9702 · 15.1The mole

The mole and Avogadro constant

0:00A single glass of water contains roughly ten to the power twenty-five water molecules. That's a one followed by twenty-five zeros. If you counted one molecule every second, you'd still be counting long after the Sun had burned out. And yet, in the next few videos, we're going to write equations about gases where the number of particles really matters. So physicists need a sensible way to count things that are far too small and far too numerous to count one by one.

0:24That way is the mole. By the end of this video you'll know why amount of substance is its own SI base quantity, what one mole actually means, and how to move confidently between mass, moles, and number of particles. Early in the course you met the SI base quantities: mass in kilograms, length in metres, time in seconds, current in amperes, and temperature in kelvin. Every other unit, like the newton or the joule, is built out of these.

0:49Amount of substance is another SI base quantity, and its base unit is the mole, written m-o-l. Why does it need its own base quantity? Because it measures something none of the others can: how many particles you have. Mass tells you how heavy a sample is, but a kilogram of hydrogen and a kilogram of lead contain completely different numbers of atoms. Amount of substance is about the count, not the weight.

1:12You already count in bundles in everyday life. Eggs come in dozens, so a dozen means twelve. Paper comes in reams, so a ream means five hundred sheets. A mole is exactly the same idea: a bundle with a fixed number of particles in it. The only difference is that the bundle is enormous, because particles are so tiny. Here's the key idea. One mole of any substance is the amount that contains a number of particles equal to the Avogadro constant, N-A.

1:38The Avogadro constant is six point zero two times ten to the twenty-three per mole. Notice the unit: per mole. The Avogadro constant is really telling you "six point zero two times ten to the twenty-three particles for every mole." That unit is what lets it convert between moles and particles. And notice the word "any". One mole of helium atoms, one mole of water molecules, and one mole of electrons all contain the same number of particles.

2:06What changes from substance to substance is the mass of that mole, not the count. The particles can be atoms, molecules, ions or electrons, so you should always be clear which particle you mean. For gases in this chapter, we'll usually be counting molecules. Where does such a strange number come from? It was chosen so that one mole of a substance has a mass in grams equal to its relative molecular mass.

2:28That is a very convenient choice, as you're about to see. Since 2019 the mole has been defined by fixing the Avogadro constant at an exact value, and to three significant figures that value is the six point zero two times ten to the twenty-three you'll use in every calculation.

Mass, moles and particles

2:45In practice you can't count molecules, but you can weigh a sample. So we need a bridge between mass and moles. That bridge is molar mass, the mass of one mole of a substance. Molar mass is usually quoted in grams per mole, and it's numerically equal to the relative molecular mass. Water has a relative molecular mass of eighteen, so its molar mass is eighteen grams per mole. Helium is four grams per mole.

3:09Oxygen gas, which is made of O-two molecules, is thirty-two grams per mole. So we have two simple relationships. The number of moles n equals the mass m divided by the molar mass, capital M. And the number of particles, capital N, equals n times the Avogadro constant. Think of it as a chain with moles in the middle. Divide mass by molar mass to get moles. Multiply moles by the Avogadro constant to get particles.

3:35To go back the other way, you just reverse each step. How many molecules are there in 36 grams of water? The molar mass of water is 18 grams per mole. Step one: find the number of moles. n equals m over M, which is 36 divided by 18. That's 2.0 moles. Step two: find the number of molecules. N equals n times N-A, which is 2.0 times 6.02 times ten to the twenty-three.

4:04That gives 1.20 times ten to the twenty-four molecules. Notice both steps are just unit logic. Grams divided by grams per mole leaves moles. Moles multiplied by "per mole" leaves a pure number of molecules.

Mass of one particle

4:15Now something you'll need in the kinetic theory video: the mass of a single molecule. What is the mass of one helium atom? Helium's molar mass is 4.0 grams per mole. Here's the reasoning. One mole has a mass of 4.0 grams, and one mole contains N-A atoms. So the mass of one atom is the molar mass divided by the Avogadro constant. But first, convert to SI units. 4.0 grams per mole is 4.0 times ten to the minus three kilograms per mole.

4:44Then divide: 4.0 times ten to the minus three, divided by 6.02 times ten to the twenty-three. That gives 6.64 times ten to the minus twenty-seven kilograms. That's an astonishingly tiny mass, which is exactly why we need such a huge number of atoms to make a sensible amount of stuff. Mistake one: mixing grams and kilograms. Molar masses are usually given in grams per mole, but physics equations use SI units.

5:10If a later formula needs a mass in kilograms, like the mass of a molecule in kinetic theory, convert the molar mass to kilograms per mole first. Getting this wrong makes your answer a thousand times too big. Mistake two: confusing molecules with atoms. One mole of oxygen gas contains 6.02 times ten to the twenty-three O-two molecules, but each molecule has two atoms, so it contains twice that many oxygen atoms.

5:36Always check which particle the question is asking about. Mistake three: thinking a mole is a mass. It isn't. A mole is a number of particles. One mole of helium and one mole of oxygen contain the same number of particles, but the oxygen is eight times heavier, because each molecule is eight times heavier. Mistake four: dropping the unit of the Avogadro constant. It's per mole, not just a number.

6:01Keeping the unit in your working is the easiest way to catch whether you should be multiplying or dividing.

Worked gas calculation and recap

6:07Questions on the mole rarely appear on their own. They're usually one step inside a bigger gas question. Here's a typical one. "A cylinder contains 2.4 times ten to the twenty-four molecules of oxygen gas. The molar mass of oxygen is 32 grams per mole. Calculate the mass of oxygen in the cylinder." Work along the chain backwards. From particles to moles, divide by N-A. 2.4 times ten to the twenty-four, divided by 6.02 times ten to the twenty-three, is 3.99 moles.

6:38From moles to mass, multiply by the molar mass. 3.99 times 32 is about 128 grams, which is 0.128 kilograms. A quick sanity check: 2.4 times ten to the twenty-four is roughly four times the Avogadro constant, so we have roughly four moles, and four times 32 grams is about 128 grams. Estimating like this catches most power-of-ten slips. You'll also see the reverse, where you're given a mass and asked for the number of molecules to put into an equation.

7:10The chain is exactly the same. Let's pull it together. Amount of substance is an SI base quantity, and its unit is the mole. It measures how many particles you have, not how heavy they are. One mole of any substance contains a number of particles equal to the Avogadro constant, six point zero two times ten to the twenty-three per mole. Molar mass is the mass of one mole, usually in grams per mole, and numerically equal to the relative molecular mass.

7:37To convert, the number of moles is mass divided by molar mass, and the number of particles is moles times the Avogadro constant. The mass of one particle is the molar mass divided by the Avogadro constant, in kilograms per mole if you want kilograms. In the next video, we'll use exactly these ideas in the equation of state for an ideal gas, where you'll see both the number of moles and the number of molecules appear.