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 + 𝑝 𝑉 - 𝑇 𝑆

Applying ^Quasistatic

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

whence the equations of state

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

Proof of minimization

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

𝑑 𝑆 r e s = - đ 𝑄 s y s 𝑇 đ 𝑊 r e s = 𝑝 𝑑 𝑉 s y s = - đ 𝑊 s y s

Noting that since the chemical makeup is fixed $𝑑 𝐸 s y s = đ 𝑄 s y s + đ 𝑊 s y s$ by the First law of thermodynamics, it follows form 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 = - 𝑑 𝐸 s y s - đ 𝑊 s y s 𝑇 + 𝑑 𝑆 s y s = - 𝑑 𝐸 s y s + 𝑝 𝑑 𝑉 s y s 𝑇 + 𝑑 𝑆 s y s ⟹ 0 \leq - 𝑑 𝐸 s y s - 𝑝 𝑑 𝑉 s y s + 𝑇 𝑑 𝑆 s y s ⟹ 0 \geq 𝑑 (𝐸 s y s + 𝑝 𝑉 s y s - 𝑇 𝑆 s y s) = 𝑑 𝐺 s y s

hence $𝐺 s y s$ is minimized.

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