INTERNAL ENERGY OF MONOATOMIC GASES

In the monoatomic gases the particles posses only translation degree of freedom and they don't possess rotational or vibrational degree of freedom. Therefore the net internal energy of the system is equal to translational contribution which can be calculated from the translational partition function.

Etotal = Etrans = RT² ( dlnq trans)v  .....(1)

                                           dT

We know that, qtrans= (2πmkT)³/₂ v .....(2)

                                            h³

Take log on both sides of equation (2)

lnq trans= 3/2ln (2πmk) + 3/2 lnT + lnV -

                   3lnh.............. (3)

(∂lnq trans)v = 0 + 3/2 (1/T) + 0−0 =  3 

       ∂T                                                     2T

Substitute the value of the above differential in equation (1)

Etotal = Etrans = RT²   3   = 3/2 RT .......(4)

                                     2T

This is the internal of a monoatomic gas depends only on temperature.

Heat capacity of monoatomic gas at constant volume is, 

   ∈v = (  ∂E ) v = 3 R

               ∂T         2