HigginsWhitaker_AICHEJ_2012

C7 leads to 2 6 6 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 2

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Unformatted text preview: 1 :::::: :::::: N31 :::::: :::::: NT 2 NT ;NÀ1 3 3 R1 N1N 6 7 6 R2 7 7 N 2 N 76 76 76 R 3 7 7 76 76 : 7 7 76 7 76 76 : 7 7 56 7 4 RN À 1 5 NTN RN 23 0 67 607 67 ¼ 607 ðC2Þ 67 67 4:5 0 and if some of the species undergoing reaction are charged species (ions), we need to impose conservation of charge18 in addition to conservation of atomic species. This is done in terms of the additional axiomatic statement given by A¼N X Axiom III : N e A RA ¼ 0 (C3) A¼1 in which NeA represents the electronic charge associated with molecular species A. In terms of matrix representation, Axiom III can be added to Eq. C1 to obtain a combined representation for conservation of atomic species and conservation of charge. This combined representation is given by 2 N11 6 N21 6 6 N31 6 6: 6 6: 6 4 NT 1 Ne1 N12 N22 N32 : : : Ne2 N13 : :::::: :::::: :::::: :::::: :::::: :::::: :::::: N1;NÀ1; N2;NÀ1 : : : : NeNÀ1 32 Here, the elements in the last row of the (T þ 1) Â N matrix take on the values associated with the charge on species 1,2,…,N as illustrated by AIChE Journal February 2012 Vol. 58, No. 2 (C5) As an example of competing reactions in a redox system,19 we consider a mixture consisting of ClOÀ , H3Oþ, 2 Cl2, H2O, ClOÀ , and ClO2. The visual representation for the 3 atomic/electronic matrix is given by Molecular Species and Charge ! ClOÀ H3 Oþ Cl2 2 2 chlorine 1 0 2 62 oxygen 6 1 0 6 hydrogen 4 0 3 0 charge À1 þ1 0 H2 O 0 1 2 0 ClOÀ 3 1 ClO2 3 1 (C6) 27 7 7 05 3 0 À1 0 and use of this result with Eq. C4 leads to 2 1 2 0 0 1 3 2 0 0 0 1 2 1 3 0 1 2 0 À1 6 6 Axiom II and III : 6 4 þ1 0 0 À1 0 2 36 6 76 76 76 56 6 6 4 23 0 607 67 ¼6 7 405 3 RClOÀ 2 RH3 Oþ 7 7 7 RCl2 7 7 RH2 O 7 7 7 RClOÀ 5 3 RClO2 ðC7Þ 0 At this point, we follow the developments given in this article, and we search for the row reduced echelon form of the atomic/electronic matrix. We begin with 2 3 1 0 2 0 1 1 62 1 0 1 3 27 7 (C8) Ae ¼ 6 40 3 0 2 0 05 À1 þ1 0 0 À1 0 and apply a series of elementary row operations to find the row reduced echelon form given by 2 6 AÃ ¼ 6 e 4 3 23 N1N R1 0 N2N 76 R2 7 6 7 76 7 0 : 76 R3 7 6 7 76 7 607 : 76 : 7 ¼ 6 7 76 7 6:7 : 76 : 7 6 7 76 7 4:5 : 54 RN À 1 5 0 NeN RN (C4) nonionic species 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 3 5=3 4=3 2=3 4=3 7 7 À1=3 À1=6 5 À1 À2 (C9) Use of this result in Eq. C7 leads to 2 6 6 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 2 À 3 RClO2 6 RH Oþ 5=3 4=3 6 3 2=3 4=3 76 RCl2 76 À1=3 À1=6 56 RH2 O 6 À1 À2 4 RClOÀ 3 RClO2 Published on behalf of the AIChE 3 23 7 0 7 7 607 7¼6 7 7 405 7 5 0 (C10) DOI 10.1002/aic 551 We now follow the type of analysis given in this article, and we apply the obvious column/row partition to obtain 32 2 3 1000 RClOÀ 2 6 0 1 0 0 76 R þ 7 6 7 6 H3 O 7 7 6 76 4 0 0 1 0 5 4 RCl2 5 RH2 O 0001 |fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl} no...
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This document was uploaded on 01/26/2014.

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