= Low-temperature Gibbs free-energy expansion
{title2=$G(T,p)=G(0,p)-\int_0^T S(t,p)\,dt$}
At fixed <pressure> and material content, $S=-(\partial G/\partial T)_p$. If the equilibrium <entropy> tends to zero, the <Gibbs free energy> approaches its zero-<temperature> value with zero slope. When $C_p=aT^r+o(T^r)$ with $r>0$ and zero residual <entropy>, integrating $dS=C_p\,dT/T$ gives $S=(a/r)T^r+o(T^r)$ and $G=G(0,p)-aT^{r+1}/[r(r+1)]+o(T^{r+1})$. Nonzero residual <entropy> $S_0$ instead contributes $-S_0T$ to $G$; a horizontal limiting tangent requires the zero-<entropy> hypothesis.
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