Maxwell relationsMaxwell's relations are a set of

equations in thermodynamics which are

derivable from the symmetry of second

derivatives and from the definitions of

the thermodynamic potentials. ese

relations are named for the nineteenth-

century physicist James Clerk Maxwell.

Equations

The four most commonMaxwell relations

Derivation

Derivation based on Jacobians

General Maxwell relationships

See also

e structure of Maxwell relations is a

statement of equality among the second

derivatives for continuous functions. It

follows directly from the fact that the

order of differentiation of an analytic function of two variables is irrelevant (Schwarz theorem). In the case of Maxwell

relations the function considered is a thermodynamic potential and xi and xj are two different natural variables for that

potential:

Schwarz' theorem (general)

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. e substantial increase in the entropy will be verified according to the relations satisfied by the laws of

thermodynamics

Flow chart showing the paths between the Maxwell relations. P:pressure, T: temperature, V: volume, S: entropy, α: coefficient ofthermal expansion, κ: compressibility, CV: heat capacity atconstant volume, CP: heat capacity at constant pressure.

Contents

Equations

e 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):

Maxwell's relations (common)

where the potentials as functions of their natural thermal and mechanical variables are the internal energy U(S, V),

enthalpy H(S, P), Helmholtz free energy F(T, V) and Gibbs free energy G(T, P). e thermodynamic square can be used as

a mnemonic to recall and derive these relations. e usefulness of these relations lies in their quantifying entropy

changes, which are not directly measurable, in terms of measurable quantities like temperature, volume, and pressure.

Maxwell relations are based on simple partial differentiation rules, in particular the total differential of a function and

the symmetry of evaluating second order partial derivatives.

DerivationDerivation of the Maxwell relation can be deduced from thedifferential forms of the thermodynamic potentials:The differential form of internal energy U is

This equation resembles total differentials of the form

It can be shown that for any equation of the form

that

Consider, the equation . We can now immediately see that

e four most common Maxwell relations

Derivation

Since we also know that for functions with continuous second derivatives, themixed partial derivatives are identical (Symmetry of second derivatives), thatis, that

we therefore can see that

and therefore that

Derivation of Maxwell Relation from Helmholtz Free energy

The differential form of Helmholtz free energy is

From symmetry of second derivatives

and therefore that

The other two Maxwell relations can be derived from differential form ofenthalpy and the differential form of Gibbs free energy

in a similar way. So all Maxwell Relationships abovefollow from one of the Gibbs equations.

Extended derivationCombined form first and second law of thermodynamics,

(Eq.1)U, S, and V are state functions. Let,

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'sthermodynamical relation.

Maxwell's first relation

Allow x = S and y = V and one gets

Maxwell's second relationAllow x = T and y = V and one gets

Maxwell's third relationAllow x = S and y = P and one gets

Maxwell's fourth relationAllow x = T and y = P and one gets

Maxwell's fifth relationAllow x = P and y = V

= 1

Maxwell's sixth relationAllow x = T and y = S and one gets

= 1

If we view the first law of thermodynamics,

as a statement about differential forms, and take the exterior derivative of this equation, we get

since . is leads to the fundamental identity

e physical meaning of this identity can be seen by noting that the two sides are the equivalent ways of writing the

work done in an infinitesimal Carnot cycle. An equivalent way of writing the identity is

e Maxwell relations now follow directly. For example,

Derivation based on Jacobians

e critical step is the penultimate one. e other Maxwell relations follow in similar fashion. For example,

e 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. e 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 re-expressed using the relationship

which are sometimes also known as Maxwell relations.

Table of thermodynamic equations

Thermodynamic equations

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General Maxwell relationships

See also