Thermodynamic potential

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−𝑇𝑆

Applying ^Quasistatic

𝑑𝐹=−𝑆𝑑𝑇−𝑝𝑑𝑉+∑𝑖𝜇𝑖𝑑𝑁𝑖

whence the equations of state

𝜕𝐹𝜕𝑇=−𝑆𝜕𝐹𝜕𝑉=−𝑝𝜕𝐹𝜕𝑁𝑖=𝜇𝑖

Proof of minimization

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

𝑑𝑆res=−đ𝑄sys𝑇

Noting that since the volume and chemical makeup are fixed 𝑑𝐸sys =đ𝑄sys by the First law of thermodynamics, it follows from the ^P4 that

0≤𝑑𝑆=𝑑𝑆res+𝑑𝑆sys=−đ𝑄sys𝑇+𝑑𝑆sys⟹0≤−𝑑𝐸sys+𝑑(𝑇𝑆sys)⟹0≥𝑑(𝐸sys−𝑇𝑆sys)=𝑑𝐹sys

hence 𝐹sys is minimized.

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