ChemE Chemical Kinetics & Reactor Design - Spring 2019 Solution to Homework Assignment 2
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1 ChE 39 - Chicl iics & Rcor Dsig - Sprig 9 Soluio o Howor ssig. Dvis progrssio of sps h icluds ll h spcis dd for ch rcio. L M C. CH C CH3 H C CH3 H C CH3H 4 Us h hrodyic o clcul h rgis of h vrious sgs of h rcio. C CH 4 + C : CH 3 + H + C : CH 3 + H + C : CH 3 H + C : G = = 445 J/ol G = = 43 J/ol G = = +46 J/ol G = = 99 J/ol Th civio rgy for h firs lry sp is oid fro h r cos. E / RT ,5/ T E R E 54,5 54,5 R 54, J ol 453 J / ol Th firs civio rrir p is 453 J/ol ov h firs lvl; h firs p is = +8 J/ol. For h scod lry sp h civio rgy for h rvrs rcio is E 4,3 R 4, J ol 34 J / ol Th scod civio rrir p is 34 J/ol ov h hird lvl; h scod p is = +8 J/ol. I h forwrd dircio, h scod civio rrir p is 8 43 = 3 J/ol. Th civio rgy for h hird rcio is ssud o zro. rgy - CH 3 + H + C CH 3 + H + C -4 CH 3 H + C -6 CH 4 + C rcio coordi
2 - -. To s h hory, w should plo h so h r quio is srigh li. If h cul sys dvis fro h proposd chis d ssupios, dviios fro h srigh li will ovious. To oi quio of h for y = x + ivr h r quio d uliply oh sids y. W should plo / o h x xis d // o h y xis. Th slop is / d h ircp is /. /r slop = / ircp = / / ohr xprssio yilds srigh li. Muliply h prvious xprssio y. I his cs, w would plo o h x xis d // o h y xis. Th slop is / d h ircp is /. /r slop = / ircp = /
3 Sr wih h rrhius forul for r coss d h logrih of ch sid. E xp RT l E l R T Thus h civio rgy c clculd fro h slop of plo of l vrsus /T. Rhr h us si-log grph ppr, w will prpr l of /T d l d h us sdrd grph ppr. Th r plod low. T C /T sc l l /T Th slop of h fid li c clculd fro h wo pois whr h li irscs h ordr:.3,.5 d.4, 6. E R E slop ris ru R J 4.9 ol J / ol To clcul h prxpoil, w clcul h prxpoil ch prur d clcul h vrg. T sc = xp[e /RT] Th vrg vlu is 6 3 sc. Bcus is xrly ssiiv o h civio rgy d h civio rgy is sowh ucri, hr will cosidrl vriio i his vlu.
4 Th gric rcio is proi durd proi ssu rrhius hvior for h r cos: = xpe /RT. To rch h s poi i h rcio, h rio of rcio i is ivrsly proporiol o h rio of r coss. x ius 9C 3 ius C C 9C C 9C xp E xp E / R373 / R365 Furhr ssu h h prxpoil is idpd of prur. x ius 9C 3 ius C xp E xp E.33 x.333 / R373 / R365 E xp R 373 4, J / ol xp 8.34 J ol 4 ius 365 No h h Gis rgy of h rcio is o dd Bgi wih ss lc o N, d[ 4 N] r of crio y uro sorpio r of dcy Th r of crio is rid y h uro flux. I his cs i is cos; r Th dcy of N os ol sc 3 6. N 36 sc N os hr N is firs ordr, ol hr 4 N r of dcy 4 [ N] For firs-ordr rcio h hlf lif is l /. Clcul. l / l 5 hr 4.6 hr Driv igrd r quio for [ N ]. 4 4 [ N] d[ r [ d[ r [ 4 d[ N] 4 4 N] 4 4 N] N] N] r [ 4 N] l 4 [ N] 4 r [ N]
5 - 5 - r l 4 [ r r 4 [ 4 [ [ N] 4 N] N] N] r r r r 4 Clcul [ N ] = 3 hrs. [ 4 N] 6. 4 ol N / hr / hr 4.6 / hr3 hr ol 4 N Chc h rsul. =, 4 r [ N]. This is corrc. Thr is o rdiociv N wh h uro ord srs. =, 4 v r [ N] cos fr log i > h sys rchs sdy s. Th r of forio quls h r of dcy. 6. Rwri h firs rcio s wo forwrd rcios d dd h rcios. su : N N M M N N 3 N N B Us h scod d hird rcios o wri r quio for N. M M N d[n] [N][] 3 [N][N] W d xprssios for h cocrios of h irdis d N. pply h sdy-s pproxiio o N. d[n] [N][] [N][N] 3 [N][] [N][N] 3 pply h sdy-s pproxiio o. d[] [ ][M] [] [M] [N][] 3[N][N] 3 Eq shows h h hird d fourh rs i q 3 ccl. Solv q 3 for [].
