= Coefficient of thermal expansion
{title2=$\alpha=V^{-1}(\partial V/\partial T)_{P,N}$}
= Thermal expansion coefficient
{synonym}
The volumetric coefficient of <thermal expansion> is the fractional <volume (thermodynamics)> increase per <temperature> increase at fixed <pressure> and composition:
$$
\alpha=\frac1V\left(\frac{\partial V}{\partial T}\right)_{P,N}
=-\frac1\rho\left(\frac{\partial\rho}{\partial T}\right)_{P,N},\qquad
\rho=\frac mV.
$$
The <mass density> form requires fixed <mass> $m$. Its units are $\mathrm{K^{-1}}$; $\alpha_T$ is another notation for this coefficient. For small <temperature> changes $\theta$ at fixed <pressure>,
$$
V(T_0+\theta)\simeq V(T_0)(1+\alpha(T_0)\theta),\qquad
\rho(T_0+\theta)\simeq\rho(T_0)(1-\alpha(T_0)\theta).
$$
For a finite change in a single <thermodynamic phase>, integrate the possibly varying coefficient: $\ln(V_2/V_1)=\int_{T_1}^{T_2}\alpha(T,P,N)\,dT$. A constant-coefficient approximation gives an exponential, while the displayed linear formula additionally requires $|\alpha\theta|\ll1$.
For an <ideal gas> at fixed particle number,
$$
\alpha=\frac1T,\qquad
\alpha(300\,\mathrm K)=\frac1{300}\,\mathrm{K^{-1}}\simeq3.333\times10^{-3}\,\mathrm{K^{-1}}.
$$
Integrating $1/T$ recovers the exact fixed-<pressure> <thermal expansion> ratio $V_2/V_1=T_2/T_1$. Combining expansion with <isothermal compressibility> gives the local response of any smooth <simple compressible system> of fixed composition,
$$
\frac{dV}{V}=\alpha\,dT-\kappa_T\,dP,\qquad
\left(\frac{\partial P}{\partial T}\right)_{V,N}=\frac\alpha{\kappa_T}.
$$
The ratio has units $\mathrm{Pa\,K^{-1}}$. For the one-<mole> <ideal gas> state in <isothermal compressibility>, it is $P_0/T_0=333.33\,\mathrm{Pa\,K^{-1}}$: a $1\,\mathrm K$ increase at fixed <volume (thermodynamics)> raises the <pressure> by about $333.33\,\mathrm{Pa}$. <Maxwell relations> identify this same ratio as the <entropy> slope with respect to <volume (thermodynamics)> and give $C_P-C_V=TV\alpha^2/\kappa_T$.
Under the <Boussinesq approximation>, the small fractional <mass density> change produces <buoyancy> while <mass density> is otherwise treated as constant. A positive <temperature> excess $\theta$ gives <reduced gravity> $g'=g\alpha_T\theta$ in the convention used in <natural ventilation>. The coefficient therefore converts a <heat> budget into a buoyancy budget. The same measurable expansion response supplies <entropy> changes and <heat capacity> differences through <Maxwell relations>, connecting fluid motion to equilibrium <thermodynamics>.
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