Constant volume Specific heat and internal energy relationship in nonideal gas.

[kntsy]

Hi,
1.
\color{blue}dq=dU+PdV
so
C_V = \left(\frac {\partial q}{\partial T}\right)_V= \left(\frac {\partial U}{\partial T}\right)_V
leads to
dU=C_{V}dT
so
\color{red}\left(\frac {\partial U}{\partial V}\right)_T = 0
meaning that internal energy of NONideal gas is a sole function of T?
2.
As
\color{blue}dq=dU+PdV
so
\left(\frac {\partial U}{\partial V}\right)_T= \left(\frac {\partial V}{\partial T}\right)_P \left(C_P-C_V\right) - P
why? even for ideal gas the change in volume results in change in internal energy:
\color{red}\left(\frac {\partial U}{\partial V}\right)_T \not= 0??

thanks for answering.

 

[kntsy]

help please...