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 BYNCND. 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 BYNCND. 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 Parksou during the Fall '11 term at FSU.
 Fall '11
 ParkSou
 Dynamics

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