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Thermodynamics filled in class notes_Part_63

Thermodynamics filled in class notes_Part_63 - 133 4.9...

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Unformatted text preview: 133 4.9. PARTIAL MOLAR PROPERTIES Note that these expressions do not formally involve partial molar properties since P and T are not constant. Take now the appropriate partial molar derivatives of G for an ideal mixture of ideal gases to get some useful relations: G = H − T S, ∂G ∂H ∂S = −T ∂ni T ,P,nj ∂ni T ,P,nj ∂ni (4.347) . (4.348) T ,P,nj Now from the definition of an ideal mixture hi = hi (T, P ), so one has N nk hk (T, P ), H= (4.349) k =1 ∂H ∂ni = T ,P,nj N ∂ ∂ni N = k =1 nk hk (T, P ) , (4.350) k =1 ∂nk hk (T, P ), ∂ni (4.351) =δik N δik hk (T, P ), = (4.352) k =1 = hi (T, P ). (4.353) Here, the Kronecker delta function δki has been again used. Now for an ideal gas one further has hi = hi (T ). The analysis is more complicated for the entropy, in which N S= k =1 N = k =1 N nk so − R ln T,k Pk Po nk so − R ln T,k yk P Po , − R ln nk so − R ln T,k P Po nk = so T,k P Po k =1 N = k =1 − R ln , (4.354) (4.355) nk N q =1 N −R nk ln k =1 nq , nk N q =1 (4.356) nq , (4.357) CC BY-NC-ND. 18 November 2011, J. M. Powers. 134 CHAPTER 4. MATHEMATICAL FOUNDATIONS OF THERMODYNAMICS ∂S ∂ni T ,P,nj N ∂ = ∂ni k =1 ∂ −R ∂ni N = k =1 P Po nk so − R ln T,k N nk N q =1 T ,P,nj ∂nk ∂ni k =1 nq nk N q =1 nk ln nq , (4.358) , (4.359) P Po so − R ln T,k =δik −R ∂ ∂ni so T,i = N nk ln T ,P,nj − R ln k =1 P Po N ∂ −R ∂ni nk nk ln T ,P,nj N q =1 k =1 nq . (4.360) Evaluation of the final term on the right side requires closer examination, and in fact, after tedious but straightforward analysis, yields a simple result which can easily be verified by direct calculation: ∂ ∂ni N nk ln T ,P,nj k =1 nk N q =1 = ln nq ni N q =1 . nq (4.361) So the partial molar entropy is in fact ∂S ∂ni T ,P,nj = so − R ln T,i = so − R ln T,i = so − R ln T,i P Po P Po Pi Po − R ln ni N q =1 nq , (4.362) − R ln yi , (4.363) , (4.364) = si (4.365) Thus, one can in fact claim for the ideal mixture of ideal gases that g i = hi − T si . 4.9.4 (4.366) Relation between mixture and partial molar properties A simple analysis shows how the partial molar property for an individual species is related to the partial molar property for the mixture. Consider, for example, the Gibbs free energy. The mixture-averaged Gibbs free energy per unit mole is g= CC BY-NC-ND. 18 November 2011, J. M. Powers. G . n (4.367) ...
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