Phase Transitions

A phase transition is a discontinuity in a thermodynamic quantity or its derivative as a function of a state variable. Phase transitions are classified by Ehrenfest according to which derivative of the Gibbs free energy is discontinuous.

10.1 Classification of Phase Transitions

For a first-order transition at temperature $T_c$The latent heat is:

$L = T_c \Delta S = T_c \left(S_{\text{phase} 2} - S_{\text{phase} 1}\right)$

The Clausius—Clapeyron equation governs the slope of the coexistence curve:

$\frac{dP}{dT} = \frac{L}{T_c \Delta v}$

Where $\Delta v = v_2 - v_1$ is the change in specific volume.

10.2 Van der Waals Equation and Critical Phenomena

The van der Waals equation of state modifies the ideal gas law to account for intermolecular forces:

$\left(P + \frac{a}{v^2}\right)(v - b) = k_B T$

Where $a$ accounts for attractive interactions and $b$ for the finite molecular volume. The critical point $(T_c, P_c, v_c)$ satisfies:

$\frac{\partial P}{\partial v}\bigg|_{T_c} = 0, \qquad \frac{\partial^2 P}{\partial v^2}\bigg|_{T_c} = 0$

Solving gives:

$T_c = \frac{8a}{27bk_B}, \qquad P_c = \frac{a}{27b^2}, \qquad v_c = 3b$

Near the critical point, define the reduced variables $\tilde{T} = T/T_c$, $\tilde{P} = P/P_c$, $\tilde{v} = v/v_c$ to obtain the universal form:

$\left(\tilde{P} + \frac{3}{\tilde{v}^2}\right)(3\tilde{v} - 1) = 8\tilde{T}$

The order parameter $\phi = (v_{\text{gas} - v_{\text{liquid})/(v_c)}}$ vanishes as:

$\phi \propto (T_c - T)^{\beta}$

Where $\beta = 1/2$ is the mean-field critical exponent (van der Waals prediction).

10.3 Critical Exponents

Near a second-order phase transition, thermodynamic quantities follow power laws characterized by critical exponents:

Where $t = (T - T_c)/T_c$ is the reduced temperature and $h$ is the conjugate field.