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.

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.

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).

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.

• Confusing first-order and second-order transitions: First-order transitions have discontinuous first derivatives of $G$, meaning latent heat and volume change. Second-order transitions have continuous first derivatives but discontinuous second derivatives (diverging susceptibility).
• Assuming mean-field exponents are exact: The van der Waals equation predicts $\beta = 1/2$, but real fluids and the 3D Ising model give $\beta \approx 0.326$. Mean-field theory breaks down near $T_c$ due to fluctuations.
• Forgetting that the Clausius-Clapeyron equation only applies to first-order transitions: At a second-order transition, $\Delta S = 0$ and $\Delta v = 0$, so the equation is undefined (0/0), and the Ehrenfest equations must be used instead.
• Misidentifying the order parameter: The choice of order parameter depends on the system. For the liquid-gas transition, it is the density difference; for a magnet, it is the magnetisation; for a superfluid, it is the macroscopic wavefunction amplitude.

Near a second-order transition, the free energy can be expanded as a power series in the order parameter $\phi$:

$F(T, \phi) = F_0(T) + a(T)\phi^2 + b(T)\phi^4 + \cdots$

For $T > T_c$, $a(T) > 0$ and the minimum is at $\phi = 0$. For $T < T_c$, $a(T) < 0$ and the minima are at $\phi = \pm \sqrt{-a/(2b)}$. The simplest choice is $a(T) = a_0(T - T_c)$ with $a_0 > 0$, giving:

$\phi = \pm \sqrt{\frac{a_0}{2b}(T_c - T)^{1/2}} \propto (T_c - T)^\beta$

with $\beta = 1/2$, recovering the mean-field result. The free energy at the minimum is:

$F(T) = F_0(T) - \frac{a_0^2}{4b}(T_c - T)^2$

The heat capacity jumps at $T_c$: $\Delta C = \frac{a_0^2 T_c}{2b}$, which is the mean-field prediction $\alpha = 0$.

The Clausius-Clapeyron equation can be integrated if $L$ and $\Delta v$ are treated as approximately constant over a small temperature range:

$P(T) = P_0 + \frac{L}{\Delta v}\ln\left(\frac{T}{T_0}\right)$

For water near $100^\circ$C, using $L = 2.26 \times 10^6$ J/kg, $\Delta v = 1.672$ m$^3$/kg, $T_0 = 373.15$ K, $P_0 = 1.013 \times 10^5$ Pa:

$P(T) \approx 1.013 \times 10^5 + 1.35 \times 10^6 \cdot \ln\left(\frac{T}{373.15}\right) \text{ Pa}$

At $T = 374.15$ K (1 K above boiling), $P \approx 1.013 \times 10^5 + 1.35 \times 10^6 \cdot \ln(1.00268) \approx 1.049 \times 10^5$ Pa, or about 1.036 atm — consistent with the linear approximation of 0.036 atm/K.

• Clausius-Clapeyron links latent heat to the coexistence curve slope: A larger latent heat $L$ or smaller volume change $\Delta v$ produces a steeper $dP/dT$, meaning the boiling point is more sensitive to pressure.
• Critical exponents satisfy scaling relations: $\alpha + 2\beta + \gamma = 2$ (Rushbrooke), $\gamma = \beta(\delta - 1)$ (Widom), and $\gamma = \nu(2 - \eta)$ (Fisher) connect the four exponents, reducing independent parameters.
• The order parameter distinguishes phases: Below $T_c$, $\phi \neq 0$ (ordered phase); above $T_c$, $\phi = 0$ (disordered phase). The continuity or discontinuity of $\phi$ classifies the transition order.
• The van der Waals equation predicts mean-field exponents: Real systems often show different exponents (e.g., 3D Ising $\beta \approx 0.326$ vs mean-field $\beta = 1/2$) due to fluctuations near $T_c$.
• Latent heat vanishes at the critical point: As $T \to T_c$, $\Delta S \to 0$ and the distinction between phases disappears, creating a continuous transition.

• Weather and climate: Phase transitions of water (evaporation, condensation, freezing) drive weather patterns and are central to climate models.
• Materials science: Understanding solid-liquid phase transitions controls casting, welding, and crystal growth processes.
• Superconductivity: The superconducting transition is a second-order phase transition characterised by the vanishing of electrical resistance and expulsion of magnetic fields.
• Cryogenics: The lambda transition in helium-4 determines the properties of superfluid helium used in low-temperature experiments.
• Food industry: Controlling phase transitions (freezing, melting, crystallisation) determines texture and shelf life of food products.