Maxwell relations

For electromagnetic equations, see Maxwell's equations.
Maxwell's relations are a set of equations in thermodynamics which are derivable from the definitions of the thermodynamic potentials. The Maxwell relations are statements of equality among the second derivatives of the thermodynamic potentials. They follow directly from the fact that the order of differentiation of an analytic function of two variables 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:
where the partial derivatives are taken with all other natural variables held constant. It is seen that for every thermodynamic potential there are possible Maxwell relations where n is the number of natural variables for that potential.
These relations are named for the nineteenthcentury physicist James Clerk Maxwell.
Contents
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 ):
where the potentials as functions of their natural thermal and mechanical variables are:
  The internal energy
  The Enthalpy
  The Helmholtz free energy
  The Gibbs free energy
The thermodynamic square can be used as a tool to recall and derive these relations.
Derivation of the Maxwell relations
Derivation of the Maxwell relations can be deduced from the differential forms of the thermodynamic potentials:
These equations resemble total differentials of the form
And indeed, it can be shown that for any equation of the form
that
Consider, as an example, the equation . We can now immediately see that
Since we also know that for functions with continuous second derivatives, the mixed partial derivatives are identical (Symmetry of second derivatives), that is, that
we therefore can see that
and therefore that
Each of the four Maxwell relationships given above follows similarly from one of the Gibbs equations
Extended derivation of the Maxwell relations
Maxwell relations are based on simple partial differentiation rules.
Combined form first and second law of thermodynamics,
 TdS = dU + PdV (Eq.1)
U, S, and V are state functions. Let,
 U = U(x,y)
 S = S(x,y)
 V = V(x,y)
Substitute them in Eq.1 and one gets,
And also written as,
comparing the coefficient of dx and dy, one gets
Differentiating above equations by y, x respectively
 (Eq.2)
 and
 (Eq.3)
U, S, and V are exact differentials, therefore,
Subtract eqn(2) and (3) and one gets
 Note: The above is called the general expression for Maxwell's thermodynamical relation.
 Maxwell's first relation
 Allow x = S and y = V and one gets
 Maxwell's second relation
 Allow x = T and y = V and one gets
 Maxwell's third relation
 Allow x = S and y = P and one gets
 Maxwell's fourth relation
 Allow x = T and y = P and one gets
 Maxwell's fifth relation
 Allow x = P and y = V
 = 1
 Maxwell's sixth relation
 Allow x = T and y = S and one gets
 = 1
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 singlecomponent 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:
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 reexpressed using the relationship
which are sometimes also known as Maxwell relations.
See also
 Table of thermodynamic equations
 Thermodynamic equations
 Thermodynamic potentials
External links
 http://theory.ph.man.ac.uk/~judith/stat_therm/node48.html a partial derivation of Maxwell's relations
Categories: Thermodynamics
 Fundamental physics concepts
 James Clerk Maxwell
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