High School Chemistry

The big picture

Pour lemonade over ice and you are holding four kinds of matter at once: solid ice, liquid water, dissolved sugar, and the aluminum of the can. A chemist looks at that glass and sorts it before saying one word about reactions, because almost every question in chemistry begins with "what kind of stuff is this?" This lesson hands you that sorting habit, and every later lesson leans on it.

Chemistry is the study of matter and the changes it goes through. Matter is anything that has mass and takes up space, from the air you breathe to the water you drink to this screen. Because there is so much variety in matter, chemists sort it into a few clear groups so it is easier to understand.

Key idea: Every sample of matter is either a pure substance, which has one fixed recipe, or a mixture, which is two or more substances simply stirred together.

Pure substances and mixtures

A pure substance always has the same makeup. It comes in two kinds. An element cannot be broken into anything simpler by ordinary chemistry. There are about 118 elements, and each has its own one or two letter symbol, such as oxygen (O), gold (Au), or iron (Fe). A compound is two or more elements chemically joined in a fixed ratio, such as water (H₂O) or table salt (NaCl). A compound behaves nothing like the elements inside it: sodium is a soft metal that reacts violently with water, and chlorine is a poisonous green gas, yet together they make ordinary, edible salt.

A mixture is a physical blend of two or more substances that keep their own identities and can be mixed in any amounts. A homogeneous mixture (also called a solution) looks the same throughout, like salt water or clean air. A heterogeneous mixture has parts you can point to, like a bowl of cereal in milk or sand stirred into water. Because the parts of a mixture are not chemically bonded, you can pull them apart by physical means such as filtering, evaporating, or using a magnet.

Two questions that sort any sample

You never have to guess. Ask two questions in order, and the answer falls out.

1. Can I separate it by physical means (filtering, boiling, a magnet, tweezers)? If yes, it is a mixture. Then ask whether the parts are visible: visible parts means heterogeneous, uniform throughout means homogeneous.
2. If no, it is a pure substance. Now ask: can chemistry break it down into simpler substances? If yes, it is a compound. If nothing can break it down, it is an element.

Run it on brass. Brass can be melted and its copper and zinc separated by physical methods, and its recipe varies from batch to batch, so brass is a mixture, and a homogeneous one because it looks uniform. Run it on water. No filtering or boiling splits water into anything simpler, so it is pure; but passing electricity through it (electrolysis) does break it into hydrogen and oxygen, so water is a compound. Run it on helium. Nothing physical separates it, and no chemistry breaks it down, so helium is an element.

Key idea: Physical separation means mixture. Chemical breakdown means compound. Neither one works on an element.

Atom, molecule, and formula unit

Three words get swapped by mistake all year, so pin them down now. An atom is a single particle of an element: one helium atom, one iron atom. A molecule is two or more atoms bonded together into one particle: O₂ is a molecule of two oxygen atoms, and H₂O is a molecule of three atoms (two H and one O).

Ionic compounds such as NaCl do not come in molecules at all. Solid salt is an endless three-dimensional grid of Na⁺ and Cl^- ions, so chemists write the smallest whole-number ratio and call it a formula unit. "NaCl" therefore means one sodium ion for every chloride ion, not a two-atom molecule floating on its own.

A quick check: a bottle of oxygen gas contains molecules, each made of 2 atoms. So 5 molecules of O₂ hold 10 atoms of oxygen. Counting atoms and counting molecules give different numbers for the same sample, and mixing them up is the single most common arithmetic error in early chemistry.

The states of matter

Matter usually shows up in one of three familiar states. A solid keeps a fixed shape and volume because its particles are packed tightly and locked in place. A liquid keeps a fixed volume but takes the shape of its container because its particles can slide past each other. A gas has neither a fixed shape nor a fixed volume; its particles spread out to fill whatever space they are given. Adding heat or taking it away drives changes between these states, such as ice melting or water boiling.

Why those three? It is a tug-of-war between two things: the attraction pulling particles together, and the motion energy pushing them apart. In a solid the attraction wins, so particles only vibrate in place. In a liquid the two are roughly matched, so particles stay in contact but tumble past one another. In a gas the motion energy wins outright, so particles fly apart and spend most of their time far from any neighbor. Heating a sample adds motion energy, which is exactly why heating moves matter up the list from solid to liquid to gas.

Key idea: State is not about what a substance is; it is about how much motion energy its particles have compared with how strongly they attract each other.

Properties and changes

A physical property can be observed without changing what the substance is, such as color, density, or melting point. A chemical property describes how a substance reacts to form something new, such as how easily it burns or rusts. The same split applies to changes.

In a physical change, the substance is the same before and after: melting ice gives liquid water, still H₂O. In a chemical change, brand new substances form: burning wood makes ash and gases that you cannot easily turn back into wood. Good clues that a chemical change has happened are a color change, bubbles of a new gas, a solid appearing out of two liquids, or heat and light being given off.

Physical properties come in two flavors, and telling them apart is genuinely useful. An extensive property depends on how much you have: mass, volume, and length all double if you take twice as much. An intensive property does not: color, melting point, boiling point, and density stay the same whether you test a drop or a bucket. That is why intensive properties identify a substance and extensive ones do not. Knowing a sample weighs 40 g tells you nothing about what it is; knowing its density is 19.3 g/cm³ narrows it to gold almost immediately.

Separating mixtures

Because a mixture is only a physical blend, you can undo it with physical tools that exploit a difference in properties. Filtration traps a solid on paper while a liquid passes through, which is how you would recover sand from sandy water. Evaporation boils off a liquid and leaves a dissolved solid behind, which is how salt is harvested from seawater.

Distillation boils a liquid, then cools the vapor back to liquid in a fresh container, separating substances that boil at different temperatures, such as pure water from salt water. A magnet can pull iron filings out of a sand-and-iron mixture. Notice that none of these methods works on a compound: no amount of filtering or boiling turns water back into hydrogen and oxygen, because those are held by chemical bonds, not just mixed together.

Worked example: classify and separate

Suppose you are handed a cloudy glass of muddy salt water and asked to classify it and recover both the mud and the salt.

• Classify: The mud is visible and settles, so the sample is a heterogeneous mixture overall. The salt-and-water part, however, is a homogeneous mixture (a solution).
• Step 1, filter: Pour it through filter paper. The mud (an insoluble solid) stays on the paper; clear salt water passes through.
• Step 2, evaporate: Gently heat the clear salt water until the water leaves as vapor. Solid salt is left in the dish.

Two physical steps, chosen from two different property differences (particle size, then boiling point), fully separate the sample. Because every step was physical, the salt you recover is the same salt you started with.

Worked example 2: physical or chemical?

Question: Sort each of these as a physical change or a chemical change, and say what evidence decides it: (a) a copper roof turning green, (b) dry ice disappearing into fog, (c) an iron nail bending, (d) hydrogen peroxide fizzing on a cut.

Solution: (a) Chemical. The green layer is copper carbonate, a new substance with a new color; the copper cannot be polished back out. (b) Physical. Dry ice is solid CO₂ becoming gaseous CO₂. Same substance, new state, so it is sublimation. (c) Physical. A bent nail is still iron; only its shape changed. (d) Chemical. The fizz is oxygen gas being made from the peroxide, so a new substance appeared.

Notice that (b) and (d) both bubble. Bubbles alone never settle the question; you have to ask whether the gas is a new substance or the same substance in a new state.

Where people get stuck: treating "looks uniform" as proof of purity. Filtered air, sea water, stainless steel, and vinegar all look perfectly uniform, and every one of them is a mixture. Uniform appearance only earns the label homogeneous. To move from homogeneous to pure you need a second piece of evidence: a fixed composition and a sharp, single melting or boiling point. Sea water boils over a range of temperatures and leaves a residue; pure water boils at one temperature and leaves nothing.

Common misconceptions

• "Dissolving is a chemical change." Dissolving salt in water is physical: the salt is still salt, and evaporating the water brings it back unchanged.
• "If it looks uniform, it must be a pure substance." Air and salt water look uniform but are homogeneous mixtures, not pure substances. Uniform appearance means homogeneous, not pure.
• "A compound is just a really well-mixed mixture." A compound has a fixed ratio and new properties, and it can only be separated by chemistry. A mixture has variable amounts and keeps its parts' properties.
• "Bubbles always mean a chemical change." Boiling water bubbles, but that is a physical change (liquid to gas). Bubbles signal a chemical change only when a brand-new gas is being produced, as when vinegar meets baking soda.
• "An atom and a molecule are the same thing." An atom is one particle of an element; a molecule is two or more atoms bonded into a single particle. One molecule of O₂ contains 2 atoms, so 5 molecules contain 10 atoms.
• "Gases are not matter because they weigh nothing." Gases have mass. A basketball inflated to high pressure really does weigh more on a sensitive balance than the same ball flat, because you added more matter.

Try it

Classify each sample as element, compound, homogeneous mixture, or heterogeneous mixture: (1) brass, (2) carbon dioxide, (3) tungsten, (4) Italian salad dressing that separates on standing, (5) sugar dissolved in tea.

Answers: (1) homogeneous mixture (copper and zinc, variable recipe), (2) compound (fixed 1:2 ratio of C to O), (3) element (nothing simpler), (4) heterogeneous mixture (you can see the layers), (5) homogeneous mixture (uniform, but the sugar can be recovered by evaporating).

Recap

• Matter is anything with mass and volume. Sort it with two questions: can physical means separate it, and can chemistry break it down?
• Pure substances are elements (nothing simpler) or compounds (fixed ratio, new properties, separable only by chemistry).
• Mixtures are heterogeneous (visible parts) or homogeneous (uniform), and both have variable recipes.
• An atom is one particle of an element; a molecule is two or more bonded atoms; ionic solids are counted in formula units such as NaCl.
• Solid, liquid, and gas differ in how particle attraction compares with particle motion energy, not in what the substance is.
• Physical changes keep the substance's identity; chemical changes make new substances, flagged by new color, new gas, a new solid, or released heat and light.
• Intensive properties (density, melting point) identify a substance; extensive properties (mass, volume) only tell you how much you have.

Sources

1. OpenStax. (2019). Phases and classification of matter (Section 1.2). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Physical and chemical properties (Section 1.3). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Phases and classification of matter (Section 1.2). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. LibreTexts. (n.d.). Matter and energy (Chapter 3). In Introductory Chemistry. chem.libretexts.org
5. LibreTexts. (n.d.). The chemical world (Chapter 1). In Introductory Chemistry. chem.libretexts.org
6. PhET Interactive Simulations. (n.d.). States of matter: Basics [Simulation]. University of Colorado Boulder. phet.colorado.edu
7. Khan Academy. (n.d.). States of matter and intermolecular forces [Unit]. In Chemistry. khanacademy.org

Why units are not optional

In 1999 NASA lost the Mars Climate Orbiter, a spacecraft that cost well over 100 million dollars, because one engineering team reported thrust in pound-force seconds while the navigation software expected newton-seconds. The arithmetic was flawless. The units were not. Every number in chemistry carries a unit, and a number without its unit is not an answer, it is a rumor.

Science runs on careful measurement, and measurement needs agreed units. Chemists use the SI system, the modern metric system. A few base units cover most of what you will do: 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. In everyday lab work you will also lean on the gram (g) and the liter (L).

Key idea: A measurement is a number and a unit. Carry the unit through every line of arithmetic and it will tell you whether you set the problem up correctly.

Mass is not weight

These two words are used interchangeably in daily life and they are not the same thing. Mass is how much matter a sample contains, measured in kilograms or grams, and it does not change when you move the sample. Weight is the force gravity pulls on that mass with, measured in newtons, and it changes with where you stand.

Take a 60 kg student. On Earth, where gravity gives 9.8 newtons of pull per kilogram, the weight is 60 kg x 9.8 N/kg = 588 N. On the Moon, where the pull is only about 1.6 N/kg, the same student weighs 60 kg x 1.6 N/kg = 96 N, roughly one sixth as much. The mass never budged: it is still 60 kg of matter on the Moon.

This matters in the lab because a balance compares your sample against known masses and gives the same reading anywhere in the universe, while a spring scale measures force and would read low on the Moon. Chemists always want mass, which is why laboratory balances are built the way they are. When a chemistry problem says "weigh out 5.00 g," it means "find 5.00 g of mass."

Key idea: Mass is a property of the stuff. Weight is a force that depends on where the stuff is.

Prefixes scale the unit

Metric prefixes multiply a base unit by a power of ten, which lets you handle very large and very small amounts without a page full of zeros.

Prefix	Symbol	Meaning
kilo	k	1000 (103)
centi	c	0.01 (10-2)
milli	m	0.001 (10-3)
micro	µ	0.000001 (10-6)

So 1 kg is 1000 g, 1 cm is 0.01 m, and 1 mL is 0.001 L. A handy fact to remember: 1 mL is exactly the same volume as 1 cubic centimeter (cm³).

Converting units by cancelling

The safe way to convert is never to "move the decimal point and hope." Instead, multiply by a fraction that equals 1, arranged so the unit you want to lose sits on the bottom and cancels. Chemists call this dimensional analysis, and it is the workhorse method for the rest of this course.

Worked example 1 (one step). Convert 45.0 mm to meters. Since 1000 mm = 1 m, the fraction (1 m / 1000 mm) equals 1.

45.0 mm x (1 m / 1000 mm) = 0.0450 m

Read the cancelling out loud: mm on top, mm on the bottom, they cross out, and m survives. If you had accidentally written the fraction upside down you would have got mm²/m, an absurd unit, and the mistake would have announced itself before you touched a calculator.

Worked example 2 (two steps). Convert 2.50 kg to milligrams. There is no single kg-to-mg fact to memorize, so bridge through grams.

2.50 kg x (1000 g / 1 kg) x (1000 mg / 1 g) = 2 500 000 mg = 2.50 x 10^6 mg

The kg cancels against kg, the g cancels against g, and mg is left standing. Chaining fractions like this means you only ever have to remember the small local facts (1 kg = 1000 g, 1 g = 1000 mg), never the long-distance ones.

Key idea: Set up every conversion so the unwanted unit cancels. If the surviving unit is the one the question asked for, your setup is right.

Temperature scales

Chemists use both Celsius and Kelvin. To go from Celsius to Kelvin, add 273: K = °C + 273 (the precise value is 273.15). Water freezes at 0 °C (273 K) and boils at 100 °C (373 K) at ordinary pressure. Kelvin never goes negative because 0 K is absolute zero, the coldest anything can possibly be.

Worked example. Body temperature is 37 °C. In kelvin that is 37 + 273 = 310 K. Going the other way, liquid nitrogen boils at 77 K, which is 77 - 273 = -196 °C. Notice that a change of one degree Celsius is exactly the same size as a change of one kelvin; only the starting point of the scale moved. That is why you add and subtract 273 rather than multiplying by anything.

Kelvin exists because several chemistry formulas, especially the gas laws in Module 5, break if temperature can be negative or zero at an arbitrary place. Doubling the temperature from 10 °C to 20 °C does not double anything physical, but doubling it from 283 K to 566 K does. Whenever a formula in this course contains a T, assume kelvin unless told otherwise.

Density connects mass and volume

Density is how much mass is packed into a given volume. You find it by dividing mass by volume, usually in grams per milliliter (g/mL) for liquids or grams per cubic centimeter for solids:

density = mass ÷ volume

Worked example. A metal block has a mass of 54.0 g and a volume of 20.0 cm³. Its density is 54.0 g ÷ 20.0 cm³ = 2.70 g/cm³. That value matches aluminum, so the block is probably aluminum. You can also work backward: if a liquid has a density of 0.80 g/mL, then 50.0 mL of it has a mass of 50.0 mL × 0.80 g/mL = 40.0 g. Because density is a ratio, it stays the same no matter how big your sample is, which makes it a helpful fingerprint for identifying substances.

Rearranging the density equation

The single relationship density = mass ÷ volume answers three kinds of question, depending on which quantity is unknown. It helps to hold all three forms in mind:

• Unknown density: d = m ÷ V
• Unknown mass: m = d × V
• Unknown volume: V = m ÷ d

Worked example (find volume). Gold has a density of 19.3 g/cm³. What volume does a 96.5 g gold nugget occupy? Use V = m ÷ d = 96.5 g ÷ 19.3 g/cm³ = 5.00 cm³. Check by multiplying back: 5.00 cm³ × 19.3 g/cm³ = 96.5 g, which matches, so the answer is right.

Worked example (will it float?). An object floats if it is less dense than the liquid it sits in. A block of oak has a density of about 0.75 g/cm³, and water is 1.00 g/cm³. Since 0.75 is less than 1.00, oak floats. Ice (0.92 g/cm³) is also less dense than liquid water, which is why ice cubes float and icebergs poke above the sea.

Multi-step unit conversion with density

Worked example. A chemistry problem asks for the mass, in kilograms, of 2.5 L of ethanol, whose density is 0.789 g/mL. Work in careful steps and watch the units cancel:

2.5 L × (1000 mL ÷ 1 L) = 2500 mL

2500 mL × (0.789 g ÷ 1 mL) = 1972.5 g

1972.5 g × (1 kg ÷ 1000 g) = 1.97 kg

The liters became milliliters, density turned milliliters into grams, and the last factor turned grams into kilograms. Rounding to the two significant figures allowed by 2.5 L gives about 2.0 kg. Chaining conversion factors this way keeps large multi-unit problems from becoming guesswork.

Measuring the volume of an odd-shaped solid

You can calculate the volume of a cube, but not of a pebble. For irregular solids, chemists use water displacement: the object pushes aside exactly its own volume of water.

Worked example. A metal pebble has a mass of 78.6 g. A graduated cylinder reads 20.0 mL of water; after the pebble is dropped in, it reads 30.0 mL. Find the density and identify the metal.

1. Volume of the pebble = 30.0 mL - 20.0 mL = 10.0 mL, which is 10.0 cm³.
2. Density = mass / volume = 78.6 g / 10.0 cm³ = 7.86 g/cm³.
3. Iron has a density of about 7.87 g/cm³, so the pebble is almost certainly iron.

Where people get stuck: reading the final volume instead of the difference. The 30.0 mL is water plus pebble; only the 10.0 mL increase belongs to the pebble. A second trap is using this method on something that floats or dissolves, which makes the reading meaningless. Push a floating object under with a thin pin and the volume reading is still honest; a dissolving one is a lost cause.

Common misconceptions

• "Heavier objects are always denser." A large foam block can outweigh a small steel bolt yet still be far less dense. Density depends on mass and volume together, not mass alone.
• "Kelvin readings can be negative on a cold day." Kelvin starts at absolute zero and never goes negative. A cold night might be -10 °C, which is still 263 K.
• "A milliliter and a cubic centimeter are different sizes." They are exactly equal: 1 mL = 1 cm³. This lets you switch between liquid and solid volume units freely.
• "Density changes if you cut the sample in half." Halving a sample halves both its mass and its volume, so the ratio, and thus the density, is unchanged.
• "Mass and weight mean the same thing." Mass (kg) is how much matter there is; weight (N) is the gravitational force on it. Your mass is identical on the Moon; your weight is about one sixth as large.

Try it

A rectangular block of wood measures 4.0 cm by 5.0 cm by 10.0 cm and has a mass of 152 g. Find its density and say whether it floats in water.

Answer: Volume = 4.0 x 5.0 x 10.0 = 200 cm³. Density = 152 g / 200 cm³ = 0.76 g/cm³. Since 0.76 is less than water's 1.00 g/cm³, it floats.

Recap

• The SI base units chemists use most are the meter, kilogram, second, kelvin, and mole, plus the working units gram and liter.
• Prefixes scale a unit by powers of ten: kilo = 1000, centi = 0.01, milli = 0.001, micro = 0.000001. Also, 1 mL = 1 cm³ exactly.
• Convert by multiplying by fractions equal to 1, arranged so the unwanted unit cancels; the surviving unit checks your setup.
• Mass is the amount of matter and never changes with location; weight is a force and does.
• K = °C + 273. Kelvin has no negative values because it starts at absolute zero, and gas-law formulas require it.
• Density = mass / volume. Rearranged: m = d x V and V = m / d.
• Density is intensive, so it identifies a substance and predicts floating, and cutting a sample in half does not change it.

