function. Sports drinks advertise "electrolytes" precisely because sweating loses these dissolved ions.

Worked example 6 (a fuller preparation). How would you prepare 250 mL of 0.150 M potassium chloride solution, and how many chloride ions would it contain? First find the moles of KCl needed: moles = molarity × liters = 0.150 mol/L × 0.250 L = 0.0375 mol. Convert to a mass using KCl's molar mass (39.10 + 35.45 = 74.55 g/mol): 0.0375 mol × 74.55 g/mol = 2.80 g. So you weigh out 2.80 g of KCl, dissolve it in some water in a 250 mL volumetric flask, and add water to the 250 mL mark. Because each KCl provides one Cl^- ion, the solution contains 0.0375 mol of chloride ions, or 0.0375 mol × 6.022 × 10²³ = 2.26 × 10²² chloride ions. This weaves together molarity, molar mass, and the mole from earlier modules.

Common misconceptions

Three misconceptions recur with solutions. First, students define molarity as moles per liter of solvent; it is moles per liter of solution, the total final volume, so you add solvent up to the mark rather than adding a full liter of solvent plus the solute. Second, students think diluting changes the number of moles of solute; dilution adds only solvent, so the moles stay exactly the same and only the volume grows, which is the very reason M₁V₁ = M₂V₂ holds. Third, students dismiss "like dissolves like" as an empty rhyme; it reflects genuine molecular physics, since polar solvents stabilize polar and ionic solutes through electrostatic attraction while nonpolar substances lack those interactions, which is why oil and water separate.

Recap

A solution is a homogeneous mixture of a solute in a solvent, and an aqueous solution uses water, whose polarity lets it dissolve polar and ionic substances according to "like dissolves like." Molarity, moles of solute per liter of solution, is the workhorse concentration unit, chosen because its use of moles connects seamlessly to stoichiometry; the denominator is total solution volume, not solvent volume. To prepare a solution you compute moles = molarity times liters, convert to a mass, and dilute up to the mark. To dilute a stock solution you use M₁V₁ = M₂V₂, which follows from the fact that dilution changes volume but not moles of solute, and you always add acid to water for safety. These concentration skills round out the quantitative toolkit and, combined with the reaction stoichiometry of Module 5, underlie techniques such as titration used throughout chemistry.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 3.3 "Molarity" and Chapter 11 "Solutions and Colloids." https://openstax.org/books/chemistry-2e/pages/3-3-molarity
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 4 "Types of Chemical Reactions and Solution Stoichiometry."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapters 4 and 13 (aqueous reactions and properties of solutions).
4. Tro, N. J. Chemistry: A Molecular Approach, 5th ed., Pearson, 2020, Chapter 14 "Solutions."
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 13 "Solutions." https://chem.libretexts.org

Every chemical reaction involves not only a rearrangement of atoms but also a change in energy, and this final lesson studies that energy. Thermochemistry is the study of the energy, usually in the form of heat, that flows during chemical and physical changes. It answers deeply practical questions: why do fuels warm us, why does an instant cold pack chill a sprain, why does some food give more energy than others, and why must certain reactions be heated constantly to keep them going? Energy is defined as the capacity to do work, and heat (q) is specifically energy transferred from one object to another because of a temperature difference. Heat always flows spontaneously from the hotter object to the colder one, never the reverse on its own, until the two reach the same temperature. The SI unit of energy is the joule (J). You will also meet the calorie, defined so that 1 cal = 4.184 J exactly; note that the food "Calorie" written with a capital C on nutrition labels is actually a kilocalorie, 1000 of these small calories, which is why a candy bar can list "250 Calories." This lesson develops where chemical energy is stored, the crucial distinction between exothermic and endothermic processes, the meaning of the enthalpy change, and the specific-heat equation for calculating heat flow.

Where chemical energy is stored: bonds

Chemical energy is stored in the bonds between atoms, and one principle underlies all of thermochemistry: breaking bonds absorbs energy, and forming bonds releases energy. This may feel backward at first, because we often think of breaking things as violent and energetic, but it is firmly true and worth pausing on. A chemical bond exists because the bonded arrangement is more stable (lower in energy) than the separated atoms, so pulling the atoms apart requires an input of energy, exactly as lifting a weight against gravity requires effort. Conversely, when atoms come together to form a bond, they fall to that lower-energy state and release the difference as energy, just as a dropped weight releases energy.

