Dr Mike Lyons Room 3.2 Chemistry Building School of Chemistry Trinity College Dublin. Course Summary.

Transcription

1 SF Chem. Chemial Kineis /. Dr Mie Lyons Room 3. Chemisry Building Shool of Chemisry Triniy College Dublin. melyons@d.ie Course Summary. Cona shor bu swee. 5 Leures in oal (4 his wee, nex wee, 3 uorials nex wee). We revise quaniaive aspes of JF ineis and disuss some new more advaned opis and inrodue he mahemaial heory of hemial ineis. Topis inlude: Leure -. Quaniaive hemial ineis, inegraion of rae equaions, zero, firs, seond order ases, rae onsan. Graphial analysis of rae daa for rae onsan and half life deerminaion for eah ase. Dependene of rae on emperaure. rrhenius equaion and aivaion energy. Kineis of omplex mulisep reaions. Parallel and onseuive reaions. Conep of rae deermining sep and reaion inermediae. Leure 3,4. Enzyme ineis (Mihaelis-Menen ase) and surfae reaions involving adsorbed reaans (Langmuir adsorpion isoherm). Leure 5. Theory of hemial reaion raes : bimoleular reaions. Simple Collision Theory & ivaed Complex Theory.

2 Reommended reading. Burrows e al Chemisry 3, OUP Chaper 8. pp P.W. ins J. de Paula.The elemens of physial hemisry. 4h ediion. OUP (5). Chaper, pp.9-56; Chaper, pp P.W. ins and J. de Paula. Physial Chemisry for he Life Sienes. s ediion. OUP (5). Par II eniled The ineis of life proesses (Chapers 6,7,8) is espeially good. Boh of hese boos by well esablished auhors are learly wrien wih an exellen syle and boh provide an exellen basi reamen of reaion ineis wih emphasis on biologial examples. These boos are se a jus he righ level for he ourse and you should mae every effor o read he reommended hapers in deail. lso he problem shees will be based on problems a he end of hese hapers! P.W. ins, J. de Paula. Physial Chemisry. 8h Ediion. OUP (6). Chaper, pp ; Chaper 3, pp ; Chaper 4, pp more advaned and omplee aoun of he ourse maerial. Muh of haper 4 is JS maerial. M.J. Pilling and P/W. Seains. Reaion Kineis. OUP (995). Modern exboo providing a omplee aoun of modern hemial reaion ineis. Good on experimenal mehods and heory. M. Robson Wrigh n inroduion o hemial ineis. Wiley (5) noher modern ineis exboo whih does as i saes in he ile, i.e. provide a readable inroduion o he subje! Well worh browsing hrough. SF Chemial Kineis. Leure -. Quaniaive Reaion Kineis.

3 Reaion Rae: The Cenral Fous of Chemial Kineis The wide range of reaion raes. Reaion raes vary from very fas o very slow : from femoseonds o enuries! femoseond (fs) = -5 s = / 5 s! 3

4 Pioseond ( - s) ehniques Femoseond ( -5 s)ehniques ins de Paula, Elemens Phys. Chem. 5 h ediion, Chaper, seion., pp.- Reaions sudies under onsan emperaure ondiions. Mixing of reaans mus our more rapidly han reaion ours. Sar of reaion pinpoined auraely. Mehod of analysis mus be muh faser han reaion iself. Chemial reaion ineis. Chemial reaions involve he forming and breaing of hemial bonds. Reaan moleules (H, I ) approah one anoher and ollide and inera wih appropriae energy and orienaion. Bonds are srehed, broen and formed and finally produ moleules (HI) move away from one anoher. How an we desribe he rae a whih suh a hemial ransformaion aes plae? reaans produs H( g) I( g) HI( g) Thermodynamis ells us all abou he energei feasibiliy of a reaion : we measure he Gibbs energy DG for he hemial Reaion. Thermodynamis does no ell us how quily he reaion will proeed : i does no provide inei informaion. 4

