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.

Extensive properties

The parameter  whose value change on division known as extensive properties and these are depends on the mass (size, quantity) of the system.

1. Volume  
2. Number of moles     
3. Mass
4. Mole 
5. Free Energy (G)
6. Entropy (S)      
7. Enthalpy (H)   
8. Internal energy (E&U)   
9. Heat capacity
10. K.E.
11. P.E.
12. Gibbs free energy (G)
13. Resistance
14. Conductance

Extensive and Intensive properties:

1: Extensive properties are additive but intensive properties are non additive.

2: Ratio of two extensive property gives an intensive property.

3: An extensive property can be converted into intensive property by defining it per mole/per gram/ per liter

Intensive properties:

After specifying the parameter of the system, when system is divided in parts the parameter whose value remains unchanged due to division is known as Intensive parameter or properties. the value of intensive  is independent of the  mass (size or quantity) of the system.

1. Refractive index
2. Surface tension
3. Viscosity
4. Molar Mass
5. Density
6. Free energy per mole
7. Specific heat capacity
8. Molar heat capacity
9. Free energy per mole
10. Pressure (P)
11. Temperature (T)
12. Boiling point
13. freezing point etc
14. Molar enthalpy
15. Molar conductivity
16. Equivalent conductivity
17. Molarity, Normality, Mole fraction, %w/w, %V/V
18. EMF of cell