Thermal expansion coefficient

Remarks

For a solid one also uses the linear coefficient $\alpha_L = \frac{1}{L} \frac{dL}{dT}$ of a length $L$; for an isotropic solid with small expansion, $\alpha \approx 3 \alpha_L$. By a Maxwell relation, $\alpha V = -(\partial S / \partial p)_T$, where $S$ is the entropy.

Example: for an ideal gas, $V = nRT/p$, so $(\partial V / \partial T)_p = nR/p = V/T$ and $\alpha = 1/T$. At $T = 300\,\mathrm{K}$ this gives $\alpha \approx 3.3 \times 10^{-3}\,\mathrm{K}^{-1}$.

Non-example of the usual sign: liquid water at atmospheric pressure between $0\,^\circ\mathrm{C}$ and $4\,^\circ\mathrm{C}$ has $\alpha < 0$. The definition still applies; only the expectation $\alpha > 0$ fails, and nothing in thermodynamic stability forces $\alpha$ to be positive.