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. These relations are named for the nineteenth-century physicist James Clerk Maxwell. ∂ ∂ x j ( ∂ Φ ∂ x i ) = ∂ ∂ x i ( ∂ Φ ∂ x j ) {displaystyle {frac {partial }{partial x_{j}}}left({frac {partial Phi }{partial x_{i}}} ight)={frac {partial }{partial x_{i}}}left({frac {partial Phi }{partial x_{j}}} ight)} + ( ∂ T ∂ V ) S = − ( ∂ P ∂ S ) V = ∂ 2 U ∂ S ∂ V + ( ∂ T ∂ P ) S = + ( ∂ V ∂ S ) P = ∂ 2 H ∂ S ∂ P + ( ∂ S ∂ V ) T = + ( ∂ P ∂ T ) V = − ∂ 2 F ∂ T ∂ V − ( ∂ S ∂ P ) T = + ( ∂ V ∂ T ) P = ∂ 2 G ∂ T ∂ P {displaystyle {egin{aligned}+left({frac {partial T}{partial V}} ight)_{S}&=&-left({frac {partial P}{partial S}} ight)_{V}&=&{frac {partial ^{2}U}{partial Spartial V}}\+left({frac {partial T}{partial P}} ight)_{S}&=&+left({frac {partial V}{partial S}} ight)_{P}&=&{frac {partial ^{2}H}{partial Spartial P}}\+left({frac {partial S}{partial V}} ight)_{T}&=&+left({frac {partial P}{partial T}} ight)_{V}&=&-{frac {partial ^{2}F}{partial Tpartial V}}\-left({frac {partial S}{partial P}} ight)_{T}&=&+left({frac {partial V}{partial T}} ight)_{P}&=&{frac {partial ^{2}G}{partial Tpartial P}}end{aligned}},!} The differential form of internal energy U isU, S, and V are state functions.Let, 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. These relations are named for the nineteenth-century physicist James Clerk Maxwell. The 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: 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.The substantial increase in the entropy will be verified according to the relations satisfied by the laws of thermodynamics 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 U(S, V), enthalpy H(S, P), Helmholtz free energy F(T, V) and Gibbs free energy G(T, P). The thermodynamic square can be used as a mnemonic to recall and derive these relations. The 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.