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1 Chap 8 Experimental Kinetics and Gas Reactins Table f Cntents 8. Rate Equatin 8.2 Order f Reactin 8.3 Reversible First-Order Reactins 8.4 Cnsecutive First-Order Reactins 8.5 Micrscpic Reversibility and Detailed Balance 8.6 Effect f Temperature, 8.7 Mechanisms f Chemical Reactins 8.8 Relatin between Rate Cnstants fr the Frward and Backward Reactins 8.9 Bimlecular Reactins 8.0 Unimlecular and Trimlecular Reactins 8. Unbranched Chain Reactins 8.2 Branched Chain Reactins 2006/03/4

2 Rate f Equatin 3 () N (2) H2 ( g) O2( g) H2O() l 2 ( g) + H ( g) NH ( g) rg m / kj ml () T P. (2) 2006/03/4 2

3 Rate Equatin f Chemical Reactin (rate equatin f chemical reactin) dx /dt Rate f cnsumptin f reactant R: -d [R]/dt Rate f frmatin f prduct P: d [P]/dt 2006/03/4 3

4 Experimental Kinetics and Gas Reactins (b) (a) The definitin f (instantaneus) rate as the slpe f the tangent drawn t the curve shwing the variatin f any prduct cncentratin with time. (b) Fr negative slpes in reactants, the sign is changed when reprting the rate, s all reactin rates are psitive. (a) 2006/03/4 4

5 Rate f Reactin i, B i i, i. ν i (, the extent f reactin). i n i n i + mle n i i, n i ν i i., R P, t0 n R (0) n P (0) tt n R (t) n P (t) nr(t) - nr(0) ξ - α 0 i νb n P(t) - n P(0) β 2006/03/4 5

6 Rate f Reactin i dn i ν i d, ν i i. : dξ dnb ν t B dn i dt ν i dξ dt ξ & d ξ dt ν i dn dt i 2006/03/4 6

7 2006/03/4 7 Rate f Equatin :, + 2B X, :,. : ( ) [ ] t B t n t ξ V υ d d ν d V d ν d d i i i i [ ] [ ] [ ] t X t B - t - υ d d d d d d 2 [ ] ( ) [ ] ( ) [ ] t V V B t B t V B V υ i i i d d ν d d ν d d ν i i i +

8 Experimental Kinetics and Gas Reactins Example 8. Rate f cnversin and rate f reactin The reactin H 2 (g) + Br 2 (g) 2HBr(g) is carried ut in a L reactin vessel. The change in the amunt f Br 2 in 0.0 s is ml. (a) What is the rate f cnversin d /dt? (b) What is the rate f reactin? (c) What are the values f d[h 2 ]/dt, d[br 2 ]/dt, and d[hbr]/dt? ns: (a) (b) (c) dξ ml - 0. ml s dt 0 0. s - dξ 0. ml s υ V dt L d [ H ] ml L - s dt d [ Br ] ml L - s dt d [ HBr] ml L - s dt ml L 2006/03/4 8 s

9 Experimental Kinetics and Gas Reactins Example 8.2 Dependence f Reactin Rate n the chemical equatin In discussin the rate f chemical reactin, it is imprtant t knw hw the stichimetric equatin is written as 2 + B 2C and + ½ B C ns: ccrding t the first stichimetric equatin υ - 2 [ ] d [] B d [ C] d dt - dt dt ccrding t the secnd stichimetric equatin 2 υ - [ ] 2 d [] B d [ C] d dt - dt dt 2006/03/4 9

10 Experimental Kinetics and Gas Reactins t () (2) ( ) (IR UV-VIS ESR NMR ESC ) 2006/03/4 0

11 Experimental Kinetics and Gas Reactins Quenching Methd: ( ) chemical reactin is quenched if the cnditins are suddenly changed (such as by sudden dilutin r cling f the reactin mixture) s that its rate is reduced t clse t zer. The technique is used t determine the cmpsitin f a reactin mixture at an arbitrary stage f a reactin. Flash Phtlysis technique( ): reactin is induced by expsure f the mixed reactants t a brief phtlytic flash f light and the ensuing reactin is mnitred by spectrscpic technique. In this way the reactins lasting nly picsecnds can be studied. 2006/03/4

12 Experimental Kinetics and Gas Reactins Real-time analysis: nalysis f the cmpsitin f the reactin mixture as the reactin prceeds. Flw methd : flwing reactins are mixed tgether and then pass alng an utlet tube. Observing the mixture cmpsitin at increasing distances alng the tube crrespnds t mnitring the reactin at different times after mixing. Stpped-flw technique is fr the study f fast reactins. Slutins f the reactants are impelled int a mixture chamber as a pistn is withdrawn suddenly t a stp. The cmpsitin in the chamber is then mnitred. 2006/03/4 2

The lcatin f the spectrmeter crrespnds t different times after initiatin.

