Thermodynamics filled in class notes_Part_62

Thermodynamics filled in class notes_Part_62 - 131 4.9...

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Unformatted text preview: 131 4.9. PARTIAL MOLAR PROPERTIES 4.9 Partial molar properties 4.9.1 Homogeneous functions In mathematics, a homogeneous function f (x1 , . . . , xN ) of order m is one such that f (λx1 , . . . , λxN ) = λm f (x1 , . . . , xN ). (4.329) f (λx1 , . . . , λxN ) = λf (x1 , . . . , xN ). (4.330) If m = 1, one has Thermodynamic variables are examples of homogeneous functions. 4.9.2 Gibbs free energy Consider an extensive property, such as the Gibbs free energy G. One has the canonical form G = G(T, P, n1 , n2 , . . . , nN ). (4.331) One would like to show that if each of the mole numbers ni is increased by a common factor, say λ, with T and P constant, that G increases by the same factor λ: λG(T, P, n1 , n2 , . . . , nN ) = G(T, P, λn1 , λn2 , . . . , λnN ). (4.332) Differentiate both sides of Eq. (4.332) with respect to λ, while holding P , T , and nj constant, to get G(T, P, n1 , n2 , . . . , nN ) = ∂G ∂G d(λn1 ) + ∂ (λn1 ) nj ,P,T dλ ∂ (λn2 ) = ∂G ∂ (λn1 ) nj ,P,T n1 + nj ,P,T ∂G d(λn2 ) +···+ dλ ∂ (λnN ) ∂G ∂ (λn2 ) nj ,P,T n2 + · · · + nj ,P,T ∂G ∂ (λnN ) d(λnN ) , dλ (4.333) nN , (4.334) nj ,P,T This must hold for all λ, including λ = 1, so one requires G(T, P, n1, n2 , . . . , nN ) = ∂G ∂n1 n1 + nj ,P,T ∂G ∂n2 nj ,P,T n2 + · · · + ∂G ∂nN nN , nj ,P,T (4.335) N = i=1 ∂G ∂ni ni . (4.336) nj ,P,T Recall now the definition partial molar property, the derivative of an extensive variable with respect to species ni holding nj , i = j , T , and P constant. Because the result has units per CC BY-NC-ND. 18 November 2011, J. M. Powers. 132 CHAPTER 4. MATHEMATICAL FOUNDATIONS OF THERMODYNAMICS mole, an overline superscript is utilized. The partial molar Gibbs free energy of species i, g i is then ∂G , (4.337) gi ≡ ∂ni nj ,P,T so that N G= g i ni . (4.338) i=1 Using the definition of chemical potential, Eq. (4.319), one also notes then that N G(T, P, n1 , n2 , . . . , nN ) = µi ni . (4.339) i=1 The temperature and pressure dependence of G must lie entirely within µi (T, P, nj ), which one notes is also allowed to be a function of nj as well. Consequently, one also sees that the Gibbs free energy per unit mole of species i is the chemical potential of that species: g i = µi . (4.340) Using Eq. (4.338) to eliminate G in Eq. (4.308), one recovers an equation for the energy: N U = −P V + T S + 4.9.3 µi ni . (4.341) i=1 Other properties A similar result also holds for any other extensive property such as V , U , H , A, or S . One can also show that N V ni = i=1 N U= ni i=1 N H= ni i=1 N A= ni i=1 N ni S= i=1 CC BY-NC-ND. 18 November 2011, J. M. Powers. ∂V ∂ni nj ,A,T ∂U ∂ni nj ,V,S ∂H ∂ni nj ,P,S ∂A ∂ni nj ,T,V ∂S ∂ni nj ,U,T , (4.342) (4.343) , (4.344) , (4.345) . (4.346) ...
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This note was uploaded on 11/26/2011 for the course EGN 3381 taught by Professor Park-sou during the Fall '11 term at FSU.

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