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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