Statistical Thermodynamics and Phase Equilibria Review

Introduction

This unit built a bridge from molecules to measurable properties. Ensembles gave the rules for counting states under different constraints; partition functions turned those counts into energies, entropies and chemical potentials; heat capacities revealed which molecular motions are active at each temperature; and phase equilibria followed from the requirement that chemical potentials match. This review draws the threads together so you can move confidently between the microscopic and macroscopic descriptions.

Core explanation

Ensembles and their potentials. An ensemble is a collection of imagined copies of a system sharing fixed constraints. In the microcanonical ensemble (fixed N, V, E) all accessible microstates are equally likely, and entropy is S = k ln Ω. In the canonical ensemble (fixed N, V, T) the system exchanges energy with a heat bath; the partition function Q = Σ exp(−Eᵢ/kT) gives A = −kT ln Q. In the grand canonical ensemble (fixed μ, V, T) particles are exchanged too, and the grand partition function Ξ gives pV = kT ln Ξ. Each ensemble pairs naturally with one thermodynamic potential, and the potentials are linked by Legendre transforms such as A = U − TS.

Ensemble equivalence. For macroscopic systems, relative fluctuations scale roughly as 1/√N, which is about 10⁻¹² for a mole. Average properties are therefore the same whichever ensemble is used, and we choose the one that is most convenient. Equivalence fails near phase transitions, where fluctuations become large.

From molecules to partition functions. For an ideal gas the molecular partition function factorises: q = q trans q rot q vib q elec, and Q = qᴺ/N! for indistinguishable molecules. Spectroscopic data supply the energy levels: bond lengths give rotational constants, vibrational wavenumbers give vibrational spacings, and electronic term energies give the electronic contribution.

Heat capacities. Each mode contributes to C V according to how its level spacing compares with kT. Translation contributes 3/2 R per mole at all ordinary temperatures. Rotation of a linear molecule adds R once T greatly exceeds the rotational temperature, which is only a few kelvin for most molecules. Vibration contributes up to R per mode but is largely frozen at room temperature for stiff bonds, because vibrational temperatures are often above 1000 K. Solids follow the Einstein and Debye models: the Debye model predicts C V ∝ T³ at low temperature and approaches the Dulong–Petit value of 3R at high temperature. The fluctuation formula C V = ⟨(ΔE)²⟩/(kT²) shows that heat capacity measures the spread of energies in the canonical ensemble.

Chemical potential and equilibrium. Statistically, μ = −kT ln(q/N) for an ideal gas, making clear that chemical potential reflects how many states are available per particle. Reaction equilibrium follows from Σνᵢμᵢ = 0, giving K in terms of molecular partition functions. Phase coexistence follows from μ(α) = μ(β), which leads to the Clapeyron equation, phase diagrams and the phase rule.

Mixtures and phase separation. The regular-solution model combines ideal mixing entropy with an interaction parameter χ. When χ exceeds 2, the Gibbs energy of mixing develops a double minimum and the mixture splits into two phases. Inside the spinodal the curvature is negative and separation is spontaneous; between spinodal and binodal the mixture is metastable and needs nucleation.

Step-by-step reasoning

To predict a property from molecular data:

1. Choose the ensemble matching the experimental constraints. 2. Build the partition function from the energy levels. 3. Obtain the potential (for example A = −kT ln Q). 4. Differentiate to find U, S, p, μ or C V. 5. Apply equality of chemical potentials to find equilibria and phase boundaries.

Visual explanation

Picture a flow chart: energy levels feed into a partition function box; arrows lead from it to the thermodynamic potential, then fan out to heat capacity, pressure and chemical potential. The chemical-potential arrow ends at a phase diagram, where crossing curves of μ against T for solid, liquid and gas mark the transitions.

Real-world analogy

Statistical thermodynamics works like a census. Counting individual households (microstates) with the right weighting produces the national statistics (thermodynamic properties). You do not need to follow each household; the averages are reliable because the population is so large.

Real-world example

Cryogenic engineers rely on heat-capacity data to design cooling systems for superconducting magnets. At a few kelvin, the heat capacities of metals are tiny because lattice vibrations are frozen out (the Debye T³ law) and only the electrons contribute, so even small heat leaks cause large temperature rises.

Why?

Why can one function, the partition function, predict so much? Because it contains the complete list of accessible energy states weighted by their Boltzmann probabilities. Every thermodynamic quantity is an average over that distribution or a measure of its spread, so each can be obtained by differentiating ln Q.

Common misconception

"Heat capacity is simply a fixed property of a substance." In fact, heat capacity varies strongly with temperature, because different molecular motions become active as kT grows to match their level spacings; hydrogen gas, for example, shows clear steps as rotation and then vibration switch on.

Worked example

Question: Estimate the molar C V of gaseous N₂ at 298 K, given a rotational temperature of about 2.9 K and a vibrational temperature of about 3400 K.

Reasoning: Translation contributes 3/2 R. Since 298 K ≫ 2.9 K, rotation is fully active and contributes R. Since 298 K ≪ 3400 K, vibration is almost entirely frozen.

Answer: C V ≈ 5/2 R ≈ 20.8 J K⁻¹ mol⁻¹, in good agreement with measurement.

Quick check

1. Which ensemble has the Helmholtz energy as its natural potential, and what is held fixed? Answer: The canonical ensemble, with the number of particles, the volume and the temperature held fixed.

Exam focus

Be able to match each ensemble to its fixed variables and potential, estimate heat capacities by deciding which modes are active, and derive phase-equilibrium conditions from equal chemical potentials. Always justify "active" or "frozen" by comparing T with the characteristic temperature.

Advanced insight

Phase transitions reveal the limits of simple ensemble reasoning. The heat capacity of a first-order transition shows a delta-function spike from latent heat, while continuous transitions show divergences or cusps linked to growing fluctuations. Near a critical point the fluctuation formula implies very large energy fluctuations, so ensemble equivalence and mean-field models both break down, and renormalisation-group theory is needed.

Summary

Ensembles fix the constraints, partition functions count the states, and derivatives of ln Q give every thermodynamic property. Heat capacities report which motions are thermally accessible and measure energy fluctuations. Chemical potentials link molecules to equilibrium: equal μ across phases defines phase diagrams, while interaction parameters, curvature of the Gibbs energy and nucleation barriers explain phase separation and metastability.

Practice questions

1. Write the expression for entropy in the microcanonical ensemble and state what Ω represents. Answer: S = k ln Ω, where Ω is the number of microstates consistent with the fixed N, V and E. 2. Why is the vibrational contribution to C V of H₂ negligible at room temperature? Answer: Its vibrational temperature is roughly 6000 K, far above 298 K, so almost all molecules are in the ground vibrational state and energy cannot be absorbed by vibration. 3. State the Debye prediction for the heat capacity of a crystal at low and high temperature. Answer: C V is proportional to T³ at low temperature and approaches the Dulong–Petit value of 3R per mole of atoms at high temperature. 4. In the regular-solution model, what happens when χ exceeds 2, and why? Answer: The Gibbs energy of mixing develops two minima, so the homogeneous mixture becomes unstable at some compositions and separates into two liquid phases, because the unfavourable interaction energy outweighs the mixing entropy.