Thermodynamic potential

Gibbs free energy

The Gibbs free energy 𝐺(𝑇,𝑝,⃗𝐍) is the Thermodynamic potential minimized by a system held at constant temperature and pressure. It may be thought of as measuring the work other than pressure-volume work obtainable from a closed thermodynamic system in an isothermal, isobaric process.

𝐺=𝐸+𝑝𝑉−𝑇𝑆

Applying ^Quasistatic

𝑑𝐺=−𝑆𝑑𝑇+𝑉𝑑𝑝+∑𝑖𝜇𝑖𝑑𝑁𝑖

whence the equations of state

𝜕𝐺𝜕𝑇=−𝑆𝜕𝐺𝜕𝑝=𝑉𝜕𝐺𝜕𝑁𝑖=𝜇𝑖

Proof of minimization

Consider a system whose volume is free in contact with a heat and pressure reservoir. Then

𝑑𝑆res=−đ𝑄sys𝑇đ𝑊res=𝑝𝑑𝑉sys=−đ𝑊sys

Noting that since the chemical makeup is fixed 𝑑𝐸sys =đ𝑄sys +đ𝑊sys by the First law of thermodynamics, it follows form the ^P4 that

0≤𝑑𝑆=𝑑𝑆res+𝑑𝑆sys=−đ𝑄sys𝑇+𝑑𝑆sys=−𝑑𝐸sys−đ𝑊sys𝑇+𝑑𝑆sys=−𝑑𝐸sys+𝑝𝑑𝑉sys𝑇+𝑑𝑆sys⟹0≤−𝑑𝐸sys−𝑝𝑑𝑉sys+𝑇𝑑𝑆sys⟹0≥𝑑(𝐸sys+𝑝𝑉sys−𝑇𝑆sys)=𝑑𝐺sys

hence 𝐺sys is minimized.

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