Physics · Kinetic Theory · NEET
A degree of freedom is one independent way a molecule can store energy - for example moving along x, y, or z. The law says that in thermal equilibrium each such independent mode holds, on average, the same amount of energy: (1/2)kBT per molecule, where kB is the Boltzmann constant and T is the absolute temperature. It does not matter whether the mode is translation or rotation; every active mode gets the same equal share. Add up all f modes and one molecule has (f/2)kBT.
Equipartition counts each 'squared term' in the energy expression as one mode worth (1/2)kBT. Translation and rotation each add only one squared term (a kinetic term like (1/2)mvx^2 or (1/2)Iw^2), so they give (1/2)kBT each. But a vibration stores energy in TWO forms: kinetic energy AND potential energy of the bond (like a spring). That is two squared terms, so one vibrational mode gives 2 x (1/2)kBT = kBT. This is why vibration counts double.
Both forms exist and you must not mix them. Per molecule use kB: average energy per active degree of freedom = (1/2)kBT, total per molecule = (f/2)kBT. Per mole use R: energy per degree of freedom = (1/2)RT, and internal energy of one mole = (f/2)RT. The link is R = NA x kB, where NA is Avogadro's number. For n moles, U = (f/2)nRT.
No - and this is the surprising, powerful part. The energy per degree of freedom, (1/2)kBT, depends only on temperature, not on the mass of the molecule or the type of gas. A heavy molecule and a light molecule at the same T have the same average energy per mode. What changes between gases is the NUMBER of active degrees of freedom f (3 for monatomic, 5 for a rigid diatomic, etc.), not the energy each one holds.
The average thermal energy for a mono-atomic gas is: (kB is the Boltzmann constant and T the absolute temperature)
A gas mixture consists of 2 moles of O2 and 4 moles of Ar at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
In thermal equilibrium at absolute temperature T, the total energy of a molecule is shared equally among all its active degrees of freedom, with each degree of freedom having average energy (1/2)kBT. Each vibrational mode counts as two degrees of freedom because it has both kinetic and potential energy terms.
It is (1/2)kBT per molecule, or (1/2)RT per mole, where kB is the Boltzmann constant, R is the universal gas constant and T is the absolute temperature. This value is the same for every active degree of freedom, regardless of the gas.
For a rigid diatomic gas, f = 5 (3 translational + 2 rotational), so the internal energy of n moles is U = (5/2)nRT. If vibration is active, f = 7 and U = (7/2)nRT.
Any molecule has 3 translational degrees of freedom (motion along x, y, z). Equipartition gives (1/2)kBT to each, so the translational kinetic energy is 3 x (1/2)kBT = (3/2)kBT. This part is the same for all gases, monatomic or not.
Since U = (f/2)nRT, the molar heat capacity at constant volume is CV = (f/2)R. Then CP = CV + R (Mayer's relation). For monatomic gas CV = (3/2)R, for diatomic CV = (5/2)R, matching experiment closely.
