Mathematical Methods for Physics and Engineering - Matematica.NET

PARTIAL DIFFERENTIATION5.11 Thermodynamic relationsThermodynamic relations provide a useful set of physical examples of partialdifferentiation. The relations we will derive are called Maxwell’s thermodynamicrelations. They express relationships between four thermodynamic quantities describinga unit mass of a substance. The quantities are the pressure P , the volumeV , the thermodynamic temperature T and the entropy S of the substance. Thesefour quantities are not independent; any two of them can be varied independently,but the other two are then determined.The first law of thermodynamics may be expressed asdU = TdS− PdV, (5.44)where U is the internal energy of the substance. Essentially this is a conservationof energy equation, but we shall concern ourselves, not with the physics, but ratherwith the use of partial differentials to relate the four basic quantities discussedabove. The method involves writing a total differential, dU say, in terms of thedifferentials of two variables, say X and Y , thus( ) ( )∂U∂UdU = dX + dY , (5.45)∂XY∂YXand then using the relationship∂ 2 U∂X∂Y =∂2 U∂Y ∂Xto obtain the required Maxwell relation. The variables X and Y aretobechosenfrom P , V , T and S.◮Show that (∂T/∂V) S = −(∂P/∂S) V .Here the two variables that have to be held constant, in turn, happen to be those whosedifferentials appear on the RHS of (5.44). And so, taking X as S and Y as V in (5.45), wehaveand find directly thatTdS− PdV= dU =( ) ∂U= T and∂SV( ) ∂UdS +∂SV( ) ∂UdV ,∂VS( ) ∂U= −P.∂VSDifferentiating the first expression with respect to V and the second with respect to S, andusing∂ 2 U∂V∂S =∂2 U∂S∂V ,we find the Maxwell relation( ) ( )∂T ∂P= − . ◭∂VS∂SV176