Derive the thermodynamic equation Cp-Cv=[p+(partialE/partialV)T](partialV/partialT)p?

Question

and evaluate this difference for an ideal gas and a gas that obeys the van der Waals equation

Answer 1

Start with the definition of heat capacity at constant pressure
Cp ≡ (∂H/∂T)p ((..)p indicates derivation at constant pressure)
Substitute the enthalpy by its definition
H ≡ E + pV
Therefore:
Cp = (∂E/∂T)p + (∂(pV)/∂T)p = (∂E/∂T)p + p×(∂V/∂T)p
Because
dE = (∂E/∂V)t dV + (∂E/∂T)v dT = (∂E/∂V)t dV + Cv dT
The partial derivative of E with respect to temperature at constant pressure is:
(∂E/∂T)p = (∂E/∂V)t (∂V/∂T)p + Cv

Join both expression together:
Cp - Cv = (∂E/∂V)t (∂V/∂T)p + p×(∂V/∂T)p = [(∂E/∂V)t + p]×(∂V/∂T)p
q.e.d.

For further calculations the is a useful relation to evaluate the partial derivative of the energy from an equation of state.
As shown in my answer to your last question (see link):
(∂E/∂V)t = T×(∂p/∂T)v - p

For an ideal gas
pV = RT <=> p = (RT)/V <=> V = (RT)/p
(∂E/∂V)t = T×R/V - p = 0
(∂V/∂T)p = R/p
Cp - Cv = [p + 0] × R/p = R

For a Van der Waals gas
(p + a/V²) = (RT)/(V-b)
As shown before (see link):
(∂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])