Constant Volume and Pressure Specific heat - Physics

Constant Volume specific heat:-

Using 1^st law of thermodynamics,
                ?U = Q – W
                      =Q          , Where, W = P?V = Px0 = 0. Cos ?V = 0
                ?U = Cvn?T
                => Cv =  ?U/n?T   …………… (I)

Cv = Constant Volume Spacific hit.

For Monoatomic ideal gas:-

                    U = 3/2 nRT

                  ?U = 3/2 nR?T ------------- (II)

From (I) and (II),

          Cv = 1/n?T  3/2 nR?T

    => Cv   = 3/2 R

R = Monoatomic Gas.

Constant Pressure specific heat:-

         Cp = Cv + R

               = 3/2 R + R

               = 5/2 R

Proof:- Cp = Cv + R

The molecule specific heat at constant pressure is defined by,

                   Cp = Q/n?T

               =>  Q = Cpn?T ................... (I)

Using 1st law of thermodynamics

                ?U = Q - W

           => ?U + W = Q

          => ?U + P?V = Cpn?T , Where, W = P?V  ............................... (II)

Form ideal gas law,

             PV = nRT

         => P?V = nR?T ........................... (III)

From (II) and (III)

           ?U + nR?T = Cpn?T

     => ?U/?T + nR?T/?T = Cpn?T/?T  [ Devided by '?T' ]

     => ?U/?T + nR = Cpn

     => Cp = 1/n  ?U/?T  +  R

     => Cp = Cv + R _(proved)