# Determine the entropy $S(U,V,N)$ from the Helmholtz free energy $F(T,V,N)$ for a Van der Waals fluid

The entropy $$S$$ of a Van der Waals fluid is: $$S(U,V,N) = S_0 + NR\ln\left[\left(\frac{V}{N}-b\right)\left(\frac{U}{N}+a\frac{N}{V}\right)^c\ \right]$$ I understand how to get from the equation of the entropy $$S(U,V,N)$$ to the equation of $$F(T,V,N)$$, but I don't know how to do the inverse process. I'm going to put my procedure that I followed: $$\begin{eqnarray} U(S,V,N) &=& N\left[\left(\frac{V}{N}-b\right)^{-1/c}\exp\left(\frac{S-S_0}{NRc}\right) - a\frac{N}{V}\right] \end{eqnarray}$$ By the definition of temperature $$T$$: $$T(S,V,N) =\left(\frac{\partial U}{\partial S}\right)_{V,N} = \frac{1}{Rc}\left(\frac{V}{N}-b\right)^{-1/c}\exp\left(\frac{S-S_0}{NRc}\right)$$ then clearing $$S$$ in the equation: $$S(T,V,N) = S_0 + NRc\cdot\ln\left[RcT\ \left(\frac{V}{N}-b\right)^{1/c}\right]$$ $$S(T,V,N) = S_0 + NR\cdot\ln\left[(RcT)^c\ \left(\frac{V}{N}-b\right)\right]$$ then if $$T$$ is replaced in $$U(S,V,N)$$: $$U(T,V,N) = N\left[RcT - a\frac{N}{V}\right]$$ Thus $$F(T,V,N)$$ is given by: $$F(T,V,N) = U-TS = RcTN - a\frac{N^2}{V} - T\left(S_0 + NR\cdot\ln\left[(RcT)^c\ \left(\frac{V}{N}-b\right)\right]\right)$$

My problem is this: I know that $$\displaystyle S = -\left(\frac{\partial F}{\partial T}\right)_{V,N}$$, but if I derivate $$S$$: $$S(T,V,N) = -\left(\frac{\partial F}{\partial T}\right)_{V,N} = S_0 + NR\ln\left[(RcT)^c\left(\frac{V}{N}-b\right)\right]$$ And I don't know how to continue to get to my first equation. Is there something wrong?

• Formula for entropy you obtained by using the defnition of $T$ as derivative do $U$ wrt $S$ is already the entropy as a function of the $T,V,N$ variables. Passing through $F$ and taking its derivative wrt $T$ looks as a useless round path. – GiorgioP Jan 12 at 8:09
• I think it is more of an extra check. You wanted to find an expression for $F$ and you found it, what is the problem? – WarreG Jan 12 at 8:22
• Welcome to Physics SE! However, I'm afraid that I am voting to close this question on the grounds that it is not clear what you are asking. You've gone around in a circle, starting with an expression for $S$ and ending with an expression for $S$ (which already appeared earlier in your derivation). Can you clarify your question? Do take a moment to read our advice on what is on-topic and off-topic here. – LonelyProf Jan 12 at 12:38
• I fix it I think – El borito Jan 12 at 20:25

Namely, $$T$$ appears in your last equation of $$S$$ instead of $$U$$. Why not use your sixth equation, U(T,V,N)=N[RcT−aNV], to get rid of T in terms of U. Hope it helps.
• But you can't use your last equation, only $F(T,V,N)$ – El borito Jan 14 at 0:02