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Maxwell Relations 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. See also Table of thermodynamic equations Thermodynamic equations Retrieved from "https://en.wikipedia.org/w/index.php?title=Maxwell_relations&oldid=830544811" This page was last edited on 15 March 2018, at 14:11. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization..
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