Thermodynamic Potentials
=================================

:Date: 28 July 2019

Dependency
-------------

:doc:`Legendre-Transformation`


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

.. math::
	\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*}

**Step 2** Helmholtz energy, Enthalpy, Gibbs energy are **simply** :doc:`Legendre-Transformation` of Internal Energy

.. math::
	\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*}