LAW OF EQUIPARTITION OF ENERGY
We have seen in Section 9.2.1 that the average kinetic energy of a molecule moving in x direction is
Similarly, when the motion is in y direction,
According to kinetic theory, the average kinetic energy of system of molecules in thermal equilibrium at temperature T is uniformly distributed to all degrees of freedom (x or y or z directions of motion) so that each degree of freedom will get 1/2 kT of energy. This is called law of equipartition of energy.
Average kinetic energy of a monatomic molecule (with f=3) =
Average kinetic energy of diatomic molecule at low temperature (with f = 5)
Average kinetic energy of a diatomic molecule at high temperature (with f =7)
Average kinetic energy of linear triatomic molecule (with f = 7) =
Average kinetic energy of nonlinear tri atomic molecule (with f = 6) =
Meyer’s relation CP − CV = R connects the two specific heats for one mole of an ideal gas.
Equipartition law of energy is used to calculate the value of CP − CV and the ratio between them γ = CP / CV.
Here γ is called adiabatic exponent.
Average kinetic energy of a molecule
For one mole, the molar specific heat at constant volume
Average kinetic energy of a diatomic molecule at low temperature = 5/2kT
Total energy of one mole of gas
(Here, the total energy is purely kinetic)
For one mole Specific heat at constant volume
Energy of a diatomic molecule at high temperature is equal to 7/2RT
Note that the CV and CP are higher for diatomic molecules than the mono atomic molecules. It implies that to increase the temperature of diatomic gas molecules by 1°C it require more heat energy than monoatomic molecules.
Note that according to kinetic theory model of gases the specific heat capacity at constant volume and constant pressure are independent of temperature. But in reality it is not sure. The specific heat capacity varies with the temperature.
Find the adiabatic exponent γ for mixture of μ 1 moles of monoatomic gas and μ2 moles of a diatomic gas at normal temperature.