Sources

1. OpenStax. (2019). Measurements (Section 1.4). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Mathematical treatment of measurement results (Section 1.6). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Measurements (Section 1.4). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. National Institute of Standards and Technology. (n.d.). SI units. NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov
5. National Institute of Standards and Technology. (n.d.). Metric (SI) prefixes. Office of Weights and Measures. nist.gov
6. LibreTexts. (n.d.). Measurement and problem solving (Chapter 2). In Introductory Chemistry. chem.libretexts.org
7. PhET Interactive Simulations. (n.d.). Density [Simulation]. University of Colorado Boulder. phet.colorado.edu

The last digit is a guess, and that is the point

Hold a ruler marked in whole centimeters against a pencil. The tip clearly falls past the 4 but before the 5, closer to 4 than to 5. You would write 4.3 cm. The 4 you know; the 3 you estimated by eye. Every measuring instrument ever built works this way: certain digits, then exactly one estimated digit, then nothing honest to say.

Every measurement has some uncertainty, and significant figures (sig figs) are how we honestly show how precise a value really is. The significant figures are all the digits you know for sure plus one final estimated digit. Writing more digits than your ruler or balance can justify is a false claim of precision, so learning these rules keeps you honest.

Key idea: Significant figures are not a rounding rule invented to annoy you. They are a promise about how much of your number the instrument actually supports.

Swap that ruler for one marked in millimeters and the same pencil now reads 4.27 cm: the 4 and the 2 are certain, the 7 is estimated. A better instrument buys you one more digit, never an unlimited supply. And a calculator, which knows nothing about your equipment, will happily hand you 4.2666666667 - a number with ten digits of confidence you have not earned.

Counting significant figures

• Every nonzero digit counts. 24.7 has three.
• Zeros between nonzero digits count. 1005 has four.
• Leading zeros never count; they only place the decimal. 0.0034 has two.
• Trailing zeros count only if a decimal point is written. 2.50 has three, but 250 is ambiguous (treat it as two unless it is written 250. or 2.50 x 10²).
• Exact counted numbers (like 12 eggs) and defined conversions (100 cm per 1 m) have unlimited sig figs and never limit an answer.

Worked example: count them. Work through these one rule at a time.

• 507.0 has four. The 5 and 7 are nonzero, the middle 0 is sandwiched between them, and the final 0 sits after a written decimal point.
• 0.00560 has three. The three leading zeros only park the decimal; the 5, the 6, and the trailing 0 all count.
• 8000 has one as written, because nothing tells you those zeros were measured. Write 8.000 x 10^3 if you really measured four digits.
• 1.20 x 10^4 has three. In scientific notation only the front part carries sig figs, which is exactly why chemists like it.

Key idea: Zeros are the only hard part. Ask whether each zero is holding a place (not significant) or reporting a measurement (significant).

Scientific notation removes the guesswork

Chemistry is full of numbers that are absurdly large or absurdly small, and writing them out invites both typos and ambiguity. Scientific notation writes any number as a single digit, a decimal point, the remaining digits, and a power of ten.

• 93 000 000 miles becomes 9.3 x 10^7 miles. The decimal moved 7 places left, so the exponent is +7.
• 0.000045 g becomes 4.5 x 10^-5 g. The decimal moved 5 places right, so the exponent is -5.
• 602 200 000 000 000 000 000 000 becomes 6.022 x 10^23, a number you will meet properly in Lesson 11.

The rule for the sign is worth saying out loud: moving the decimal left makes the exponent go up, because you shrank the front number and must pay it back. Moving it right makes the exponent go down. And notice how the ambiguity vanishes: 8000 could mean one sig fig or four, but 8.0 x 10^3 means exactly two, with no argument possible.

Sig figs in calculations

Multiplication and division: the answer keeps as many sig figs as the measurement with the fewest. Example: 4.56 × 1.4 = 6.384 on the calculator, but 1.4 has only two sig figs, so the answer is 6.4.

Addition and subtraction: the answer keeps as many decimal places as the value with the fewest decimal places. Example: 12.11 + 0.3 = 12.41 on the calculator, but 0.3 has only one decimal place, so the answer is 12.4.

Round only at the very end of a multi-step problem so small rounding errors do not pile up.

Why two different rules? Because multiplication spreads relative error while addition lines up decimal places. When you add 12.11 and 0.3, the second value is silent about its hundredths digit, so the sum has to be silent there too. When you multiply 4.56 by 1.4, the weakest factor knows only two digits, so the product cannot know more.

Worked example: addition. Add 24.681 g + 0.9 g + 6.53 g. The calculator says 32.111 g. The value 0.9 g has just one decimal place, and it is the fewest, so the sum is 32.1 g.

Worked example: both rules in one problem. A student weighs an empty beaker at 45.62 g, then the beaker plus liquid at 78.91 g, and measures the liquid's volume as 40.0 mL. Find the density.

1. Subtract to get the liquid's mass. 78.91 g - 45.62 g = 33.29 g. Both values carry two decimal places, so the difference keeps two decimal places: 33.29 g, which is four sig figs.
2. Divide to get the density. 33.29 g / 40.0 mL = 0.83225 g/mL on the calculator. Now the multiplication-division rule applies: 40.0 has three sig figs and 33.29 has four, so the fewest is three.
3. Report. Density = 0.832 g/mL.

Watch what happened in step 1: subtracting two four-sig-fig numbers gave a four-sig-fig answer here, but if the beaker had read 45.62 g and the full beaker 45.98 g, the difference would be 0.36 g, only two sig figs from two four-digit measurements. Subtracting nearly equal numbers destroys precision, which is why chemists avoid designing experiments that depend on tiny differences of big numbers.

Dimensional analysis

Dimensional analysis (the factor-label method) converts units by multiplying by fractions equal to one, called conversion factors, set up so the unwanted units cancel. To convert 3.50 kg to grams:

3.50 kg × (1000 g ÷ 1 kg) = 3500 g

The kilograms cancel, leaving grams. You can chain several factors in one line. To convert 90.0 km/h to meters per second:

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

Always write the units, cancel them like you cancel in fractions, and check that what is left is exactly the unit the question asked for. If the units come out right, the arithmetic almost always follows.

Accuracy versus precision

These two words are not the same. Accuracy is how close a measurement is to the true value. Precision is how close repeated measurements are to each other. A dartboard makes the difference clear: darts clustered in the bullseye are accurate and precise; darts clustered tightly in one corner are precise but not accurate; darts scattered all over are neither. Significant figures communicate precision, while accuracy depends on a correctly calibrated instrument.

A fully worked significant-figure problem

Worked example. A student measures a rectangular metal sheet as 12.4 cm by 3.2 cm and a mass of 21.06 g, then reports the density per square centimeter of area. Track the sig figs at every stage.

• Area: 12.4 cm × 3.2 cm = 39.68 cm² on the calculator. The value 3.2 has only two sig figs, so area = 40. cm² (two sig figs).
• Mass per area: 21.06 g ÷ 39.68 cm² = 0.5307 g/cm² using the unrounded area. Limited by the two sig figs of 3.2, the answer is 0.53 g/cm².

Notice the key habit: carry extra digits through the middle of the calculation (use 39.68, not 40) and round only the final answer. Rounding too early would have shifted the last digit.

A worked rounding walk-through

Round 0.024856 to three significant figures. The three significant digits are 2, 4, and 8. The next digit is 5 (followed by more), so round the 8 up to 9, giving 0.0249. In scientific notation that is 2.49 × 10^-2, which makes the three sig figs unmistakable and removes any ambiguity about the leading zeros.

Common misconceptions

• "More decimal places always means a better answer." Writing 6.384 when your data supports only two sig figs overstates precision. Report 6.4 instead.
• "Leading zeros are significant." In 0.0034 the zeros only locate the decimal point; only the 3 and 4 count, so it has two sig figs.
• "Round after every step." Rounding mid-calculation lets errors accumulate. Keep guard digits and round once, at the end.
• "Accurate and precise mean the same thing." A scale that always reads 2 grams heavy is precise (repeatable) but not accurate (wrong value).
• "Conversion factors limit sig figs." Defined relationships such as 1 m = 100 cm are exact, so they never cut down your answer. Only measurements do.

Where people get stuck: rounding the intermediate number and then using the rounded value. In the density example above, rounding 33.29 to 33.3 first would give 33.3 / 40.0 = 0.8325, which still rounds to 0.832 - lucky this time. But 12.4 x 3.2 rounded to 40 and then divided into 21.06 gives 0.5265, which rounds to 0.53, while the honest route gives 0.5307, also 0.53. The two agree until one day they do not, and you will not be able to tell which day that is. The safe habit is simple: keep every digit your calculator shows until the final line, then round once.

Try it

(a) How many sig figs are in 0.03080? (b) Compute 6.02 x 0.35 and report it properly. (c) Compute 105.7 - 0.62 and report it properly.

Answers: (a) four - the leading zeros are placeholders, but the 3, the internal 0, the 8, and the trailing 0 all count. (b) 6.02 x 0.35 = 2.107, and 0.35 has two sig figs, so report 2.1. (c) 105.7 - 0.62 = 105.08, but 105.7 has only one decimal place, so report 105.1.

Recap

• A measurement's significant figures are the certain digits plus one estimated digit; a better instrument buys one more digit, not unlimited digits.
• Count sig figs by asking of each zero: placeholder (not significant) or measured (significant)?
• Scientific notation, such as 6.022 x 10^23, removes all ambiguity about trailing zeros and tames very large and very small numbers.
• Multiplication and division: keep the fewest sig figs. Addition and subtraction: keep the fewest decimal places.
• Exact counts and defined conversions have unlimited sig figs and never limit an answer.
• Carry extra digits through the middle of a problem and round only at the end.
• Accuracy is closeness to the true value; precision is repeatability. A biased instrument can be precise and still wrong.
• Dimensional analysis converts units by cancelling; if the surviving unit is the one you wanted, the setup is right.

Sources

1. OpenStax. (2019). Measurement uncertainty, accuracy, and precision (Section 1.5). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Mathematical treatment of measurement results (Section 1.6). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Measurement uncertainty, accuracy, and precision (Section 1.5). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. LibreTexts. (n.d.). Measurement and problem solving (Chapter 2). In Introductory Chemistry. chem.libretexts.org
5. National Institute of Standards and Technology. (n.d.). SI units. NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov
6. National Institute of Standards and Technology. (n.d.). Fundamental physical constants. CODATA internationally recommended values. physics.nist.gov
7. Khan Academy. (n.d.). Chemistry [Course]. khanacademy.org

How anyone knows what is inside an atom

Nobody has ever seen a proton. So how did we find out? In 1909 Hans Geiger and Ernest Marsden, working for Ernest Rutherford, fired positively charged alpha particles at a sheet of gold foil only a few thousand atoms thick. Almost all of them sailed straight through, as expected. But a tiny fraction, roughly one in several thousand, bounced back sharply, as if it had struck something small, hard, and positively charged. Rutherford said it was like firing a shell at tissue paper and having it come back at you.

That single result killed the earlier picture of an atom as a soft ball of positive jelly with electrons stuck in it, and replaced it with the picture you are about to learn: nearly all the mass and all the positive charge squeezed into a minuscule nucleus, with the rest of the atom empty.

All matter is built from atoms, and every atom shares the same basic parts. At the center is a tiny, dense nucleus that holds positively charged protons and neutral neutrons. Around the nucleus, in a cloud, move negatively charged electrons, which have almost no mass. The atom is mostly empty space: if the nucleus were the size of a marble, the electron cloud would stretch across a whole stadium.

The numbers behind that picture: a typical atom is about 10^-10 m across, while its nucleus is about 10^-15 m across. Divide and the atom is roughly 100 000 times wider than its nucleus. Since volume goes as the cube of width, the nucleus occupies about a thousand-billionth of the atom's space, yet holds more than 99.9 percent of its mass.

Key idea: Protons decide which element you have, neutrons decide which isotope, and electrons decide how it behaves chemically.

Particle	Charge	Relative mass	Location
Proton	+1	1	Nucleus
Neutron	0	1	Nucleus
Electron	-1	about 1/1836	Electron cloud

Atomic number and mass number

The atomic number (Z) is the number of protons, and it defines the element. Every carbon atom has 6 protons; change that number and you have a different element. In a neutral atom, the number of electrons equals the number of protons. The mass number (A) is the total count of protons plus neutrons. So the number of neutrons is just A minus Z.

Three short equations do all the bookkeeping in this lesson:

protons = Z    neutrons = A - Z    charge = protons - electrons

That last one is worth staring at. A neutral atom has equal protons and electrons, so the charge comes out zero. Take an electron away and the positives outnumber the negatives, giving a positive ion. Add an electron and you get a negative ion. The nucleus never changes in ordinary chemistry, so the proton count, and therefore the element, is safe no matter how many electrons come and go.

Worked example: counting particles in an ion

Question: A particle has 20 protons, 20 neutrons, and 18 electrons. Name it and give its charge.

Solution: Z = 20, and element 20 is calcium, so this is a calcium particle. The mass number A = 20 + 20 = 40, making it calcium-40. Charge = protons - electrons = 20 - 18 = +2. It is the calcium ion Ca²⁺. Notice it lost two electrons, not two protons; losing protons would have turned it into a different element entirely.

Question: A particle has 8 protons and 10 electrons. What is it?

Solution: Element 8 is oxygen. Charge = 8 - 10 = -2, so this is the oxide ion, O^2-. It gained two electrons.

Isotopes

Isotopes are atoms of the same element (same number of protons) that have different numbers of neutrons, and therefore different masses. Carbon has three natural isotopes: carbon-12 (6 protons, 6 neutrons), carbon-13 (6 protons, 7 neutrons), and carbon-14 (6 protons, 8 neutrons). They act the same in chemical reactions because chemistry depends on electrons, not neutrons.

Atomic mass is a weighted average

The atomic mass printed on the periodic table is the average mass of an element's atoms, weighted by how common each isotope is. Chlorine is about 75% chlorine-35 and 25% chlorine-37. Its atomic mass is therefore close to:

(0.75 × 35) + (0.25 × 37) = 26.25 + 9.25 = 35.5 amu

That is why the table lists chlorine as about 35.5 amu even though no single chlorine atom has that exact mass. Mass here is measured in atomic mass units (amu), defined so that one carbon-12 atom weighs exactly 12 amu.

Reading an isotope symbol

Isotopes are often written with the mass number on top and the atomic number on the bottom, or simply as name-mass, such as oxygen-18. To decode any isotope you only need two counts. For oxygen-18: oxygen's atomic number is 8, so there are 8 protons; a neutral atom then has 8 electrons; and neutrons = mass number minus atomic number = 18 - 8 = 10 neutrons. For potassium-40 (Z = 19): 19 protons, 19 electrons, and 40 - 19 = 21 neutrons. The element name and the atomic number always agree, so if a problem gives you the name, you already know the proton count.

Worked example: weighted average atomic mass

Copper occurs as two isotopes: copper-63 with a mass of 62.93 amu at 69.17% abundance, and copper-65 with a mass of 64.93 amu at 30.83% abundance. A weighted average multiplies each isotope's mass by its fractional abundance, then adds:

(0.6917 × 62.93) + (0.3083 × 64.93)

= 43.53 + 20.02 = 63.55 amu

The periodic table lists copper as 63.55 amu, which matches. Note that the average sits closer to 63 than to 65 because copper-63 is the more abundant isotope. A weighted average always leans toward the most common isotope, which is a good sanity check on your arithmetic.

Worked example: running the average backwards

Question: Boron has two isotopes, boron-10 (10.013 amu) and boron-11 (11.009 amu). The periodic table gives boron's atomic mass as 10.81 amu. What are the two abundances?

Solution: Call the fraction of boron-10 x. The two fractions must add to 1, so the fraction of boron-11 is 1 - x. Set up the weighted average and solve:

10.013x + 11.009(1 - x) = 10.81

10.013x + 11.009 - 11.009x = 10.81

-0.996x = -0.199   so   x = 0.199 / 0.996 = 0.200

So boron is about 20.0% boron-10 and 80.0% boron-11. Check it against common sense: 10.81 sits much nearer 11 than 10, so the heavier isotope should dominate, and it does, four to one. Measured values are 19.9% and 80.1%, so our arithmetic lands almost exactly on the real world.

Worked example: build an atom from its counts

An atom has 17 protons, 18 neutrons, and 17 electrons. Identify it. The atomic number equals the proton count, 17, which is chlorine. The mass number is 17 + 18 = 35, so this is chlorine-35. Because protons and electrons are equal, the atom is neutral (no overall charge).

Where people get stuck: confusing the mass number with the atomic mass. The mass number is a count, so it is always a whole number and belongs to one specific isotope: chlorine-35 has mass number 35. The atomic mass is a weighted average over all the isotopes in a natural sample, so it is almost never whole: chlorine's is 35.45 amu. No chlorine atom weighs 35.45 amu, just as no family has 2.3 children. Averages describe collections, not individuals.

Common misconceptions

• "Isotopes are different elements." Isotopes of one element have the same number of protons, so they are the same element; only the neutron count (and mass) differs.
• "Atomic mass equals the number of protons." The atomic number is the proton count. Atomic mass is the weighted-average mass of the isotopes and is usually not a whole number.
• "The mass number is on the periodic table." The table shows the average atomic mass, not the whole-number mass number of any one isotope.
• "Electrons add noticeably to an atom's mass." An electron is only about 1/1836 the mass of a proton, so essentially all of an atom's mass is in its nucleus.
• "An ion is a different element." Ions form by gaining or losing electrons, never protons. Na and Na⁺ are both sodium; only the electron count changed.
• "Radioactive isotopes react differently in chemistry." Carbon-14 bonds exactly like carbon-12, because bonding is done by electrons. Carbon-14 simply also decays, which is what makes it useful for dating.

Try it

(a) How many protons, neutrons, and electrons are in a neutral atom of iron-56 (Z = 26)? (b) Silver has two isotopes: silver-107 (106.905 amu, 51.84%) and silver-109 (108.905 amu, 48.16%). Estimate its atomic mass.

Answers: (a) 26 protons, 56 - 26 = 30 neutrons, and 26 electrons because it is neutral. (b) (0.5184 x 106.905) + (0.4816 x 108.905) = 55.42 + 52.45 = 107.87 amu, which is exactly what the periodic table shows for silver. It lands just below the midpoint of 107 and 109, as it should, because the lighter isotope is slightly more common.

Recap

• Atoms contain protons (+1) and neutrons (0) in a dense nucleus, with electrons (-1, about 1/1836 the mass of a proton) in a surrounding cloud.
• The atom is about 100 000 times wider than its nucleus, so it is overwhelmingly empty space - which is how Rutherford's alpha particles mostly passed straight through gold foil.
• Atomic number Z = protons, and it defines the element. Mass number A = protons + neutrons, so neutrons = A - Z.
• Charge = protons - electrons. Ions differ from atoms only in their electron count.
• Isotopes share Z but differ in neutrons, so they differ in mass while behaving identically in chemical reactions.
• The atomic mass on the periodic table is the abundance-weighted average of the isotope masses, in amu defined by carbon-12 = 12 amu exactly.
• A weighted average always leans toward the most abundant isotope, which makes it easy to sanity-check.

Sources

1. OpenStax. (2019). Atomic structure and symbolism (Section 2.3). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Evolution of atomic theory (Section 2.2). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Early ideas in atomic theory (Section 2.1). In Chemistry 2e. Rice University. openstax.org
4. National Institute of Standards and Technology. (n.d.). Atomic weights and isotopic compositions with relative atomic masses. Physical Measurement Laboratory. nist.gov
5. LibreTexts. (n.d.). Atoms and elements (Chapter 4). In Introductory Chemistry. chem.libretexts.org
6. PhET Interactive Simulations. (n.d.). Build an atom [Simulation]. University of Colorado Boulder. phet.colorado.edu
7. PhET Interactive Simulations. (n.d.). Isotopes and atomic mass [Simulation]. University of Colorado Boulder. phet.colorado.edu

Fireworks are evidence

Drop a little sodium into a flame and the flame turns a hard, unmistakable yellow. Copper gives blue-green, lithium gives red, potassium gives lilac. Not a smear of color either - each element gives its own exact shades, the same ones every time, which is how fireworks chemists mix a show and how astronomers name the elements in a star they will never visit.

That is the clue. If electrons could sit at any energy at all, heated atoms would glow in every color smoothly, like a blacksmith's iron. Instead they emit a few sharp colors. The only explanation is that an atom's electrons are allowed to hold certain energies and no others. Heat kicks an electron up to a higher allowed energy; when it drops back, the leftover energy leaves as light of one exact color. Sodium's famous yellow at 589 nm is a single, specific fall between two specific levels.

Key idea: Electron energy levels are not a filing convention chemists invented. They are forced on us by the fact that atoms emit only a few sharp colors.

An atom's chemistry is decided almost entirely by its electrons, and electrons are not scattered at random. They occupy energy levels, often called shells, numbered 1, 2, 3, and so on outward from the nucleus. Lower-numbered shells sit closer to the nucleus, are lower in energy, and fill first. Each shell holds only so many electrons: the first holds up to 2, the second up to 8, and the third up to 8 for the main-group elements you will focus on here.