5 Basi ideas in reaion ineis. Chemial reaion ineis deals wih he rae of veloiy of hemial reaions. We wish o quanify The veloiy a whih reaans are ransformed o produs The deailed moleular pahway by whih a reaion proeeds (he reaion mehanism). These objeives are aomplished using experimenal measuremens. We are also ineresed in developing heoreial models by whih he underlying basis of hemial reaions an be undersood a a mirosopi moleular level. Chemial reaions are said o be aivaed proesses : energy (usually hermal (hea) energy) mus be inrodued ino he sysem so ha hemial ransformaion an our. Hene hemial reaions our more rapidly when he emperaure of he sysem is inreased. In simple erms an aivaion energy barrier mus be overome before reaans an be ransformed ino produs. Reaion Rae. Wha do we mean by he erm reaion rae? The erm rae implies ha somehing hanges wih respe o somehing else. How may reaion raes be deermined? R The reaion rae is quanified in erms of he hange in onenraion of a reaan or produ speies wih respe o ime. This requires an experimenal measuremen of he manner in whih he onenraion hanges wih ime of reaion. We an monior eiher he onenraion hange direly, or monior hanges in some physial quaniy whih is direly proporional o he onenraion. The reaan onenraion dereases wih inreasing ime, and he produ onenraion inreases wih inreasing ime. The rae of a hemial reaion depends on he onenraion of eah of he pariipaing reaan speies. The manner in whih he rae hanges in magniude wih hanges in he magniude of eah of he pariipaing reaans is ermed he reaion order. Reaan onenraion R d Ne reaion rae Unis : mol dm -3 s - [P] ime [R] Produ onenraion dp 5

6 Geomeri definiion of reaion rae. Rae expressed as angen line To onenraion/ime urve a a Pariular ime in he reaion. dr [ ] R d P R Reaion Raes and Reaion Soihiomery O 3 (g) + NO(g) NO (g) + O (g) d O rae = - 3 d[no] = - d[no ] = + d[o ] = + Reaion rae an be quanified by monioring hanges in eiher reaan onenraion or produ onenraion as a funion of ime. 6

7 H O (aq) H O (l) + O (g) The general ase. Why do we define our rae in his way? removes ambiguiy in he measuremen of reaion raes in ha we now obain a single rae for he enire equaion, no jus for he hange in a single reaan or produ. a bb Rae pp qq B d d R a b dq d q p P 7

8 The reaion rae (reaion veloiy) R is quanified in erms of hanges in onenraion [J] of reaan or produ speies J wih respe o hanges in ime. The magniude of he reaion rae hanges as he reaion proeeds. R J Rae, rae equaion and reaion order : formal definiions. J D lim D D J J dj H ( g) O ( g) H O( g) d R H do dh O Noe : Unis of rae :- onenraion/ime, hene R J has unis mol dm -3 s -. J denoes he soihiomeri oeffiien of speies J. If J is a reaan J is negaive and i will be posiive if J is a produ speies. Rae of reaion is ofen found o be proporional o he molar onenraion of he reaans raised o a simple power (whih need no be inegral). This relaionship is alled he rae equaion. The manner in whih he reaion rae hanges in magniude wih hanges in he magniude of he onenraion of eah pariipaing reaan speies is alled he reaion order. Reaion rae and reaion order. The reaion rae (reaion veloiy) R is quanified in erms ofhanges in onenraion [J] of reaan or produ speies J wih respe o hanges in ime. The magniude of he reaion rae hanges (dereases) as he reaion proeeds. Rae of reaion is ofen found o be proporional o he molar onenraion of he reaans raised o a simple power (whih need no be inegral). This relaionship is alled he rae equaion. The manner in whih he reaion rae hanges in magniude wih hanges in he magniude of he onenraion of eah pariipaing reaan speies is alled he reaion order. Hene in oher words: he reaion order is a measure of he sensiiviy of he reaion rae o hanges in he onenraion of he reaans. 8

