UMMAY OF TOICHIOMETIC ELATION AND MEAUE OF EACTION' POGE AND COMPOITION FO MULTIPLE EACTION UPDATED 0/4/03 - AW APPENDIX A. In case of multple reactons t s mportant to fnd the number of ndependent reactons. Ths ermnes for how many speces the change n the number of moles may occur ndependently,.e for how many speces we must montor the change n the number of moles caused by reactons n order to compute the composton of the whole system (.e to compute the changes n the number of moles of all other speces). Usually the problem s stated as reactons between speces: A 0;,.. where s the stochometrc coeffcent of speces n reacton (ecall > 0 products, < 0 reactants). Then: a) Fnd the number of ndependent reactons. b) Fnd the molar etents of the ndependent reactons ( ś X,,...) n terms of the measured changes n the number of moles,, of key speces. ( F F o,,,...) c) Epress the changes n moles of the remanng - speces, (,,...) n terms of changes n moles of the selected key speces. d) Determne the etents of the remanng (dependent) reactons n terms of the calculated etents (f so desred). Frst we wll ntroduce the classcal procedure and then show a short-cut method whch accomplshes all of the aboe n a sngle step.
Classcal Method: Consder the matr of stochometrc coeffcents of all orgnal reacton equatons and speces (.e. the (' X ) matr): orgnal ( X )...... 5... 5 5 and fnd ts rank,.e, the largest nonzero ermnant. Use for eample Gaussan elmnaton (see for eample E. Kreyszg "Adanced Engneerng Mathematcs" Fourth Edton, Wley, N.Y. 979, Chapter 7-6). Ths procedure wll generate - rows of zeros when s the rank of the matr and the number of ndependent reactons. All reactons for whch rows of zeros are generated are the dependent ones. The rest s pcked as a set of ndependent reactons for whch the matr of stochometrc coeffcents s: new ( X)............ Please understand that reactons hae now been renumbered, snce some were deleted as beng dependent, and the elements of the new matr of stochometrc coeffcents do not necessarly correspond n order one to one to the elements of the old matr. Now usng the law of stochometry we can wrte: T ( ) X ś ( X)X ( ) () where T s the transpose of the matr.e., s matr wth columns and rows nterchanged.
Epanded ew of eq () s: th row.................................... X& X&...... X&...... X& (a) Consderng now ust the -th row of the ector, and rememberng that matr multplcaton s rows nto columns, we get: X ś X ś... ś F { number of moles of produced n the reacton system 3 Σ oer all ndependent reactons X ν ś 3 X number of moles of produced by reacton (b) We can sole equaton () and we wll show a formal soluton n terms of Cramer's rule. (Ths of course s not practcal and actual soluton should proceed by Gaussan elmnaton or Gauss Jordan technque). The soluton of () s: where T T ( ) X& () () s the nerse matr of ( T ) how to obtan t by Gauss Jordan technque]. [see Kreyszg or some other lnear algebra tet to fnd nce the alue of the ermnant s unchanged f rows and columns are nterchanged, the soluton n terms of Cramer's rule s: ( ) () X & F,,... (3) 3
where ()......... ν (4)......... and () F s the ( ) where the -th row s replaced by F,, etc. ( )......... ν... (5)...... Ths soles parts (a), of fndng the number of ndependent reactons and (b), of relatng etents of ndependent reactons to measured changes n moles of key speces. We hae related etents, X, to the changes n moles of speces, F,,.... If < we need to epress the changes n moles of the remanng - speces n terms of the changes n moles of the frst speces (part c). tochometry dctates that eq (b) s ald for all speces, not ust for the frst speces for whch we hae used t so far. Hence, we can wrte k k X& ; k,,..., (6) Ths ges the change n moles of the remanng - speces n terms of the now known etents. We can proceed a step further and substtute equaton (3) nto equaton (6) to obtan: k k ( ) () F (7) 4
After rearrangement we recognze the followng: 0 k () k () () F ( ) k (8) The left hand sde of the aboe equaton s ust an epanson n mnors of the ermnant on the rght hand sde when the epanson s done about the elements of the last ( ) column. If we deelop ths ermnant wth respect to the frst row (the alue of the ermnant s properly zero ndcatng that F k, k,,..., s lnearly dependent on the frst s ) we get: ' k () ( ) ) 0 (9) k whch fnally yelds k ( ) ) k () n terms of the frst k ; k,..., where ( ) ) s ( ) k F s : k, k wth the -th column replaced by,.... k (0) throw ( ) ) k... k k k......... Note that n prong eq (8) and (9) one uses the property that the alue of a ermnant must be multpled by (-) each tme two rows or two columns are nterchanged. Our results can be summarzed as follows: The etents are obtaned from eq (3) whch now becomes () () () ( ) () F X& ( ) ;,... (3a) (3a) s () wth row and column deleted. The change n moles for the non-key speces s gen by: 5
