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Thermodynamics filled in class notes_Part_100

# Thermodynamics filled in class notes_Part_100 - 207 6.1...

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Unformatted text preview: 207 6.1. SUMMARY OF MULTIPLE REACTION EXTENSIONS or N Kc,j = ρi νij , j = 1, . . . , J. (6.13) i=1 For isochoric reaction, the evolution of species concentration i due to the combined eﬀect of J reactions is given by ≡ ωi ˙ dρi = dt J νij aj T βj N −E j RT exp j =1 ν′ ρk kj k =1 ≡kj (T ) forward reaction 1 1 − Kc,j N k =1 ν ρk kj , reverse reaction ≡rj =(1/V )dζj /dt i = 1, . . . , N. (6.14) The extension to isobaric reactions is straightforward, and follows the same analysis as for a single reaction. Again, three intermediate variables which are in common usage have been deﬁned. First one takes the reaction rate of the j th reaction to be rj ≡ aj T βj N −E j RT exp ν′ ρk kj k =1 ≡kj (T ) forward reaction 1 1 − Kc,j N , νkj ρk k =1 reverse reaction j = 1, . . . , J, (6.15) or rj = aj T βj exp −E j RT ≡kj (T ), Arrhenius rate N ν′ ρk kj k =1 N 1 − Kc,j forward reaction k =1 reverse reaction law of mass action ν ′′ ρk kj , j = 1, . . . , J, (6.16) = 1 dζj (6.17) . V dt Here ζj is the reaction progress variable for the j th reaction. Each reaction has a temperature-dependent rate function kj (T ), which is kj (T ) ≡ aj T βj exp −E j RT , j = 1, . . . , J. (6.18) CC BY-NC-ND. 18 November 2011, J. M. Powers. 208 CHAPTER 6. THERMOCHEMISTRY OF MULTIPLE REACTIONS The evolution rate of each species is given by ωi , deﬁned now as ˙ J ωi ≡ ˙ νij rj , i = 1, . . . , N. (6.19) j =1 The multi-reaction extension for mole change in terms of progress variables is J dni = νij dζj , i = 1, . . . , N. (6.20) j =1 One also has N dG|T,P = µi dni , i=1 N J i=1 ∂G ∂ζj = ∂ζk , ∂ζj (6.23) νik µi i=1 k =1 J N = (6.22) k =1 J N ζp νik dζk , µi = (6.21) νik δkj , µi i=1 (6.24) j =1 N = µi νij , (6.25) i=1 = −αj , j = 1, . . . , J. (6.26) In a very similar fashion to that shown for a single reaction, one can further sum over all reactions and prove that mixture mass is conserved, element mass and number are conserved. A similar expression is obtained for temperature changes. Example 6.1 Show that element mass and number are conserved for the multi-reaction formulation. CC BY-NC-ND. 18 November 2011, J. M. Powers. ...
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