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Thermodynamics filled in class notes_Part_101

# Thermodynamics filled in class notes_Part_101 - 6.1 SUMMARY...

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Unformatted text preview: 6.1. SUMMARY OF MULTIPLE REACTION EXTENSIONS 209 Start with Eq. (6.14) and expand as follows: J dρi dt φli = dρi dt = φli i=1 νij rj , (6.28) φli νij rj , (6.29) j =1 d (φli ρi ) = dt J j =1 N J φli νij rj , φli νij rj , (6.31) N = φli ρi (6.30) i=1 j =1 J N d dt (6.27) J d (φli ρi ) = dt N νij rj , j =1 j =1 i=1 i=1 =ρl e dρl e dt J N = rj j =1 φli νij , (6.32) i=1 =0 dρl e dt = 0, l = 1, . . . , L, l = 1, . . . , L, (6.34) l = 1, . . . , L. d (Ml ρl e ) = 0, dt dρl e = 0, dt (6.33) (6.35) It is also straightforward to show that the mixture density is conserved for the multi-reaction, multicomponent mixture: dρ = 0. (6.36) dt The proof of the Clausius-Duhem relationship for the second law is an extension of the single reaction result. Start with Eq. (5.460) and operate much as for a single reaction model. dS |U,V 1 − T = N µi dni i=1 ≥ 0, (6.37) irreversible entropy production dS dt U,V V =− T =− V T N µi dni 1 ≥ 0, dt V (6.38) µi dρi ≥ 0, dt (6.39) i=1 N i=1 CC BY-NC-ND. 18 November 2011, J. M. Powers. 210 CHAPTER 6. THERMOCHEMISTRY OF MULTIPLE REACTIONS V =− T V =− T V =− T V =− T V =− T N J νij rj ≥ 0, (6.40) µi νij rj ≥ 0, (6.41) µi i=1 j =1 N J i=1 j =1 J N µi νij ≥ 0, rj j =1 i=1 J N ν′ ρi ij kj j =1 i=1 J N ν′ ρi ij kj j =1 i=1 (6.42) 1 1− Kc,j 1 1− Kc,j N N ν ρi ij i=1 µi νij ≥ 0, i=1 N ν ρi ij (6.43) 1 Kc,j RT ln i=1 N ν ρi ij i=1 ≥ 0, (6.44) J = −RV N j =1 ν′ ρi ij kj 1− i=1 N 1 Kc,j ν ρi ij 1 Kc,j ln i=1 N ν ρi ij i=1 ≥ 0. (6.45) Note that Eq. (6.42) can also be written in terms of the aﬃnities (see Eq. (6.10)) and reaction progress variables (see Eq. (6.17) as dS dt 1 T = U,V J αj j =1 dζj ≥ 0. dt (6.46) Similar to the argument for a single reaction, if one deﬁnes N R′j = kj R′′ = j ρi ′ νij , (6.47) i=1 kj Kc,j N ρi ′′ νij , (6.48) i=1 then it is easy to show that rj = R′j − R′′ , j and dS dt J = RV U,V j =1 R′j − R′′ ln j (6.49) R′j R′′ j ≥ 0. (6.50) Since kj (T ) > 0, R > 0, and V ≥ 0, and each term in the summation combines to be positive semi-deﬁnite, one sees that the Clausius-Duhem inequality is guaranteed to be satisﬁed for multi-component reactions. CC BY-NC-ND. 18 November 2011, J. M. Powers. ...
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