Every reaction breaks some bonds in the reactants and forms new bonds in the products. Whether the reaction gives off or absorbs energy overall depends entirely on the balance between the energy absorbed to break the old bonds and the energy released in forming the new ones. If forming the new bonds releases more energy than breaking the old bonds cost, the reaction has a net release of energy. If breaking the old bonds cost more than forming the new ones returns, the reaction has a net absorption of energy. This single accounting idea explains the two great categories of reactions below.

Exothermic and endothermic processes

To describe energy flow we speak of the system (the reaction itself) and the surroundings (everything else, including the container, the air, and any thermometer). Reactions divide into two types by which way heat flows between them:

• An exothermic process releases heat to its surroundings, which therefore warm up (the prefix exo means "out"). Combustion, explosions, neutralizations, and most bond-forming reactions are exothermic. In an exothermic reaction, the products store less chemical energy than the reactants, and the difference is released to the surroundings as heat. Burning wood and the setting of concrete are exothermic; you can feel the warmth.
• An endothermic process absorbs heat from its surroundings, which therefore cool down (endo means "in"). Melting ice, evaporating water, and the dissolving reaction inside an instant cold pack are endothermic. Here the products store more chemical energy than the reactants, and the extra energy is drawn in from the surroundings, which is exactly why a cold pack feels cold: it is pulling heat out of your skin.

Enthalpy change

At constant pressure, which is the usual situation for a reaction open to the atmosphere in a beaker, the heat absorbed or released by a reaction is called its enthalpy change, written with the Greek capital delta as delta H. The sign of delta H carries the entire directional meaning and must be read with care:

• A negative delta H means the reaction is exothermic: energy leaves the system and enters the surroundings.
• A positive delta H means the reaction is endothermic: energy enters the system from the surroundings.

For example, burning one mole of methane, the main component of natural gas, has a delta H of about -890 kJ. That large negative number tells you the reaction is strongly exothermic and releases a great deal of heat, which is precisely why methane is such an excellent fuel for heating homes and cooking food. In general, the magnitude of delta H tells you how much energy is involved and the sign tells you which direction it flows. A common misreading to avoid: a negative delta H does not mean energy is lost or destroyed. Energy is always conserved; the negative sign simply means the energy has been transferred out of the system to the surroundings as heat.

Specific heat and calculating heat flow

Not all substances respond to heat in the same way. Pour the same amount of heat into a gram of water and a gram of iron, and the iron will get much hotter. The property that captures this is the specific heat (c) of a substance, defined as the amount of energy needed to raise the temperature of 1 gram of it by 1 degree Celsius. Water has an unusually large specific heat of 4.184 J/(g·degree C), meaning it takes a lot of energy to warm water even a little, and it releases a lot of energy when it cools. This high specific heat has enormous consequences: it is why water resists changes in temperature, why large lakes and coastal climates are milder and more stable than inland deserts, and why your body, which is mostly water, can regulate its temperature effectively. A common misconception is that a high specific heat makes something heat up quickly; the opposite is true, since a high specific heat means the substance needs a lot of energy to change temperature and therefore heats and cools slowly.

The heat gained or lost by a sample as its temperature changes is calculated with a single central equation:

q = m × c × ΔT

where q is the heat, m is the mass, c is the specific heat, and delta T is the temperature change, always calculated as the final temperature minus the initial temperature. Because delta T is final minus initial, its sign takes care of direction automatically: a rising temperature gives a positive delta T and a positive q (heat absorbed), while a falling temperature gives a negative delta T and a negative q (heat released).

Worked example 1. How much heat is needed to warm 50.0 g of water from 20.0 degrees C to 80.0 degrees C? First find the temperature change: delta T = 80.0 - 20.0 = 60.0 degrees C. Then apply the formula:

q = 50.0 g × 4.184 J/(g·degree C) × 60.0 degrees C = 12,552 J ≈ 12.6 kJ

The positive result means the water absorbed about 12.6 kilojoules of energy from its heat source. Worked example 2 (cooling). How much heat is released when 100.0 g of water cools from 50.0 degrees C to 25.0 degrees C? Here delta T = 25.0 - 50.0 = -25.0 degrees C, a negative value. Then q = 100.0 g × 4.184 J/(g·degree C) × (-25.0 degrees C) = -10,460 J, about -10.5 kJ. The negative sign correctly signals that 10.5 kJ of heat is released to the surroundings rather than absorbed, because the water is cooling. This single equation lets you quantify the heat behind everything from a warming cup of coffee to the cooling of a car engine.

