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)