Introduction to the Equipartition of Energy
The equipartition of energy theorem states: each degree of freedom contributes on average ½ kT per molecule to the stored energy of a gas. k is Boltzman’s constant, i.e., k = R/ N = 1.38 x 10-23 T/K. R = the universal gas constant and N = Avogadro number
Thus for a monatomic molecule of an ideal gas which has three degrees of freedom per molecule kinetic energy = 3/2 kT (for one molecule of gas):
The equipartition result is as follows:
Equipartition energy for diatomic gases
Similarly, a molecule of diatomic gas has 5 degrees of freedom per molecule. Therefore, kinetic energy per molecule is 5/2 kT and mean kinetic energy per mole is 5/2 kT × N (where N is number of molecules in one mole; N is called the Avogadro number). Further the number of molecules per cc at NTP in all gases is the same and is independent of the nature of gas. This number is called Loschmidt’s number.
We say that the different ‘modes’ (translational, rotational or vibrational modes) are excited, i.e. they ‘come into play’, and contribute to the heat capacity of a gas in different temperature ranges. At low temperatures, only translational modes are excited – i.e., cV is 3/2 R per mole, and the rotational and vibrational modes are said to be ‘frozen out’.
The measured value of cV around room temperature (~300 K) is 5/2 R: both translational (3 modes) and vibrational modes (2 modes) contribute.
Total Translational energy
The total translational energy of all the molecules of a gas is
The average kinetic energy of a molecule is
Experimentally, the contributions of the degrees of freedom to energy can be observed in measurements of cV.
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