Degrees of freedom

In the section headed the 'Kinetic theory of matter' we considered monatomic gases. We must now extend the ideas to cover gases of higher atomicity, that is, molecules with more than one atom per molecule.

If we go back to the kinetic theory formula, PV = 1/3[mNc²] = RT, you can see that

RT = 2/3 [average kinetic energy of the molecules] since kinetic energy = ½ mNc².

Therefore:

Average kinetic energy of the gas molecules = 3/2 RT

We have considered the motion of these molecules to be in three directions, in other words we say that the molecule has three degrees of freedom. It is therefore sensible to suppose that one-third of the total energy is associated with each degree of freedom, and this is known as Boltzmann's law of equipartition of energy. Thus each degree of freedom has an amount of energy ½ RT associated with it.

If a gas has its temperature raised at constant volume the energy input is the increase in kinetic energy of the gas molecules. So for a unit mass and for a rise in temperature dT we have:
C_VdT = kinetic energy increase = 3/2 RdT

Where C_V is the specific heat capacity of the gas at constant volume.

But C_P - C_V = R, so for a monatomic gas:

C_V = 3/2 R    and C_P = 5/2 R    (monatomic gas)

Now consider a diatomic molecule. In addition to three translational degrees of freedom it can also rotate about three axes X, Y and Z (Figure 1).

The energy associated with axis X is very small however, and so we say that the molecule has five degrees of freedom.

If we assume that the energy associated with each rotational degree of freedom is the same as that for each translational degree of freedom then the total energy of the molecule will be 5 x ½ RT = 5/2 RT. (The vibrational energy of the molecule is insignificant except at very high temperatures.)

Using the same argument as for the monatomic gas, we have for the two principal molar specific heats of a diatomic gas:

C_V = 5/2 R    and    C_P = 7/2 R    (diatomic molecule)

at temperatures around room temperature.

We therefore have for the ratio C_P/C_V. (γ):

For a monatomic gas, γ = 5/3 = 1.66

For a diatomic gas, γ = 7/5 = 1.40

For more complex molecules we have that:

C_P/C_V = γ = 1+ 2/n

where n is the total number of degrees of freedom.

The table below gives the value of γ for a number of common gases.

Gas	Value of γ
Air	1.41
Ammonia	1.31
Argon	1.66
Carbon dioxide	1.30
Carbon monoxide	1.40
Helium	1.66
Hydrogen	1.41
Oxygen	1.40

These results show very good agreement with theory.

The preceding section has dealt with one mole of a gas. If we now consider a sample of gas containing n moles we have:

2/3 N (½ mc²) = nRT and this gives ½ mc² = 3/2[R/L]T


where L is the Avogadro constant.
The quantity R/L is known as the Boltzmann constant (k) and its value can be shown to be: 1.38x10^-23 J K^{- 1}

The average translational energy of a single molecule is therefore: 3/2 kT