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

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.

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.

• I'm afraid I don't understand this answer at all. How can I rewrite them in terms of $S$ for example? – Spine Feast Oct 6 '15 at 14:30
• @DepeHb: well, how do relate temperature, energy and entropy ? – Adam Oct 6 '15 at 14:31
• $T = \left( \frac{\partial U}{\partial S} \right)_{V,N}$ – Spine Feast Oct 6 '15 at 14:39
• @DepeHb: indeed. that should give you a way to get the first part of the result. – Adam Oct 6 '15 at 14:50
• So from this I get $\left( \frac{\partial U}{\partial S} \right)_{V,N} = U(S,V,N)/c_V N$, and $U(S,V,N) = f(N,V)e^{S/Nc_V}$, which is part of the result. Now I suppose I'll get something from p as well, but what about $\mu$? It doesn't appear anywhere. – Spine Feast Oct 6 '15 at 14:53