Why do lower shells fill first? Because opposite charges attract, and an electron near the nucleus is held tightly - it sits in a deep energy well. Nature always settles into the lowest energy arrangement available, the same reason water runs downhill and a dropped ball ends up on the floor. An electron parked in shell 1 has given up more energy than one parked in shell 3, so shell 1 gets taken first.

Subshells and orbitals

Inside a shell, electrons live in subshells labeled s, p, d, and f. An orbital is a region where an electron is likely to be found, and each orbital holds at most 2 electrons. The s subshell has one orbital (2 electrons), and the p subshell has three orbitals (6 electrons). Electrons fill the lowest-energy subshells first, a rule called the Aufbau principle.

Writing electron configurations

An electron configuration lists which subshells are filled and how many electrons are in each. For example:

• Hydrogen (1 electron): 1s¹
• Carbon (6 electrons): 1s² 2s² 2p²
• Sodium (11 electrons): 1s² 2s² 2p⁶ 3s¹

The superscripts always add up to the total number of electrons in the atom.

Read a configuration like a receipt: 1s2 means "the s subshell of shell 1, holding 2 electrons." Sodium's 1s2 2s2 2p6 3s1 adds to 2 + 2 + 6 + 1 = 11, which had better equal sodium's atomic number of 11 - and it does. That check catches almost every configuration mistake you will ever make.

Long configurations get tedious, so chemists abbreviate with the previous noble gas in square brackets. Neon is 1s2 2s2 2p6, so sodium can be written [Ne] 3s1. Potassium (19 electrons) is [Ar] 4s1. The bracket stands for a full, unreactive core, and what follows it is exactly the part that does chemistry.

Two rules that keep electrons apart

Two extra rules finish the picture, and both come from the same source: electrons repel each other, so they spread out where they can.

• Pauli exclusion principle. An orbital holds at most 2 electrons, and those 2 must have opposite spin, usually drawn as one arrow up and one arrow down. That is why the cap is 2 and not 3 or 10.
• Hund's rule. When several orbitals in one subshell have the same energy, electrons occupy them singly, all with parallel spins, before any orbital gets a second electron. Two electrons crammed into one orbital repel each other more than two electrons in separate orbitals.

Nitrogen shows both. It has 7 electrons, so 2p holds 3 of them. Hund's rule puts one in each of the three p orbitals rather than pairing two up:

2p: [ up ] [ up ] [ up ] not [ up down ] [ up ] [   ]

Oxygen, with one more electron, is forced to pair: 2p: [ up down ] [ up ] [ up ]. Those two unpaired electrons in oxygen are the reason liquid oxygen is attracted to a magnet, a startling demonstration you can watch in any first-year chemistry video.

Valence electrons drive bonding

The electrons in the outermost shell are the valence electrons, and they are the ones that take part in bonding. For the main-group elements, the number of valence electrons matches the group number on the periodic table: Group 1 has 1, Group 2 has 2, and Groups 13 through 18 have 3 through 8. Atoms are especially stable when their outer shell is full, which for most elements means 8 valence electrons.

This is the octet rule, and it explains why the noble gases in Group 18, which already have full outer shells, barely react at all. The drive to reach a full octet is the single most useful idea for predicting how atoms bond.

The filling order

Subshells do not always fill in a simple outward march, because their energies overlap. The order that works for the elements in this course is: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p. Notice that 4s fills before 3d. A memory aid is the diagonal rule, where you list the subshells in rows and read along diagonals. For the first 20 elements you rarely go past 4s, so the pattern stays manageable.

Worked example: configuration of phosphorus

Phosphorus has 15 electrons. Fill the subshells in order, giving each its capacity until the electrons run out:

• 1s holds 2 (13 left), 2s holds 2 (11 left), 2p holds 6 (5 left), 3s holds 2 (3 left), 3p takes the last 3.

So phosphorus is 1s² 2s² 2p⁶ 3s² 3p³. Check the total: 2 + 2 + 6 + 2 + 3 = 15, correct. The outermost shell is the third, holding 3s² 3p³ = 5 valence electrons, which matches phosphorus's position in Group 15.

Worked example: counting valence electrons

Take calcium, element 20. Its configuration is 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² (total 20). The highest shell number present is 4, and it contains 4s², so calcium has 2 valence electrons, exactly what Group 2 predicts. To reach an octet, calcium finds it far easier to lose those 2 outer electrons than to gain 6, which is why calcium forms a 2+ ion.

Worked example: sulfur, two ways

Question: Write the full and shorthand configurations for sulfur (Z = 16), give its valence count, and predict the ion it forms.

Solution: Fill in order and keep a running total.

1. 1s takes 2 (14 remaining), 2s takes 2 (12), 2p takes 6 (6), 3s takes 2 (4), 3p takes the last 4.
2. Full: 1s2 2s2 2p6 3s2 3p4. Check: 2 + 2 + 6 + 2 + 4 = 16. Correct.
3. Shorthand: the noble gas before sulfur is neon (10 electrons), so sulfur is [Ne] 3s2 3p4.
4. Valence: shell 3 is outermost and holds 2 + 4 = 6 valence electrons, matching Group 16.
5. Prediction: gaining 2 electrons reaches 8; losing 6 would be far harder. So sulfur forms S^2-.

Where people get stuck: the 4s-before-3d swap looks like a typo, but it is real. Energy, not shell number, sets the filling order, and the 4s subshell happens to dip just below 3d in energy for the elements where it matters. That is exactly why potassium and calcium sit in the fourth row above the transition metals: their last electrons went into 4s, not 3d. When you write configurations in filling order and then read valence electrons off the highest shell number, both facts stay straight.

Common misconceptions

• "Shells fill strictly 1, 2, 3, 4 with no overlap." Energies overlap, so 4s fills before 3d. Follow the filling order, not just the shell number.
• "Valence electrons are all the electrons in the atom." Valence electrons are only those in the outermost (highest-numbered) shell, and they are the ones involved in bonding.
• "Every atom wants exactly 8 valence electrons." The octet rule is a strong guide, but hydrogen and helium are full with just 2 because the first shell holds only 2.
• "An orbital and a shell are the same thing." A shell contains subshells, which contain orbitals; each orbital holds at most 2 electrons.
• "Electrons orbit the nucleus like planets." The Bohr picture of neat circular orbits gets the energy levels right but the paths wrong. An orbital is a region of probability, not a track.
• "Emitting light means the atom lost an electron." In a flame test the electron only drops back down to a lower level and releases the difference as light. The atom keeps all its electrons.

Try it

(a) Write the full electron configuration of chlorine (Z = 17) and give its valence-electron count. (b) Write the shorthand configuration of potassium (Z = 19). (c) How many unpaired electrons does carbon have?

Answers: (a) 1s2 2s2 2p6 3s2 3p5, totalling 17, with 2 + 5 = 7 valence electrons (Group 17, so it needs just one more to reach an octet - which is why chlorine is so reactive). (b) [Ar] 4s1. (c) Carbon is 1s² 2s² 2p², and by Hund's rule those two p electrons occupy separate orbitals, so carbon has 2 unpaired electrons.

Recap

• Atoms emit only a few sharp colors, which proves electrons may hold only certain allowed energies.
• Electrons occupy shells that fill from the inside out because an electron close to the nucleus sits at lower energy, and systems settle at the lowest energy available.
• The filling order is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p - energy order, not shell order, which is why 4s comes before 3d.
• Each orbital holds at most 2 electrons with opposite spins (Pauli); s has 1 orbital, p has 3.
• Within one subshell, electrons spread out singly before pairing (Hund), because they repel each other.
• The superscripts in a configuration must sum to the atomic number - always check this.
• Valence electrons live in the highest-numbered shell, equal the group number for main-group elements, and drive all bonding through the octet rule.

Sources

1. OpenStax. (2019). Electronic structure of atoms (electron configurations) (Section 6.4). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). The Bohr model (Section 6.2). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Electronic structure of atoms (electron configurations) (Section 3.4). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. OpenStax. (2019). The Bohr model (Section 3.2). In Chemistry: Atoms First 2e. Rice University. openstax.org
5. LibreTexts. (n.d.). Electrons in atoms and the periodic table (Chapter 9). In Introductory Chemistry. chem.libretexts.org
6. Khan Academy. (n.d.). Atomic structure and properties [Unit]. In Chemistry. khanacademy.org
7. National Institute of Standards and Technology. (n.d.). Periodic table of the elements. Physical Measurement Laboratory. nist.gov

A table that predicted the future

When Dmitri Mendeleev published his table in 1869, he did something braver than sorting known elements: he left holes. He was so confident in the pattern that where no element fit, he insisted one must exist and had simply not been found. For the hole below silicon he predicted an element of atomic weight near 72, grey, with a density around 5.5 g/cm³. Seventeen years later Clemens Winkler isolated germanium: atomic weight 72.6, grey, density about 5.3 g/cm³.

Gallium and scandium filled two more of his holes within a decade. A table that only organizes is a filing cabinet; a table that predicts is a theory. This lesson is about the pattern behind those predictions, and it is a pattern you can read straight off the electron configurations from the last lesson.

The periodic table lists the elements in order of increasing atomic number, laid out in a grid that reveals repeating patterns. A vertical column is a group (or family); elements in a group have the same number of valence electrons and behave in similar ways. A horizontal row is a period; going across a period, a shell is being filled. Because properties repeat at regular intervals, the arrangement is called periodic.

Key idea: Elements in the same group act alike because they have the same number of valence electrons. The table is really a map of electron configurations.

Regions of the table

Metals take up the left and center: they are shiny, conduct heat and electricity, and tend to lose electrons. Nonmetals sit in the upper right and tend to gain or share electrons. A staircase of metalloids (such as silicon) divides the two and have properties in between. A few named families are worth knowing: Group 1 alkali metals are soft and very reactive, Group 2 alkaline earth metals are reactive but a bit less so, Group 17 halogens are reactive nonmetals, and Group 18 noble gases are nearly unreactive.

Three key trends

Atomic radius is the size of an atom. Going down a group, atoms get larger because each period adds a new shell. Going across a period from left to right, atoms get smaller because the growing positive charge in the nucleus pulls the same shell in tighter.

Reactivity follows a pattern too. Metals get more reactive going down a group because their outer electrons are farther from the nucleus and easier to lose. Nonmetals get more reactive going up and to the right because they pull in electrons more strongly. Fluorine is the most reactive nonmetal for this reason.

Electronegativity measures how strongly an atom pulls on shared bonding electrons. It increases across a period and decreases down a group. Fluorine, in the upper right (setting aside the noble gases), is the most electronegative element. These trends move together in a sensible way, and remembering the top-right corner of the table as small and highly electronegative will carry you a long way.

Why the trends happen

Two competing effects explain almost every trend. The first is nuclear charge: more protons pull electrons in more strongly. The second is shielding: inner-shell electrons partly block the outer electrons from feeling the full nuclear pull, and each new shell adds distance. Across a period, protons increase while the shell stays the same, so the pull wins and atoms shrink. Down a group, each step adds a full new shell, so distance and shielding win and atoms grow. Every trend below is just these two ideas applied.

Worked example: rank atoms by size

Put lithium (Li), fluorine (F), and cesium (Cs) in order of increasing atomic radius. Li and F are both in period 2, and since size shrinks left to right, F is smaller than Li. Cs sits far down Group 1, so it has many more shells and is by far the largest. Increasing order: F < Li < Cs. The reasoning combines the across-period trend (F smaller than Li) with the down-group trend (Cs much larger).

Worked example: compare electronegativity

Which is more electronegative, oxygen (O) or sulfur (S)? Both are in Group 16, but oxygen sits above sulfur. Electronegativity decreases down a group because the bonding electrons are farther from the nucleus and more shielded, so oxygen is more electronegative. This is why oxygen pulls electrons so strongly in bonds and forms compounds with almost every other element.

On the Pauling scale, which is the one you will meet again in the bonding lessons, fluorine is 3.98, oxygen is 3.44, sulfur is 2.58, lithium is 0.98, and cesium is 0.79. The numbers sweep from the bottom-left corner to the top-right corner exactly as the trend says they should, which is worth checking for yourself on any periodic table that prints them.

A fourth trend: ionization energy

Ionization energy is the energy needed to pull one electron off an atom. It follows the same logic as electronegativity: it increases across a period (electrons are held tighter) and decreases down a group (outer electrons are farther out and easier to remove). This is exactly why metals, on the left, give up electrons easily (low ionization energy) while nonmetals, on the right, hold theirs tightly.

Worked example: why magnesium stops at 2+

Ionization energies can be measured one electron at a time, and the numbers for magnesium tell a story you can read.

Electron removed	Energy needed (kJ/mol)
1st	738
2nd	1451
3rd	7733

The second electron costs about twice the first, which is unremarkable: once the atom is positive, it holds the rest more tightly. But the third costs more than five times the second. Something changed.

Look at the configuration: magnesium is [Ne] 3s2. The first two electrons come out of the 3s valence subshell. The third would have to be dragged out of the filled neon core, which sits closer to the nucleus and is complete. That is why magnesium forms Mg²⁺ and never Mg³⁺. The huge jump in the table is the octet rule showing up as a measured number, in kilojoules, rather than as a rule you were asked to believe.

Predicting ion charges from position

For main-group elements, the group tells you the charge the ion will take, because atoms take the cheapest route to a full outer shell.

• Group 1 (1 valence electron): lose 1, forming 1+. Na becomes Na⁺.
• Group 2 (2 valence): lose 2, forming 2+. Mg becomes Mg²⁺.
• Group 13 (3 valence): lose 3, forming 3+. Al becomes Al³⁺.
• Group 15 (5 valence): gain 3, forming 3-. N becomes N^3-.
• Group 16 (6 valence): gain 2, forming 2-. O becomes O^2-.
• Group 17 (7 valence): gain 1, forming 1-. Cl becomes Cl^-.
• Group 18: already full, so no ion forms under ordinary conditions.

Key idea: Count valence electrons. If fewer than four, losing them is cheaper; if more than four, gaining a few is cheaper. Either way the destination is the nearest noble-gas configuration.

Where people get stuck: assuming an atom "wants" a charge. Nothing wants anything. What happens is that some arrangements sit at lower energy than others, and the measured ionization energies above show exactly how much energy each step costs. Magnesium forms 2+ because the first two electrons are cheap and the third is ruinously expensive, not because magnesium prefers the number two.

Common misconceptions

• "Atoms get bigger across a period because you are adding particles." You add protons and electrons to the same shell, so the stronger nuclear pull actually makes atoms smaller across a period.
• "The period number is the same as the group number." Periods are rows (they give the number of occupied shells); groups are columns (they give the valence-electron count for main-group elements).
• "Noble gases are the most electronegative because they are in the top right." Noble gases already have full shells and generally do not attract more electrons, so fluorine, not neon, is the most electronegative element.
• "Metalloids are a type of metal." Metalloids lie along the staircase and have properties between metals and nonmetals; they are their own category.
• "A bigger atom must have a bigger nucleus, so it must hold electrons harder." Extra shells add distance and shielding faster than extra protons add pull, so bigger atoms lose electrons more easily, not less.
• "Elements in a group have the same number of electrons." They have the same number of valence electrons. Sodium has 11 electrons and potassium has 19, but both have exactly one in their outer shell.

Try it

(a) Rank Mg, Ba, and Ca by increasing atomic radius. (b) Which has the higher first ionization energy, sodium or chlorine? (c) What ion does aluminum form, and why?

Answers: (a) All three are Group 2, and radius grows down a group, so Mg < Ca < Ba. (b) Chlorine - both are in period 3, and ionization energy rises left to right as the nucleus pulls harder on the same shell. (c) Aluminum is in Group 13 with 3 valence electrons, so losing all three is cheaper than gaining five; it forms Al³⁺, reaching neon's configuration.

Recap

• The table orders elements by atomic number into groups (columns, same valence count) and periods (rows, filling one shell).
• Metals sit left and center, nonmetals upper right, and metalloids along the staircase between them.
• Two competing effects explain every trend: nuclear charge pulls electrons in, while extra shells and shielding push them out.
• Atomic radius grows down and to the left; electronegativity and ionization energy grow up and to the right, peaking at fluorine.
• Successive ionization energies show a huge jump once the valence shell is empty, which is why Mg stops at 2+.
• Group number predicts ion charge for main-group elements: 1+, 2+, 3+ on the left, 3-, 2-, 1- on the right, nothing for Group 18.
• Mendeleev's table predicted undiscovered elements, which is the strongest evidence that the pattern is real and not just tidy.

Sources

1. OpenStax. (2019). Periodic variations in element properties (Section 6.5). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). The periodic table (Section 2.5). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). The periodic table (Section 3.6). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. International Union of Pure and Applied Chemistry. (n.d.). Periodic table of elements. iupac.org
5. National Institute of Standards and Technology. (n.d.). Periodic table of the elements. Physical Measurement Laboratory. nist.gov
6. Royal Society of Chemistry. (n.d.). Periodic table. rsc.org
7. LibreTexts. (n.d.). Electrons in atoms and the periodic table (Chapter 9). In Introductory Chemistry. chem.libretexts.org

A violent reaction that makes something you eat

Drop a pea-sized piece of sodium metal into a jar of chlorine gas and it bursts into a blinding yellow flame. What is left in the jar is table salt, safe enough to sprinkle on chips. Two dangerous elements went in; a completely harmless compound came out. Understanding why that happens is the whole point of this lesson, and the answer is about electrons changing owners.

Atoms are most stable when their outer shell is full, and one way to get there is to gain or lose electrons. When an atom does this, it becomes an ion, a particle with an electric charge because its protons and electrons no longer balance.

Key idea: An ionic bond is not a stick joining two balls. It is the electrical attraction between a positive ion and a negative ion, pulling in every direction at once.

Cations and anions

A cation is a positive ion, formed when an atom loses one or more electrons. Metals do this readily because they have only a few valence electrons and low pull on them. Sodium loses one electron to become Na⁺. An anion is a negative ion, formed when an atom gains electrons. Nonmetals do this because they are only a few electrons short of a full shell. Chlorine gains one electron to become Cl^-. An easy memory aid: a cation is "paw-sitive," and gaining electrons (which are negative) makes an anion negative.

Predicting ion charges from the group

For the main-group elements, the periodic table tells you the likely charge.

Group	Typical ion charge	Example
1	+1	Na+
2	+2	Mg2+
13	+3	Al3+
15	-3	N3-
16	-2	O2-
17	-1	Cl-

Ions are not the same size as their atoms

Losing or gaining electrons changes an ion's size, and in a direction that makes sense once you say it out loud. A cation is smaller than its parent atom: sodium loses its single 3s electron, so its whole outer shell disappears and the remaining electrons are pulled in tighter by the unchanged 11 protons. A sodium atom is about 186 pm across; Na⁺ is about 102 pm, barely half.

An anion is larger than its parent atom: chlorine gains an electron, so the same 17 protons now have 18 electrons to hold, the crowding pushes the cloud outward, and the ion swells. A chlorine atom is about 99 pm; Cl^- is about 181 pm.

Both new ions are also isoelectronic with a noble gas, meaning they have exactly the same electron count and configuration. Na⁺ has 10 electrons, matching neon. Cl^- has 18, matching argon. That is the destination every main-group ion is heading for.

The ionic bond

When a metal meets a nonmetal, the metal hands over electrons and the nonmetal takes them. Now you have a positive cation and a negative anion, and opposite charges attract. That electrostatic attraction is the ionic bond. Sodium gives an electron to chlorine, and Na⁺ and Cl^- lock together as NaCl. Ionic compounds form rigid crystal patterns called lattices, have high melting points, and conduct electricity when melted or dissolved in water because their ions are then free to move.

Why the transfer actually happens

Here is a surprise worth sitting with. Handing an electron from sodium to chlorine, on its own, does not release energy - it costs energy.

• Pulling one electron off a sodium atom costs +496 kJ per mole (that is sodium's ionization energy).
• Attaching that electron to a chlorine atom releases -349 kJ per mole (chlorine's electron affinity).
• Running total: +496 - 349 = +147 kJ per mole. Uphill. It should not happen.

So why does the jar of sodium and chlorine catch fire? Because the story does not stop with one pair of ions. Millions of Na⁺ and Cl^- ions then snap together into a crystal, and building that lattice releases about -787 kJ per mole. Add it all up: +147 - 787 = about -640 kJ per mole released. That enormous downhill step is the flame you see.

Key idea: Ionic compounds form because the lattice is so stable, not because a single atom "wants" to lose an electron. The crystal pays the bill.

