🥵Thermodynamics Unit 11 Review

A critical point is the specific thermodynamic state where the boundary between two phases disappears entirely. Beyond this point, you can't distinguish liquid from gas. Critical exponents then quantify how thermodynamic properties diverge or vanish as you approach this point, and scaling laws tie those exponents together in surprisingly universal ways.

Critical points in phase transitions

At most temperatures and pressures, liquids and gases are clearly different: they have different densities, different enthalpies, different entropies. But if you heat a liquid under pressure along the liquid-gas coexistence curve, you eventually reach a point where those differences shrink to zero. That's the critical point.

The critical point occurs at a unique combination of temperature $T_c$ $T_{c}$ , pressure $P_c$ $P_{c}$ , and molar volume $V_c$ $V_{c}$ for each substance.
- Water: $T_c = 647.096 \text{ K}$ , $P_c = 22.064 \text{ MPa}$
- Carbon dioxide: $T_c = 304.13 \text{ K}$ , $P_c = 7.38 \text{ MPa}$
Above the critical point, the substance exists as a supercritical fluid with no distinct liquid or gas phase.
At the critical point itself, density, enthalpy, and entropy of the liquid and gas phases converge to identical values.
The phase transition at the critical point is continuous (second-order): there is no latent heat, no abrupt jump in density. Properties change smoothly but with dramatic fluctuations.

On a P-V diagram, the critical point sits at the top of the coexistence dome. The critical isotherm has an inflection point there, meaning both $\left(\frac{\partial P}{\partial V}\right)_T = 0$ and $\left(\frac{\partial^2 P}{\partial V^2}\right)_T = 0$ at the critical point. This is why quantities like compressibility diverge: the system becomes infinitely "soft" to density fluctuations.

Critical points in phase transitions, Phase Changes | Physics

Critical exponents of thermodynamic quantities

As you approach the critical point, thermodynamic quantities don't just change; they diverge or vanish following power laws. The exponents in those power laws are the critical exponents.

The key variable is the reduced temperature:

$t = \frac{T - T_c}{T_c}$

This dimensionless quantity measures how far you are from the critical temperature. As $t \to 0$ , you're approaching the critical point.

Here are the standard critical exponents and what they describe:

Exponent	Quantity	Power-law behavior	Physical meaning
$\alpha$	Specific heat $C_v$	$C_v \propto \lvert t \rvert^{-\alpha}$	Heat capacity diverges near $T_c$
$\beta$	Order parameter (e.g., $\Delta\rho$ )	$\Delta\rho \propto \lvert t \rvert^{\beta}$	Density difference vanishes below $T_c$
$\gamma$	Isothermal compressibility $\kappa_T$	$\kappa_T \propto \lvert t \rvert^{-\gamma}$	Compressibility diverges near $T_c$
$\delta$	Critical isotherm	$\lvert P - P_c \rvert \propto \lvert \rho - \rho_c \rvert^{\delta}$	Pressure-density relation exactly at $T_c$
$\nu$	Correlation length $\xi$	$\xi \propto \lvert t \rvert^{-\nu}$	Fluctuation range diverges near $T_c$
$\eta$	Correlation function at $T_c$	$G(r) \propto r^{-(d-2+\eta)}$	How correlations decay with distance at $T_c$
A few things to notice. Exponents $\alpha$ , $\gamma$ , and $\nu$ appear with negative signs in the power law, meaning those quantities diverge as $t \to 0$ . The order parameter exponent $\beta$ is positive, meaning the order parameter vanishes as you approach $T_c$ from below. The exponent $\delta$ describes behavior exactly at $T_c$ (not as a function of $t$ ), so it plays a slightly different role.

For the 3D Ising universality class (which covers liquid-gas transitions), the approximate values are $\alpha \approx 0.11$ , $\beta \approx 0.33$ , $\gamma \approx 1.24$ , $\delta \approx 4.79$ , $\nu \approx 0.63$ , and $\eta \approx 0.04$ . These differ significantly from the mean-field values ( $\alpha = 0$ , $\beta = 0.5$ , $\gamma = 1$ , $\delta = 3$ , $\nu = 0.5$ , $\eta = 0$ ), which is why mean-field theory fails quantitatively near critical points.

Critical points in phase transitions, Interpreting Phase Diagrams | Introduction to Chemistry

Universality in critical phenomena

One of the most striking results in statistical physics is universality: systems that look completely different at the microscopic level can have identical critical exponents.

A liquid near its liquid-gas critical point and a ferromagnet near its Curie temperature have entirely different microscopic physics. Yet they share the same critical exponents. The reason is that near a critical point, the correlation length $\xi$ diverges, and the system's behavior becomes dominated by long-range collective fluctuations rather than short-range microscopic details.

What determines which universality class a system belongs to? Only three things matter:

Spatial dimensionality $d$ of the system (2D vs. 3D, for example)
Symmetry of the order parameter (scalar for liquid-gas, vector for magnets with continuous spin symmetry, etc.)
Range of interactions (short-range vs. long-range)

Everything else, the lattice structure, the exact form of the interaction potential, whether you're dealing with atoms or spins, is irrelevant to the critical exponents.

This is why studying simplified models is so powerful. The Ising model (discrete spins on a lattice with nearest-neighbor interactions) captures the critical behavior of real liquid-gas transitions in 3D, because both belong to the same universality class: 3D systems with a scalar order parameter and short-range interactions.

Scaling behavior near critical points

The critical exponents aren't all independent. Scaling laws are exact relations that connect them, reducing the number of independent exponents.

The four major scaling relations:

Rushbrooke inequality (an equality for systems obeying scaling): $\alpha + 2\beta + \gamma = 2$
Widom relation: $\gamma = \beta(\delta - 1)$
Josephson (hyperscaling) relation: $\nu d = 2 - \alpha$
Fisher relation: $\gamma = (2 - \eta)\nu$

You can verify these with the 3D Ising values above: $0.11 + 2(0.33) + 1.24 \approx 2.01$ , which is consistent with Rushbrooke within rounding.

These relations have a deep origin. Renormalization group (RG) theory, developed by Kenneth Wilson in the 1970s, provides the theoretical framework. The RG approach shows that near a critical point, the system looks statistically the same at all length scales (self-similarity), and this scale invariance is what produces the power-law behavior and constrains the exponents.

A practical consequence: if you measure any two independent critical exponents experimentally, the scaling laws predict all the others. This gives you a powerful consistency check on experimental data and a way to classify systems into universality classes.

Note that the Josephson relation is the only one involving the spatial dimension $d$ explicitly. It holds as an equality only below the upper critical dimension ( $d = 4$ for the Ising class). Above $d = 4$ , mean-field exponents take over and hyperscaling breaks down.

🥵Thermodynamics Unit 11 Review