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Atoms are small. Really, really small. In fact, one hydrogen atom has a mass of just -24 grams! This can make chemical calculations involving individual atoms quite tricky. To solve this problem, we measure quantities of atoms, particles, or molecules in units called moles. Moles are based on a number called the Avogadro constant.

• We'll start by defining mole and the Avogadro constant.
• After that, we'll learn how you use the Avogadro constant in various equations.
• This will include learning about molar mass and finding out how to calculate both the number of atoms in a substance and the mass of one atom.

## The mole and Avogadro's constant

Imagine you are going to the supermarket. On your list: one dozen eggs, two pints of milk, and a baker's dozen bread rolls. These are all specific quantities. If you buy a dozen eggs, you'll know that you'll end up with precisely twelve. Two pints of milk is 1136.5 millilitres, whilst a baker's dozen is thirteen. There should be no confusion about how many eggs or bread rolls or how much milk you need to buy.

Well, another way of specifying quantities is the mole.

The mole is a chemical unit used to represent 6.02214076 × 1023 entities. This number is known as the Avogadro constant and has the symbol mol.

An entity is another word for a particle. It can refer to an atom, electron, ion, or molecule.

If we say we have one mole of hydrogen atoms, we know that we have precisely 6.02214076 × 1023 hydrogen atoms. If we say that we have two moles of oxygen molecules, we know that we have 2 × 6.02214076 × 1023 = 24 oxygen molecules. And if we say that we have 9.853 moles of methane molecules, we know that we have 9.853 × 6.02214076 × 1023 = Think of a mole as just another quantity. Just like a pair means two, or half a dozen means six, a mole means 6.02214076 x 1023.

Let's have a closer look at that number we mentioned before: 6.02214076 × 1023. As we said, this is known as the Avogadro constant, or simply just Avogadro's constant.

Avogadro's constant is the number of entities in a mole of any substance. It is equal to 6.02214076 × 1023.

We tend to shorten Avogadro's constant to 6.022 x 1023.

Amedeo Avogadro was an 18th and 19th-century scientist from the Kingdom of Sardinia, which is now a part of Italy. He is most famous for his theory about the volume of gases, known as Avogadro's law. This law states that two samples of the same volume of any ideal gases contain an equal number of molecules, provided they are kept at the same temperature and pressure. The Avogadro constant was first estimated in 1865 by Josef Loschmidt, but the term Avogadro's constant was only invented in 1909 by the physicist Jean Perrin, who named it in Avogadro's honour.

Now that we know about moles and Avogadro's constant, we can look at some of the equations linking them. First of all, we'll explore the relationship between moles, mass numbers, and Avogadro's constant.

### Moles, molar mass, and Avogadro's constant

You might be looking at Avogadro's constant and thinking that it is a fairly odd number. Where did it come from? Scientists must have chosen it for some particular reason - they didn't just pick a random value out of the blue! In fact, Avogadro's constant, which we know is just the number of entities in a mole, is exactly equal to the number of carbon atoms in 12.0 g of carbon-12. This means that one mole of carbon-12 atoms has a mass of exactly 12.0 g.

You might notice something. Carbon-12 atoms have a relative atomic mass of 12.0; 12.0 is also the mass of one mole of these atoms. This leads us on to our next important point: the mass of one mole of any substance is equal to its relative atomic mass, or relative molecular mass in grams. We can also call the mass of one mole of a substance its molar mass.

Molar mass is the mass of one mole of a substance. It is measured in g mol-1. Similarly, molar volume is the volume occupied by one mole of a gas. It is measured in dm3 mol-1.

Confused about the difference between relative atomic mass, relative molecular mass and molar mass? We'd recommend you check out "Relative Atomic Mass" for a more in-depth look at the first two terms, but here's an overview of the differences:

• Relative atomic mass measures the average mass of one atom of an element, compared to 1/12th of the mass of a carbon-12 atom. It is unitless.
• Relative molecular mass measures the average mass of one molecule of a species, also compared to 1/12th of the mass of a carbon-12 atom. Once again, it is unitless.
• Molar mass is the mass of one mole of a substance, whether it be an element or a molecule. It is measured in g·mol-1.
• The relative atomic/molecular mass, and molar mass of a species, are the same numerically. For example, the relative atomic mass of carbon-12 is exactly 12, whilst the molar mass - the mass of one mole of carbon-12 atoms - is 12 g·mol-1.

So, to find molar mass, you take a substance's relative atomic mass or relative molecular mass, and add g·mol-1 to the end.

Take methane, CH4. It has a relative molecular mass of 12.0 + 4(1.0) = 16.0. Therefore, methane has a molar mass of 16.0 g·mol-1. Or, in other words, 6.022 x 1023 molecules of methane has a mass of 16.0 g.

Notice how in this example, we multiplied the relative molecular mass of methane, 16.0, by the number of moles, 1, to find its mass? This leads us to a useful bit of maths. There's a handy equation we can use to relate molar mass, number of moles, and mass:

$\mathrm{number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{mass}}{\mathrm{molar}\mathrm{mass}}$

Remember - molar mass and relative atomic or molecular mass are the same numerically. Therefore, you might also see this equation written as $\mathrm{number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{mass}}{{\mathrm{A}}_{\mathrm{r}}\mathrm{or}{\mathrm{M}}_{\mathrm{r}}}$

Have a go at the following question.

Let's say that we have 34.5g of sodium, Na. How many moles of Na do we have?

