# Maxwell’s Relations

[latexpage]

### 5 Common Thermodynamic Potentials

1. Internal energy, $U =$ the capacity to do work and the capacity to release heat
• Natural variables: $S,V, \{ N_i \}$
• $S$ = entropy of the system
• $V$ = volume of the system
• $N_i$ = number of particles of type $i$ in the system
• Formula: $U = \int {(Tds - PdV + \sum _{i}{\mu_i dN_i}})$  where $\mu _i =$ the chemical potential of $i$
2. Helmholz free energy, $A =$ the capacity to do mechanical and non-mechanical work
• Natural variables: $T,V,\{ N_i \}$
• $T =$ temperature of the system
• Formula: $A = U - TS$
3. Enthalpy, $H =$ the capacity to do non-mechanical work and the capacity to release heat
• Natural variables: $S,P, \{ N_i \}$
• $P = pressure$
• Formula: $H = U + PV$
4. Gibbs free energy, $G =$ the capacity to do non-mechanical work
• Natural variables: $T,P, \{ N_i \}$
• Formula: $G = H - TS = U + PV -TS$
5. Landau (or grand) potential, $\Omega =$ the characteristic state function for the  grand canonical ensemble (i.e., the statistical ensemble used to represent the possible states of a mechanical system of particles maintained in thermodynamic equilibrium with a reservoir)
• Natural varaiables: $T,V, \{ \mu_i \}$
• Formula: $\Omega = U - TS - \sum_{i}{\mu_i N_i}$

### Four Most Common Relations

Note: $P =$ pressure ($\mathrm{Pa = J/m^3}$), $V =$ volume ($\mathrm{m^3}$), $S =$ entropy ($\mathrm{J/K}$), $T =$ temperature ($\mathrm{K}$)

1. $U_{SV} = (\frac{\partial T}{\partial V})_S= - (\frac{\partial P}{\partial S})_V = \frac{\partial ^2 U}{\partial S \partial V}$
2. $H_{SP} = (\frac{\partial T}{\partial P})_S= - (\frac{\partial V}{\partial S})_P = \frac{\partial ^2 H}{\partial S \partial P}$
3. $A_{TV} = (\frac{\partial S}{\partial V})_T= - (\frac{\partial P}{\partial T})_V = \frac{\partial ^2 A}{\partial T \partial V}$
4. $G_{TP} = - (\frac{\partial S}{\partial P})_T= - (\frac{\partial V}{\partial T})_P = \frac{\partial ^2 G}{\partial T \partial P}$

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