Thermodynamic Potentials

Date:: 28 July 2019

Dependency

What is the problem ?

The Key of Thermodynamics
The Essence of Enternal Energy, Helmholtz energy, Enthalpy, Gibbs energy

Derivation

Step 1 Fundamental Equation For Internal Energy

\[\begin{split}\begin{align*} dE & = TdS + \sum\limits_{i=1}^r F_i dx_i + \sum\limits_{j=1}^s \mu_j dN_j \\ & where\\ & T: \text{Temperature}\\ & S: \text{Entropy}\\ & F_i: \text{Generalized Forces such as Pressure} P\\ & x_i: \text{Generalized Coordinates such as Volume} V\\ & \mu_j: \text{Chemical Potentials}\\ & N_j: \text{Particles Numbers}\\ & and \\ & S,{x_i},{N_j} \text{ are Natural Variables of } E \end{align*}\end{split}\]

Step 2 Helmholtz energy, Enthalpy, Gibbs energy are simply Legendre Transformation of Internal Energy

\[\begin{split}\begin{align*} \text{Internal Energy} & E(S,{x_i},{N_j})\\ \text{Helmholtz energy} & F(T,{x_i},{N_j})\equiv E-TS \\ \text{Enthalpy} & H(S,{F_i},{N_j})\equiv E- \sum\limits_{i=1}^r F_i x_i\\ \text{Gibbs energy} & G(T,{F_i},{N_j})\equiv E-TS -\sum\limits_{i=1}^r F_i x_i\\ \end{align*}\end{split}\]