How to calculate gas equation (not necessarily ideal) when the internal energy depends solely on the temperature?

I know that for an ideal gas it can be shown through the equation of state and the Maxwell relationships, that the energy of the gas depends only on temperature. But my question is how can I calculate a state equation when I know that the energy of specific real gas depends only on the temperature?

• What’s the problem with using the ideal gas equation, if there are no indications of nonideality? Aug 8 at 16:10
• This is the question I saw. you know dependency of the energy in T only and you need to find the state equation e.g. P=... not only for ideal gas so you cant use ideal gas equation state. Aug 8 at 16:13

1 Answer

It is unnecessary to use Maxwell relations to show that if a gas obeys the equation of state of the perfect gas, the energy depends only on the temperature (and the number of moles).

Indeed, from $$pV=nRT$$, one gets $$\frac{p}{T}=\left.\frac{\partial{S}}{\partial{V}} \right|_{U,n}= \frac{nR}{V}$$ which can be integrated to give $$S=nR\log V + f(U,n)$$ Therefore, the inverse temperature $$\frac{1}{T}=\left.\frac{\partial{S}}{\partial{U}} \right|_{V,n}= \left.\frac{\partial{f}}{\partial{U}} \right|_{V,n}(U,n).$$ From this last equation, by using the inverse function theorem, we get $$U(n,T)$$ using the inverse function theorem. Notice that unless additional information is provided, it is impossible to obtain the unknown function $$f(U,n)$$.

If we know that the internal energy is a function of $$T$$ (and n) but does not depend on $$V$$, this implies that $$S(U,V,n)=\phi(V,n)+f(U,n)$$ (the dependence of the internal energy on $$T$$ and $$n$$ can be derived analogously to the previous case). However, we cannot go beyond the result that. $$\frac{p}{T}=\left.\frac{\partial{S}}{\partial{V}} \right|_{U,n}= \left.\frac{\partial{\phi}}{\partial{V}} \right|_{n}.$$ There is no way to get the unknown function $$\phi(V,n)$$ without additional information.

• Thanks you very much! Aug 8 at 17:57