Chemistry, Physics

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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