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# Thermodynamic potentials

In thermodynamics, four quantities, measured in units of energy, are called thermodynamic potentials:

 Internal energy U The energy needed to create a system Helmholtz free energy F = U - TS Also represented by A Enthalpy H = U + PV Gibbs free energy G = U + PV - TS

where T = temperature, S = entropy, P = pressure, V = volume

 Contents

## Differential definitions

The following differential relations hold for the four potentials:

 dU = TdS - PdV dF = - SdT - PdV dH = TdS + VdP dG = - SdT + VdP

If we write the above four equations generally as

$\left.\right.d\Phi=Adx+Bdy$

Then it is seen that

$A=\left(\frac{\partial \Phi}{\partial x}\right)_y$
$B=\left(\frac{\partial \Phi}{\partial y}\right)_x$

yielding expressions for T, P, S, and V in terms of derivatives of the potentials

$+T=\left(\frac{\partial U}{\partial S}\right)_V =\left(\frac{\partial H}{\partial S}\right)_P$
$-P=\left(\frac{\partial U}{\partial V}\right)_S =\left(\frac{\partial F}{\partial V}\right)_T$
$+V=\left(\frac{\partial H}{\partial P}\right)_S =\left(\frac{\partial G}{\partial P}\right)_T$
$-S=\left(\frac{\partial G}{\partial T}\right)_P =\left(\frac{\partial F}{\partial T}\right)_V$

Furthermore, mathematically we have

$\left(\frac{\partial}{\partial y} \left(\frac{\partial \Phi}{\partial x}\right)_y \right)_x = \left(\frac{\partial}{\partial x} \left(\frac{\partial \Phi}{\partial y}\right)_x \right)_y$

which gives:

$\left(\frac{\partial A}{\partial y}\right)_x = \left(\frac{\partial B}{\partial x}\right)_y$

which are known as Maxwell's relations

## Chemical reactions

Changes in these quantities are useful for assessing the degree to which a chemical reaction will proceed. The relevant quantity depends on the reaction conditions, as shown in the following table. Δ denotes the change in the potential and at equilibrium the change will be zero.

Constant V Constant P ΔU ΔH ΔF ΔG

Most commonly one considers reactions at constant P and T, so the Gibbs free energy is the most useful potential in studies of chemical reactions.