9 Woring ou a rae equaion. = 5. x -3 s - NO5 ( g) 4NO ( g) O ( g) T = 338 K Iniial rae deermined by evaluaing angen o onenraion versus ime urve a a given ime. Rae Equaion. Iniial rae is proporional o iniial onenraion of reaan. ( rae) ( rae) NO5 N O 5 d N O rae = rae onsan 5 N O 5 x yb Produs soihiomeri empirial rae equaion (obained from experimen) oeffiiens Reaion order deerminaion. Vary [], eeping [B] onsan and measure rae R. Vary [B], eeping [] onsan and measure rae R. d R x rae onsan B d y B, = reaion orders for he reaans (go experimenally) Rae equaion an no in general be inferred from he soihiomeri equaion for he reaion. Log R Log R Slope = Log [] Slope = Log [B] 9

10 Differen rae equaions imply differen mehanisms. H X X HX I, Br, Cl H H d R I H I / HBr H Br HBr Br Cl HI d HI R H Br HBr HCl d HCl R H Cl / The rae law provides an imporan guide o reaion mehanism, sine any proposed mehanism mus be onsisen wih he observed rae law. omplex rae equaion will imply a omplex mulisep reaion mehanism. One we now he rae law and he rae onsan for a reaion, we an predi he rae of he reaion for any given omposiion of he reaion mixure. We an also use a rae law o predi he onenraions of reaans and produs a any ime afer he sar of he reaion. Inegraed rae equaion. Burrows e al Chemisry 3, Chaper 8, pp ins de Paula 5 h ed. Seion.7,.8, pp.7-3 Many rae laws an be as as differenial equaions whih may hen be solved (inegraed) using sandard mehods o finally yield an expression for he reaan or produ onenraion as a funion of ime. We an wrie he general rae equaion for he proess Produs as d F() where F() represens some disin funion of he reaan onenraion. One ommon siuaion is o se F() = n where n =,,, and he exponen n defines he reaion order wr he reaan onenraion. The differenial rae equaion may be inegraed one o yield he soluion = () provided ha he iniial ondiion a zero ime whih is = is inrodued.

11 Zero order ineis. The reaion proeeds a he same rae R regardless of onenraion. Rae equaion : d R when inegrae using iniial ondiion () unis of rae onsan : mol dm -3 s - slope = - R half life R / when / / diagnosi plo / slope Firs order ineis. Iniial onenraion d rae produs Firs order differenial rae equaion. d Iniial ondiion Firs order reaion ( rae) ( rae) = firs order rae onsan, unis: s - Solve differenial equaion Via separaion of variables ( ) e exp Reaan onenraion as funion of ime.

12 Firs order ineis. Half life / / / u / s ln /.693 ( ) e exp () u exp Mean lifeime of reaan moleule e Firs order ineis: half life. In eah suessive period of duraion / he onenraion of a reaan in a firs order reaion deays o half is value a he sar of ha period. fer n suh periods, he onenraion is (/) n of is iniial value. / u a a u / / u.5 u.5 u.5 ln /.693 half life independen of iniial reaan onenraion

13 Seond order ineis: equal reaan onenraions. d separae variables inegrae P dm 3 mol - s - slope = half life / / / as / / rae varies as square of reaan onenraion s order ineis nd order ineis u ( ) u( ) () e u( ) e () u( ) 3

14 s and nd order ineis : Summary. Reaion Differenial rae equaion Conenraion variaion wih ime Diagnosi Equaion Half Life d Produs () exp ln ( ) ln ln / d () Produs () / ln () Slope = - /() / s order Diagnosi Plos. Slope = nd order n h order ineis: equal reaan onenraions. d n Half life n P separae variables inegrae n n n n n =,,3,.. rae onsan obained from slope n slope n n / n n ln ln ln n n / n ln / slope n n / n as / n as / reaion order n deermined from slope ln 4

15 Summary of inei resuls. Rae equaion n P / d R Reaion Order 3 n Inegraed expression Unis of Half life / mol dm -3 s - s - ln ln 3 dm 3 mol - s - dm 6 mol - s - 3 n n n n n n n Seond order ineis: Unequal reaan onenraions. rae equaion da db dp R ab iniial ondiions a a b b a b inegrae using parial fraions F a, b B P / b a half life ln a slope = dm 3 mol - s - Pseudo firs order ineis when b >>a / ln ln b b pseudo s order rae onsan F b a b b a a a, b ln 5

