# The mole and Avogadro’s number

Any body is characterized by its mass. Because the mass of different particles (molecules, atoms, electrons, ions) is not the same, the number of particles in the same amount of mass is different for each substance.

For example, because the mass of an oxygen molecule mO2 is 16 times bigger than the mass of a hydrogen molecule mH2, the number of oxygen molecules NO2 contained in 1 kg of oxygen gas is 16 times less than the number of hydrogen molecules NH2 contained in 1 kg of hydrogen gas. We can write the relationship between the number and mass of oxygen and hydrogen molecules, for the same mass of gas, as:

$N_{O_2} \cdot m_{O_2} = N_{H_2} \cdot m_{H_2} = 1 \text{ kg} \tag{1}$

From (1), we can write the number of molecules ratio between oxygen and hydrogen, for the same amount of gas:

$N_{O_2} = N_{H_2} \cdot \frac{m_{H_2}}{m_{O_2}} = \frac{N_{H_2}}{16} \tag{2}$

Observation: The mass of the oxygen and hydrogen molecules are taken from The Periodic Table of Elements.

### Mole definition

Because atoms and molecules are very small (microscopic), weighing or counting them directly is not possible. Therefore, mostly in chemistry, there is a need to define a new unit of measurement for the amount of substance, which contains the same number of particles (molecules, atoms, electrons, ions), regardless of the type of substance. This unit of measurement is called the mole.

The mole is defined as the amount of substance of a system which contains as many fundamental units as the number of atoms contained in 12 g of Carbon 12 (symbol 12C). The fundamental units can be: molecules, atoms, ions, nuclei, electrons or formula units.

The mole is a measurement unit in the SI base, with the symbol mol.

Every time we use the mole as a unit of measurement, we also have to specify the fundamental unit (e.g 2.5 mol of H2O molecules).

### Molar volume definition

The volume occupied by 1 mol of substance is called molar volume. It’s usually symbolized as Vm.

From experimental data, at standard temperature and pressure (stp) conditions (t0 = 0 °C, p0 = 1.013·105 N/m2), it’s proven that all gases have the same molar volume Vm0 = 22.41 L = 22.41·10-3 m3.

### Molar mass definition

The mass of a mole of substance is called molar mass. In other words, the molar mass is defined as the mass of a mole of that substance, measured in grams per mole [g/mol or g·mol-1]. To calculate the molar mass of a substance we need to know the atomic mass of the elements within the substance.

Image: Periodic Table of Elements – Iron

For example, the atomic mass of Iron is 55.845 amu (atomic mass units) which means that the molar mass of 1 mol of Fe is 55.845 g.

The same applies for other elements, the molar mass of 1 mol of carbon is 12.0107 g.

Observation: The atomic mass of the chemical elements is taken from The Periodic Table of Elements.

The number of atoms contained in 1 mol of carbon-12 (which has the molar mass of 12 g) is called the Avogadro’s number. The symbol for Avogadro’s number is NA and it is equal with 6.0221367·1023 mol-1.

Any 1 mol of any substance contains 6.022·1023 fundamental units. A fundamental unit can be atoms (e.g. iron, Fe), molecules (e.g. oxygen, O2) or formula units (e.g. water, H2O)

Since the number of Avogadro is the same for all substances, it is a universal constant.

The total number of fundamental units N  (atoms or molecules) contained in ν moles of substance is given by:

$N = \nu \cdot N_A \tag{3}$

The molar mass of a substance μ is equal with the product between the mass of the fundamental unit m  (atom or molecule) and Avogadro’s number NA.

$\mu = m \cdot N_A \tag{4}$

From (3) and (4) we can observe that Avogadro’s number is the link between the macroscopic and microscopic units of substance.

