Thermodynamics filled in class notes_Part_96

# Thermodynamics filled in class notes_Part_96 - 199 5.6 SOME...

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Unformatted text preview: 199 5.6. SOME CONSERVATION AND EVOLUTION EQUATIONS Now take Eq. (4.387) and perform some straightforward operations on it: dS |U,V N 1 − T = ≥ 0, µi dni i=1 (5.460) irreversible entropy production U,V =− V T =− dS dt V T V =− T N µi dni 1 ≥ 0, dt V (5.461) µi dρi ≥ 0, dt (5.462) i=1 N i=1 N β µi νi aT exp i=1 N −E RT ≡k (T ) ν′ ρk k k =1 forward reaction ≡r =− V T N N ′ νk ρk i=1 1− 1 1− Kc µi νi k (T ) V = − k (T ) T ν′ ρk k k =1 ρkνk ≥ 0,(5.463) k =1 N reverse reaction N ρkνk N k =1 N 1 Kc 1 1 − Kc ≥ 0, k =1 (5.464) N ρkνk µi νi i=1 k =1 ≥ 0, (5.465) =−α Change the dummy index from k back to i, V = k (T ) T N ρi i=1 ′ νi 1 1− Kc N ρ i νi i=1 α ≥ 0, (5.466) V r α, T α dζ = . T dt (5.467) = (5.468) Consider now the aﬃnity α term in Eq. (5.466) and expand it so that it has a more useful form: N α=− i=1 N µi νi = − =− g i νi , i=1 N g o + RT ln T,i i=1 (5.469) Pi Po νi , (5.470) CC BY-NC-ND. 18 November 2011, J. M. Powers. 200 CHAPTER 5. THERMOCHEMISTRY OF A SINGLE REACTION N =− N g o νi T,i i=1 = ∆G −RT νi Pi Po ln i=1 , o N −∆Go = RT RT − i=1 =ln KP N ln KP − ln = RT i=1 N 1 ln + ln KP i=1 = −RT = −RT ln = −RT ln = −RT ln N 1 KP Po RT i=1 PN i=1 (5.472) (5.473) , νi Pi Po , (5.474) νi Pi Po , νi Kc 1 Kc νi Pi Po , νi Pi Po ln (5.471) N (5.475) νi ρi RT Po i=1 , N ρi νi (5.476) . (5.477) i=1 Equation (5.477) is the common deﬁnition of aﬃnity. Another form can be found by employing the deﬁnition of Kc from Eq. (5.270) to get Po RT α = −RT ln − PN i=1 o ∆G + ln RT = −RT νi exp Po RT Po RT = −∆Go − RT ln − − N ∆Go RT PN i=1 PN ρi νi , (5.478) i=1 νi N ρi νi , (5.479) i=1 N i=1 νi ρi νi . (5.480) i=1 To see clearly that the entropy production rate is positive semi-deﬁnite, substitute Eq. (5.477) into Eq. (5.466) to get dS dt U,V V = k (T ) T N ρi 1 1− Kc ′ νi i=1 N ρi νi i=1 −RT ln 1 Kc N ρi νi i=1 ≥ 0, (5.481) N = −RV k (T ) ρi i=1 ′ νi 1− CC BY-NC-ND. 18 November 2011, J. M. Powers. 1 Kc N ρ i νi i=1 ln 1 Kc N ρi νi i=1 ≥ 0. (5.482) ...
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