HigginsWhitaker_AICHEJ_2012

63c elementary chemical reaction rate equation ii rii

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Unformatted text preview: r, we believe that Eq. 57 should be clearly identified as a stoichiometric result based on the application of Eq. 41. We can summarize the results for the schemata given by Eqs. 52 as follows: Elementary chemical kinetic schema II: kII (56) ð52cÞ Elementary chemical kinetic schema I: kI Br2 À 2Br (58a) ! I Elementary chemical reaction rate equation I: RBr2 ¼ ÀkI cBr2 Published on behalf of the AIChE (58b) February 2012 Vol. 58, No. 2 AIChE Journal rI  kI cBr2 Elementary reference reaction rate I: I RBr2 Elementary stoichiometry I: (58c) RI ¼ À Br 2 RV Br ¼ À RV 2 Br 2 (58d) Elementary stoichiometry V: (62d) We begin our analysis of Eqs. 58 through 62 by listing the net molar rate of production of all five species in terms of the elementary rates of production according to SCHEMA II Elementary chemical kinetic schema II: RBr2 ¼ RI 2 þ RII 2 þ RIII2 þ RIV2 þ RV 2 Br Br Br Br Br (63b) RHBr ¼ RI þ RII þ RIII þ RIV þ RV HBr HBr HBr HBr HBr ð59aÞ (63a) RH2 ¼ RI 2 þ RII 2 þ RIII2 þ RIV þ RV2 H H H H2 H kII Br þ H2 À HBr þ H ! (63c) Elementary chemical reaction rate equation II: RII ¼ ÀkII cBr cH2 Br Elementary reference reaction rate II: rII  kII cBr cH2 ð59bÞ RH ¼ RI þ RII þ RIII þ RIV þ RV H H H H H Elementary stoichiometry II: RII ¼ RII 2 ; Br H RII ¼ ÀRII ; RII ¼ ÀRII Br HBr Br H RBr ¼ RI þ RII þ RIII þ RIV þ RV Br Br Br Br Br ð59cÞ ð59dÞ SCHEMA III Elementary chemical kinetic schema III: (63d) (63e) At this point, we can use the reference reaction rates that appear in Eqs. 58 through 62 to express the net rates of production as 1 RBr2 ¼ ÀrI þ 0 À rIII þ 0 þ rV (64a) 2 RH2 ¼ 0 À rII þ 0 þ rIV þ 0 RIII ¼ ÀkIII cH cBr2 H ð60bÞ Elementary reference reaction rate III: rIII  kIII cH cBr2 ð60cÞ Elementary stoichiometry III: RIII ¼ RIII2 ; H Br RIII ¼ ÀRIII ; RIII ¼ ÀRIII H HBr H Br ð60dÞ (64d) RBr ¼ 2rI À rII þ rIII þ rIV À rV Elementary chemical reaction rate equation III: (64c) RH ¼ 0 þ rII À rIII À rIV þ 0 ð60aÞ (64b) RHBr ¼ 0 þ rII þ rIII À rIV þ 0 kIII ! H þ Br2 À HBr þ Br (64e) In matrix form, the net rates of production given by Eqs. 64 can be expressed as 2 SCHEMA IV Elementary chemical kinetic schema IV: H þ HBr kIV À ! H2 þ Br ð61aÞ 32 RBr2 À1 6 RH2 7 6 0 6 76 6 RHBr 7 ¼ 6 0 6 76 4 RH 5 4 0 RBr 2 0 À1 0 À1 0 1 1 1 À1 1 À1 À1 À1 1 1 1=2 0 0 0 À1 32 3 rI 7 6 rII 7 7 76 7 6 rIII 7 7 76 5 4 rIV 5 rV (65) Elementary chemical reaction rate equation IV: RIV ¼ ÀkIV cH cHBr H ð61bÞ rIV  kIV cH cHBr (61c) Elementary reference reaction rate IV: Elementary stoichiometry IV: RIV ¼ RIV ; RIV ¼ ÀRIV ; H HBr H H2 RIV ¼ ÀRIV H Br ð61dÞ SCHEMA V Elementary chemical kinetic schema V: kV 2Br À Br2 ! (62a) Elementary chemical reaction rate equation V: RV ¼ ÀkV c2 Br Br (62b) Elementary reference reaction rate V: AIChE Journal rV  kV c2 Br February 2012 Vol. 58, No. 2 (62c) which is just a special case of Eq. 45. The row/row partition illustrated by Eqs. 47 and 48 can be applied to this matrix equation to obtain 2 3 rI À1 0 À1 0 1=2 6 rII 7 RBr2 6 7 4 RH2 5 ¼ 4 0 À1 0 1 0 5 6 rIII 7 6 7 4 rIV 5 0 1 1 À1 0 RHBr |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} r ffl V stoichiometric matrix 2 3 2 3 3 rI ! !6r 7 0 1 À1 À1 0 6 II 7 RH 6r 7 ¼ 2 À1 1 1 À1 6 III 7 RBr |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 rIV 5 Bodenstein matrix rV Published on behalf of the AIChE (66) 2 DOI 10.1002/aic (67) 545 At this point, we impose the condition of local reaction equilibrium given by RH ¼ 0 ; Local reaction equilibrium: RBr ¼ 0 Finally, we can use the representations for the concentration of cBr and cH given by Eq. 73 to obtain (after some algebra) RHBr This allows us to express Eq. 67 as 2 3 rI !6r 7 ! 0 1 À1 À1 0 6 II 7 6 rIII 7 ¼ 0 7 2 À1 1 1 À1 6 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 4 rIV 5 Bodenstein matrix rV (69) and a series of elementary row operations leads to 2 1 0 3 rI ! !6r 7 0 0 0 À1=2 6 II 7 6 rIII 7 ¼ 0 6 7 0 1 À1 À1 0 4 rIV 5 rV (70) The reference reaction rates are given in Eq. 58 through Eq. 62, and we summarize those results as 2 2 3 (71) Use of these representations in Eq. 70 provides 1 À kV c2 ¼ 0 Br 2 (72b) and this leads to the following expressions for the concentration of the Bodenstein products: pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffi kII cH2 2k1 =kV cBr2 cH ¼ ; ðkIII cBr2 þ kIV cHBr Þ cBr ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffi 2kI =kV cBr2 (73) At this point we make use of Eq. 66 to obtain an expression for the net rate of production of hydrogen bromide that is given by 2 ½RHBr Š ¼ ½ 0 1 1 À1 3 rI 6 rII 7 6 7 0 Š 6 rIII 7 6 7 4 rIV 5 r...
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This document was uploaded on 01/26/2014.

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