Worked example 3 (comparing substances). Iron has a specific heat of about 0.449 J/(g·degree C), roughly one-ninth that of water. How much heat raises 50.0 g of iron by 60.0 degrees C, and how does it compare with the water in example 1? q = 50.0 g × 0.449 J/(g·degree C) × 60.0 degrees C = 1347 J, about 1.35 kJ. This is far less than the 12.6 kJ needed for the same mass of water over the same temperature rise, which is a direct numerical illustration of why water's high specific heat makes it so resistant to temperature change and such a good coolant and heat reservoir.

Measuring heat by calorimetry

How do chemists actually measure the heat of a reaction? The technique is calorimetry, and its logic rests on conservation of energy: the heat released by a reaction is absorbed by its surroundings, so if you run the reaction in an insulated container of water and measure the water's temperature change, you can calculate the heat using q = m times c times delta T for the water. A simple coffee-cup calorimeter, essentially an insulated cup with a thermometer, is enough for many reactions in solution. Worked example 4. A reaction is carried out in 200.0 g of water, and the water's temperature rises from 22.0 degrees C to 27.5 degrees C. How much heat did the reaction release? The water absorbed q = 200.0 g × 4.184 J/(g·degree C) × (27.5 - 22.0) degrees C = 200.0 × 4.184 × 5.5 = 4602 J, about 4.6 kJ. Because the water gained this heat, the reaction released it, so the reaction is exothermic with about -4.6 kJ of heat released. Calorimetry is how the enthalpy values printed in reference tables, including the -890 kJ for burning methane, were originally determined.

Energy and changes of state

Thermochemistry also governs the physical changes of state you met in Module 1. Melting a solid and boiling a liquid both require energy input (they are endothermic), because energy is needed to overcome the attractions holding the particles together, while freezing and condensing release that energy (they are exothermic). A striking feature is that during a change of state the temperature does not rise even as heat is added; all the incoming energy goes into rearranging the particles rather than speeding them up. This is why a pot of boiling water stays at 100 degrees C no matter how high you turn the burner, and why an ice bath holds steady at 0 degrees C until all the ice has melted. It also explains why sweating cools you: evaporating sweat is endothermic and draws heat away from your skin, and why a steam burn is so severe, because condensing steam releases a large amount of heat onto the skin. These everyday phenomena are direct consequences of the bond-and-attraction energy accounting at the heart of this lesson.

Common misconceptions

Three misconceptions are worth naming. First, students read a negative delta H as energy being lost or destroyed; energy is conserved, and a negative delta H means the system releases energy to the surroundings, transferring it out rather than annihilating it. Second, students think breaking chemical bonds releases energy; breaking bonds always absorbs energy, while forming bonds releases it, and a reaction is exothermic only when the bonds formed release more than the bonds broken absorbed. Third, students believe a high specific heat makes a substance heat up quickly; in fact it makes the substance heat and cool slowly, which is why water is famously resistant to temperature change.

Recap

Thermochemistry tracks the heat that flows in chemical and physical changes, measured in joules. Chemical energy lives in bonds, with bond breaking absorbing energy and bond forming releasing it, so the net energy of a reaction depends on the balance of the two. Exothermic reactions release heat and have a negative delta H, while endothermic reactions absorb heat and have a positive delta H, with the magnitude giving the amount of energy and the sign giving its direction. The equation q = m times c times delta T quantifies heat flow, where the temperature change is final minus initial, and water's unusually high specific heat explains its powerful moderating effect on temperature. This lesson completes General Chemistry I, tying the atomic and molecular ideas of the earlier modules to the energy that drives chemical change, and it sets the stage for the deeper study of thermodynamics, equilibrium, and kinetics in a second course.

Sources

1. Flowers, P., et al. Chemistry 2e, OpenStax (Rice University), 2019, Chapter 5 "Thermochemistry." https://openstax.org/books/chemistry-2e/pages/5-introduction
2. Zumdahl, S. S., and Zumdahl, S. A. Chemistry, 10th ed., Cengage Learning, 2018, Chapter 6 "Thermochemistry."
3. Brown, T. L., LeMay, H. E., Bursten, B. E., et al. Chemistry: The Central Science, 14th ed., Pearson, 2018, Chapter 5 "Thermochemistry."
4. National Institute of Standards and Technology (NIST), Chemistry WebBook (thermochemical data). https://webbook.nist.gov/chemistry/
5. Averill, B., and Eldredge, P. General Chemistry (LibreTexts), Chapter 5 "Energy Changes in Chemical Reactions." https://chem.libretexts.org