= Internal energy of a van der Waals gas
{title2=$U(V,T)=h(T)-a/V$}
For fixed <amount of substance>, thermodynamic consistency gives
$$
\left(\frac{\partial U}{\partial V}\right)_T
=T\left(\frac{\partial P}{\partial T}\right)_V-P.
$$
Indeed, the <first law of thermodynamics> gives $U_V|_T=T S_V|_T-P$, and the <Helmholtz free energy> <Maxwell relation> gives $S_V|_T=P_T|_V$. For the fixed-amount convention $P=RT/(V-b)-a/V^2$ with temperature-independent $a,b$, substitution gives $U_V|_T=a/V^2$. Integration yields $U=h(T)-a/V$. If the dilute-limit reference is $U\to cT$ as $V\to\infty$ for every $T$, then $h(T)=cT$.
The same formula may be interpreted molarly if $V,U,a,b$ all use that convention. Mixing molar and total quantities changes the amount factors. The attraction controls the volume correction to <internal energy>, while excluded volume alone does not contribute to this derivative when $a,b$ are independent of temperature.
Back to article page