16 Temp Effes in Chemial Kineis. ins de Paula Elemens P Chem 5 h ediion Chaper, pp.3-34 Burrows e al Chemisry 3, Seion 8.7, pp

17 Energy Van Hoff expression: dln K DU dt RT P Sandard hange in inernal energy: DU E E R K P d d ln d ln DU ln dt dt dt RT P R E d ln E dt RT d ln E dt RT DU TS E Reaion oordinae P This leads o formal definiion of ivaion Energy. E Temperaure effes in hemial ineis. Chemial reaions are aivaed proesses : hey require an energy inpu in order o our. Many hemial reaions are aivaed via hermal means. The relaionship beween rae onsan and emperaure T is given by he empirial rrhenius equaion. The aivaion energy E is deermined from experimen, by measuring he rae onsan a a number of differen E exp emperaures. The rrhenius equaion is used o onsru an rrhenius plo of ln versus /T. The aivaion energy is deermined from he slope of his plo. d R d ln d ln RT / T dt ln Pre-exponenial faor E Slope R T RT 7

18 8

19 In some irumsanes he rrhenius Plo is urved whih implies ha he ivaion energy is a funion of emperaure. Hene he rae onsan may be expeed o vary wih emperaure aording o he following expression. at m E exp RT We an relae he laer expression o he rrhenius parameers and E as follows. E ln ln a mlnt RT d ln m E dt T RT E E mrt E RT RT E mrt Hene m m E E at e exp exp RT RT at e m m Svane ugus rrhenius Conseuive Reaions. Moher / daugher radioaive deay. 8 Po 4 5 Pb s 3 4 Bi 6 da a dx a x dp x s 4 3 oupled LDE s define sysem : a( ) a exp a x( ) X P Mass balane requiremen: p a a x The soluions o he oupled equaions are : p( ) a a exp exp exp a exp exp We ge differen inei behaviour depending on he raio of he rae onsans and 9

20 energy Conseuive reaion : Case I. Inermediae formaion fas, inermediae deomposiion slow. Case I. TS I TS II X P DG I DG II I : fas DG I << DG II II : slow rds X P reaion o-ordinae Sep II is rae deermining sine i has he highes aivaion energy barrier. The reaan speies will be more reaive han he inermediae X. X P u v w a a x a p a Normalised onenraion Case I. / =. Inermediae X Reaan Iniial reaan more reaive han inermediae X. Produ P u = a/a v = x/a w = p/a Conenraion of inermediae signifian over ime ourse of reaion.

21 energy Conseuive reaions Case II: Inermediae formaion slow, inermediae deomposiion fas. Case II. X P X P ey parameer Inermediae X fairly reaive. [X] will be small a all imes. TS I I : slow rds II : fas DG I TS II DG I >> DG II DG II Sep I rae deermining sine i has he highes aivaion energy barrier. X P reaion o-ordinae X P Case II. Inermediae X is fairly reaive. Conenraion of inermediae X will be small a all imes.. normalised onenraion normalised onenraion Reaan Produ P Inermediae X u=a/a v=x/a w=p/a = X P = u=a/a v=x/a w=p/a = / = Inermediae onenraion is approximaely onsan afer iniial induion period.

