Key Concepts of Thermodynamic Potentials

Why This Matters

Thermodynamic potentials are the bridge between the microscopic chaos of particle motion and the macroscopic properties you can actually measure in a lab. You're being tested on your ability to choose the right potential for the right constraints—whether a system is held at constant temperature, pressure, volume, or particle number completely changes which potential minimizes at equilibrium. Master this, and you'll understand why reactions proceed, when phases transform, and how energy flows through any system.

These potentials aren't just abstract math—they're Legendre transforms of each other, trading one natural variable for its conjugate. This means you need to know which variables are held fixed, which potential applies, and how the partition function connects to each one. Don't just memorize definitions; know what physical situation each potential describes and why minimizing it tells you about spontaneity and equilibrium.

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Potentials at Fixed Temperature and Volume

When you control temperature and volume, the Helmholtz free energy becomes your workhorse. The system exchanges heat with a reservoir but cannot expand or contract.

Internal Energy (U)

• Total microscopic energy—the sum of all kinetic and potential energies of particles in the system, serving as the foundation for all other potentials
• First law connection: $dU = \delta Q - \delta W$, meaning changes come from heat added or work done on the system
• Statistical mechanics link: calculated from the partition function via $U = -\frac{\partial \ln Z}{\partial \beta}$ where $\beta = 1/k_BT$

Helmholtz Free Energy (F)

• Definition: $F = U - TS$, representing the maximum useful work extractable at constant T and V
• Spontaneity criterion: $dF < 0$ for spontaneous processes when temperature and volume are fixed
• Partition function bridge: $F = -k_BT \ln Z$, making it the most direct connection between statistical mechanics and thermodynamics

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Potentials at Fixed Temperature and Pressure

Most real experiments—chemistry in open beakers, biological systems, atmospheric processes—occur at constant pressure. Here, the system can exchange both heat and do expansion work against the atmosphere.

Enthalpy (H)

• Definition: $H = U + PV$, accounting for internal energy plus the "pressure-volume work" needed to establish the system
• Constant pressure heat flow: $\Delta H = Q_P$, making it the go-to quantity for calorimetry experiments
• Thermodynamic cycles: essential for analyzing engines, refrigerators, and any process where pressure stays fixed during heat transfer

Gibbs Free Energy (G)

• Definition: $G = H - TS = U + PV - TS$, representing maximum non-expansion work at constant T and P
• Chemical equilibrium: $\Delta G = 0$ at equilibrium; $\Delta G < 0$ indicates spontaneous reactions under lab conditions
• Phase transitions: determines which phase is stable—the phase with lowest G wins at any given T and P

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Potentials for Open Systems

When particles can enter or leave your system, you need potentials that account for chemical potential $\mu$. The grand canonical ensemble describes systems exchanging both energy and particles with a reservoir.

Grand Potential (Ω)

• Definition: $\Omega = F - \mu N = -k_BT \ln \Xi$, where $\Xi$ is the grand canonical partition function
• Open system criterion: $d\Omega < 0$ for spontaneous processes at constant T, V, and chemical potential $\mu$
• Fluctuating particle number: essential for gases in contact with reservoirs, adsorption phenomena, and quantum systems where particle number isn't conserved

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Quick Reference Table

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Self-Check Questions

1. Which two potentials both apply to systems at constant temperature, and what distinguishes when you'd use each one?

2. You're analyzing a chemical reaction in an open beaker at room temperature. Which thermodynamic potential determines spontaneity, and why would using Helmholtz free energy give you the wrong answer?

3. Compare and contrast the Helmholtz free energy and the grand potential: what natural variables does each use, and how does this connect to whether particle number is fixed?

4. If you're given a partition function $Z$, write the expressions for both internal energy $U$ and Helmholtz free energy $F$. Why is $F$ often more useful in statistical mechanics calculations?

5. An FRQ describes a gas in thermal equilibrium with a heat bath, confined to a rigid container, but able to exchange particles through a semipermeable membrane. Which potential should you minimize to find equilibrium, and what variables are held constant?