Finding $U(S,V,N)$ for ideal gas, given $pV=NkT$ and $U(T)=N c_V T$

Question

Given the experimental results for the ideal gas, I want to recover the internal energy expressed in its natural variables as:

$$U(S,V,N) = \alpha e^{\frac{S}{N c_V}} V^{\frac{c_V-c_p}{c_V}} N^{\frac{c_p}{c_V}}$$

with $\alpha$ as some constant and $c_V,c_p$ the (intensive) heat capacities. My starting point are the two equations:

$$pV=(c_P-c_V)NT,$$ $$U=c_V NT.$$

These completely characterize the system, and so does any thermodynamic potential expressed in its 'natural' variables, so one should follow from the other. To recover the experimental facts from $U(S,V,N)$ is simple enough, but I'm interested in the other direction. Unfortunately I was unable to find this derivation anywhere - I found the formula mentioned on wikipedia but I can't get my hands on the books cited.

So, how to do it? I'm not at all interested in any statistical physics considerations, only pure thermodynamics. Preferably without any elaborate tricks, I believe this must be doable in a methodical manner with just straightforward (if tedious?) math, i.e. using with all the multivariable calculus identities (Jacobians etc.), and this is the method I'd like to see.

Answer 1

You need to rewrite the equations of the ideal gas to the energy U as a function of its natural variables, or to its derivatives. If you know how to related $p$ and $T$, to $U$, a simple mathematical operation can give you the expected result.

Edit as the OP solves the problem:

Using $\partial U/\partial S=T$ and the third equation of the OP's question, we find $$U=f(N,V)e^{\frac{S}{c_V N}}.$$

Using $\partial U/\partial V=-p$ and the third equation of the OP's question, we find $$U=f(N)V^{\frac{c_V-c_p}{c_V}}e^{\frac{S}{c_V N}}.$$

Finally, using the fact that the energy is extensive : $$U(\lambda S,\lambda V, \lambda N)=\lambda U(S,V,N),$$ one finds that $f(N)\propto N^{\frac{c_p}{c_V}}$, which allows us to show that using only the thermodynamic equations of the ideal gas, we can find the energy as a function of the entropy, volume and number of particle.