Specific Heat Cv

Introduction:

Molar specific heat capacity of a gas at constant volume Cv is defined as the quantity of heat required to raise the temperature of one mole of a gas through 1 K keeping its volume constant.
Specific heat Cv of ,monoatomic , diatomic are discussed here.

Specific Heat Cv of Monoatomic Gases:

Monoatomic gases like argon helium have three degrees of freedom.
We know , kinetic energy per molecule , per degree of freedom is `1/2` kT
Therefore kinetic energy  per molecule with three degrees of freedom is `3/2` kT
Total kinetic energy of one mole of the monoatomic gas is given by
 E =`3/2` kT* N
    = `3/2` RT, where N is Avogadro number
 `(dE)/(dT) = 3/2R`
If dE is small amount of heat required to raise the temperature of 1 mole of the gas at constant volume, through a temperature dT
dE = 1 x Cv x dT
Cv = = `(dE)/(dT) = 3/2 R`
As R = 8.32 J mol^-1K^-1
Cv = `3/2` 8.31
Cv = 12.465 J mol^-1 K^-1
Cp is molar specific heat at constant pressure.
Then Cp –Cv = R
Cp = CV +R
     = `3/2`  R +R = `5/2` R = `5/2`  x 8.32 = 20.775 J mol^-1K^-1

Specific Heat Cv of Diatomic Gases:

In diatomic gases like hydrogen , oxygen , nitrogen etc, a molecule has five degrees of freedom. Hence the total energy associated with one mole of diatomic gas is
     E = 5 x `1/2` kT x N = `5/2` RT
Also Cv = `(dE)/(dT)`  = `d/(dT) (5/2 RT)`
              = `5/2` R
Cv = `5/2` x 8.31 = 20.775 Jmol^-1K^-1 
But Cp = Cv +R
            = `5/2` R +R
            = `7/2` R
            = `7/2` x 8.31 = 29.0.85 J mol^-1K^-1