To calculate the number of moles of our sample of Na, we need to know its mass and its molar mass, which is the same numerically as its relative atomic mass. Well, Na has a relative atomic mass of 23.0. To find the number of moles, we divide mass by relative atomic mass:

$\mathrm{number}\mathrm{of}\mathrm{moles}=\frac{34.5}{23}=1.5\mathrm{mol}$

We, therefore, have 1.5 moles of Na.

Here's another example.

A reaction yields 2.4 moles of water, H2O. What is the mass of this water in grams?

In this example, we know the number of moles of water produced. We can also work out its relative molecular mass: 2(1.0) + 1(16.0) = 18.0. This is the same numerically as its molar mass. We can use these values to find mass by rearranging the equation we used above:

$\mathrm{mass}=\mathrm{number}\mathrm{of}\mathrm{moles}×\mathrm{molar}\mathrm{mass}$

Plugging our values into the equation, we get the following:

$\mathrm{mass}=2.4×18=43.2\mathrm{g}$

### Moles, number of particles, and Avogadro's constant

Let's now look at the relationship between the number of moles, number of particles, and Avogadro's constant. We briefly met this when we first introduced you to moles up above, but we'll explore it again.

We know that one mole of any substance contains 6.022 x 1023 entities. This is just Avogadro's constant. Two moles of a substance would therefore contain twice as many entities: 2 x 6.022 x 1023 =

Find the number of oxygen molecules present in 88.0 g of oxygen, O2.

What information do we know? Well, we know the mass of oxygen, and we can work out its relative molecular mass: 2 x 16.0 = 32.0. We can use these values to find the number of moles.

$\mathrm{number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{mass}}{{\mathrm{M}}_{\mathrm{r}}}\phantom{\rule{0ex}{0ex}}\mathrm{number}\mathrm{of}\mathrm{moles}=\frac{88.0}{32}=2.75\mathrm{mol}$

We can now use the number of moles and Avogadro's constant to find the number of molecules:

$\mathrm{number}\mathrm{of}\mathrm{molecules}=\mathrm{number}\mathrm{of}\mathrm{moles}×\mathrm{Avogadro}\text{'}\mathrm{s}\mathrm{constant}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{number}\mathrm{of}\mathrm{molecules}=2.75×6.022×{10}^{23}=1.66×{10}^{24}$

### Relative atomic mass, the mass of one particle, and Avogadro's constant

Do you remember at the beginning, when we quoted the mass of a single hydrogen atom as -24 grams? Now let's learn how we worked that value out.

Remember: one mole of a substance - or to be precise, 6.022 x 1023 of its entities - has a mass equal to its relative atomic or relative molecular mass. As we learned, 6.022 x 1023 atoms of carbon have a mass of 12.0 g. If we divide this mass by the number of carbon atoms, we can find the mass of one atom. Here's the equation:

$\mathrm{mass}\mathrm{of}\mathrm{one}\mathrm{entity}=\frac{\mathrm{molar}\mathrm{mass}}{\mathrm{Avogadro}\text{'}\mathrm{s}\mathrm{constant}}$

Take hydrogen. One mole of hydrogen atoms has a molar mass numerically equal to its relative atomic mass, 1.0. If we sub that value into the equation, we get the following:

$\mathrm{mass}\mathrm{of}\mathrm{one}\mathrm{H} \mathrm{atom}=\frac{1.0}{6.022×{10}^{23}}=1.66×{10}^{-24}\mathrm{g}$

That's it! We hope you've now got a good understanding of moles, Avogadro's constant, and how to use these values in equations.

## Avogadro Constant - Key takeaways

• A mole is a chemical quantity used to represent 6.02214076 × 1023 entities. This number is known as Avogadro's constant and is the number of atoms in 12g of carbon-12.
• Molar mass is the mass of one mole of a substance. It is measured in g mol-1 and is numerically equal to its relative atomic or relative molecular mass.
• $\mathrm{number}\mathrm{of}\mathrm{moles}=\frac{\mathrm{mass}}{\mathrm{molar}\mathrm{mass}}$.
• $\mathrm{number}\mathrm{of}\mathrm{entities}=\mathrm{number}\mathrm{of}\mathrm{moles}×\mathrm{Avogadro}\text{'}\mathrm{s}\mathrm{constant}$.
• $\mathrm{mass}\mathrm{of}\mathrm{one}\mathrm{entity}=\frac{\mathrm{molar}\mathrm{mass}}{\mathrm{Avogadro}\text{'}\mathrm{s}\mathrm{constant}}$.
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Avogadro's constant is a quantity used in chemistry to represent the number of particles in a mole. It has a value of 6.02214076 × 1023, meaning that a mole of any substance contains exactly 6.02214076 × 1023 entities.

How do you calculate the number of atoms using Avogadro's constant?

To calculate the number of atoms in a substance, multiply the number of moles by Avogadro's constant. For example, 1.5 moles of carbon atoms contain 1.5 x 6.022 x 1023 = 9.033 x 1023 atoms.

How do you work out moles using Avogadro's constant?

Avogadro's constant is equal to the number of entities in one mole of a substance. This means that if you know the number of entities, you can calculate the number of moles. This would equal the number of entities divided by Avogadro's constant, which is 6.022 x 1023. You can also work out the number of moles using a substance's relative atomic or relative molecular mass, and its mass in grams. Here, number of moles equals mass divided by relative atomic or molecular mass.

What is the numerical value of Avogadro's constant?

Avogadro's constant equals 6.02214076 × 1023, although we often shorten it to 6.022 × 1023.

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