Maxwell Relations - General Maxwell Relationships

General Maxwell Relationships

The above are not the only Maxwell relationships. When other work terms involving other natural variables besides the volume work are considered or when the number of particles is included as a natural variable, other Maxwell relations become apparent. For example, if we have a single-component gas, then the number of particles N is also a natural variable of the above four thermodynamic potentials. The Maxwell relationship for the enthalpy with respect to pressure and particle number would then be:

$\left(\frac{\partial \mu}{\partial P}\right)_{S, N} = \left(\frac{\partial V}{\partial N}\right)_{S, P}\qquad= \frac{\partial^2 H }{\partial P \partial N}$

where μ is the chemical potential. In addition, there are other thermodynamic potentials besides the four that are commonly used, and each of these potentials will yield a set of Maxwell relations.

Each equation can be re-expressed using the relationship

$\left(\frac{\partial y}{\partial x}\right)_z = 1\left/\left(\frac{\partial x}{\partial y}\right)_z\right.$

which are sometimes also known as Maxwell relations.

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