Unformatted text preview: 129 4.8. MIXTURES WITH VARIABLE COMPOSITION Obviously, by its deﬁnition, µi is on a per mole basis, so it is given the appropriate overline
notation. In Eq. (4.300), the independent variables and their conjugates are
x1
x2
x3
x4
xN +2 =
=
=
= V,
S,
n1 ,
n2 ,
.
.
.
= nN , ψ1 = −P,
ψ2 = T,
ψ3 = µ1 ,
ψ4 = µ2 ,
.
.
.
ψN +2 = µN . (4.301)
(4.302)
(4.303)
(4.304)
(4.305) Equation (4.300) has 2N +1 − 1 Legendre functions. Three are in wide usage: the extensive
analog to those earlier found. They are
H = U + P V,
A = U − T S,
G = U + P V − T S. (4.306)
(4.307)
(4.308) A set of nontraditional, but perfectly acceptable additional Legendre functions would be
formed from U − µ1 n1 . Another set would be formed from U + P V − µ2 n2 . There are
many more, but one in particular is sometimes noted in the literature: the socalled grand
potential, Ω. The grand potential is deﬁned as
N Ω ≡ U − TS − µi ni . (4.309) i=1 Diﬀerentiating each deﬁned Legendre function, Eqs. (4.3064.309), and combining with
Eq. (4.300), one ﬁnds
N dH = T dS + V dP + µi dni , (4.310) i=1
N dA = −SdT − P dV +
dG = −SdT + V dP +
dΩ = −P dV − SdT − µi dni , (4.311) µi dni , (4.312) ni dµi . (4.313) i=1
N i=1
N i=1 Thus, canonical variables for H are H = H (S, P, ni). One ﬁnds a similar set of relations as
CC BYNCND. 18 November 2011, J. M. Powers. 130 CHAPTER 4. MATHEMATICAL FOUNDATIONS OF THERMODYNAMICS before from each of the diﬀerential forms:
T= ∂U
∂S P=−
V = =
V ,ni ∂U
∂V ∂H
∂P S,ni =
S,ni ∂H
∂S P ,ni =− ∂A
∂V ∂G
∂P T ,ni =− ∂G
∂T S=− ∂A
∂T V ,ni ni = − ∂Ω
∂ µi , (4.314)
∂Ω
∂V =− T ,ni , (4.315) T ,µi , (4.316)
∂Ω
∂T V ,T,µj µi = ∂U
∂ni =
S,V,nj =− P ,ni , . ∂H
∂ni (4.317) V ,µi (4.318) =
S,P,nj ∂A
∂ni =
T ,V,nj ∂G
∂ni (4.319)
T ,P,nj Each of these induces a corresponding Maxwell relation, obtained by cross diﬀerentiation.
These are
∂T
∂V
∂T
∂P
∂P
∂T
∂V
∂T
∂ µi
∂T
∂ µi
∂P
∂ µl
∂nk =
S,ni =
V ,ni ∂P
∂S , (4.320) V ,ni ∂V
∂S P ,ni ∂S
∂V T ,ni , (4.321) , (4.322) P ,ni =− ∂S
∂P T ,ni P ,nj =− ∂S
∂ni V ,nj =
T ,nj =
T ,P,nj ∂S
∂V
∂ni
∂ µk S,ni =− =
T ,µi =
V ,T,µj ,j =k CC BYNCND. 18 November 2011, J. M. Powers. ,
, ∂V
∂ni (4.324) V ,nj ∂ µk
∂nl (4.323) T ,P,nj ∂P
∂T
∂nk
∂ µi , (4.325)
, , (4.326)
(4.327) V ,µi (4.328)
V ,T,µj ,j =i ...
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 Fall '11
 ParkSou
 Dynamics, Gate, Trigraph, Legendre polynomials, J. M. Powers, ni dµi, µi dni

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