First-Order and Continuous Phase Transitions

Introduction

Not all phase changes resemble boiling. Some have a coexistence interval and latent heat; others change their structure continuously while heat capacity or susceptibility becomes anomalous. Classifying a transition requires attention to the thermodynamic limit and to which derivatives of free energy change. Order parameters provide a physical description of the new phase, while response functions show how strongly it reacts to small perturbations.

Core explanation

For a pure substance at fixed T and p, Gibbs energy G is continuous across an equilibrium first-order transition, because phase chemical potentials are equal at coexistence. Its first derivatives can jump: S = −(∂G/∂T) p and V = (∂G/∂p) T differ between phases. A nonzero entropy jump gives latent heat ΔH = TΔS for a reversible transition. Boiling and ordinary melting are familiar first-order examples. Their coexistence curve has a Clapeyron slope determined by ΔS/ΔV.

A continuous transition has no latent heat at its transition point in the ideal thermodynamic-limit classification, and the relevant order parameter changes continuously from zero or one phase value. First derivatives of the appropriate free energy are continuous in a simple second-order picture, while second derivatives such as heat capacity, compressibility or magnetic susceptibility may jump, diverge or show more complicated singular behaviour. Modern critical phenomena need not fit a neat integer-order Ehrenfest category, so it is safer to describe measured derivative behaviour than to apply a label mechanically.

An order parameter captures a broken symmetry or structural distinction. Magnetisation can serve for a ferromagnetic ordering transition in zero applied field; density difference between liquid and vapour tends to zero at their critical endpoint. At a first-order transition, an order parameter may jump. At a continuous transition, it approaches zero smoothly while fluctuations and correlation length can grow strongly.

Finite systems round transitions. A finite simulation or nanoparticle has a smooth partition function rather than a true nonanalytic thermodynamic singularity under ordinary conditions. A sharp peak in heat capacity may indicate an impending bulk transition or simply a finite collection of levels, such as a Schottky anomaly. Distinguishing them requires size scaling, latent heat, hysteresis and structural evidence rather than peak shape alone.

Step-by-step reasoning

Identify the potential and constraints, then ask whether two phases coexist with a jump in entropy or volume. Check for latent heat ΔH = TΔS. Identify a physically meaningful order parameter and whether it changes abruptly or continuously. Inspect response functions and finite-size effects. State which observations support a classification and which remain ambiguous.

Visual explanation

Draw G against T for two phases as curves crossing with different slopes at a first-order transition; their lower envelope has a kink. Beneath, plot entropy with a jump. Next draw a continuous-transition order parameter smoothly approaching zero while a heat-capacity curve peaks or diverges near the critical temperature.

Real-world analogy

A light switch abruptly flips between off and on, while a dimmer can change brightness smoothly. A first-order order parameter may jump like a switch; a continuous one may vary like a dimmer. The analogy is limited because thermodynamic transitions arise from collective molecular behaviour, and finite systems can blur an apparent switch.

Real-world example

Water boiling below its critical temperature absorbs latent heat while liquid and vapour coexist at the saturation pressure. At the liquid–vapour critical point, their density difference vanishes and the boundary terminates. The distinction helps interpret why boiling has a phase interface below the critical point but not a liquid–vapour coexistence line above it.

Why?

Phase transitions occur when the dominant equilibrium state changes as a control variable changes. For first-order transitions, two distinct minima can exchange stability, yielding a slope discontinuity in free energy and a latent heat. For continuous transitions, one minimum can change character smoothly while collective fluctuations make response derivatives unusually large.

Common misconception

Every heat-capacity peak is not proof of a phase transition; independent two-level excitations can make broad peaks. Also, “continuous” does not mean every thermodynamic derivative is smooth or finite. A critical response may diverge in the infinite-system limit while the order parameter itself approaches zero continuously.

Worked example

At a first-order transition temperature T t = 300 K, suppose molar entropy changes by ΔS m = +40 J mol⁻¹ K⁻¹ on moving from α to β. The reversible latent enthalpy is ΔH m = T tΔS m = 300 × 40 = 12,000 J mol⁻¹ = 12.0 kJ mol⁻¹. Since ΔS m is nonzero, the G-versus-T slopes differ by −40 J mol⁻¹ K⁻¹ at coexistence. A truly continuous transition at the same T would not have this finite latent heat.

Quick check

1. Why can two first-order coexisting phases have equal G m but different S m? Answer: Equality of G m is the coexistence condition. Entropy is the negative temperature derivative of G m, and the two phase curves can cross with different slopes, giving a finite entropy jump and latent heat.

Exam focus

Specify the thermodynamic limit when discussing true singularities. Use first derivatives of G for entropy and volume under T,p control. Connect latent heat to TΔS, not to a jump in G itself. Name an order parameter and distinguish a broad two-level peak from evidence of collective transition.

Advanced insight

Landau free-energy models describe transition order by how minima in an order-parameter landscape change. Their mean-field predictions can be altered by fluctuations near a critical point. Finite-size scaling of response peaks helps infer the underlying bulk behaviour from simulations or small samples.

Summary

First-order transitions have coexisting phases with equal free energy but discontinuous entropy or volume and often latent heat. Continuous transitions lack a finite latent heat while order parameters and response functions can show critical behaviour. Finite systems smooth singularities, so classification needs more than one observed peak.

Practice questions

1. A transition has ΔH m = 6.0 kJ mol⁻¹ at 250 K. What is ΔS m for the stated direction? Answer: ΔS m = ΔH m/T = 6000/250 = 24 J mol⁻¹ K⁻¹, indicating a first-order entropy change under the stated reversible transition. 2. Why does a finite simulation's heat-capacity peak not by itself prove a bulk critical point? Answer: Finite partition functions give rounded responses, and discrete-level Schottky effects can also make peaks. Size dependence and structural or fluctuation evidence are needed. 3. Give one possible order parameter for liquid–vapour criticality. Answer: The difference between coexisting liquid and vapour densities. It decreases toward zero as the critical endpoint is approached from below.