Question

state the law of equipartition of energy of dynamic system and use it to find value of ratio of two specific heat capacity of a monatomic and diatomic gas molecules​

Answer 1

Boltzmann's law of equipartition of energy states that the energy of a molecule (gas) is equally partitioned (divided) into different degrees of freedom that the molecule has.  The energy per partition (degree of freedom) is equal to 1/2 k T  per molecule  or 1/2 * RT  per mole.

     T = absolute temperature, k = Boltzmann's constant,

     R = Universal gas constant

   A diatomic molecule has 1 degree of freedom each for translational movement in x, y and z directions. But the molecule's vibrational energy for vibration about Center of mass is very small. Because the vibrations are very small. Further, there are three degrees of rotational energy. The diatomic molecule can rotate about its axis of line joining the atoms. This rotational energy is too small, as the moment of inertia is too little. It can rotate about two axes perpendicular to the axis joining the atoms. Thus the remaining two rotational degrees of freedom contribute to an energy of  2 * 1/2 * RT.

    Total energy of diatomic gas per mole = Q = 5/2 R T

      dQ/dT  at constant volume = Cv = 5/2 R

When we measure Cp (specific heat at constant pressure), the gas expands and does work. So it absorbs additionally energy equal to RT per mole. 

   So Cp = Cv + R

     Cp = 7/2  for a diatomic gas.

Cp / Cv = gamma = 7/5  for a diatomic gas

This is an approximation for the actual value.  The real values measured range from 1.3 to 1.5.