Helmholtz free energy

The Helmholtz free energy $𝐹 (𝑇, 𝑉, ⃗ 𝐍)$ is the Thermodynamic potential minimized by a system held at constant temperature and volume. It may be thought of as measuring the useful work obtainable from a closed thermodynamic system in an isothermal process.

𝐹 = 𝐸 - 𝑇 𝑆

Applying ^Quasistatic

𝑑 𝐹 = - 𝑆 𝑑 𝑇 - 𝑝 𝑑 𝑉 + \sum 𝑖 𝜇 𝑖 𝑑 𝑁 𝑖

whence the equations of state

𝜕 𝐹 𝜕 𝑇 = - 𝑆 𝜕 𝐹 𝜕 𝑉 = - 𝑝 𝜕 𝐹 𝜕 𝑁 𝑖 = 𝜇 𝑖

Proof of minimization

Consider a system of constant volume in thermal contact with a heat reservoir. Then

𝑑 𝑆 r e s = - đ 𝑄 s y s 𝑇

Noting that since the volume and chemical makeup are fixed $𝑑 𝐸 s y s = đ 𝑄 s y s$ by the First law of thermodynamics, it follows from the entropy maximum principle that

0 \leq 𝑑 𝑆 = 𝑑 𝑆 r e s + 𝑑 𝑆 s y s = - đ 𝑄 s y s 𝑇 + 𝑑 𝑆 s y s ⟹ 0 \leq - 𝑑 𝐸 s y s + 𝑑 (𝑇 𝑆 s y s) ⟹ 0 \geq 𝑑 (𝐸 s y s - 𝑇 𝑆 s y s) = 𝑑 𝐹 s y s

hence $𝐹 s y s$ is minimized.

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