( ) k () k ; k,... (0) hortcut Method: Ths method ges all the nformaton n one sngle sweep. Consder the augmented matr T F orgnal ( )........................ Use Gauss-Jordan technque on ths augmented matr to generate 's on the dagonal and zeros elsewhere by normalzng the pot element and by generatng. The procedure stops when we cannot generate any more 's on the dagonal and the rows below the last are flled wth zeros. emember, we can always nterchange rows, keepng track to renumber our reactons accordngly, to get the rows wth zeros on the bottom. When we are fnshed wth the Gauss-Jordan procedure the augmented matr has the form: k 6
rows wth ' s on dagonal dagonal untary matr ermnes No. of ndependent reactons 64748 0 0...0 00...0 0 0...0... 0 0 0... Coeffcents for eactons 64748 λ... λ λ λ λ 3... λ... λ...... λ 3... α α α etents rows of zeros 0 0 0...0 0 0 0...0... 0 0 0...0 0...0 0...0... 0...0... β β β narants In the upper left hand corner the ( X ) matr wth 's on the dagonal and zeros eerywhere else ermnes the rank and the number of ndependent reactons. The frst rows on the top n the rght hand corner, to the rght of the partton, ge the etents: X& α ;,... () where α hae been ermned by the procedure. By comparson wth eq (3a) we see that the α generated by the procedure must be ( ) () α ( ) ;,,..., ;,,..., 7
The lower rght corner,.e the last - rows to the rght of the partton, defnes the ( - ) narants of the system: k β k 0; k,,... () These lnear combnatons of ' s (narances) must be zero. Otherwse the last - rows would not hae a 0 0 relaton and the system would be mpossble and hae no soluton, meanng that a mass balance was olated wth a wrong stochometrc coeffcent someplace. Comparson of equaton () and equaton (0) ges: ( ) k β k ; k,,..., ;,,..., (3) () Note that β k hae been calculated by the Gauss-Jordan procedure. Fnally, the ( X( ) matrof λ' s, squeezed between the dagonal ( X ) dentty matr and the partton, yelds the coeffcents for epressng the etents of dependent reactons n terms of the ndependent ones: X& k λ X& k k, ;,..., (4) Let us document ths by an eample: PECIE 3 4 C CO C O O O CO CO CO C O CO CO ( ) ( ) ( 3) ( 4) () 0 ( ) 0 () 3 0 orgnal 0 0 0 8
Use short-cut method. Wrte the transpose matr: T orgnal 0 0 0 T Then proceed wth the augmented matr ( orgnal ) as follows: 0 0 3 0 4 0 0 0 3 0 4 0 0 0 0 0 3 0 0 0 4 etents narants There are (two) ndependent reactons. Ther etents n terms of molar changes of the frst two speces are:?x ( F F 0 ) ( F 0 F )?X ( F 0 F ) ( F 0 F ) There are ndependent reactons and ther etents are gen aboe. There are - 4 - two narants. The changes n the number of moles of speces 3 and 4 can be epressed n terms of the change n moles of speces and usng the two narants as shown below: 3 ( ) ( F F 0 F F 0 ) 4 ( F 0 F ) F 0 F ( ) 9
The etent of the thrd reacton s: etent of the dependent reacton { X ś 3 X ś X ś.e the thrd reacton s the sum of the frst two dded by two. Ths n ths smple case s obous by nspecton. Check the classcal method results. By ths method we would hae frst ermned and would hae taken the frst two reactons as ndependent. The X X matr of stochometrc coeffcents usng the frst two components as the key ones, s: 0 ; () 0 () () ( ) ; ( ) ; ( ) 0 () ( ) 0 ( ) Then α ( ) α ( ) ; a ( ) 3 0 ; α ( ) 4 0 Hence:?X?X 0 QED checks QED Now: β 3 0 β 3 4 4 0 ; β 4 0 0 ; β 4 3 0; 4 0 checks Once we know the number of ndependent reactons and ther etents we can wrte for all speces,,..., 0
F F o ν? X ;,,... (5) F tot F F tot,o ν? X ; ν ν (6) Another hortcut Method: Another approach to the aboe problem s to count all the chemcal elements present n the reacton system under consderaton (say total number of elements s E) and form the matr of coeffcents E so that ν s the number of atomc speces present n the chemcal speces. The matr E s a (EX) matr. One should fnd the rank of matr E, say E. Then the number of ndependent reactons s equal to the total number of chemcal speces mnus the rank of the matr E,.e. E (7)