# Equation of state for ideal gas from Helmholtz free-energy

Starting from the definition of Helmholtz free energy: $$F:=U-TS$$ (where $$U$$ is the internal energy , $$T$$ temperature and $$S$$ entropy) we derive in few steps the following relation: $$F=-T\int \frac{U}{T^2}\mathrm d T+ \text{constant} \tag{1}$$

Now, we know also that Maxwell relations holds so at $$T=\text{constant}$$ we have: $$P=-\frac{\partial F}{\partial V} \tag{2}$$

In ideal gas the internal energy have the following form: $$U=\frac{3}{2} NT \tag{3}$$

If i substitute $$(3)$$ in $$(1)$$ and put the result in $$(2)$$ i should find the classical equation of state for ideal gas: $$PV=NT \tag{4}$$ ...but from calculation i don't find this. Where is the error in my steps? It could be in the value of the constant ?

$$p = - \frac{\partial F}{\partial V} = -C'(V) T.$$

Using (4) in this equation we obtain $$C(V) = N * ln(V)+ constant$$

• 1) is a general relation? Commented Apr 26, 2020 at 12:54
• I would say yes. From Maxwell relations we have "S= -∂F/∂T" for V=const. Insert in the definition of F, put F under the derivative wrt T and obtain : -U/T^2 = ∂/∂T ( F/T). Then simply integrate. No other assumption in this derivation Commented Apr 26, 2020 at 13:00

In 1) there is additive "constant" of integration. The integration is only over $$T$$, the terms may depend also on volume $$V$$ which can be arbitrary. Therefore the "constant" in that integration over $$T$$ can be actually a function of $$V$$:

$$F(T,V) = -T\int \frac{U}{T^2}dT + C(V)T.$$

Since the first term, for an ideal gas, does not depend on volume, the only part relevant for calculating pressure from $$F$$ is the second term:

$$p = - \frac{\partial F}{\partial V} = -C'(V) T.$$

The conclusion is, we cannot infer the familiar equation of state of ideal gas $$p = nc_V T / V$$ just from knowing $$U = nc_V T$$. The above result suggests large class of functions of $$C(V)$$ is consistent with $$U=nc_VT$$. But we did find at least that pressure must be proportional to temperature $$T$$.

In Callen there is a rationale for this - the equation $$U=nc_VT$$ is not the fundamental form, that is, $$U$$ is not expressed as function of its natural parameters $$S,V$$. If it was, we should be able to derive the equation of state from it.

• Yes, i suppose that the trick was in the constant. But if i start from P=NT/V we have that -dF/dV = NT/V. Integrati h we find F=...+C. We cant deduce the value of the constant from this relation and the previuos? Commented Apr 26, 2020 at 23:44
• Both integrations introduce one unknown function: the first integration introduced function $C(V)$, the second integration introduced another function $D(T)$. There is no constant in numerical sense, these are functions. Commented Apr 26, 2020 at 23:51
• But a posteriori we can deduce the form of this function ,right?(see what i’ve added) Commented Apr 27, 2020 at 12:59
• Yes, that seems possible. Commented Apr 27, 2020 at 19:45