This also explains the properties. To melt NaCl you must break that whole lattice apart against 787 kJ per mole of attraction, which is why salt melts at 801 degrees C while candle wax melts in your hand. And because each ion is held by many neighbors in every direction, ionic solids are hard but brittle: shove one layer sideways and suddenly like charges face each other, and the crystal splits cleanly.

Worked example: predict the formula of an ionic compound

Combine magnesium and chlorine. Magnesium is in Group 2, so it forms Mg²⁺. Chlorine is in Group 17, so it forms Cl^-. A compound must be electrically neutral overall, so the positive and negative charges have to cancel. One Mg²⁺ supplies +2, and each Cl^- supplies -1, so you need two chloride ions: (+2) + 2(-1) = 0. The formula is MgCl₂.

A quick shortcut, the crossover method, is to take each ion's charge number and use it as the other ion's subscript: Mg gets subscript 1 (from Cl's charge of 1) and Cl gets subscript 2 (from Mg's charge of 2), giving MgCl₂. Always check that the charges truly cancel.

Worked example: aluminum oxide

Aluminum forms Al³⁺ (Group 13) and oxygen forms O^2- (Group 16). To balance +3 and -2, find the least common multiple of 3 and 2, which is 6. You need two Al³⁺ (total +6) and three O^2- (total -6), so the formula is Al₂O₃. Verify: 2(+3) + 3(-2) = +6 - 6 = 0. The crossover method gives the same result: aluminum takes subscript 2 and oxygen takes subscript 3.

Polyatomic ions

Some ions are groups of atoms carrying an overall charge, called polyatomic ions. Common ones include nitrate (NO₃^-), sulfate (SO₄^2-), hydroxide (OH^-), and ammonium (NH₄⁺), the one common positive polyatomic ion. They bond ionically just like single-atom ions. When you need more than one of a polyatomic ion, wrap it in parentheses: calcium nitrate is Ca(NO₃)₂, because one Ca²⁺ needs two NO₃^- to balance.

Worked example: three more formulas, step by step

The method never changes: write both ions with charges, then find the smallest whole-number ratio that adds to zero.

1. Sodium oxide. Na⁺ and O^2-. One oxide needs two sodiums: 2(+1) + 1(-2) = 0. Formula Na₂O.
2. Lithium nitride. Li⁺ and N^3-. Three lithiums per nitride: 3(+1) + 1(-3) = 0. Formula Li₃N.
3. Aluminum sulfate. Al³⁺ and SO₄^2-. The least common multiple of 3 and 2 is 6, so take two Al³⁺ (+6) and three sulfates (-6): 2(+3) + 3(-2) = 0. Because there is more than one polyatomic ion, it needs parentheses: Al₂(SO₄)₃.

Count the atoms in that last formula to be sure you can read it: 2 aluminum, then 3 sulfates each containing 1 sulfur and 4 oxygens, giving 3 sulfur and 12 oxygen. Nine atoms of one kind hiding behind one small subscript is exactly why parentheses matter.

Where people get stuck: writing AlSO43 or Al2SO43. Without parentheses, the subscript 3 attaches only to the oxygen and the formula now describes a different substance. The rule is simple: if you need more than one copy of a polyatomic ion, the whole ion goes in brackets and the subscript goes outside. If you need exactly one, no brackets - sodium nitrate is NaNO₃, never Na(NO₃).

Common misconceptions

• "Cations are negative because they come from cats." A cation is positive; it forms when an atom loses electrons. Anions are negative from gaining electrons.
• "Ionic compounds are made of molecules." Ionic solids are giant lattices of ions, not discrete molecules. We use a formula unit (the smallest whole-number ratio) instead.
• "Solid salt conducts electricity." In the solid, ions are locked in place. Salt conducts only when melted or dissolved, when the ions are free to move.
• "You add subscripts and charges together." Subscripts count atoms; charges must cancel to zero. Keep the two ideas separate when writing formulas.
• "An ion has a different number of protons than its atom." Only electrons move. Na and Na⁺ both have 11 protons; changing protons would change the element.
• "Charges appear in the finished formula." The charges are working notes. NaCl and MgCl₂ are written without any charge signs because the compound as a whole is neutral.

Try it

Write the formula for (a) potassium sulfide, (b) calcium phosphate given phosphate is PO₄^3-, and (c) ammonium chloride.

Answers: (a) K⁺ and S^2-, so two potassiums per sulfide: K₂S. (b) Ca²⁺ and PO₄^3-, least common multiple 6, so three calciums (+6) and two phosphates (-6): Ca₃(PO₄)₂ - the main mineral in your bones. (c) NH₄⁺ and Cl^- balance one to one, so NH₄Cl with no parentheses needed.

Recap

• An ion is a charged particle: cations form by losing electrons, anions by gaining them. Only electrons move, never protons.
• Main-group charges follow the group: +1, +2, +3 on the left and -3, -2, -1 on the right, each aiming at a noble-gas configuration.
• Cations shrink (a whole shell is gone) and anions swell (more electrons for the same nuclear pull).
• An ionic bond is the electrostatic attraction between oppositely charged ions throughout a lattice, not a single link between a pair.
• Transfer alone costs energy (+147 kJ/mol for Na and Cl); the lattice energy (about -787 kJ/mol) is what makes the reaction go.
• That same lattice energy explains high melting points, brittleness, and conduction only when molten or dissolved.
• Build formulas so the total charge is zero, and bracket any polyatomic ion you need more than one of.

Sources

1. OpenStax. (2019). Ionic bonding (Section 7.1). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Ionic and molecular compounds (Section 2.6). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Ionic bonding (Section 4.1). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. OpenStax. (2019). Strengths of ionic and covalent bonds (Section 7.5). In Chemistry 2e. Rice University. openstax.org
5. LibreTexts. (n.d.). Chemical bonding (Chapter 10). In Introductory Chemistry. chem.libretexts.org
6. Khan Academy. (n.d.). Chemical bonds [Unit]. In Chemistry. khanacademy.org
7. Britannica. (n.d.). Chemical bonding. Encyclopaedia Britannica. britannica.com

Why two atoms stop moving apart

Bring two hydrogen atoms slowly together. Far apart, they ignore each other. Closer in, each nucleus starts attracting the other atom's electron, and the energy of the pair falls - the system is getting more stable. Push closer still and the two positive nuclei start shoving each other, so the energy shoots back up. In between sits a sweet spot: 74 picometers apart, the lowest energy the pair can reach. That distance is the bond length of H₂, and the 436 kJ per mole you would have to pay to pull them apart again is its bond energy.

That is what a covalent bond is: not glue, not a stick, but two nuclei sharing a pool of electrons and sitting at the distance where the total energy is lowest.

Ionic bonds work when a metal can hand electrons to a nonmetal. But when two nonmetals meet, neither one wants to give up electrons, so they solve the problem a different way: they share. A shared pair of electrons is a covalent bond, and the group of atoms held together this way is a molecule.

Key idea: Metal plus nonmetal means electrons transfer (ionic). Nonmetal plus nonmetal means electrons are shared (covalent). Both roads lead to full outer shells.

Sharing to reach an octet

Two hydrogen atoms each have one electron and each need two for a full first shell, so they share one pair and form H₂. Oxygen atoms need more, so two oxygen atoms share two pairs (a double bond) to make O₂. Nitrogen atoms share three pairs (a triple bond) in N₂. In each case, sharing lets both atoms count the shared electrons toward a full outer shell.

More shared pairs means a shorter, stronger bond, and the numbers show it clearly:

Molecule	Bond	Length (pm)	Energy to break (kJ/mol)
H2	single	74	436
O2	double	121	498
N2	triple	110	945

Look at nitrogen: 945 kJ/mol is one of the strongest bonds in ordinary chemistry, and it explains something you can see. Air is 78 percent N₂ and it just sits there, refusing to react, while the 21 percent O₂ rusts your bike. Breaking that triple bond to make fertilizer costs so much energy that the industrial process for doing it, developed by Haber and Bosch, consumes roughly one percent of the world's energy supply. A number in a table, and a fact about global agriculture, are the same fact.

Drawing Lewis structures

A Lewis structure is a bookkeeping drawing that shows every valence electron, either as a shared pair (a line) or as an unshared lone pair (two dots). Four steps do it every time.

1. Add up the valence electrons from every atom.
2. Put the least electronegative atom in the middle (hydrogen is never central, since it can hold only 2 electrons).
3. Join the outer atoms to the center with single bonds, then spend the remaining electrons as lone pairs on the outer atoms first.
4. If the central atom is still short of 8, pull a lone pair in from a neighbor to make a double or triple bond.

Worked example: water, H₂O. Valence electrons: oxygen 6, plus 1 from each hydrogen, giving 6 + 1 + 1 = 8. Oxygen goes in the middle. Two O-H single bonds use 4 electrons, leaving 4. Hydrogen is already full with 2 each, so both remaining pairs go on oxygen as lone pairs. Final count: oxygen sees 4 bonding electrons plus 4 lone-pair electrons = 8. Each hydrogen sees 2. Everyone is satisfied, and all 8 electrons are accounted for.

Worked example: ammonia, NH₃. Valence electrons: nitrogen 5 plus three hydrogens at 1 each = 8. Nitrogen is central. Three N-H bonds use 6, leaving 2, which become one lone pair on nitrogen. Nitrogen sees 6 + 2 = 8. That single lone pair is why ammonia is a base, as you will see in Lesson 16.

Worked example: carbon dioxide, CO₂. Valence electrons: carbon 4 plus two oxygens at 6 each = 16. Carbon is central. Two single bonds use 4, leaving 12, which as three lone pairs on each oxygen gives every oxygen its octet - but carbon is stuck at 4 electrons. So pull one lone pair from each oxygen into the bond, making two double bonds. Now carbon has 8, each oxygen has 4 bonding plus 4 lone = 8, and the total is still 16. The structure is O=C=O.

Polar and nonpolar bonds

Sharing is not always equal. If the two atoms have different electronegativities, the more electronegative atom pulls the shared electrons closer to itself. This creates a polar covalent bond, with a slightly negative end and a slightly positive end. If the two atoms are identical or very close in electronegativity, the sharing is even and the bond is nonpolar. The bond in H₂ is nonpolar because both atoms are the same; the bond in H-Cl is polar because chlorine pulls harder. Bonding is really a spectrum, from perfectly shared, to unevenly shared, to fully transferred (ionic).

Comparing the two kinds of compounds

Property	Ionic compounds	Covalent compounds
Made of	Metal + nonmetal	Nonmetal + nonmetal
Smallest unit	Formula unit in a lattice	Molecule
Melting point	High	Usually lower
Conducts when dissolved	Usually yes	Usually no

These differences all trace back to one idea: ionic compounds are held together by strong attractions between charged ions spread through a whole lattice, while covalent compounds are separate molecules with weaker attractions between them.

The seven diatomic elements

Seven elements never travel alone in nature; they always pair up as diatomic molecules: hydrogen (H₂), nitrogen (N₂), oxygen (O₂), fluorine (F₂), chlorine (Cl₂), bromine (Br₂), and iodine (I₂). A common memory trick is "Have No Fear Of Ice Cold Beer," where each first letter cues one element. This matters when you balance equations later: the reactant is O₂, not a lone O.

Worked example: counting shared pairs with the octet rule

Predict the bonding in a molecule of fluorine, F₂. Each fluorine atom has 7 valence electrons and needs just 1 more for an octet. Neither atom will give up electrons (both are highly electronegative nonmetals), so they share one pair. That single shared pair counts toward both atoms' octets: each fluorine now sees its own 6 unshared electrons plus the 2 shared, for a full 8. The result is a single covalent bond, F-F. The number of bonds an atom forms usually equals the number of electrons it needs to complete its octet: 1 for a halogen, 2 for oxygen, 3 for nitrogen, 4 for carbon.

Worked example: is the bond polar?

Compare the bonds in Cl₂ and HCl. In Cl₂, both atoms are chlorine with identical electronegativity, so the electrons are shared equally and the bond is nonpolar. In HCl, chlorine (electronegativity about 3.0) pulls harder than hydrogen (about 2.1). The difference of roughly 0.9 pulls the shared electrons toward chlorine, so chlorine carries a small negative charge and hydrogen a small positive charge, making it a polar covalent bond. As a rough guide, electronegativity differences below about 0.4 are nonpolar, from about 0.4 to 1.7 are polar covalent, and above about 1.7 tend toward ionic.

A polar molecule is not the same as a polar bond

This is the step that separates a good answer from a great one. A molecule can be stuffed with polar bonds and still be nonpolar overall, because the pulls can cancel like a tug-of-war with evenly matched teams.

Carbon dioxide, O=C=O. Each C=O bond is strongly polar; oxygen (3.44) far outpulls carbon (2.55). But the molecule is linear, so one oxygen pulls left exactly as hard as the other pulls right. The two pulls cancel and CO₂ is a nonpolar molecule made of polar bonds.

Water, H-O-H. The O-H bonds are polar in the same way. But water is bent, at about 104.5 degrees, because oxygen's two lone pairs push the hydrogens down to one side. Now both pulls point roughly the same way and they add instead of cancelling. Water is a strongly polar molecule.

That one geometric difference is why water dissolves salt and sugar, why it has a startlingly high boiling point for such a light molecule, and ultimately why life is wet. Shape decides.

Key idea: To judge a molecule's polarity you need two things: the polarity of each bond, and the molecule's shape. Symmetric shapes cancel; lopsided ones do not.

Where people get stuck: counting bonds instead of counting electrons. When you check an octet, a double bond contributes four electrons to that atom's count, not two, and a lone pair contributes two even though it is bonding nothing. In O=C=O the carbon looks like it only has "two bonds," which sounds wrong for carbon - until you count electrons and find 4 + 4 = 8. Count electrons, never lines.

Common misconceptions

• "Covalent means the atoms give away electrons." Giving away electrons is ionic. Covalent bonding is sharing, so both atoms count the shared electrons.
• "Every covalent bond is polar." Bonds between identical atoms (H₂, O₂, Cl₂) are nonpolar because the sharing is perfectly even.
• "Oxygen exists as single O atoms in air." The oxygen you breathe is O₂, a diatomic molecule with a double bond, not lone oxygen atoms.
• "Molecular compounds conduct electricity like salt water." Most molecular compounds have no ions, so their solutions usually do not conduct.
• "A molecule with polar bonds must be a polar molecule." CO₂ has two very polar bonds that point in opposite directions and cancel, leaving a nonpolar molecule. Shape decides.
• "A double bond is two separate single bonds." It is one bond made of two shared pairs, and it is shorter and stronger than a single bond, not twice as long.

Try it

(a) How many valence electrons go into a Lewis structure of CH₄, and how many lone pairs does carbon end up with? (b) Is the C-H bond polar? (c) Is CH₄ a polar molecule?

Answers: (a) Carbon 4 plus four hydrogens at 1 each = 8 electrons. All 8 are used in four C-H bonds, so carbon has zero lone pairs and a full octet. (b) Carbon is 2.55 and hydrogen 2.20 on the Pauling scale, a difference of only 0.35, so the bond is essentially nonpolar. (c) No. Even if those bonds were slightly polar, methane's four bonds point symmetrically toward the corners of a tetrahedron, so every pull is cancelled by the others.

Recap

• A covalent bond is two nuclei sharing electrons at the separation where the pair's energy is lowest - 74 pm and 436 kJ/mol for H₂.
• Nonmetal plus nonmetal means sharing; metal plus nonmetal means transfer.
• Sharing one, two, or three pairs gives single, double, and triple bonds; more pairs means shorter and stronger, which is why N₂ at 945 kJ/mol is so unreactive.
• Lewis structures: count all valence electrons, put the least electronegative atom in the middle, bond, fill outer atoms, then form multiple bonds if the center is short.
• Count electrons rather than lines: a double bond gives an atom four electrons, a lone pair gives two.
• Unequal electronegativity makes a bond polar; identical atoms make it nonpolar.
• Molecular polarity needs bond polarity and shape. Linear CO₂ cancels out; bent water does not, which is why water dissolves so much.
• Seven elements occur as diatomic molecules, and covalent compounds generally melt lower and conduct less than ionic ones.

Sources

1. OpenStax. (2019). Covalent bonding (Section 7.2). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Lewis symbols and structures (Section 7.3). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Molecular structure and polarity (Section 7.6). In Chemistry 2e. Rice University. openstax.org
4. OpenStax. (2019). Covalent bonding (Section 4.2). In Chemistry: Atoms First 2e. Rice University. openstax.org
5. OpenStax. (2019). Lewis symbols and structures (Section 4.4). In Chemistry: Atoms First 2e. Rice University. openstax.org
6. PhET Interactive Simulations. (n.d.). Molecule shapes [Simulation]. University of Colorado Boulder. phet.colorado.edu
7. LibreTexts. (n.d.). Chemical bonding (Chapter 10). In Introductory Chemistry. chem.libretexts.org

One letter apart, worlds apart

Sodium chloride is table salt. Sodium chlorate is a weedkiller. The names differ by three letters and the formulas by three oxygen atoms, and confusing them in a stockroom would be a serious mistake. That is why chemistry has an agreed naming system: so that a name written in Toronto is read identically in Tokyo, with no guessing.

Nomenclature is the set of rules for naming compounds so that every chemist reads the same name the same way. The rules are different for ionic and molecular compounds, so first decide which type you have: a metal with a nonmetal is ionic, while two nonmetals are molecular.

Key idea: Before you name anything, classify it. Ionic names use charges and never prefixes; molecular names use prefixes and never charges. Getting the classification right makes the rest almost automatic.

Here is the decision, in order:

1. Does the formula start with a metal (or with NH₄)? Then it is ionic. Go to the ionic rules.
2. Is the metal one with a fixed charge (Groups 1 and 2, plus Al, Zn, Ag)? Then no Roman numeral. Otherwise work out the charge and write one.
3. Are both parts nonmetals? Then it is molecular. Use prefixes.

Naming ionic compounds

Name the cation (metal) first, unchanged, then the anion (nonmetal) with its ending changed to -ide. So NaCl is sodium chloride and MgO is magnesium oxide. To write a formula from a name, balance the charges so the compound comes out neutral. Magnesium is Mg²⁺ and chloride is Cl^-, so it takes two chlorides per magnesium: MgCl₂.

Many transition metals can form more than one charge, so a Roman numeral in the name shows which one. Iron can be Fe²⁺ or Fe³⁺: FeCl₂ is iron(II) chloride and FeCl₃ is iron(III) chloride.

Polyatomic ions

A polyatomic ion is a charged group of atoms that stays together as a unit. A few common ones are worth memorizing.

Name	Formula
Nitrate	NO3-
Sulfate	SO42-
Carbonate	CO32-
Hydroxide	OH-
Ammonium	NH4+

When a formula needs more than one polyatomic ion, put it in parentheses: calcium nitrate is Ca(NO₃)₂, because Ca²⁺ needs two nitrate ions to balance.

A few more show up constantly, and they are worth adding to the list: phosphate (PO₄^3-), acetate (C₂H₃O₂^-), hydrogen carbonate or bicarbonate (HCO₃^-), and cyanide (CN^-).

The -ate and -ite pattern

Polyatomic names look arbitrary until you spot the system. Two names built on the same element differ by exactly one oxygen, and the endings tell you which is which: -ate has one more oxygen than -ite, while the charge stays the same.

• Sulfate SO₄^2- and sulfite SO₃^2-. Same 2- charge, one oxygen apart.
• Nitrate NO₃^- and nitrite NO₂^-. Same 1- charge.
• Chlorate ClO₃^- and chlorite ClO₂^-, plus perchlorate ClO₄^- (one more still) and hypochlorite ClO^- (one fewer still). The bleach in your laundry cupboard is sodium hypochlorite, NaClO.

Learn one member of each pair and the ending gives you the other, which turns a memorization job into a much smaller one.

Naming molecular compounds

For two nonmetals, use prefixes to show how many of each atom is present: mono (1), di (2), tri (3), tetra (4), penta (5). The first element keeps its name (dropping mono if it would start the name), and the second gets the -ide ending. So CO is carbon monoxide, CO₂ is carbon dioxide, and N₂O₄ is dinitrogen tetroxide. Notice that ionic names never use these prefixes; the charges already fix the ratio. Mixing up the two systems is the single most common naming mistake, so always check the type first.

Worked example: name an ionic compound with a variable metal

Name Fe₂O₃. Oxygen is reliably O^2-, and there are three of them, for a total negative charge of -6. That -6 must be balanced by two iron atoms, so together the irons carry +6, which means each iron is +3. Iron can be +2 or +3, so the name needs a Roman numeral: iron(III) oxide. The trick with variable metals is to work backward from the known anion charge to find the metal's charge.

