Thermodynamic Potentials

Thermodynamic Potentials

A thermodynamic potential is a scalar, extensive quantity that can be used to represnet the thermodynamic state of a system. The first thermodynamic potential you are typically introduced to is the internal energy and from the first and second laws of thermodynamics we know that, in a system that can only do PV work, infinitessimal changes in internal energy can be calculated using: $$ \textrm{d}E = T\textrm{d}S - P\textrm{d}V $$ New themodynamic potentials are introduced by considering a system in contact with a reservoir. Some important thermodynamic potentials include: $$ \begin{aligned} \textrm{Enthalpy} & \qquad H = E + PV \\ \textrm{Helmholts free energy} & \qquad F = E - TS \\ \textrm{Gibbs free energy} & \qquad G = E + PV - TS \\ \textrm{Grand potential} & \qquad \Omega = E - TS - \sum_i \mu_i N_i \end{aligned} $$ The differentials of these quantities are: $$ \begin{aligned} \textrm{Enthalpy} & \qquad \textrm{d}H = T\textrm{d}S + V\textrm{d}P \\ \textrm{Helmholts free energy} & \qquad \textrm{d}F = -P\textrm{d}V - S\textrm{d}T \\ \textrm{Gibbs free energy} & \qquad \textrm{d}G = V\textrm{d}P - S\textrm{d}T \\ \textrm{Grand potential} & \qquad \textrm{d}\Omega = -P\textrm{d}V - S\textrm{d}T - \sum_i N_i \textrm{d}\mu_i \end{aligned} $$ It is possible to use these results, together with the fact that all thermodynamic potentials are exact differentials, to relate all the thermodynamic variables to partial derivatives of thermodynamic potentials.

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