Question

The difference for specific heats for an ideal gas,

Evaluate the difference in specific heats for gases obeying (1) the Van Der Waals and (2) The Dieterici Equations of State. Comment on the results for the difference in specific heat for these gases compared with the ideal gas.

Answer 1

For a Van der Waals gas
(p + a/V²) = (RT)/(V-b)
As shown before
(∂E/∂V)t = a/V²
The partial derivative of the volume you get by implicit differentiation
(p + a/V²) - (RT)/(V-b) = 0
-R/(V-b) +[-2a/V³ + (RT)/(V-b)²] (∂V/∂T)p = 0
=>
(∂V/∂T)p = [R/(V-b)] / [(RT)/(V-b)² -2a/V³]

Cp - Cv = [p+a/V²] × [R/(V-b)] / [(RT)/(V-b)² -2a/V³]
= [(RT)/(V-b)] × [R/(V-b)] / [(RT)/(V-b)² -2a/V³]
= R / (1 - [2a×(V-b)²]/[RTV³])