To put it simply, a mole is a number, as:

• a dozen is 12
• a pair is 2
• a mole is 6.022·1023

### Mole summary

The mole is:

• a unit of measurement used in chemistry
• a bridge between the atom or molecule (microscopic) and the amount of substance used in laboratories (macroscopic)
• the mass of substance containing the same number of fundamental units as there are atoms in exactly 12 g of carbon-12
• Avogadro’s number 6.022·1023 of anything
• a way of counting the atoms of a substance by weighing it

A mole of substance:

• has a specific mass
• occupies a given volume

Example of moles for mono-atomic substances

 Substance Number of atoms Molar mass [g] Molar volume [L] 1 mol of Xenon, Xe 6.022·1023 131.293 22.41 (gas) 1 mol of Helium, He 6.022·1023 4.002602 22.41 (gas) 1 mol of Neon, Ne 6.022·1023 20.1707 22.41 (gas)

As you can see, 1 mol of any substance has the same number of atoms and the same molar volume (for gases). The molar mass varies from substance to substance and depend on the atomic mass of the element (see The Periodic Table of Elements).

Example of moles for bi-atomic substances

The molar mass of a bi-atomic substance is the twice the atomic mass of the element. For example, an oxygen molecule O2 is made up from two oxygen atoms. The atomic mass of an oxygen atom is approx. 16 g which makes the molar mass of 1 mol of oxygen 32 g.

 Substance Number of molecules Molar mass [g] Molar volume [L] 1 mol of Hydrogen, H2 6.022·1023 1.00794 · 2 ≅ 2 22.41 (gas) 1 mol of Oxygen, O2 6.022·1023 15.9994 · 2 ≅ 32 22.41 (gas) 1 mol of Nitrogen, N2 6.022·1023 14.0067 · 2 ≅ 28 22.41 (gas)

### Examples of moles for poly-atomic substances (compounds)

The molar mass of a poly-atomic substance (also called formula unit) is the sum of the atomic mass of each atom and molecule. For example, the molar mass of 1 mol of water (H2O) is the sum between the atomic masses of all the constituent atoms: 1.00794 · 2 + 15.9994 ≅ 18 g.

 Substance Number of formula units Molar mass [g] Molar volume [L] 1 mol of Water, H2O 6.022·1023 1.00794 · 2 + 15.9994 ≅ 18 22.41 (vapor) 1 mol of Salt, NaCl 6.022·1023 22.98976 + 35.453 ≅ 58 22.41 (gas) 1 mol of Methane, CH4 6.022·1023 12.0107 + 1.00794 · 4 ≅ 16 22.41 (gas)

### Chemistry examples involving moles

Example 1. How many atoms we have in 2.7 moles of Krypton (Kr) ?

We know that 1 mol of any substance has 6.022·1023 atoms. Therefore in 2.7 moles of Krypton we are going to have: 2.7 · 6.022·1023 = 16.2594·1023 atoms of Krypton.

$2.7 \text{ mol} \cdot \frac{6.022 \cdot 10^{23} \text{ atoms}}{1 \text{ mol}} = 16.2594 \cdot 10^{23} \text{ atoms}$

Example 2. How many moles there are in 1000 g (1 kg) of Iron (Fe) ?

From The Periodic Table of Elements we know that 1 mol of Iron has 55.93319 g. To find out the how many moles of Iron we have in 1 kg, we calculate the ratio: 1000 / 55.93319 = 17.8785 moles of Iron.

$1000 \text{ g} \cdot \frac{1 \text{ mol Fe}}{55.93319 \text{ g}} = 17.8785 \text{ mol}$

Example 3. How many atoms there are in 2 g of Lead (Pb) ?

From The Periodic Table of Elements we know that 1 mol of Lead has 207.2 g. Also we know that 1 mol of any substance has 6.022·1023 atoms. Writing the numbers down, gives:

$2 \text{ g} \cdot \frac{1 \text{ mol Pb}}{207.2 \text{ g}} \cdot \frac{6.022 \cdot 10^{23} \text{ atoms}}{1 \text{ mol}}=5.81 \cdot 10^{21} \text{ Pb atoms}$

Example 4. What is the mass of 1 atom of Titanium (Ti) ?

From The Periodic Table of Elements we know that 1 mol of Titanium has 47.867 g. Also we know that 1 mol of any substance has 6.022·1023 atoms. Writing the numbers down, gives:

$\frac{47.867 \text{ g}}{6.022 \cdot 10^{23} \text{ atoms}} = 7.9487 \cdot 10^{-23} \text{ g}$

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