22 X P Fas Slow Rae Deermining Sep Reaan deays rapidly, onenraion of inermediae speies X is high for muh of he reaion and produ P onenraion rises gradually sine X--> P ransformaion is slow. Rae Deermining Sep X P Slow Reaan deays slowly, onenraion of inermediae speies X will be low for he duraion of he reaion and o a good approximaion he ne rae of hange of inermediae onenraion wih ime is zero. Hene he inermediae will be formed as quily as i is removed. This is he quasi seady sae approximaion (QSS). Fas Parallel reaion mehanism. We onsider he inei analysis of a onurren or parallel reaion sheme whih is ofen me in real siuaions. single reaan speies an form wo disin produs. We assume ha eah reaion exhibis s order ineis. Iniial ondiion : =, a = a ; x =, y =. Rae equaion: R da a a a a a a( ) a exp exp Half life: / ln ln ll of his is jus an exension of simple s order ineis. X Y, = s order rae onsans We an also obain expressions for he produ onenraions x() and y(). dx a a exp x( ) a a x( ) dy y( ) a exp a y( ) exp a a exp Final produ analysis yields rae onsan raio. exp exp x( ) Lim y( )

23 Parallel Mehanism: >> normalised onenraion a(). x() = / / y() v. 979 u() v() w() w. 98 Theory. v w( ) = Parallel Mehanism: >> Theory normalised onenraion v. w( ) a() x() y() = w( ).99 u() v() w() v( ).99 3

24 onenraion Reahing Equilibrium on he Marosopi and Moleular Level NO N O 4 N O 4 (g) olourless NO (g) brown Chemial Equilibrium : a inei definiion. Counless experimens wih hemial sysems have shown ha in a sae of equilibrium, he onenraions of reaans and produs no longer hange wih ime. This apparen essaion of aiviy ours beause under suh ondiions, all reaions are mirosopially reversible. We loo a he dinirogen eraoxide/ nirogen oxide equilibrium whih ours in he gas phase. Conenraions vary wih ime NO N O 4 Kinei regime Conenraions ime invarian NO N eq O4 eq Equilibrium sae NO N O 4 (g) NO (g) olourless Kinei analysis. R NO4 R NO Valid for any ime Equilibrium: brown R R N O 4 NO eq NO eq K N O 4 eq Equilibrium onsan ime N O 4 4

25 Firs order reversible reaions : undersanding he approah o hemial equilibrium. ' B Rae equaion da a b Iniial ondiion a a b Mass balane ondiion a b a Inrodue normalised variables. u a a v b a u v u v Rae equaion in normalised form du u d Soluion produes he onenraion expressions u Reaion quoien Q v exp exp Q v u exp exp Firs order reversible reaions: approah o equilibrium.. Kinei regime onenraion Reaan Produ B u () v () Equilibrium K Q v u (+') 5

26 Undersanding he differene beween reaion quoien Q and Equilibrium onsan K. pproah o Equilibrium Q < Q() 8 6 Equilibrium Q = K = = (+') Q v u exp exp Q K K u v Kinei versus Thermodynami onrol. In many hemial reaions he ompeiive formaion of side produs is a ommon and ofen unwaned phenomenon. I is ofen desirable o opimize he reaion ondiions o maximize seleiviy and hene opimize produ formaion. Temperaure is ofen a parameer used o modify seleiviy. Refer o JCE papers dealing wih his opi given as exra handou. Operaing a low emperaure is generally assoiaed wih he idea of high seleiviy (his is ermed inei onrol). Conversely, operaing a high emperaure is assoiaed wih low seleiviy and orresponds o Thermodynami onrol. Time is also an imporan parameer. a given emperaure, alhough he ineially onrolled produ predominaes a shor imes, he hermodynamially onrolled produ will predominae if he reaion ime is long enough. 6

27 Shor reaion imes ssume ha reaion produ P is less sable han he produ P. lso is formaion is assumed o involve a lower aivaion energy E. Temperaure effe. Kinei onrol. ssume ha energy of produs P and P are muh lower han ha of he reaan R hen E, <<E,- and E, << E,-. low emperaure one negles he fraion of moleules having an energy high enough o re-ross he energy barrier from produs o reaans. Under suh ondiions he produ raio [P ]/[P ] is deermined only by he aivaion barriers for he forward R P reaion seps. Thermodynami onrol. high emperaure he available hermal energy is onsidered large enough o assume ha energy barriers are easily rossed. Thermodynami equilibrium is reahed and he produ raio [P ]/[P ] is now deermined by he relaive sabiliy of he produs P and P. P K DG DG exp P K RT j * DG j exp RT P DG DG exp P RT * * 7