6 - 6 - [ ][M] [] [M] [] / [ ] / 4 Susiu q io. d[n] [N][] [N][] [N][] 5 Susiu q 4 io q 5. d[n] / / [N][] fr you wor xrcis 8 d rflc o xrcis 6, you y qusio if h rsul of h sdy-s pproxiio o N - q - is sufficily good pproxiio o jusify cclig hs rs i q 3. This ws h qusio posd y Tvi Ny Corll ChE 4. Th is, wh if [N][] =,, d 3 [N][N] =,,? lhough i is good pproxiio h [N][] = 3 [N][N], h diffrc w hs wo rs igh coprl o h diffrc w h firs wo rs i q 3. Th swr is providd y opic w will covr x w - chi rcios. Rcios d 3 for cycl. For vry occurrc of rcio hr is xcly o occurrc of rcio 3, o ii h ol ur of rdicls, d N os, i h cycl. So h pproxiio i q is vry good; r h h pproxiio of pr-quiliriu for rcio. Bu h rsul i q 4 lds o ohr qusio - is i vlid o ssu sdy-s pproxiio for? Th pproxiio is d[]/ =, u his lds o q 4, which shows [] is proporiol o [ ]. is produc; i icrss s h rcio procds; d[ ]/ = ½d[N]/. Us q 4 o clcul xprssio for d[]/. / d [ ] d / d[] [] / Th y is h rio / is sll. Th olr Gis rgy of rcio for is 464 J/ol. rcio prur of 6, / / is ou. I is good pproxiio h d[]/ =. 7. Wri r quio fro h scod rcio. d[p] pply h pr-quiliriu pproxiio o h firs rcio. [Q] [Q] Susiu h xprssio for io q. d[p] [Q] B gi, gi wih h r quio fro h scod rcio. d[p] pply h sdy-s pproxiio o C.
7 - 7 - d [Q] [Q] 4 gi, susiu h xprssio for io q. d[p] [Q] 8. Sr wih h diffril r quio for, d h spr d igr. l B Wri ss lc o h closd vssl. ss i h vssl iiilly, B, C, ss i h vssl lr B C Bcus iiilly h rcor cois oly, B, = C, =., B C Covr fro uis of ss o uis of olr cocrio y dividig ch r y h vssl volu d ch olculr wigh. No h, B d C ch hv h s olculr wigh. W hus rriv [ ] Th r of cosupio of is slow coprd o h r of ircovrsio w B d C. Usig h driig logy, dris slowly io B. Ts B d C r cocd y lrg pip, so hy quicly rch quiliriu. Th digr low ssus h C is hrodyiclly lowr h B, so hr is or C h B quiliriu. Th is, w ssud >. B C
8 - 8 - W hus ssu pos quiliriu. Th r h B covrs o C quls h r h C covrs o B. Susiu h xprssio for ov d h xprssio for drivd i pr io h ss lc. So suds glcd o ssu pos-quiliriu d isd solvd h full diffril quio: d Th full soluio is B] [ Nor Rizvi d Rsh Hood oh Corll ChE 9 sd isighful qusio: If w c ssu posquiliriu, such h =, h w should l o ccl hs rs fro h r quio for, s such d This xprssio for is diffr fro h xprssio for drivd ov, y oh usd pos-quiliriu ssupio. Which si is r? d i grl, wh c quiliriu ssupios usd o ccl opposig rs i diffril r quio? To xplor which si is r, w sr wih h xprssio for oid wih o ssupios, d pply h fcs h «d «, s giv i h xrcis s. Th xpoils xp + will dcrs o zro uch fsr h h xpoils xp. Wih hs ssupios, h xc xprssio for siplifis s follows.
9 - 9 - W oi h xprssio drivd y susiuig igrd r quio for io h ss lc. This is h r si. So why is i lss ccur o us h pos-quiliriu pproxiio o ccl rs i h r quio for? Th y is h, or. How sll is h diffrc w d? Th diffrc is pproxily qul o h riig r,. For xpl, cosidr =,,, =,,, d =. I is good pproxiio h, u i is o good pproxiio h + =. I his cs, h rror is fcor of. r cosidr ohr xplio. s ov, if w ssu pos-quiliriu, such h =, h hs rs ccl fro h r quio for, s such d Rcll h diffril r quio for, Wih h pos-quiliriu pproxiio = pplid o h r quio for B, w hv h rsul d This pproxiio yilds h icorrc rsul h ll h rcs o for B; o C. I chicl ss, h sligh diffrc i h dyic lc of h quiliriu w B d C is pproxily qul o h r h covrs o B. So wh of h pr-quiliriu pproxiio, for xpl, i h sris rcio, B C such h «d «? Th pr-quiliriu pproxiio givs us. Is i vlid o ccl hs rs i h r quio for, such s d? This rsol pproxiio, u o s ccur s firs igrig h r quio for, siilr o pr, Susiu his xprssio for io h diffril r quio for, d h igr. So, cclig h rs d would rsol pproxiio prhps wihi fcor of, dpdig o h rliv sizs of, d. Copyrigh Corll Uivrsiy 9. ll righs rsrvd. Rproducio or rrsissio of his docu, i whol or i pr, i y r, wihou h wri cos of T.M. Duc, is violio of copyrigh lw.
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