Enthalpy

Enthalpy $𝐻 (𝑆, 𝑝, ⃗ 𝐍)$ is the Thermodynamic potential minimized by a system held at constant pressure. It may be thought of as measuring the heat obtainable from a closed thermodynamic system in an isobaric process.

𝐻 = 𝐸 + 𝑝 𝑉

applying ^Quasistatic

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

whence the equations of state

𝜕 𝐻 𝜕 𝑆 = 𝑇 𝜕 𝐻 𝜕 𝑝 = 𝑉 𝜕 𝐻 𝜕 𝑁 𝑖 = 𝜇 𝑖

Proof of minimization

Consider a system whose volume is free in contact with a pressure reservoir. Then
$0 \leq 𝑑 𝑆 = 𝑑 𝑆 r e s + 𝑑 𝑆 s y s = đ 𝑄 r e s 𝑇 r e s + 𝑑 𝑆 s y s = 𝑑 𝐸 r e s - đ 𝑊 r e s 𝑇 r e s + 𝑑 𝑆 s y s = - 𝑑 𝐸 s y s - 𝑝 𝑑 𝑉 s y s 𝑇 r e s + 𝑑 𝑆 s y s$

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Enthalpy

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