Question

For the theory of equipartition of energy the energy associated with the each molecule is f/ 2 KT if the gas has n number of moles , now how we can convert the moles into molecules and the energy associated with total molecule is found to be n×na×f/2KT. Here na×K=R(gas constant) how? And total energy associated with total molecule is n ×R×T×f/2.

Answer 1

The theorem of equipartition of energy states that molecules in thermal equilibrium have the same average energy associated with each independent degree of freedom of their motion and that the energy is
The equipartition result

serves well in the definition of kinetic temperature since that involves just the translational degrees of freedom, but it fails to predict the specific heats of polyatomic gases because the increase in internal energy associated with heating such gases adds energy to rotational and perhaps vibrational degrees of freedom. Each vibrational mode will get kT/2 for kinetic energy and kT/2 for potential energy - equality of kinetic and potential energy is addressed in the virial theorem. Equipartition of energy also has implication for electromagnetic radiation when it is in equilibrium with matter, each mode of radiation having kT of energy in the Rayleigh-Jeans law.For the translational degrees of freedom only, equipartition can be shown to follow from the Boltzmann distribution.



Internal energy in general includes both kinetic energy and potential energy associated with the molecular motion. But the potential energy is associated with intermolecular forces and is presumed to be zero in an ideal gas where the only molecular interactions are the perfectly elastic collisions between molecules. Therefore the internal energy of an ideal gas is entirely kinetic energy.
While steam at 100 degrees Celsius is not strictly an ideal gas, the diagram illustrates the fact that the phase change to the gaseous state leaves only the kinetic portion of the internal energy. For a monoatomic ideal gas this internal energy is given by
If rotation and vibrational kinetic energies are involved (polyatomic molecules) then
Study of the specific heats of gases provides evidence of whether or not rotation and vibration play a significant role in the molecular kinetic energy.