Worked example: write a formula from a name

Write the formula for ammonium sulfate. Ammonium is the polyatomic cation NH₄⁺ and sulfate is SO₄^2-. To balance +1 against -2, you need two ammonium ions per sulfate: 2(+1) + (-2) = 0. Because you need more than one polyatomic ion, wrap ammonium in parentheses: (NH₄)₂SO₄. Notice the sulfate needs no parentheses because only one is present.

Worked example: name a molecular compound

Name P₂O₅. Both are nonmetals, so use prefixes. There are two phosphorus atoms (di) and five oxygen atoms (penta), and the second element takes the -ide ending: diphosphorus pentoxide. The a in penta is dropped before oxide to keep the name easy to say. There is no Roman numeral here because molecular naming uses prefixes instead of charges.

Worked example: a transition metal with a polyatomic ion

Question: Name Cu(NO₃)₂.

Solution: Copper is a transition metal, so it may need a Roman numeral. Work from the part you know for certain.

1. Nitrate is NO₃^-, and there are two of them, so the negative total is 2(-1) = -2.
2. The compound must be neutral, so the single copper must be +2.
3. Copper can be +1 or +2, so the numeral is required: copper(II) nitrate.

Notice the numeral is II, not 2, and there is no space before the bracket. Also notice that the "2" in the formula is the number of nitrates, while the "II" in the name is the copper's charge. Those two numbers happen to match here, and often they do not.

Worked example: from name to formula, in reverse

Question: Write the formula for chromium(III) sulfide.

Solution: The Roman numeral hands you the cation directly: Cr³⁺. Sulfide is the -ide form of sulfur, a Group 16 element, so it is S^2-. Balance +3 against -2 using the least common multiple 6: two chromiums give +6, three sulfides give -6. The formula is Cr₂S₃. Check: 2(+3) + 3(-2) = 0.

Question: Write the formula for magnesium hydroxide (the active ingredient in milk of magnesia).

Solution: Magnesium is Group 2, so Mg²⁺. Hydroxide is OH^-. One magnesium needs two hydroxides, and since there is more than one polyatomic ion it takes parentheses: Mg(OH)₂.

Where people get stuck: reaching for a Roman numeral by habit. Sodium is always +1, calcium is always +2, aluminum is always +3, and zinc and silver are reliably +2 and +1, so "sodium(I) chloride" is wrong, not just clumsy. The numeral exists to resolve an ambiguity, and where there is no ambiguity it does not belong. The reverse trap is leaving it off a metal that genuinely varies: "iron oxide" is incomplete, because FeO and Fe₂O₃ are different compounds with different colors, formulas, and uses.

Common misconceptions

• "Use prefixes like di and tri for ionic compounds." Prefixes are only for molecular (two-nonmetal) compounds. Ionic names never use them because the charges already fix the ratio.
• "The Roman numeral shows how many atoms there are." The Roman numeral shows the charge on the metal, not the number of atoms. Iron(III) means Fe³⁺, not three irons.
• "Every metal needs a Roman numeral." Only metals that can have more than one charge (many transition metals) need one. Group 1 and 2 metals have fixed charges.
• "Polyatomic ions break apart, so you never need parentheses." A polyatomic ion stays together as a unit, so when you need more than one you enclose it in parentheses, as in Ca(NO₃)₂.
• "-ate and -ite are different charges." They differ by one oxygen atom only. Sulfate and sulfite are both 2-; nitrate and nitrite are both 1-.
• "The Roman numeral is copied from a subscript." You have to work it out from the anion charge. In Cu(NO₃)₂ the numeral II happens to match the subscript, but in Fe₂O₃ the numeral is III and no subscript is 3 on the iron.

Try it

Name (a) K₂SO₄ and (b) SF₆. Write formulas for (c) lead(II) iodide and (d) sodium sulfite.

Answers: (a) Potassium is fixed at +1, so no numeral: potassium sulfate. (b) Two nonmetals, so prefixes: sulfur hexafluoride (mono is dropped from the first element). (c) Pb²⁺ with I^- needs two iodides: PbI₂. (d) Sulfite is SO₃^2-, so two sodiums balance it: Na₂SO₃.

Recap

• Classify first: metal (or ammonium) plus nonmetal is ionic; two nonmetals is molecular.
• Ionic names are cation then anion with an -ide ending, and they never use count prefixes because the charges already fix the ratio.
• Add a Roman numeral only for metals with more than one possible charge, and find it by working backward from the anion's charge.
• Molecular names use mono, di, tri, tetra, penta, hexa to count atoms of each element.
• Learn the common polyatomic ions, and use the -ate/-ite pattern: -ate has one more oxygen, same charge.
• Bracket any polyatomic ion you need more than one of, as in Ca(NO₃)₂ and Mg(OH)₂.
• Going from a name back to a formula, always finish by checking that the charges sum to zero.

Sources

1. OpenStax. (2019). Chemical nomenclature (Section 2.7). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Chemical formulas (Section 2.4). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Chemical nomenclature (Section 4.3). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. OpenStax. (2019). Ionic and molecular compounds (Section 2.6). In Chemistry 2e. Rice University. openstax.org
5. International Union of Pure and Applied Chemistry. (n.d.). Periodic table of elements. iupac.org
6. LibreTexts. (n.d.). Molecules and compounds (Chapter 5). In Introductory Chemistry. chem.libretexts.org
7. National Institute of Standards and Technology. (n.d.). NIST Chemistry WebBook. Standard Reference Database 69. webbook.nist.gov

The experiment that made chemistry a science

In the 1770s Antoine Lavoisier did something obvious in hindsight and revolutionary at the time: he weighed things. He sealed tin and air inside a flask, weighed the whole thing, heated it until the metal turned to a grey powder, and weighed it again. The mass had not changed by a hair. Open the flask and air rushed in, and only then did the mass climb - by exactly the mass of the air that entered. Burning was not destroying anything. It was combining with something.

That result gave chemistry its first hard accounting rule, and every equation you balance in this lesson is that rule written down.

A chemical reaction rearranges atoms to make new substances. We describe it with a chemical equation: the reactants (starting materials) go on the left, an arrow shows the direction of change, and the products (new substances) go on the right. For example, hydrogen burning in oxygen is written H₂ + O₂ → H₂O.

Key idea: A reaction rearranges atoms; it never creates or destroys them. Balancing is just making the equation say that honestly.

Conservation of mass

The law of conservation of mass says that atoms are never created or destroyed in a reaction, only rearranged. That means the number of atoms of each element must be the same on both sides of the equation. The equation H₂ + O₂ → H₂O is not yet balanced: the left has 2 oxygen atoms but the right has only 1. A balanced equation is our way of respecting conservation of mass.

Reading a formula for atom counts

Before you can balance, you have to count, and counting means multiplying the coefficient by every subscript inside the formula.

• 3 H2O means 3 molecules, each with 2 H and 1 O, so 6 H and 3 O.
• 2 Ca(NO3)2 means 2 formula units, each with 1 Ca and 2 nitrates, so 2 Ca, 4 N, and 12 O. The 3 inside multiplies the 2 outside the bracket, then the coefficient 2 multiplies everything again.
• 4 Al2O3 means 8 Al and 12 O.

Get this right and balancing becomes bookkeeping. Get it wrong and no amount of clever coefficient-juggling will save you.

Balancing with coefficients

We balance by placing coefficients (big numbers in front of formulas) to change how many of each molecule we have. You may change coefficients, but you must never change the subscripts inside a formula, because that would turn the substance into something else.

1. Count the atoms of each element on both sides.
2. Add coefficients to make the counts match, one element at a time.
3. Save hydrogen and oxygen for last when possible, and recount at the end.

Worked example. Balance H₂ + O₂ → H₂O. Put a 2 in front of H₂O to get 2 oxygen atoms on the right, matching the O₂ on the left. That gives 2H₂O, which now has 4 hydrogen atoms, so put a 2 in front of H₂ on the left. The balanced equation is:

2 H₂ + O₂ → 2 H₂O

Check: left has 4 H and 2 O; right has 4 H and 2 O. It balances.

Second worked example. Methane burning: CH₄ + O₂ → CO₂ + H₂O. Carbon is fine (1 each). Hydrogen: the left has 4, so put a 2 in front of H₂O to get 4 on the right. Now count oxygen on the right: 2 (in CO₂) + 2 (in 2 H₂O) = 4, so put a 2 in front of O₂. The result is CH₄ + 2 O₂ → CO₂ + 2 H₂O.

Here is the before-and-after audit, which is the habit worth building. Never declare an equation balanced without writing this table out:

Atom	Left (unbalanced)	Right (unbalanced)	Left (balanced)	Right (balanced)
C	1	1	1	1
H	4	2	4	4
O	2	3	4	4

The two right-hand columns match row for row, so mass is conserved and the equation is finished.

Worked example: rusting iron

Question: Balance Fe + O₂ → Fe₂O₃.

Solution: Start with the element that appears in the fewest places, then fix oxygen last.

1. Iron: the right has 2 in Fe₂O₃. Try 2 Fe on the left.
2. Oxygen: the right now has 3, the left has 2. Neither is a multiple of the other, so use the least common multiple, 6.
3. Six oxygens on the right needs 2 Fe₂O₃ (2 x 3 = 6). Six on the left needs 3 O₂.
4. Recount iron: the right now has 2 x 2 = 4, so the left needs 4 Fe.

4 Fe + 3 O₂ → 2 Fe₂O₃

Atom	Left	Right
Fe	4	2 x 2 = 4
O	3 x 2 = 6	2 x 3 = 6

Balanced. Notice that iron had to be revisited after oxygen was fixed. That back-and-forth is normal, not a sign you did it wrong.

Worked example: propane in a barbecue

Question: Balance C₃H₈ + O₂ → CO₂ + H₂O.

Solution: For any combustion, do carbon, then hydrogen, then oxygen. Carbon: 3 on the left, so 3 CO₂. Hydrogen: 8 on the left, and each water carries 2, so 4 H₂O. Oxygen on the right is now 3 x 2 + 4 x 1 = 10 atoms, which is 5 O₂.

C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O

Atom	Left	Right
C	3	3
H	8	4 x 2 = 8
O	5 x 2 = 10	6 + 4 = 10

Handling polyatomic ions and fractions

Two tricks make harder equations easier. First, if a polyatomic ion appears unchanged on both sides, balance it as a single unit instead of atom by atom. Second, if balancing forces a fraction, clear it by multiplying every coefficient through.

Worked example (fraction cleared). Balance C₂H₆ + O₂ → CO₂ + H₂O. Balance carbon first: 2 CO₂ on the right. Balance hydrogen: 6 H on the left means 3 H₂O on the right. Now count oxygen on the right: 2(2) + 3(1) = 7 atoms, so you need 7/2 O₂. To remove the fraction, multiply every coefficient by 2:

2 C₂H₆ + 7 O₂ → 4 CO₂ + 6 H₂O

Check: left 4 C, 12 H, 14 O; right 4 C, 12 H, (8 + 6) = 14 O. Balanced.

Worked example (polyatomic ion kept whole). Balance Ca(NO₃)₂ + Na₃PO₄ → Ca₃(PO₄)₂ + NaNO₃. Nitrate and phosphate survive the reaction unchanged, so treat each as a single item rather than counting N, P, and O separately.

1. Calcium: the right needs 3, so put 3 in front of Ca(NO₃)₂.
2. Phosphate: the right has 2, so put 2 in front of Na₃PO₄.
3. Sodium: the left now has 2 x 3 = 6, so put 6 in front of NaNO₃.
4. Nitrate: the left has 3 x 2 = 6 and the right now has 6. Done.

3 Ca(NO₃)₂ + 2 Na₃PO₄ → Ca₃(PO₄)₂ + 6 NaNO₃

Counting nitrate as one unit rather than tracking 18 separate oxygen atoms turned a nasty problem into four short lines.

State symbols

Real equations usually carry a small label after each formula saying what physical state it is in: (s) solid, (l) liquid, (g) gas, and (aq) aqueous, meaning dissolved in water. Combustion of methane fully written out is:

CH₄(g) + 2 O₂(g) → CO₂(g) + 2 H₂O(g)

These labels are not decoration. In Lesson 12 the state decides whether a product falls out of solution as a visible solid, and in Lesson 17 it changes the energy released, because turning liquid water into steam costs extra energy.

Where people get stuck: the temptation to "fix" an equation by editing a subscript. If methane's equation will not balance, changing H₂O to H₂O₂ does balance the oxygen - and also replaces water with hydrogen peroxide, a bleach that would sting a cut. Subscripts are facts about the substances, decided by the bonding rules from Lessons 7 to 9. Coefficients are the only dial you may turn.

Types of reactions

Recognizing a reaction type helps you predict products. The main patterns are:

• Synthesis (combination): two or more reactants join into one product, A + B → AB, as in 2 H₂ + O₂ → 2 H₂O.
• Decomposition: one reactant splits into two or more products, AB → A + B, as in 2 H₂O₂ → 2 H₂O + O₂.
• Single replacement: one element takes another's place, A + BC → AC + B, as in Zn + 2 HCl → ZnCl₂ + H₂.
• Double replacement: two compounds swap partners, AB + CD → AD + CB, as in AgNO₃ + NaCl → AgCl + NaNO₃.
• Combustion: a fuel reacts with O₂ to give CO₂ and H₂O, as in the methane example above.

Common misconceptions

• "You can balance by changing subscripts." Changing H₂O to H₂O₂ makes a different substance. Only coefficients may be changed.
• "Balanced means equal numbers of molecules on each side." Balancing means equal numbers of each type of atom, not equal molecule counts.
• "A subscript of 1 or a coefficient of 1 must be written." A coefficient or subscript of 1 is understood and left off, as the single O in H₂O.
• "Fractions are always wrong in equations." A fraction is a valid intermediate step; you simply multiply through to clear it for the final whole-number answer.
• "Mass disappears when something burns." A log leaves a little ash because most of its mass left as CO₂ and water vapor. Weigh the gases too, as Lavoisier did, and nothing is missing.
• "A coefficient multiplies only the first atom in the formula." It multiplies everything in the formula, including whatever is inside brackets.

Try it

Balance (a) Al + CuCl₂ → AlCl₃ + Cu and (b) Ca(OH)₂ + HCl → CaCl₂ + H₂O.

Answers: (a) 2 Al + 3 CuCl₂ → 2 AlCl₃ + 3 Cu. Chlorine is the awkward one: 2 per CuCl₂ and 3 per AlCl₃, so use the least common multiple 6, giving 3 CuCl₂ and 2 AlCl₃; aluminum and copper then follow. Check: left 2 Al, 3 Cu, 6 Cl; right 2 Al, 6 Cl, 3 Cu. (b) Ca(OH)₂ + 2 HCl → CaCl₂ + 2 H₂O. Check: left 1 Ca, 2 O, 2 + 2 = 4 H, 2 Cl; right 1 Ca, 2 Cl, 4 H, 2 O.

Recap

• Reactants go on the left, products on the right, and atoms are only rearranged - Lavoisier's sealed-flask result.
• Count atoms by multiplying the coefficient through every subscript, including everything inside brackets.
• Balance with coefficients only. Changing a subscript changes the substance.
• Work one element at a time, leave hydrogen and oxygen until last, and expect to revisit earlier elements.
• Keep a polyatomic ion whole if it survives the reaction unchanged; it saves enormous effort.
• A fraction is a legal intermediate step - multiply every coefficient through to clear it.
• Always finish with an atom-count check, left column against right column, before calling it balanced.
• State symbols (s), (l), (g), (aq) carry real information you will need for solutions and energy later.
• The five patterns - synthesis, decomposition, single replacement, double replacement, combustion - let you predict products.

Sources

1. OpenStax. (2019). Writing and balancing chemical equations (Section 4.1). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Classifying chemical reactions (Section 4.2). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Writing and balancing chemical equations (Section 7.1). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. LibreTexts. (n.d.). Chemical reactions (Chapter 7). In Introductory Chemistry. chem.libretexts.org
5. PhET Interactive Simulations. (n.d.). Balancing chemical equations [Simulation]. University of Colorado Boulder. phet.colorado.edu
6. Khan Academy. (n.d.). Chemical reactions and stoichiometry [Unit]. In Chemistry. khanacademy.org
7. American Chemical Society. (n.d.). Education resources. acs.org

Counting by weighing

Walk into a hardware store and ask for 5000 nails. Nobody counts them. They weigh a handful, work out the mass of one nail, and weigh out the right total. Counting by weighing is an old, practical trick, and it is exactly what a chemist does with atoms - except a chemist is counting things far too small to see even one of.

Atoms are far too small and too many to count one at a time, so chemists count them in giant bundles using the mole. A mole is just a fixed number of things, the way a dozen is 12. One mole is Avogadro's number of particles: about 6.022 × 10²³. That enormous number is chosen so that one mole of a substance has a mass in grams equal to its atomic or molecular mass in amu. This link between the atomic scale and the gram scale is the beating heart of quantitative chemistry.

Key idea: The mole is a bridge. On one side is the world you can weigh, in grams. On the other is the world of individual atoms. Molar mass is the bridge itself.

How big is 6.022 x 10^23 really? Suppose you counted particles at one per second, without sleeping, from now on. Finishing one mole would take about 1.9 x 10^16 years, which is more than a million times the age of the universe. You are never counting these one at a time, which is precisely why the mole exists.

Since 2019 the mole has a fixed definition: one mole contains exactly 6.02214076 x 10^23 elementary entities, a number set by international agreement rather than measured. For every calculation in this course, 6.022 x 10^23 is plenty.

Atom, molecule, mole: three different questions

These three words get tangled constantly, so separate them with an example. Take a flask holding 1 mole of carbon dioxide.

• Moles of CO₂: 1 mol, by assumption.
• Molecules of CO₂: 6.022 x 10^23, because that is what a mole means.
• Atoms in total: each molecule has 3 atoms (1 C and 2 O), so 3 x 6.022 x 10^23 = 1.807 x 10^24 atoms.
• Moles of oxygen atoms: 2 mol, since each CO₂ carries two of them.

One flask, four different correct answers, depending entirely on what was asked. Read the question twice and underline the thing being counted; that single habit prevents most mole-lesson mistakes.

Molar mass

The molar mass of a substance is the mass of one mole of it, in grams per mole (g/mol). For an element, it is just the atomic mass from the periodic table. For a compound, add up the molar masses of all the atoms in the formula.

Worked example. Find the molar mass of water, H₂O. Hydrogen is about 1.008 g/mol and oxygen is about 16.00 g/mol.

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

So one mole of water weighs about 18.02 g. As a second example, the molar mass of carbon dioxide, CO₂, is 12.01 + 2(16.00) = 44.01 g/mol.

Worked example with parentheses. Find the molar mass of calcium hydroxide, Ca(OH)₂. The subscript 2 applies to the whole bracket, so there are 2 oxygens and 2 hydrogens, not one of each.

40.08 + 2(16.00 + 1.008) = 40.08 + 2(17.008) = 40.08 + 34.02 = 74.10 g/mol

Forgetting to distribute that subscript is the most common molar-mass error there is, and it quietly wrecks every calculation downstream.

Converting grams, moles, and particles

Molar mass and Avogadro's number are the two conversion factors that link the three quantities. The core relationships are:

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

Worked example. How many moles are in 36.0 g of water? Divide by the molar mass: 36.0 g ÷ 18.02 g/mol = 2.00 mol. And how many molecules is that? Multiply by Avogadro's number: 2.00 mol × 6.022 × 10²³ = 1.20 × 10²⁴ molecules. Set these up with dimensional analysis, watch the units cancel, and even long chains become routine.

Both directions, with the units on show

The safest way to avoid multiplying when you should divide is never to decide at all: write molar mass as a fraction and let the units tell you which way up it goes.

Grams to moles. How many moles are in 25.0 g of sodium chloride? Molar mass NaCl = 22.99 + 35.45 = 58.44 g/mol. You want grams to cancel, so grams must sit on the bottom of the fraction:

25.0 g NaCl x (1 mol NaCl / 58.44 g NaCl) = 0.428 mol NaCl

The unit "g NaCl" appears once on top and once on the bottom and cancels; mol survives, which is what was asked for.

Moles to grams. What is the mass of 0.250 mol of glucose, C₆H₁₂O₆? First build the molar mass: 6(12.01) + 12(1.008) + 6(16.00) = 72.06 + 12.10 + 96.00 = 180.16 g/mol. Now flip the fraction so moles cancel:

0.250 mol x (180.16 g / 1 mol) = 45.0 g glucose

Same conversion factor, used upside down. That is the entire difference between the two directions, and the units decide it for you every time.

