The derivative of Heat Capacity with respect of pressure

[Astrocyte]

Homework Statement

Show that derivative of Heat Capacity for constant V with respect of pressure for ideal gas is "zero"
(∂C_V/∂P)_T=0

Relevant Equations

C_V=T(∂S/∂T)_V, pV=nKT

I'm thinking about that for 1 hour. But, I could not do it.

 

[mitochan]

Your result for ideal gas is written as $-p(\frac{\partial V}{\partial T})_S > 0$.

Not for this homework, but for slightly modified ones, i.e.,

Show that derivative of Heat Capacity for constant P with respect of pressure for ideal gas is "zero"
(∂C_P/∂P)_T=0

or

Show that derivative of Heat Capacity for constant V with respect of volume for ideal gas is "zero"
(∂C_V/∂V)_T=0

with pV=nKT, I could prove it.

 

[Chestermiller]

Here's a hint: Start out by writing $$dU=\left(\frac{\partial U}{\partial T}\right)_VdT+\left(\frac{\partial U}{\partial V}\right)_TdV$$What are the partial derivatives in this equation equal to for the case of an ideal gas?