Thermal expansion coefficient

The thermal expansion coefficient (more precisely, the isobaric volumetric expansion coefficient) is a response function that quantifies how a system’s volume changes with temperature when the pressure is held fixed.

For fixed composition (e.g. fixed particle number $N$),

Physical interpretation

For a small quasistatic change at fixed pressure, the definition implies

So $\alpha$ is the fractional volume increase per unit temperature increase at constant pressure. Most materials have $\alpha>0$ in ordinary ranges, but $\alpha$ can be negative in anomalous regimes (thermal contraction upon heating).

In terms of the number density $n=N/V$ at fixed $N$, the same physics can be expressed as

Key relations and properties

• From an equation of state: If an equation of state is given as $p=p(T,V,N)$, then holding $p$ fixed and differentiating implicitly yields

  $$\alpha = -\frac{1}{V}\, \frac{\left(\frac{\partial p}{\partial T}\right)_{V,N}} {\left(\frac{\partial p}{\partial V}\right)_{T,N}} = \kappa_T\left(\frac{\partial p}{\partial T}\right)_{V,N},$$

  where $\kappa_T$ is the isothermal compressibility .

• Maxwell relation form: Using the Gibbs free energy differential and the Maxwell relations , one obtains

  $$\left(\frac{\partial V}{\partial T}\right)_{p,N} = -\left(\frac{\partial S}{\partial p}\right)_{T,N},$$

  hence

  $$\alpha = -\frac{1}{V}\left(\frac{\partial S}{\partial p}\right)_{T,N}.$$

• Links to heat capacities and compressibilities: For a simple compressible system,

  $$C_P - C_V = \frac{T\,V\,\alpha^2}{\kappa_T},$$

  connecting $\alpha$ with the constant-pressure heat capacity $C_P$, the constant-volume heat capacity $C_V$, and $\kappa_T$.

  The difference between isothermal and adiabatic compressibilities can likewise be written as

  $$\kappa_T - \kappa_S = \frac{T\,V\,\alpha^2}{C_P}.$$

• Stability consistency: In stable single-phase equilibrium, the identities above are consistent with $C_P \ge C_V$ and $\kappa_T \ge \kappa_S$, reflecting thermodynamic stability .