Particles to moles. How many moles are 3.011 x 10^23 molecules of ammonia? Put Avogadro's number on the bottom so molecules cancel:

3.011 x 10^23 molecules x (1 mol / 6.022 x 10^23 molecules) = 0.5000 mol

Half of Avogadro's number, half a mole. A sanity check you can do in your head, which is exactly the kind of check worth doing before you trust a calculator.

Worked example: a full grams-to-particles chain

How many oxygen atoms are in 50.0 g of carbon dioxide, CO₂? This takes three linked steps. First convert grams to moles of CO₂ using the molar mass 44.01 g/mol:

50.0 g ÷ 44.01 g/mol = 1.136 mol CO₂

Next convert moles of CO₂ to molecules with Avogadro's number:

1.136 mol × 6.022 × 10²³ = 6.84 × 10²³ molecules CO₂

Finally, each CO₂ molecule contains 2 oxygen atoms, so multiply by 2:

6.84 × 10²³ × 2 = 1.37 × 10²⁴ oxygen atoms

The chain grams → moles → molecules → atoms used one conversion factor at each arrow, and every unit cancelled cleanly into the next.

Percent composition

Molar mass also tells you what fraction of a compound's mass comes from each element, called the percent composition. For each element: percent = (mass of that element in one mole ÷ molar mass of the compound) × 100. Worked example. In water (18.02 g/mol), the two hydrogens contribute 2.016 g, so hydrogen is (2.016 ÷ 18.02) × 100 = 11.2%, and oxygen is (16.00 ÷ 18.02) × 100 = 88.8%. The two percentages add to 100%, which is a quick check.

Common misconceptions

• "A mole of any two substances weighs the same." A mole is a fixed count of particles, but the mass depends on the substance: a mole of water is 18.02 g while a mole of CO₂ is 44.01 g.
• "Molar mass and atomic mass are unrelated numbers." They are numerically equal: an element's atomic mass in amu equals its molar mass in g/mol.
• "To get particles you divide by Avogadro's number." Moles times Avogadro's number gives particles; you divide by it to go from particles back to moles.
• "You can convert grams straight to particles in one step." Grams connect to particles only through moles, so you must pass through moles using molar mass first.
• "A mole of a compound is the same as a mole of its atoms." One mole of CO₂ contains one mole of carbon atoms but two moles of oxygen atoms, and three moles of atoms in total.
• "The subscript outside a bracket only applies to the last atom." In Ca(OH)₂ the 2 applies to both the O and the H, giving 2 oxygens and 2 hydrogens.

Where people get stuck: deciding whether to multiply or divide by the molar mass. Do not decide - reason from the units. If the answer needs moles and you have grams, put grams on the bottom of the fraction so it cancels. If the answer needs grams and you have moles, put moles on the bottom. A quick reality check helps too: molar masses are usually tens to hundreds of grams, so a few grams of anything is a small fraction of a mole, and a few hundred grams is a few moles. If you calculate 1500 moles from 25 grams, you divided the wrong way round.

Try it

(a) What is the molar mass of ammonium sulfate, (NH₄)₂SO₄? (b) How many moles are in 88.0 g of CO₂? (c) How many atoms of hydrogen are in 1.00 mol of methane, CH₄?

Answers: (a) The bracket holds 2 nitrogens and 8 hydrogens: 2(14.01) + 8(1.008) + 32.06 + 4(16.00) = 28.02 + 8.06 + 32.06 + 64.00 = 132.14 g/mol. (b) 88.0 g / 44.01 g/mol = 2.00 mol. (c) Each CH₄ has 4 hydrogens, so 4 x 6.022 x 10^23 = 2.41 x 10^24 hydrogen atoms, which is 4.00 mol of hydrogen atoms.

Recap

• The mole is a count - exactly 6.02214076 x 10^23 particles - used because atoms cannot be counted individually.
• Molar mass in g/mol equals the atomic or formula mass in amu, which is what makes the mole a bridge between grams and particles.
• For a compound, add up every atom's mass, distributing any subscript outside a bracket to everything inside it.
• Grams to moles: divide by molar mass. Moles to grams: multiply. Let the units decide which.
• Moles to particles: multiply by 6.022 x 10^23. Particles to moles: divide.
• Grams and particles never connect directly; every route goes through moles.
• Atoms, molecules, and moles answer different questions about the same sample - read carefully which one is wanted.
• Percent composition = (mass of that element in one mole / molar mass) x 100, and the percentages must total 100.

Sources

1. OpenStax. (2019). Formula mass and the mole concept (Section 3.1). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Determining empirical and molecular formulas (Section 3.2). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Formula mass (Section 6.1). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. National Institute of Standards and Technology. (n.d.). CODATA value: Avogadro constant. NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov
5. National Institute of Standards and Technology. (n.d.). Atomic weights and isotopic compositions with relative atomic masses. nist.gov
6. LibreTexts. (n.d.). Chemical composition (Chapter 6). In Introductory Chemistry. chem.libretexts.org
7. Khan Academy. (n.d.). Chemical reactions and stoichiometry [Unit]. In Chemistry. khanacademy.org

A recipe you can trust with your life

A car airbag has to inflate in about 30 milliseconds with a very specific volume of gas. Too little and your head hits the wheel; too much and the bag itself injures you. The engineers get it right by weighing out an exact mass of sodium azide, because the balanced equation tells them precisely how many moles of nitrogen gas that mass will produce. This lesson is that calculation.

Stoichiometry is the part of chemistry that uses a balanced equation to figure out how much of one substance reacts with or produces how much of another. It works because the coefficients in a balanced equation tell you the exact ratio of the particles involved.

Key idea: Coefficients are a recipe in moles. Everything else in this lesson is converting into moles, using the recipe, and converting back out.

The mole ratio

Consider the balanced equation for making ammonia:

N₂ + 3 H₂ → 2 NH₃

The coefficients say that 1 molecule of N₂ reacts with 3 molecules of H₂ to make 2 molecules of NH₃. Scale that up by Avogadro's number and the same ratio holds for moles: 1 mole of N₂ reacts with 3 moles of H₂ to make 2 moles of NH₃. Any two of these amounts form a mole ratio you can use as a conversion factor, such as (2 mol NH₃ ÷ 3 mol H₂).

Moles to moles

Worked example. How many moles of NH₃ form from 6.0 mol of H₂ (with plenty of N₂)? Multiply by the mole ratio that cancels H₂:

6.0 mol H₂ × (2 mol NH₃ ÷ 3 mol H₂) = 4.0 mol NH₃

Grams to grams

Real labs measure mass, not moles, so a full stoichiometry problem often has three steps: convert grams of the given substance to moles, use the mole ratio, then convert moles of the wanted substance back to grams.

Worked example. In the reaction 2 H₂ + O₂ → 2 H₂O, how many grams of water form from 8.0 g of H₂? (Molar masses: H₂ = 2.02 g/mol, H₂O = 18.02 g/mol.)

1. Grams of H₂ to moles: 8.0 g ÷ 2.02 g/mol = 3.96 mol H₂ (about 4.0 mol).
2. Mole ratio: 4.0 mol H₂ × (2 mol H₂O ÷ 2 mol H₂) = 4.0 mol H₂O.
3. Moles of H₂O to grams: 4.0 mol × 18.02 g/mol = 72 g H₂O.

So about 72 g of water forms. Notice how every step is a dimensional-analysis conversion: grams to moles, moles to moles, moles to grams. Master that three-step path and you can solve most basic stoichiometry problems.

Worked example: propane, with every unit shown

Question: A barbecue burns propane: C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O. How many grams of carbon dioxide come from burning 22.0 g of propane?

Solution: Build the molar masses first. C₃H₈ = 3(12.01) + 8(1.008) = 36.03 + 8.06 = 44.09 g/mol. CO₂ = 44.01 g/mol.

1. Grams to moles. 22.0 g C3H8 x (1 mol C3H8 / 44.09 g C3H8) = 0.499 mol C3H8. Grams cancel.
2. Mole ratio from the coefficients. 0.499 mol C3H8 x (3 mol CO2 / 1 mol C3H8) = 1.50 mol CO2. Moles of propane cancel.
3. Moles to grams. 1.50 mol CO2 x (44.01 g CO2 / 1 mol CO2) = 66.0 g CO2. Moles cancel, grams survive.

Answer: about 66 g of CO₂ from 22 g of propane. Look at that: the products weigh more than the fuel. Nothing is wrong - the extra mass came from the oxygen in the air, which is the same discovery Lavoisier made in Lesson 10. Every gram of gas leaving a barbecue was partly air a moment ago.

Write the whole thing as one line and the cancelling becomes visible at a glance:

22.0 g C3H8 x (1 mol C3H8 / 44.09 g) x (3 mol CO2 / 1 mol C3H8) x (44.01 g / 1 mol CO2) = 66.0 g CO2

The limiting reactant

Reactions rarely start with exactly the right ratio of ingredients. The limiting reactant is the one that runs out first; it caps how much product can form. The other reactant is left over in excess. A kitchen analogy: if a recipe needs 2 slices of bread and 1 slice of cheese per sandwich, and you have 10 bread and 3 cheese, the cheese limits you to 3 sandwiches, and 4 slices of bread are left over.

Worked example. For N₂ + 3 H₂ → 2 NH₃, suppose you have 2.0 mol N₂ and 3.0 mol H₂. Which limits the reaction? Compare what each could make. From N₂: 2.0 mol × (2 mol NH₃ ÷ 1 mol N₂) = 4.0 mol NH₃.

From H₂: 3.0 mol × (2 mol NH₃ ÷ 3 mol H₂) = 2.0 mol NH₃. Hydrogen makes less, so H₂ is the limiting reactant and only 2.0 mol NH₃ can form. Nitrogen is in excess. The rule: whichever reactant yields the least product is the limiting one.

Percent yield

The amount of product predicted by stoichiometry is the theoretical yield. The amount you actually collect in the lab is the actual yield, and it is usually less because of spills, side reactions, or incomplete reactions. The percent yield compares them:

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

Worked example. If a reaction should make 72 g of water (theoretical) but you recover 63 g (actual), the percent yield is (63 ÷ 72) × 100 = 87.5%. A percent yield can never sensibly exceed 100%; if it does, something (often leftover water or impurities in the product) has inflated the measured mass.

Worked example: limiting reactant and percent yield from masses

Question: Aluminum burns in chlorine: 2 Al + 3 Cl₂ → 2 AlCl₃. A student uses 5.40 g of Al and 15.0 g of Cl₂, and collects 16.2 g of AlCl₃. Find the limiting reactant, the theoretical yield, the leftover mass of the excess reactant, and the percent yield.

Solution. Molar masses: Al = 26.98, Cl₂ = 2(35.45) = 70.90, AlCl₃ = 26.98 + 3(35.45) = 133.33 g/mol.

1. Both reactants to moles. Al: 5.40 / 26.98 = 0.200 mol. Cl₂: 15.0 / 70.90 = 0.212 mol.
2. Ask what each could make. From Al: 0.200 x (2 mol AlCl₃ / 2 mol Al) = 0.200 mol AlCl₃. From Cl₂: 0.212 x (2 mol AlCl₃ / 3 mol Cl₂) = 0.141 mol AlCl₃.
3. Smaller wins. Chlorine yields less, so Cl₂ is limiting and only 0.141 mol of AlCl₃ is possible.
4. Theoretical yield. 0.141 mol x 133.33 g/mol = 18.8 g AlCl₃.
5. Leftover aluminum. Making 0.141 mol of AlCl₃ consumes 0.141 mol of Al (a 2:2 ratio), so 0.200 - 0.141 = 0.059 mol is left, which is 0.059 x 26.98 = 1.59 g of Al unreacted.
6. Percent yield. (16.2 / 18.8) x 100 = 86.2%.

Notice step 2 carefully. There were more moles of chlorine than aluminum, and chlorine still ran out first, because the recipe demands chlorine three-to-two. Raw amounts never settle the question; only the comparison of possible products does.

Where people get stuck: reading the mole ratio off the subscripts instead of the coefficients. In 2 Al + 3 Cl₂ → 2 AlCl₃, the ratio of Cl₂ to AlCl₃ is 3 to 2, taken from the big numbers in front. The subscript 3 in AlCl₃ describes what is inside one formula unit and plays no part in the ratio. A useful discipline: before starting any stoichiometry problem, double-check the equation is balanced. An unbalanced equation gives wrong ratios and therefore a confidently wrong answer.

Common misconceptions

• "The limiting reactant is the one you have the least of." It is the one that makes the least product, which depends on the mole ratio, not just the raw amount. Always compare products, not starting amounts.
• "Mole ratios come from the subscripts." Mole ratios come from the coefficients of the balanced equation. Subscripts stay inside formulas and describe one molecule.
• "You can skip converting grams to moles." The mole ratio only works with moles, so you must convert masses to moles before applying it.
• "Percent yield above 100% just means a great reaction." Yields above 100% signal an error, usually an impure or still-wet product weighing more than the pure substance should.
• "The products must weigh the same as the reactant you started with." Mass is conserved across all reactants, and one of them is often oxygen from the air. Burning 22 g of propane gives 66 g of CO₂ plus water.
• "Once you know the limiting reactant, the excess one no longer matters." You still need it to work out how much is left over, which is a standard exam question and a real cost in industry.

Try it

For 2 H₂ + O₂ → 2 H₂O, you start with 4.0 g of H₂ and 40.0 g of O₂. (a) Which reactant is limiting? (b) What is the theoretical yield of water in grams? (Molar masses: H₂ = 2.02, O₂ = 32.00, H₂O = 18.02 g/mol.)

Answers: (a) H₂: 4.0 / 2.02 = 1.98 mol, which could make 1.98 mol of water (2:2 ratio). O₂: 40.0 / 32.00 = 1.25 mol, which could make 1.25 x 2 = 2.50 mol of water. Hydrogen makes less, so H₂ is limiting. (b) 1.98 mol x 18.02 g/mol = 35.7 g of water. Oxygen is in excess, with 1.25 - 0.99 = 0.26 mol (about 8.3 g) left unreacted.

Recap

• Coefficients in a balanced equation are a recipe in moles, and any two of them form a mole ratio you can use as a conversion factor.
• The standard path is grams → moles → (mole ratio) → moles → grams. Every arrow is a fraction whose units cancel.
• Take mole ratios from coefficients, never from subscripts, and confirm the equation is balanced before you start.
• The limiting reactant is the one that yields the least product, which is not always the one you have least of.
• Find leftovers by working out how much of the excess reactant was actually consumed and subtracting.
• Theoretical yield is what the equation predicts; actual yield is what you collect; percent yield = (actual / theoretical) x 100.
• A yield above 100% is a red flag for impurities or a wet product, not a triumph.

Sources

1. OpenStax. (2019). Reaction stoichiometry (Section 4.3). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Reaction yields (Section 4.4). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Reaction stoichiometry (Section 7.3). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. OpenStax. (2019). Reaction yields (Section 7.4). In Chemistry: Atoms First 2e. Rice University. openstax.org
5. LibreTexts. (n.d.). Quantities in chemical reactions (Chapter 8). In Introductory Chemistry. chem.libretexts.org
6. PhET Interactive Simulations. (n.d.). Reactants, products and leftovers [Simulation]. University of Colorado Boulder. phet.colorado.edu
7. Khan Academy. (n.d.). Chemical reactions and stoichiometry [Unit]. In Chemistry. khanacademy.org

You can smell across a room

Someone opens a bottle of perfume at the far end of a classroom and, without any fan or draft, you smell it a minute later. Nothing pushed those molecules toward you. They simply moved - fast, constantly, and in every direction, bouncing off air molecules until some arrived at your nose. That everyday observation is proof that particles are in permanent motion, and this lesson turns it into a model.

Why does a solid hold its shape while a gas fills a room? The answer is the kinetic molecular theory, which pictures matter as tiny particles in constant motion. Two ideas drive everything: particles are always moving, and temperature is a measure of their average speed. Heating adds energy and speeds particles up; cooling takes energy away and slows them down.

Key idea: Temperature is not "how much heat something has." It is a measure of the average kinetic energy of the particles - how fast they are typically moving.

The five assumptions behind the model

The theory is short enough to state in full, and every gas behavior in the next lesson follows from it.

1. Gases are made of enormous numbers of tiny particles separated by distances far larger than the particles themselves. A gas is mostly empty space, which is why gases can be compressed and solids cannot.
2. The particles are in constant, random, straight-line motion until they hit something.
3. Collisions are perfectly elastic: no kinetic energy is lost overall, which is why the motion never runs down.
4. Attractions between gas particles are negligible at ordinary pressures, so each particle acts independently.
5. The average kinetic energy of the particles is directly proportional to the temperature in kelvin. Double the kelvin temperature and you double the average kinetic energy.

Point 5 is the one that pays off. It explains why gas-law formulas insist on kelvin: the relationship is proportional, and a proportion needs a scale that starts at true zero. It also explains what pressure is. Pressure is not a substance pressing on the walls; it is the combined drumming of countless particle collisions. More particles, or faster particles, means more collisions per second and harder ones, so the pressure rises.

A second payoff: since kinetic energy depends on both mass and speed, lighter particles must move faster than heavier ones at the same temperature. That is why a helium balloon goes limp in a day while an air-filled one lasts a week. Helium atoms, being light and fast, find the tiny gaps in the rubber far more often.

The states as particle motion

In a solid, particles are packed closely and only vibrate in place, held by strong attractions, so a solid keeps a fixed shape and volume. In a liquid, particles are still close but have enough energy to slide past one another, so a liquid keeps its volume but flows to fit its container. In a gas, particles have so much energy that they break free of one another and zoom around with lots of empty space between them, so a gas expands to fill any container and can be squeezed into a smaller one.

Changes of state

Adding or removing heat moves matter between states. Each change has a name:

Change	From	To	Energy
Melting	Solid	Liquid	Absorbed
Freezing	Liquid	Solid	Released
Vaporizing (boiling)	Liquid	Gas	Absorbed
Condensing	Gas	Liquid	Released

Notice a pattern: going toward a gas (melting, vaporizing) absorbs energy to loosen the particles, while going toward a solid (freezing, condensing) releases energy as the particles settle. During a change of state the temperature stays constant, because the added energy goes into breaking attractions rather than speeding particles up. This is why a pot of boiling water stays at 100 °C no matter how high you turn the burner.

Two changes skip the liquid state entirely. Sublimation goes straight from solid to gas, which is what dry ice (solid CO₂) does as it fogs. The reverse, gas straight to solid, is deposition, which is how frost forms on a cold window. Both follow the same energy rule: sublimation absorbs energy, deposition releases it.

Evaporation and cooling

A liquid does not have to boil to become a gas. At any temperature, some surface particles are moving fast enough to escape into the air, a process called evaporation. Because the fastest particles leave, the ones left behind have a lower average speed, so the remaining liquid cools. This is why sweat cools your skin: the fastest-moving water molecules evaporate and carry energy away, leaving you cooler. The same idea explains why a wet towel feels cold and why rubbing alcohol feels chilly on the skin, since it evaporates even faster than water.

Reading a heating curve

Imagine slowly heating a block of ice and graphing its temperature over time. The temperature rises while the ice warms, then flattens out at 0 °C during melting, rises again through the liquid range, flattens once more at 100 °C during boiling, and finally rises as steam. The two flat stretches are the changes of state, where energy breaks attractions instead of raising temperature.

Worked reasoning. If ice starts at −10 °C and you heat it steadily, it first climbs to 0 °C, then pauses at 0 °C until every bit has melted, so a thermometer reading a steady 0 °C tells you the sample is still a mix of ice and water, not that heating has stopped.

Worked example: how much energy does the flat part cost?

Question: Take 50.0 g of ice already at 0 °C and turn it into steam at 100 °C. How much energy does each stage need? (Heat of fusion 334 J/g; specific heat of liquid water 4.184 J/g/°C; heat of vaporization 2260 J/g.)

1. Melt the ice at 0 °C. q = 50.0 g x 334 J/g = 16 700 J = 16.7 kJ. The temperature does not move at all.
2. Warm the water from 0 to 100 °C. q = m x c x deltaT = 50.0 x 4.184 x 100 = 20 920 J = 20.9 kJ.
3. Boil it at 100 °C. q = 50.0 g x 2260 J/g = 113 000 J = 113 kJ. Again the temperature does not move.

Total: about 151 kJ. Look at how the energy is distributed. Boiling alone takes 113 kJ, which is more than five times the cost of heating the same water across its entire 100-degree liquid range. Separating molecules from one another is far more expensive than making them move faster.

That single number explains a lot of ordinary life. Steam burns are far worse than boiling-water burns, because condensing steam dumps that whole 2260 J/g into your skin before the water even starts to cool. It is also why sweating works so well as a cooling system, and why a pan of water takes a long time to boil dry after it reaches 100 °C.

