8. Avogadro law

All the gas laws discussed could be formulated without referring to the molecular structure of the gases. When formulating the next gas law, the Avogadro law, it should be assumed that gas consists of separate molecules.

The masses of individual molecules are very small. Therefore, it is convenient to use relative values, not absolute values expressed in kilograms or grams, but relative values. By definition, the molecular mass of a substance is defined as the relation of the mass of a molecule of that substance to \(1 \over 12\) of the mass of the carbon atom.

From the course of chemistry it is known that the amount of grams (or kilograms) of substance, numerically equal to its molecular weight, is called a gram-molecule or mole (respectively, a kilogram-molecule or kilo-mole) and is denoted as \(\mu \). If the mass of the molecule of the substance is denoted as \(m\), and the mass of the carbon atom \(m_0\), then according to the definition

\( \mu = {m \over {{1 \over 12} m_0}} \)(1-4)

In order to find out what the mole of a given substance is equal, it is required to know the molecular composition of the substance and the atomic masses of the atoms entering the molecule. For example, the mole of carbon dioxide \(CO_2\) is 44 g/mol, because the atomic mass of oxygen is 16 and carbon is 12.

The Italian chemist Avogadro has experimentally found that under the same condition of temperature and pressure, equal volumes of different gases contain the same number of molecules (or in other words moles (the amount of substance of the gas) of different gases at the same conditions occupy the same volumes).

The law can be written as:

\( {V \over n} = k \)

where
\(V\) is the volume of the gas;
\(n\) is the amount of substance of the gas (\(m \over \mu \)measured in moles);
\(k\) is a constant for a given temperature and pressure.

In particular, at a pressure of \(p =1\;atm\) and \(t =\,0^0\;C\), the volume of one mole of any gas is equal to \(V_{0\mu} = 22.4\;liters/mol = 22.4\;m^3/kmol\).

Avogadro number.

It follows from the definition of mole that the number of molecules in a mole of any substance is the same. This number is called the Avogadro number.

The number of molecules in the mole is equal

\( N_A = {\mu \over m} \)

where
\(m\) is the mass of one molecule in grams.

Using the definition of \(\mu\) (1-4), we get:

\( N_A = {m \over {{1 \over 12} m_0m}} = {1 \over {{1 \over 12} m_0}} \)(1-5)

This value is the same for any substance.

The Avogadro number can be determined by measuring the mass of the carbon atom (or some other atom) in grams. Methods for measuring the masses of individual atoms are well developed. They are based on the displacement of the ion beam (group of atoms that bears one or more positive or negative electrical charges) by an electromagnetic field.

Measurements give a mass of carbon atom \(m_0 = 1.995 \cdot 10^{-23} \, g\)

From here, according to (1-5)

\( N_A = 6.02 \cdot 10^{23} \, mole^{-1} \)

There are other methods of determining the Avogadro number that are not related to the mass measurement of individual atoms. They all give the same result.