Maxwell relations

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Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. The Maxwell relations are statments of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of differentiation in a second derivative is irrelevant. If Φ is a thermodynamic potential and $x i$ and $x j$ are two different natural variables for that potential, then the Maxwell relation for that potential and those variables is:

$\frac{\partial }{\partial x_j}\left(\frac{\partial \Phi}{\partial x_i}\right)= \frac{\partial }{\partial x_i}\left(\frac{\partial \Phi}{\partial x_j}\right)$

where the partial derivatives are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are n(n-1)/2 possible Maxwell relations where n is the number of natural variables for that potential.

[edit] The four most common Maxwell relations

The four most common Maxwell relations are the equalities of the second derivatives of each of the four thermodynamic potentials, with respect to their thermal natural variable (temperature T or entropy S ) and their mechanical natural variable (pressure P or volume V ):

$\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V\qquad= \frac{\partial^2 U }{\partial S \partial V}$

$\left(\frac{\partial T}{\partial P}\right)_S = +\left(\frac{\partial V}{\partial S}\right)_P\qquad= \frac{\partial^2 H }{\partial S \partial P}$

$\left(\frac{\partial S}{\partial V}\right)_T = +\left(\frac{\partial P}{\partial T}\right)_V\qquad= \frac{\partial^2 A }{\partial T \partial V}$

$\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P\qquad= \frac{\partial^2 G }{\partial T \partial P}$

where the potentials as functions of their natural thermal and mechanical variables are:

$U(S,V)\,$ - The internal energy

$H(S,P)\,$ - The Enthalpy

$A(T,V)\,$ - The Helmholtz free energy

$G(T,P)\,$ - The Gibbs free energy

[edit] General Maxwell relationships

The above are by no means 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:

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

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

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