Suppose for a gas I guess an equation of state in the form $f(P,V,T)=0$. Is it possible that this hypothetical equation of state turns out to violate the second law of thermodynamics. A simple example of this kind would be very helpful.
We know that the entropy of a substance has to increase with increasing volume (at constant temperature), but, from one of the Maxwell relationships, the partial derivative of entropy with respect to volume at constant temperature is equal the partial derivative of pressure with respect to temperature at constant volume. So partial derivative of pressure with respect to temperature at constant volume must be greater than zero. If not, the proposed equation of state violates the 2nd law.