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 ?
 A: 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. 
In general the dimensionless heat capacity is not a constant but rather depends on the pressure and temperature (or equivalently, on the volume and energy). The classic example is the van der Walls equation, in which the deviations from the ideal gas equation are explicitly derived. They become more pronounced as the density increases and the temperature decreases, which makes sense because under sufficiently high pressure and/or low temperature, most gases will turn into a liquid or a solid, and these clearly don't obey the ideal gas law.
Conversely, the deviations from the ideal gas law become less pronounced as the density decreases and the temperature increases. This is because intermolecular forces play less and less of a role, since the molecules are less likely to come close enough to one another for them to take effect. Thus, your professor's comment was probably made under the assumption that we're talking about a very dilute gas that isn't near a phase boundary.
