
Constant Volume specific heat:-
Using 1st 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)