189

9.1Moles, molar mass and the Avogadro constant

According to the new definitions of SI units, one mole a quantity of a substance containing as many particles as atoms in 12 g of carbon-12 of any substance contains as many particles as atoms in 12 g of carbon-12; this is exactly 6.02214076 × 1023 particles. This number is known as the Avogadro constant the number of particles in one mole: 6.02214076 × 1023, N A, and is one of the fundamental constants of physics.

What we mean by ‘particle’ depends on the substance; it can be a single atom or a molecule. For example, the particles in carbon are single atoms, the particles in hydrogen gas (H2) are diatomic molecules and the particles in carbon dioxide gas (CO2) are triatomic molecules.

So, one mole of carbon, one mole of H2 and one mole of CO2 all contain 6.02 × 1023 particles.

| Number of particles | Consisting of
One mole of carbon | 6.02 × 1023 | 6.02 × 1023 atoms of carbon
One mole of H2 | 6.02 × 1023 | 2 × 6.02 × 1023 hydrogen atoms
One mole of CO2 | 6.02 × 1023 | 6.02 × 1023 carbon atoms and 2 × 6.02 × 1023 oxygen atoms

Table 9.1

If a substance contains N particles then the number of moles n is:

n=N N A

key point

The molar mass the mass in grams of one mole of a substance of a substance is the mass, in grams, of one mole of the substance. The unit of molar mass is g mol−1.

For the element X Z A (which contains Z protons, A protons and neutrons and Z electrons) the molar mass is approximately A g mol−1. So, for example, the molar mass of He 2 4 is 4 g mol−1 and that of U 92 238 is 238 g mol−1.

The number of moles in a quantity of m grams of a substance with molar mass µ is then n=m μ.

Check yourself 1

aWhich has more atoms: 5 moles of He 2 4 or 1 mole of Ne 10 20?

bWhich has more atoms: 1 kg of He 2 4 or 1 kg of U 92 238?

cWhich atom is heavier: an atom of He 4 2 or an atom of U 92 238 ?

190

Worked example 9.1

aEstimate the number of atoms of gold in 1.0 kg of gold (Au 79 197).

bCalculate the mass of an atom of gold.

Answer

aThe molar mass of gold is 197 g mol–1. So 1000 g of gold (= 1 kg) contains 1000 197≈5.1 mol. One mole contains 6.02 × 1023 atoms, so the number of atoms in 1 kg of gold is

6.02×10 23×5.1=3×10 24.

bOne mole of gold has a mass of 197 g and contains N A, atoms. Hence the mass of one atom is

197×10−3 6.02×10 23=3.3×10−25 kg.

Worked example 9.2

Calculate how many grams of scandium, Sc 21 45, contain the same number of particles as 8.0 g of argon, Ar 18 40.

Answer

The molar mass of argon is 40 g mol−1, so a quantity of 8.0 g of argon corresponds to 8.0 40=0.20 mol. Thus, we need 0.20 mol of scandium. This corresponds to 0.20 × 45 = 9.0 g.

Worked example 9.3

Estimate the number of water molecules in an ordinary glass of water. The molar mass of water is 18 g mol−1.

Answer

A glass contains about 200 cm3 of water, which has a mass of 200 g. So the glass contains 200 18≈11 mol or 6.02×10 23×11≈7×10 24 molecules of water.

TEST YOUR UNDERSTANDING

1How many tyres are there in one mole of Ferraris?

2Calculate the number of molecules in 28 g of hydrogen gas (H2, molar mass 2.0 g mol−1).

3A substance has a molar mass of 28 g mol−1. How many moles of the substance are there in 12 g of the substance?

4Calculate the number of moles in 6.0 g of helium gas (molar mass 4.0 g mol−1).

5Determine the number of moles in a sample of a gas that contains 2.0 × 1024 molecules.

6A substance has a molar mass of 32 g mol−1. What is the mass in grams of 3 moles of the substance?

7A sample contains 3.0 × 1024 molecules. The molar mass of the sample is 12 g mol−1. What is the mass of the sample?

8Determine the mass in grams of carbon (molar mass 12 g mol−1) that contains as many molecules as 21 g of krypton (molar mass 84 g mol−1).

9The molar mass of helium is 4.0 g mol−1. Calculate the mass of a helium atom.