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Thermodynamics filled in class notes_Part_61

# Thermodynamics filled in class notes_Part_61 - 129 4.8...

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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 non-traditional, 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 so-called 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. (4.306-4.309), 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 BY-NC-ND. 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 BY-NC-ND. 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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