Class 11 NEET Physics Thermodynamics and Kinetic Theory

A Vander Waals equation is given by:
(P+‌

a

V2

)(V−b)=RT
Where, a and b are constant
Solving above equation,
P=‌

RT

V−b

−‌

a

V2

Taking derivative of P w.r.t volume
‌

∂P

∂V

=0
‌

∂2P

∂V2

=0
So, P becomes:
‌

∂P

∂V

=−‌

RT

(V−b)2

+‌

2a

V3

=0
‌

2a

V3

=‌

RT

(V−b)2

..........(1)
‌

a

V4

=‌

RT

2V(V−b)2

..............(2)
Taking double derivative again,
‌

∂2P

∂V2

=‌

2RT

(V−b)3

−‌

6a

V4

=0
OI ‌

RT

(V−b)3

=‌

3a

V4

Put equation (2) in above equation
‌

RT

(V−b)3

=‌

3RT

2V(V−b)2

On rearranging, 3V−3b=2V
Vc=3b
VC is critical volume Use this value in equation (1)
‌

RT

4b2

=‌

2a

27b3

Tc=‌

8a

27Rb

TC is critical temperature.