# General Thermodynamic equation of state

I heard my professor saying that the equation $$PV = \frac{2}{3}U$$ is valid for any non-relativistic gas, be it Ideal or Real gas(includes quantum ideal gases). Is this true, If it is how can we prove with all generality ?

No, that equation is only for a monatomic ideal gas. It will hold to a good approximation for a real monatomic gas, but for any other type of gas the $3/2$ will be replaced by a different number, known as the dimensionless heat capacity.