I have reached a conclusion that no such z can be found. Are there any flaws in my argument? Or are there cases that aren't covered in this?
Attempt
##\log(\frac{1}{z})=\ln\frac{1}{|z|}+i\arg(\frac{1}{z})##
##-\log(z)=-\ln|z|-i\arg(z)##
For the real part...
If the integration of post 9 were right, does it imply that for $$\Delta T C_{V,ideal}=\Delta U$$ since dU/dV=0; for vdw gas, we have an extra positive term added on the dU side that makes it bigger?
$$\Delta T C_{V,vdw}=\Delta U+\frac{n^2a}{V^2}$$
Sorry I did not follow. For $$-[P-T\left(\frac{\partial P}{\partial T}\right)_V]=-[\frac{nRT}{V-nb}-\frac{n^2a}{V^2}-T\frac{nR}{V-nb}]=\frac{n^2a}{V^2}$$ right? Even dropping the n will give positive results.
So taking the partial derivatives with respect to T means Cv for vdw gas does not...
I have tried to do that, I got $$dU=C_VdT+\frac{n^2a}{V^2}dV$$ and the integration (which I am not sure if this is correct) gives me $$\Delta U=C_V\Delta T-\frac{n^2a}{V}$$ Is so far the calculation correct?
I think the C_V for van der waals gas will be larger than ideal gas since it‘s a more precise description. However, for the relationship I cannot come up with a specific equation.