Maxwell Relations

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Maxwell relations

Maxwell'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.

Contents

Equations

The four most common Maxwell relations Derivation

Derivation based on Jacobians

General Maxwell relationships

See also

Equations Flow chart showing the paths between the Maxwell relations. P: e structure of Maxwell relations is a pressure, T: temperature, V: volume, S: entropy, α: coefficient of statement of equality among the second thermal expansion, κ: compressibility, CV: heat capacity at derivatives for continuous functions. It constant volume, CP: heat capacity at constant pressure. 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 e four most common Maxwell relations

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.

Derivation 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.

Derivation Derivation of the Maxwell relation can be deduced from the diﬀerential 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 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

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 of enthalpy and the diﬀerential form of Gibbs free energy in a similar way. So all Maxwell Relationships above follow from one of the Gibbs equations.

Extended derivation Combined 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'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

Derivation based on Jacobians

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

General Maxwell relationships

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.