Degree of Freedom and Law of Equipartion of Energy:

It is equal to number of modes of energy transfer when a gaseous molecule undergoes collision.                      OR

        It represent the number of independent modes to describe the molecular motion.  

       Total degree of freedom =  3N (Where N  is Number of atom in molecule)

1- Translational degree of freedom is 3 (three) always for mono, di and t     tri atomic molecule.

2- Rotational degree of freedom is zero for mono atomic gas, 2 (two) for diatomic molecules and   3 (three) for triatomic molecule

3-Vibrational degree of freedom is also zero for mono atomic gas and 1(one) diatomic gas molecule  and for polyatomic gases Vibrational DOF is calculated individually.( f_vib= 3N- f_trans+ f_rot)

Total degree of freedom:=  f_trans + f _rot + f _{vib    &}  f_vib= 3N- f_trans + f _rot    

Molecules                N                      TDF

He                             1                         3

O₂                                    2                         6

CO₂                           3                         9

NH₃                            4                        12

PCl₅                           6                        18

Case-1

Monoatomic       Diatomic       Triatomic (linear)       Triatomic (Non linear)

f _total =3                f_total  =6          f_total =9                      f _total=9

f _trans=3                  f_trans =3            f_trans=3                     f_trans=3

f _rot  =0                      f_rot     =2           f_rot    = 2                    f_rot    =3

f _vib  =0                     f_vib     =1           f_vib    = 4                        f_vib   =3

                              q =n C_mdT

                                q_V=n C_vmdT

                                (C_m)v=(dq/dT)_v

 By FLOT dq= dU+ dW     

At constant volume  dW=0   so   dqv=dU

                         Hence   ( C_m)V= (dU/dT)_v

LAW OF EQUIPARTIAL OF ENERGY :
Average energy associate with each molecule per degree of freedom is U= 1/2KT  (where K is Boltz’s man constant.

Let degree of freedom is = f   then U is U=1/2fkT

             And U=1/2fkTN_A per molecule  we know  kN_A=R

                     U=1/2fRT  and   dU/dT=1/2fR

             And  dU/dT=C_v      hence  C_v=1/2fR

Cv=1/2f_transR +1/2f_rotR  (Where Vib degree inactive in chemistry)

       For ideal gas C_pm-C_vm=R  and  Gama= C_pm/C_vm

         

Adiabatic exponent :Adiabatic exponent (Gama) for a mixture of gas with different heat capacity is defined as :

where n1, n2 ........................ are moles of different gases

Example: Calculate change in internal energy of 10 gm of H₂ ,when it's state is changed from(300K, 1Atm) to (500 K, 2Atm) ?

Solution: For ideal gas

       

C_v for H₂ (diatomic) in low temperature range will be 5R as vibrational part is not included.