Why some substances boil so much higher than others

Water and methane have almost the same molar mass, 18.02 and 16.04 g/mol, yet water boils at 100 °C and methane at about -162 °C. Nearly 260 degrees apart, for molecules of nearly the same weight.

The difference is the attractions between molecules. Water is bent and strongly polar (Lesson 8), so each molecule grips its neighbors hard through hydrogen bonding. Methane is symmetric and nonpolar, so its molecules barely notice each other. Boiling means tearing molecules apart from each other, so the stronger the attractions, the higher the boiling point.

Key idea: Melting and boiling points measure the strength of the forces between particles, not the strength of the bonds inside them. Boiling water does not break a single O-H bond.

Where people get stuck: believing that adding heat must raise temperature. Heat and temperature are different things. Heat is energy in transit; temperature reports average particle speed. During melting or boiling, incoming energy goes entirely into pulling particles apart, so the thermometer sits still while the energy pours in. That is exactly what the flat sections of a heating curve are showing you.

Common misconceptions

• "Particles in a solid do not move at all." Solid particles still vibrate in place; they simply lack the energy to leave their fixed positions.
• "Temperature rises steadily while ice melts." During a change of state the temperature holds constant because the energy goes into breaking attractions, not into faster motion.
• "A liquid must reach its boiling point to become a gas." Evaporation happens at any temperature from the surface; boiling is just rapid vaporization throughout the liquid.
• "Heating always raises temperature." While a substance changes state, added heat converts to the energy of separating particles, so the temperature can stay flat even as you keep heating.
• "Boiling breaks the chemical bonds in a molecule." Boiling separates molecules from each other. Steam is still H₂O, with every O-H bond intact.
• "All the particles in a sample move at the same speed." Temperature gives the average. There is always a spread, which is exactly why a few surface molecules can escape and evaporation happens below the boiling point.

Try it

(a) How much energy is needed to melt 25.0 g of ice at 0 °C? (b) Two flasks are at the same temperature, one holding helium and one holding oxygen. Which gas has the faster average particle speed, and why?

Answers: (a) q = 25.0 g x 334 J/g = 8350 J = 8.35 kJ, with no temperature change at all. (b) Helium. Same temperature means the same average kinetic energy, and helium atoms are far lighter (4.00 vs 32.00 g/mol), so they must be moving faster to carry the same energy.

Recap

• Kinetic molecular theory: particles are always moving, gases are mostly empty space, collisions are elastic, and average kinetic energy is proportional to the kelvin temperature.
• Pressure is the combined effect of countless particle collisions with the container walls.
• Solids vibrate in place, liquids slide past one another, and gases move freely with large gaps between particles.
• Melting, vaporizing, and subliming absorb energy; freezing, condensing, and depositing release it.
• Temperature stays flat during a change of state because the energy goes into separating particles, not speeding them up.
• Vaporizing water costs 2260 J/g, far more than the 418 J/g needed to heat it across its entire liquid range - which is why steam burns are so severe.
• Boiling and melting points measure attractions between molecules; water beats methane by 260 degrees because water is polar and methane is not.
• Evaporation happens at any temperature and cools the liquid, because only the fastest particles escape.

Sources

1. OpenStax. (2019). Phase transitions (Section 10.3). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). The kinetic-molecular theory (Section 9.5). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). The kinetic-molecular theory (Section 8.5). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. LibreTexts. (n.d.). Liquids, solids, and intermolecular forces (Chapter 12). In Introductory Chemistry. chem.libretexts.org
5. PhET Interactive Simulations. (n.d.). States of matter: Basics [Simulation]. University of Colorado Boulder. phet.colorado.edu
6. Khan Academy. (n.d.). States of matter and intermolecular forces [Unit]. In Chemistry. khanacademy.org
7. National Institute of Standards and Technology. (n.d.). NIST Chemistry WebBook. Standard Reference Database 69. webbook.nist.gov

Why a bag of chips puffs up in the mountains

Carry a sealed bag of chips from sea level up a mountain road and it swells until it looks ready to burst. Nothing was added. The air outside got thinner, so it stopped squeezing the bag as hard, and the gas inside pushed out until the two pressures matched again. Your ears popping on a plane is the same physics, one eardrum at a time.

Gases respond in predictable ways when you change their pressure, volume, or temperature, and the gas laws capture those responses as simple relationships. Three quantities matter: pressure (P), the force the gas exerts on its container walls; volume (V), the space it fills; and temperature (T), which for gas laws must always be in kelvin.

Key idea: Every gas law is the same statement, seen from a different angle: pressure comes from particle collisions, so anything that changes how often or how hard particles hit the walls changes the pressure.

Pressure gets measured in several units, and problems switch between them without warning. These are all the same pressure:

1 atm = 760 mmHg = 760 torr = 101.325 kPa

The laws below work in any unit as long as you use the same one on both sides, because the units cancel in the ratio. Temperature is the exception: kelvin, always, no exceptions.

Boyle's law: pressure and volume

Boyle's law says that at constant temperature, pressure and volume are inversely related: squeeze a gas into half the volume and its pressure doubles. In symbols, P₁V₁ = P₂V₂.

Worked example. A gas occupies 3.0 L at 2.0 atm. If it is allowed to expand to 6.0 L at the same temperature, what is the new pressure? Rearrange to P₂ = P₁V₁ ÷ V₂ = (2.0 atm × 3.0 L) ÷ 6.0 L = 1.0 atm. Doubling the volume halved the pressure, exactly as Boyle's law predicts.

Why does that happen? Give the same number of particles twice the room and each one has twice as far to travel before it reaches a wall, so the wall gets hit half as often. Half the collision rate means half the pressure. Boyle's law is not a rule the gas obeys; it is arithmetic about collisions.

Charles's law: volume and temperature

Charles's law says that at constant pressure, the volume of a gas is directly proportional to its temperature in kelvin: heat a gas and it expands. In symbols, V₁ ÷ T₁ = V₂ ÷ T₂.

Worked example. A gas fills 300 mL at 300 K. If it is heated to 400 K at constant pressure, what is the new volume? Rearrange to V₂ = V₁ × T₂ ÷ T₁ = 300 mL × 400 K ÷ 300 K = 400 mL. The gas expanded as it warmed.

Why kelvin is required

These temperature relationships only work on the Kelvin scale, because Kelvin starts at absolute zero, where particle motion is minimal. If you used Celsius, a value of 0 °C would wrongly suggest zero volume, and negative Celsius temperatures would give impossible negative volumes. Always convert to kelvin (K = °C + 273) before using a gas law. One more useful fact: at standard temperature and pressure (STP), meaning 0 °C and 1 atm, one mole of any gas takes up about 22.4 liters.

Gay-Lussac's law: pressure and temperature

Gay-Lussac's law covers the third pairing: at constant volume, the pressure of a gas is directly proportional to its kelvin temperature. Heat a sealed rigid container and the pressure climbs. In symbols, P₁ ÷ T₁ = P₂ ÷ T₂. This is why an aerosol can carries a warning never to heat it: rising temperature raises the pressure inside until the can can rupture.

Worked example. A sealed can holds gas at 3.0 atm and 300 K. If it is heated to 450 K at constant volume, what is the new pressure? Rearrange to P₂ = P₁ × T₂ ÷ T₁ = 3.0 atm × 450 K ÷ 300 K = 4.5 atm. The pressure rose by the same factor (1.5) as the temperature.

The combined gas law

When pressure, volume, and temperature all change at once, the three laws merge into the combined gas law:

(P₁V₁) ÷ T₁ = (P₂V₂) ÷ T₂

Worked example. A gas occupies 2.0 L at 1.0 atm and 300 K. What volume does it fill at 2.0 atm and 600 K? Solve for V₂:

V₂ = (P₁V₁T₂) ÷ (T₁P₂) = (1.0 atm × 2.0 L × 600 K) ÷ (300 K × 2.0 atm)

V₂ = 1200 ÷ 600 = 2.0 L

Doubling the pressure alone would have halved the volume to 1.0 L, but doubling the temperature at the same time expanded it back, so the two effects cancelled and the volume stayed at 2.0 L. Each of Boyle's, Charles's, and Gay-Lussac's laws is just the combined gas law with one quantity held constant.

Avogadro's law and the ideal gas law

One quantity is missing from all of the above: how much gas there is. Avogadro's law supplies it - at fixed temperature and pressure, volume is proportional to the number of moles. Twice as much gas takes twice the space, which is exactly what you feel when you blow up a balloon.

Fold that in and all four relationships collapse into one equation, the ideal gas law:

PV = nRT

Here n is the number of moles and R is the gas constant, 0.0821 L·atm/(mol·K). That value of R commits you to particular units: pressure in atm, volume in litres, and temperature in kelvin. Use different units and you need a different R, which is a common source of wrong answers.

Worked example 1. How many moles of gas are in a 5.00 L cylinder at 2.00 atm and 300 K?

n = PV / RT = (2.00 atm x 5.00 L) / (0.0821 L·atm/mol·K x 300 K)

n = 10.0 / 24.63 = 0.406 mol

Check the units: atm x L on top, and L·atm/(mol·K) x K on the bottom. The atm, L, and K all cancel and mol is left on top. The setup was right.

Worked example 2: where 22.4 L comes from. What volume does 1.00 mol of any gas occupy at STP, meaning 273 K and 1.00 atm?

V = nRT / P = (1.00 x 0.0821 x 273) / 1.00 = 22.4 L

That is the molar volume quoted earlier, and now you can derive it instead of memorizing it. Notice the equation never asked which gas. One mole of hydrogen and one mole of carbon dioxide fill the same 22.4 L at STP, even though one weighs 2 g and the other 44 g, because gas volume depends on how many particles there are, not on how heavy they are.

Where people get stuck: forgetting to convert to kelvin, and it is worth seeing how badly it goes wrong. Take a gas at 300 mL and 27 °C heated to 54 °C. Celsius doubled, so it is tempting to say the volume doubles to 600 mL. In kelvin those temperatures are 300 K and 327 K, so the true answer is 300 x (327/300) = 327 mL. The Celsius shortcut overestimates by almost a factor of two. Convert first, every single time.

Common misconceptions

• "You can plug Celsius into a gas law." Gas laws need absolute temperature, so always convert to kelvin (K = °C + 273) first; using Celsius gives wrong or even negative results.
• "Boyle's law means pressure and volume rise together." They are inversely related: as one goes up the other goes down, so their product stays constant.
• "Doubling the Celsius temperature doubles the volume." Only doubling the kelvin temperature doubles the volume; going from 20 °C to 40 °C is not a doubling in kelvin.
• "A rigid sealed container has constant pressure when heated." Its volume is fixed, so by Gay-Lussac's law the pressure rises as the temperature rises.
• "Heavier gases take up more room." At the same temperature and pressure, one mole of any gas fills the same volume. Volume counts particles, not their mass.
• "You must convert pressure to atm before using Boyle's law." In the two-state laws the pressure units cancel, so mmHg on both sides is fine. Only PV = nRT ties you to specific units, because R does.

Try it

(a) A balloon holds 2.50 L at 25 °C. What is its volume at 75 °C, at constant pressure? (b) What pressure does 0.500 mol of gas exert in a 10.0 L container at 400 K?

Answers: (a) Convert first: 25 + 273 = 298 K and 75 + 273 = 348 K. Then V₂ = 2.50 x (348 / 298) = 2.92 L. Note it grew by only 17 percent, not the 200 percent the Celsius numbers might suggest. (b) P = nRT / V = (0.500 x 0.0821 x 400) / 10.0 = 16.42 / 10.0 = 1.64 atm.

Recap

• Pressure is the drumming of particle collisions on the container walls, so anything changing collision rate or force changes pressure.
• 1 atm = 760 mmHg = 760 torr = 101.325 kPa; in two-state laws any consistent unit works, but temperature must be kelvin.
• Boyle: P₁V₁ = P₂V₂ at constant T, an inverse relationship.
• Charles: V₁/T₁ = V₂/T₂ at constant P, a direct relationship.
• Gay-Lussac: P₁/T₁ = P₂/T₂ at constant V, which is why you never heat a sealed can.
• Combined: (P₁V₁)/T₁ = (P₂V₂)/T₂ when more than one thing changes.
• Ideal gas law: PV = nRT with R = 0.0821 L·atm/(mol·K), which forces atm, litres, and kelvin.
• One mole of any gas fills about 22.4 L at STP, a result you can derive from PV = nRT rather than memorize.

Sources

1. OpenStax. (2019). Relating pressure, volume, amount, and temperature: The ideal gas law (Section 9.2). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Gas pressure (Section 9.1). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Relating pressure, volume, amount, and temperature: The ideal gas law (Section 8.2). In Chemistry: Atoms First 2e. Rice University. openstax.org
4. National Institute of Standards and Technology. (n.d.). CODATA value: Molar gas constant. NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov
5. LibreTexts. (n.d.). Gases (Chapter 11). In Introductory Chemistry. chem.libretexts.org
6. PhET Interactive Simulations. (n.d.). Gas properties [Simulation]. University of Colorado Boulder. phet.colorado.edu
7. National Institute of Standards and Technology. (n.d.). SI units. NIST Reference on Constants, Units, and Uncertainty. physics.nist.gov

Getting the concentration right can matter enormously

A saline drip in a hospital is 0.9 percent sodium chloride. Not 0.5, not 2. Get it wrong in either direction and the patient's red blood cells swell or shrivel. Concentration is not a detail attached to a solution; for anything from medicine to swimming pools to your own blood, it is the whole point.

A solution is a homogeneous mixture, uniform all the way through. It has two parts: the solute is the substance being dissolved (present in the smaller amount), and the solvent is the substance doing the dissolving (present in the larger amount). In salt water, salt is the solute and water is the solvent. Water dissolves so many things that it is called the universal solvent, thanks to its polar molecules.

Key idea: Concentration is a ratio. It tells you how crowded the solute is, and says nothing on its own about how much you have.

Concentration is not amount

These two get confused constantly, so pin them apart with numbers. A test tube holding 10 mL of 2.0 M salt solution and a bucket holding 5.0 L of 2.0 M salt solution have the same concentration. Every millilitre of either one is equally salty. But the amounts of salt are wildly different:

• Test tube: 0.010 L x 2.0 mol/L = 0.020 mol of NaCl.
• Bucket: 5.0 L x 2.0 mol/L = 10 mol of NaCl, five hundred times as much.

Concentration answers "how crowded?" and amount answers "how much?" Reading a problem carefully for which one it wants is half the battle, because 2.0 M is not a quantity of anything until you also say how many litres.

How dissolving works

When an ionic solid like salt meets water, the polar water molecules surround each ion and pull it away from the crystal, spreading the ions evenly through the liquid. A helpful rule of thumb is "like dissolves like": polar solvents such as water dissolve polar and ionic substances, while nonpolar solvents dissolve nonpolar substances such as oils. That is why oil and water do not mix.

Speeding up dissolving

Three things make a solid dissolve faster: stirring (which brings fresh solvent to the surface), heating (faster-moving particles collide with the solid more), and crushing the solid into smaller pieces (more surface area is exposed to the solvent). None of these change how much can dissolve, only how quickly.

Molarity measures concentration

Concentration describes how much solute is in a given amount of solution. The most common measure in chemistry is molarity (M), defined as moles of solute per liter of solution:

molarity = moles of solute ÷ liters of solution

Worked example. What is the molarity of a solution made by dissolving 0.50 mol of NaCl in enough water to make 2.0 L of solution? Molarity = 0.50 mol ÷ 2.0 L = 0.25 M. A solution labeled 0.25 M contains 0.25 mol of solute in every liter. You can rearrange the formula to find any missing piece: moles = molarity × liters, so 3.0 L of that 0.25 M solution would contain 0.25 M × 3.0 L = 0.75 mol of NaCl.

From grams to molarity

Often you know a mass, not moles, so you convert grams to moles with the molar mass first. Worked example. What is the molarity if you dissolve 58.5 g of NaCl (molar mass 58.5 g/mol) in enough water to make 0.500 L of solution? First find moles: 58.5 g ÷ 58.5 g/mol = 1.00 mol. Then divide by the volume: 1.00 mol ÷ 0.500 L = 2.00 M. Watch the volume unit: molarity uses liters, so a volume given in milliliters must be divided by 1000 first.

Worked example: actually making a solution

Question: You need 250.0 mL of 0.150 M NaCl for a lab. How many grams do you weigh out?

Solution: Work backwards from the definition of molarity.

1. Volume to litres. 250.0 mL x (1 L / 1000 mL) = 0.2500 L.
2. Moles needed. moles = M x L = 0.150 mol/L x 0.2500 L = 0.0375 mol. Litres cancel, moles survive.
3. Moles to grams. 0.0375 mol x 58.44 g/mol = 2.19 g of NaCl.

One practical note that trips up real lab work: molarity is moles per litre of solution, not per litre of solvent. So you dissolve the 2.19 g in a little water, then top up to exactly the 250.0 mL mark. You do not add 250.0 mL of water to the salt, because the salt itself takes up some room and the final volume would come out over the mark.

Saturation and solubility

There is a limit to how much solute a solvent can hold. A solution is unsaturated when more solute can still dissolve, saturated when it holds the maximum at that temperature, and any extra solute simply settles at the bottom. The maximum amount that dissolves is the solubility, and for most solids it rises with temperature, which is why hot water dissolves more sugar than cold. Gases behave in the opposite way, dissolving better in cold liquids, which is why a warm soda goes flat faster as its dissolved CO₂ escapes.

Diluting a solution

Adding solvent to a solution spreads the same amount of solute through more volume, lowering the concentration. Because the moles of solute do not change, dilutions follow M₁V₁ = M₂V₂.

Worked example. How much water must you add to 100 mL of 6.0 M HCl to dilute it to 2.0 M? Solve for the final volume: V₂ = M₁V₁ ÷ M₂ = (6.0 M × 100 mL) ÷ 2.0 M = 300 mL. Since you start with 100 mL and need 300 mL total, you add 200 mL of water. (Safety note: always add acid to water, never water to acid, because the mixing releases heat.)

The reason M₁V₁ = M₂V₂ works is worth seeing rather than memorizing. Both sides are just "moles of solute": molarity times volume gives moles, and dilution adds only solvent, so the moles before and the moles after are the same number. The equation is a statement that nothing was added or removed except water.

Worked example (the other direction). Stockrooms keep concentrated solutions and you dilute what you need. How much 2.00 M NaOH stock do you take to make 500.0 mL of 0.100 M NaOH?

V1 = M2 V2 / M1 = (0.100 M x 500.0 mL) / 2.00 M = 25.0 mL of stock

Measure 25.0 mL of the stock, put it in a 500 mL volumetric flask, and add water up to the mark. Sanity check: the concentration dropped by a factor of 20, so the volume must have risen by a factor of 20, and 25.0 x 20 = 500. That ratio check catches an upside-down formula instantly.

Where people get stuck: thinking a dilution destroys solute. It does not. Those 25.0 mL of stock contain 0.0500 mol of NaOH before dilution (0.0250 L x 2.00 M), and the finished 500.0 mL contains 0.500 L x 0.100 M = 0.0500 mol. Identical. All that changed is how much water the same molecules are spread through. Concentration went down; amount stayed put.

Common misconceptions

• "Molarity uses milliliters." Molarity is moles per liter, so any volume in milliliters must be converted to liters (divide by 1000) before dividing.
• "The solute is always solid and the solvent always liquid." Any state can play either role; in air, gaseous oxygen is a solute dissolved in gaseous nitrogen.
• "Stirring or heating lets you dissolve unlimited solute." Those only speed up dissolving; solubility sets the maximum, and past it extra solute just settles out.
• "Diluting a solution changes the number of moles of solute." Dilution adds only solvent, so the moles of solute stay the same while the concentration drops.
• "A concentrated solution always contains more solute than a dilute one." Not necessarily. A drop of concentrated acid holds far less acid than a bathtub of dilute acid. Concentration is a ratio; amount needs a volume too.
• "To make 1 L of solution, add solute to 1 L of water." Molarity is per litre of finished solution, so you dissolve the solute and then top up to the mark.

Try it

(a) What is the molarity of a solution containing 0.400 mol of KCl in 800.0 mL of solution? (b) How many moles of solute are in 250.0 mL of 0.500 M glucose? (c) How much 6.00 M HCl is needed to make 200.0 mL of 0.300 M HCl?