28 d R dp R P dp R P R P P Shor ime pproximaion. Negle -, - erms. R R exp R P exp R P exp P P R P R P Long ime approximaion. Kinei onrol Limi. d R d P d P P P R R K K P P K R K K K R K K K K Thermodynami onrol limi. General soluion valid for inermediae imes. R R exp exp P R exp exp P R exp exp 4 4, These expressions reprodue he orre limiing forms orresponding o inei and Thermodynami onrol in he limis of shor and long ime respeively. 8

29 Deailed mahemaial analysis of omplex reaion mehanisms is diffiul. Some useful mehods for solving ses of oupled linear differenial rae equaions inlude marix mehods and Laplae Transforms. In many ases uilisaion of he quasi seady sae approximaion leads o a onsiderable simplifiaion in he inei analysis. Quasi-Seady Sae pproximaion. QSS induion period. X P Conseuive reaions =.. The QSS assumes ha afer an iniial induion period (during whih he onenraion x of inermediaes X rise from zero), and during he major par of he reaion, he rae of hange of onenraions of all reaion inermediaes are negligibly small. Normalised onenraion.5 u() w() v() P Mahemaially, QSS implies dx R R R X formaion X formaion R X removal X removal...5. = inermediae X onenraion approx. onsan QSS: a fluid flow analogy. QSS illusraed via analogy wih fluid flow. If fluid level in an is o remain onsan hen rae of inflow of fluid from pipe mus balane rae of ouflow from pipe. Reaion inermediae onenraion equivalen o fluid level. Inflow rae equivalen o rae of formaion of inermediae and ouflow rae analogous o rae of removal of inermediae. P Fluid level P 9

30 Conseuive reaion mehanisms. Rae equaions X P - u du u v d dv u v d dw v d a a v u v w u x a w p a v w Definiion of normalised variables and iniial ondiion. u v Sep I is reversible, sep II is Irreversible. Coupled LDE s an be solved via Laplae Transform or oher mehods. exp exp exp exp w exp exp Noe ha and are omposie quaniies onaining he individual rae onsans. QSS assumes ha dv d u v v ss ss uss ss duss u d u v ss ss ss exp exp Using he QSS we an develop more simple rae equaions whih may be inegraed o produe approximae expressions for he perinen onenraion profiles as a funion of ime. The QSS will only hold provided ha: he onenraion of inermediae is small and effeively onsan, and so : he ne rae of hange in inermediae onenraion wr ime an be se equal o zero. dwss v d ss exp wss exp d exp 3

31 normalised onenraion log Conenraion versus log ime urves for reaan, inermediae X, and produ P when he rae equaions are solved using he QSS. Values used for he rae onsans are he same as hose used above. QSS reprodues he onenraion profiles well and is valid. QSS will hold when onenraion of inermediae is small and onsan. Hene he rae onsans for geing rid of he inermediae ( - and ) mus be muh larger han ha for inermediae generaion ( ). X P u() v() w() normalised onenraion Conenraion versus log ime urves for reaan, inermediae X and produ P when full se of oupled rae equaions are solved wihou any approximaion. - >>, >> and - = = 5. The onenraion of inermediae X is very small and approximaely onsan hroughou he ime ourse of he experimen. X P - X.. log P u ss () v ss () w ss () normalised onenraion log Conenraion versus log ime urves for reaan, inermediae X and produ P when he Coupled rae equaions are solved using he quasi seady sae approximaion. The same values for he rae onsans were adoped as above. The QSS is no good in prediing how he inermediae onenraion varies wih ime, and so i does no apply under he ondiion where he onenraion of inermediae will be high and he inermediae is long lived. X u() v() w() P normalised onenraion X P - Conenraion versus log ime urves for reaan, inermediae X and produ P when full se of oupled rae equaions are solved wihou any approximaion. - <<,,, and - = =. The onenraion of inermediae is high and i is presen hroughou muh of he duraion of he experimen. u ss () v ss () w ss ().. log P X 3