Answers: (a) 800.0 mL = 0.8000 L, so M = 0.400 / 0.8000 = 0.500 M. (b) moles = 0.500 mol/L x 0.2500 L = 0.125 mol. (c) V₁ = (0.300 x 200.0) / 6.00 = 10.0 mL of stock, then water up to 200.0 mL. Check the ratio: concentration fell by 20 times, volume rose by 20 times.

Recap

• A solution is a homogeneous mixture: solute dissolved in solvent, uniform throughout.
• "Like dissolves like" - polar and ionic solutes dissolve in polar solvents such as water; nonpolar in nonpolar.
• Stirring, heating, and crushing speed dissolving up but never raise the maximum that will dissolve.
• Molarity = moles of solute / litres of solution. Convert millilitres to litres first, and top up to the mark rather than adding a litre of water.
• Concentration is a ratio and amount is a quantity. Two containers can share a concentration and hold hugely different amounts.
• Solubility is the ceiling at a given temperature; below it a solution is unsaturated, at it saturated.
• Most solids dissolve better hot; gases dissolve better cold, which is why warm soda goes flat.
• Dilution: M₁V₁ = M₂V₂, because both sides are the same unchanged number of moles.

Sources

1. OpenStax. (2019). Molarity (Section 3.3). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). The dissolution process (Section 11.1). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Solubility (Section 11.3). In Chemistry 2e. Rice University. openstax.org
4. OpenStax. (2019). Molarity (Section 6.3). In Chemistry: Atoms First 2e. Rice University. openstax.org
5. LibreTexts. (n.d.). Solutions (Chapter 13). In Introductory Chemistry. chem.libretexts.org
6. PhET Interactive Simulations. (n.d.). Molarity [Simulation]. University of Colorado Boulder. phet.colorado.edu
7. PhET Interactive Simulations. (n.d.). Concentration [Simulation]. University of Colorado Boulder. phet.colorado.edu

Your blood is guarded to two decimal places

Human blood sits between pH 7.35 and 7.45. Drift below 7.0 or above 7.8 and the situation becomes life-threatening. Meanwhile, a couple of feet away, your stomach happily runs at a pH near 1.5 to 3.5 - acidic enough to dissolve metal. Your body maintains both, simultaneously, all day. Understanding what that number means, and why a change of a few tenths matters so much, is what this lesson is for.

Acids and bases are two families of compounds you meet every day, from the citric acid in lemons to the ammonia in cleaners. Chemists have a clear way to define them. An acid is a substance that produces hydrogen ions (H⁺) when dissolved in water, such as hydrochloric acid, HCl. A base produces hydroxide ions (OH^-) when dissolved in water, such as sodium hydroxide, NaOH.

Key idea: pH is a logarithmic scale, so it compresses a huge range into small numbers. Each single unit is a factor of ten in the hydrogen ion concentration.

Properties you can observe

Property	Acids	Bases
Taste	Sour (like lemons)	Bitter
Feel	-	Slippery
Litmus paper	Turns blue litmus red	Turns red litmus blue
Ion produced in water	H+	OH-

Never taste or touch laboratory chemicals to test them; these properties are described here only so you understand what the categories mean.

The pH scale

The pH scale measures how acidic or basic a solution is, running from 0 to 14. It is based on the concentration of hydrogen ions. A pH of 7 is neutral (pure water). A pH below 7 is acidic, and the lower it goes, the more acidic. A pH above 7 is basic, and the higher it goes, the more basic. Each step of 1 on the scale is a tenfold change in acidity, so a pH of 3 is ten times more acidic than a pH of 4.

When the hydrogen ion concentration is a simple power of ten, the pH is easy to find: pH is the negative of the exponent. If [H⁺] = 1 × 10^-3 mol/L, then the pH is 3, which is acidic. If [H⁺] = 1 × 10^-9 mol/L, the pH is 9, which is basic.

In general the rule is pH = -log[H+], and the log key on your calculator handles concentrations that are not tidy powers of ten.

Worked example. A solution has [H⁺] = 2.5 x 10^-4 mol/L. Find the pH.

pH = -log(2.5 x 10^-4) = 3.60

Sanity-check that before trusting it. The concentration lies between 10^-4 and 10^-3, so the pH must lie between 3 and 4 - and it does. Higher concentration always means lower pH, because of the minus sign in the definition.

Worked example (backwards). Sea water has a pH of about 8.1. What is its hydrogen ion concentration? Undo the log by raising 10 to the negative pH:

[H+] = 10^-8.1 = 7.9 x 10^-9 mol/L

Compare that with pure water at 1.0 x 10^-7 mol/L. Sea water is basic, so it holds fewer hydrogen ions - about one eighth as many. The number of decimal places in a pH looks small and hides an enormous physical range.

pOH, and why the scale stops at 14

Water itself splits very slightly into H⁺ and OH^-, and in any water solution the two concentrations are locked together so that their product is always 1.0 x 10^-14. Taking logs of that relationship gives a rule worth memorizing:

pH + pOH = 14

So a solution of pH 3 has pOH 11, and a solution of pH 11 has pOH 3. In pure water, neither ion outnumbers the other, both concentrations are 1.0 x 10^-7 mol/L, and pH = pOH = 7. That is where "7 is neutral" comes from - it is not an arbitrary midpoint of the scale, it is a measured property of water.

Worked example. A cleaning solution has [OH^-] = 1.0 x 10^-2 mol/L. Its pOH is 2, so its pH is 14 - 2 = 12: strongly basic, exactly what you would expect from a drain cleaner.

Neutralization

When an acid and a base are mixed, they cancel each other in a neutralization reaction, producing water and a salt. For example:

HCl + NaOH → NaCl + H₂O

The H⁺ from the acid joins the OH^- from the base to make water (H₂O), and the leftover ions (Na⁺ and Cl^-) form a salt. The result moves toward a neutral pH of 7, which is why antacids (bases) relieve an acidic stomach.

Worked example. Balance the neutralization of sulfuric acid with sodium hydroxide. Sulfuric acid, H₂SO₄, can supply two H⁺, so it needs two NaOH:

H₂SO₄ + 2 NaOH → Na₂SO₄ + 2 H₂O

Check the atoms: 4 H and 1 S and 6 O on each side (4 O in the sulfate plus 2 O in the two hydroxides), plus 2 Na on each side. The two H⁺ pair with the two OH^- to make two waters, and the salt formed is sodium sulfate.

Strong versus weak

Acids and bases differ in strength, which is how completely they break apart in water. A strong acid such as HCl ionizes almost completely, releasing nearly all its H⁺, so it gives a very low pH. A weak acid such as acetic acid (in vinegar) ionizes only slightly, so most molecules stay intact and the pH is much closer to 7 even at the same concentration.

The same distinction applies to bases: NaOH is a strong base, while ammonia is a weak base. Strength (how completely it ionizes) is not the same as concentration (how much is dissolved).

Strong is not concentrated: the numbers

That last sentence gets nodded at and then forgotten, so here it is with real values.

• 0.10 M hydrochloric acid. HCl is strong, so essentially every molecule gives up its H⁺. That makes [H⁺] = 0.10 mol/L, and the pH is 1.00.
• 1.0 M acetic acid. Ten times as many acid molecules per litre - far more concentrated. But acetic acid is weak, and under 1 percent of its molecules ionize, giving [H⁺] of roughly 0.0042 mol/L and a pH near 2.4.

Read those two lines again. The dilute strong acid has more than twenty times the hydrogen ion concentration of the concentrated weak acid. "Concentrated" describes how much acid you put in the bottle. "Strong" describes what fraction of it actually falls apart in water. They are independent dials, and only the second one is about the chemistry of the substance itself.

Key idea: Strong or weak is a property of the substance and never changes. Concentrated or dilute is a property of the solution you made, and you control it with water.

This matters practically. Concentrated acetic acid (glacial acetic acid) will still burn you badly, because 1.0 M of a weak acid is a serious amount of acid even if little of it is ionized at any instant - as fast as H⁺ is consumed, more molecules ionize to replace it. And very dilute hydrochloric acid is mild enough that a version of it sits in your stomach right now.

Indicators

An indicator is a dye that changes color depending on pH, giving a quick read on acidity. Litmus is the classic example: red in acid, blue in base. Universal indicator goes further, shifting through a rainbow, from red in strong acid through green at neutral to purple in strong base, so its color maps to an approximate pH. In the lab, indicators reveal the exact moment an acid has been neutralized by a base.

Common misconceptions

• "A strong acid and a concentrated acid are the same thing." Strength is how completely an acid ionizes; concentration is how much is dissolved. They are independent.
• "A higher pH number means more acidic." It is the reverse: lower pH is more acidic, higher pH is more basic, and 7 is neutral.
• "Each pH step is a small change." Each unit is a tenfold change in acidity, so pH 2 is one hundred times more acidic than pH 4.
• "Neutralization always gives an exactly neutral solution." It moves toward neutral and makes water plus a salt, but the final pH depends on the particular acid and base and can land slightly off 7.
• "pH 7 is neutral because 7 is halfway between 0 and 14." It is neutral because pure water genuinely holds [H⁺] = [OH^-] = 1.0 x 10^-7 mol/L. The midpoint is a consequence, not the reason.
• "Diluting a strong acid makes it weak." Dilution changes concentration, not strength. Very dilute HCl is still a strong acid; it just has less of it.

Where people get stuck: the minus sign in pH = -log[H⁺]. Because of it, everything runs backwards from intuition: bigger [H⁺] gives a smaller pH. Build the reflex with two anchors. Lemon juice, pH about 2, is loaded with hydrogen ions. Bleach, pH about 13, has almost none. Whenever a calculation gives you a pH, ask whether the answer sits on the side of the anchor you expected before you write it down.

Try it

(a) A solution has [H⁺] = 1.0 x 10^-5 mol/L. What are its pH and pOH? (b) Which is more acidic, pH 4.0 or pH 6.0, and by what factor? (c) A solution has pOH 3.0. Is it acidic or basic?

Answers: (a) pH = 5.0, so pOH = 14 - 5 = 9.0. It is acidic. (b) pH 4.0, by a factor of 100 - two pH units means two factors of ten. (c) pH = 14 - 3 = 11, so it is basic.

Recap

• An acid produces H⁺ in water; a base produces OH^-.
• pH = -log[H⁺], so a higher hydrogen ion concentration gives a lower pH.
• Below 7 is acidic, 7 is neutral, above 7 is basic, and each unit is a tenfold change in [H⁺].
• pH + pOH = 14 in water, and pure water sits at pH = pOH = 7 because [H⁺] = [OH^-] = 1.0 x 10^-7 mol/L.
• Strong means fully ionized; weak means only slightly ionized. This is a fixed property of the substance.
• Concentrated means a lot of solute per litre. This is a property of the solution and you set it with water.
• 0.10 M HCl (pH 1.0) is far more acidic than 1.0 M acetic acid (pH about 2.4), which is dilute-but-strong beating concentrated-but-weak.
• Neutralization gives water plus a salt, and indicators such as litmus report pH by changing color.

Sources

1. OpenStax. (2019). pH and pOH (Section 14.2). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Bronsted-Lowry acids and bases (Section 14.1). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Relative strengths of acids and bases (Section 14.3). In Chemistry 2e. Rice University. openstax.org
4. OpenStax. (2019). Classifying chemical reactions (Section 4.2). In Chemistry 2e. Rice University. openstax.org
5. LibreTexts. (n.d.). Acids and bases (Chapter 14). In Introductory Chemistry. chem.libretexts.org
6. PhET Interactive Simulations. (n.d.). pH scale [Simulation]. University of Colorado Boulder. phet.colorado.edu
7. PhET Interactive Simulations. (n.d.). Acid-base solutions [Simulation]. University of Colorado Boulder. phet.colorado.edu

Two packets, opposite behavior

A chemical hand warmer and an instant cold pack look almost identical: a sealed pouch you squeeze and shake. One climbs to about 55 degrees C and stays there for hours. The other drops close to freezing in seconds. Same idea, opposite sign, and this lesson is about that sign.

Every chemical reaction involves energy, usually in the form of heat. Thermochemistry is the study of these energy changes. The key idea is that breaking bonds requires energy (you must pull atoms apart), while forming bonds releases energy (atoms settle into a more stable arrangement). Whether a reaction gives off or takes in heat overall depends on the balance between these two.

Key idea: Breaking bonds always costs energy. Forming bonds always releases it. Every enthalpy change is the difference between those two totals - nothing more mysterious than that.

Exothermic and endothermic reactions

An exothermic reaction releases heat to its surroundings, making them warmer. Burning fuel and the reactions inside a hand warmer are exothermic. Because energy leaves the reacting chemicals, we say the enthalpy change (ΔH) is negative. An endothermic reaction absorbs heat from its surroundings, making them cooler. An instant cold pack works this way. Here energy enters the chemicals, so ΔH is positive. A simple memory aid: exo means exit (heat exits), and endo means into (heat goes in).

Why the sign works

If forming the product bonds releases more energy than breaking the reactant bonds required, there is energy left over to warm the surroundings, and the reaction is exothermic. If breaking the reactant bonds costs more than forming the product bonds gives back, the reaction must pull energy in from the surroundings, and it is endothermic.

Worked example: predicting the sign from bond energies

You can put numbers on that balance. The rule is deltaH = (energy to break all reactant bonds) - (energy released forming all product bonds).

Question: For H₂ + Cl₂ → 2 HCl, is the reaction exothermic or endothermic? (Bond energies: H-H is 436 kJ/mol, Cl-Cl is 242 kJ/mol, H-Cl is 431 kJ/mol.)

1. Bonds broken. One H-H and one Cl-Cl: 436 + 242 = 678 kJ spent.
2. Bonds formed. Two H-Cl bonds, because the equation makes 2 HCl: 2 x 431 = 862 kJ released.
3. Net. deltaH = 678 - 862 = -184 kJ.

The negative sign means energy left the chemicals, so the reaction is exothermic and the flask would get warm. Notice how the coefficient mattered: forgetting to double the H-Cl bond would have given +247 kJ and the opposite conclusion. Bond-energy problems are stoichiometry problems wearing a different hat.

Calculating heat with specific heat

How much a substance heats up for a given amount of energy depends on its specific heat (c), the energy needed to raise one gram by one degree Celsius. Water has an unusually high specific heat of 4.184 J/(g·°C), which is why it resists temperature change and is used as a coolant. The heat transferred is:

q = m × c × ΔT

where q is heat in joules, m is mass in grams, c is specific heat, and ΔT is the temperature change.

Worked example. How much heat is needed to warm 50.0 g of water from 20.0 °C to 50.0 °C? Here ΔT = 50.0 - 20.0 = 30.0 °C.

q = 50.0 g × 4.184 J/(g·°C) × 30.0 °C = 6276 J ≈ 6.28 kJ

A positive result means the water absorbed that much energy. If ΔT were negative (cooling), q would come out negative, meaning heat was released instead.

Worked example: solving for a missing piece. Suppose 209 J of heat raises the temperature of a water sample by 10.0 °C. What is the mass? Rearrange q = m × c × ΔT to solve for m: m = q ÷ (c × ΔT) = 209 J ÷ (4.184 J/(g·°C) × 10.0 °C) = 209 ÷ 41.84 = 5.00 g. Any one of the four quantities can be found when the other three are known.

Activation energy and energy diagrams

Even an exothermic reaction usually needs a push to get started. The activation energy is the minimum energy the reactants must gain, often as a small input of heat, to begin reacting; it is why a match must be struck before it burns. An energy diagram plots energy along the path of a reaction.

The reactants start at one level, climb over an energy hill (whose height is the activation energy), and settle at the product level. If the products sit lower than the reactants, energy was released overall and the reaction is exothermic; if the products sit higher, energy was absorbed and the reaction is endothermic. The gap between the reactant and product levels is ΔH.

Conservation of energy

Energy is never created or destroyed, only transferred, so the heat lost by one thing is gained by another. When a hot metal is dropped into cool water, the heat released by the metal equals the heat absorbed by the water, and the two settle at one shared temperature. This bookkeeping, tracking where the joules go, is the heart of thermochemistry and the reason q for the surroundings is equal and opposite to q for the system.

Worked example: identifying a metal by calorimetry

Question: A 50.0 g piece of metal is heated to 100.0 °C and dropped into 100.0 g of water at 20.0 °C. The mixture settles at 23.5 °C. What is the metal's specific heat, and what might the metal be?

Solution: Follow the joules. The water gained exactly what the metal lost.

1. Heat gained by the water. deltaT for the water is 23.5 - 20.0 = +3.5 °C. So q = 100.0 x 4.184 x 3.5 = +1464 J.
2. Heat lost by the metal. By conservation, q = -1464 J. The metal's deltaT is 23.5 - 100.0 = -76.5 °C.
3. Solve for c. c = q / (m x deltaT) = (-1464) / (50.0 x -76.5) = 1464 / 3825 = 0.383 J/(g·°C).
4. Identify. Copper's specific heat is 0.385 J/(g·°C), so the metal is very likely copper.

Two things are worth noticing. First, the two minus signs cancelled in step 3, which is why the specific heat comes out positive as it must. Second, look at the temperatures: the metal fell 76.5 degrees while the water rose only 3.5. The water was twice the mass and had roughly eleven times the specific heat, so it soaked up the same joules with barely a shrug. That is exactly what "water resists temperature change" means, expressed as an experiment.

Where people get stuck: the belief that breaking bonds releases energy - probably because burning fuel feels like "breaking it down" and burning clearly releases energy. Say it the other way and the confusion clears: bonds are what atoms fall into, and getting out of a hole always costs energy. When methane burns, breaking the C-H and O=O bonds costs a lot; but the C=O and O-H bonds that form are so stable that they pay it back with a large surplus. The flame is the surplus, not the breaking.

Common misconceptions

• "Exothermic reactions do not need any energy to start." Most still require activation energy to begin; being exothermic only means they release more energy than they take in overall.
• "Breaking bonds releases energy." Breaking bonds always costs energy; it is forming bonds that releases it. The net ΔH compares the two.
• "A negative q or ΔH means the temperature is negative." The negative sign just means heat was released; the temperature can be perfectly ordinary.
• "A high specific heat means a substance heats up fast." The opposite is true: a high specific heat means it resists temperature change and heats up slowly, like water.
• "A catalyst makes a reaction release more energy." A catalyst lowers the activation energy, so the reaction starts more easily and goes faster. It does not change ΔH at all, because the reactants and products still sit where they sat.
• "Endothermic reactions feel hot because they involve energy." They feel cold. They pull heat out of your hand, which is precisely why a cold pack works.

Try it

(a) How much heat is released when 200.0 g of water cools from 80.0 °C to 25.0 °C? (b) A reaction breaks bonds worth 1250 kJ and forms bonds worth 1180 kJ. Is it exothermic or endothermic, and what is ΔH?

Answers: (a) ΔT = 25.0 - 80.0 = -55.0 °C, so q = 200.0 x 4.184 x (-55.0) = -46 024 J, or about -46.0 kJ. The minus sign says the water released that energy to its surroundings. (b) ΔH = 1250 - 1180 = +70 kJ, a positive value, so the reaction is endothermic and would cool its container.

Recap

• Breaking bonds always costs energy; forming bonds always releases it. ΔH is the difference: broken minus formed.
• Exothermic reactions release heat and have negative ΔH; endothermic reactions absorb heat and have positive ΔH.
• Bond-energy calculations must respect the coefficients - two HCl means two H-Cl bonds formed.
• q = m x c x ΔT links heat, mass, specific heat, and temperature change, and can be rearranged for any one of the four.
• Water's high specific heat of 4.184 J/(g·°C) means it resists temperature change, which is why it works as a coolant and as a calorimeter fluid.
• In calorimetry the heat lost by one object equals the heat gained by the other, which is enough to identify an unknown metal.
• Activation energy is the hill a reaction must climb to start, which is why an exothermic reaction can still need a match.
• A catalyst lowers that hill; it never changes ΔH.

Sources

1. OpenStax. (2019). Energy basics (Section 5.1). In Chemistry 2e. Rice University. openstax.org
2. OpenStax. (2019). Calorimetry (Section 5.2). In Chemistry 2e. Rice University. openstax.org
3. OpenStax. (2019). Enthalpy (Section 5.3). In Chemistry 2e. Rice University. openstax.org
4. OpenStax. (2019). Energy basics (Section 9.1). In Chemistry: Atoms First 2e. Rice University. openstax.org
5. OpenStax. (2019). Calorimetry (Section 9.2). In Chemistry: Atoms First 2e. Rice University. openstax.org
6. National Institute of Standards and Technology. (n.d.). NIST Chemistry WebBook. Standard Reference Database 69. webbook.nist.gov
7. PhET Interactive Simulations. (n.d.). Energy forms and changes [Simulation]. University of Colorado Boulder. phet.colorado.edu