13 Experimental Kinetics and Gas Reactins 2 The arrangement used in the flw technique fr studying reactin rates. The reactants are injected int the mixing chamber at a steady rate. The lcatin f the spectrmeter crrespnds t different times after initiatin. In the stpped-flw technique the reagents are driven quickly int the mixing chamber by the driving syringes and then the time dependence f the cncentratins is mnitred. 2006/03/4 3

14 Experimental Kinetics and Gas Reactins If reactin ges essentially t cmpletin, the rate equatin fr a reactin may be fund experimentally t be k [] [B] then it is classified as a rder in and -rder in B. In this rate equatin, k is the rate cnstant, and and is nt necessarily an integer, all are independent f cncentratin and time. The verall rder f a reactin is the sum f all the individual rders. In this case, the verall rder is /03/4 4

15 The rate law f reactin is determined experimentally, and cannt in general be inferred frm the chemical equatin fr the reactin. Fr example, a reactin with very simple stichimetry: ( g) + Br ( g) 2HBr( g) H2 2 but its rate law is cmplicated: υ [ ][ ] 3/ 2 k H2 Br2 [ Br ] + k '[ HBr] 2 In certain case the rate law des reflect the stichimetry f the reactin; but is either a cincidence r reflects a feature f the underlying reactin mechanism. 2006/03/4 5

16 First-rder Rate Equatin The rate equatin fr a first-rder reactin P [ ] d υ - dt k[ ] [ ] [ ] d r - kdt If the cncentratin f is [] at t and [] 2 at t 2, [ ] [ ] d [ ] [ ] [] 2 t - 2 k dt, r ln k t2 [] t 2 ( ) If the initial cncentratin f is [] at t 0 ln [ ] [ ] plt f ln[] versus t gives a slpe f -k, the rate cnstant has a unit f s - -t [ ] [ ] e-k, ln[ ] ln[ ] - t kt, r t k 2006/03/4 6

17 Experimental Kinetics and Gas Reactins 3 The expnential decay f the reactant in a first-rder reactin. The larger the rate cnstant, the mre rapid the decay: here k large 3k small. [ ] [ ] e- kt, 2006/03/4 7

18 First rder reactin The linear plt fr the first rder reactin: ln [ ] t kt [ ] 2006/03/4 8

19 First-rder Rate Equatin The rate equatin fr a first-rder reactin a P [ ] d υ - a dt k[ ] [ ] [ ] d If the cncentratin f is [] at t and [] 2 at t 2, [ ] [ ] If the initial cncentratin f is [] at t 0 akdt k dt plt f ln[] versus t gives a slpe f -k, the rate cnstant has a unit f s - r - [ ] [ ] [] 2 d t - 2 k dt, r ln k [] 2 t [ ] [ ] 2 ( t -t ) [ ] [ ] e-k t, ln [ ] ln [ ] - k t ln kt, r 2006/03/4 9

20 Exercise nalysing a first-rder reactin The variatin in the partial pressure f azmethane with time was fllwed at 600 K with the results given belw. Cnfirm that the decmpsitin is first rder in azmethae, CH 3 N 2 CH 3 (g) CH 3 CH 3 (g) + N 2 (g) and find the rate cnstant at 600 K. t( s) P/(trr) /03/4 20

21 Exercise t (s) P/(trr) t (s) ln (P/P ) Methd: T cnfirm that a reactin is first-rder, plt ln ([]/[] ) against time and expect a straight line. Because the partial pressure f a gas is prprtinal t its cncentratin, an equivalent prcedure is t plt ln (P/P ) vs. t and its slpe can be identified as k. 2006/03/4 2

22 Experimental Kinetics and Gas Reactins 4 The determinatin f the rate cnstant f a first-rder reactin: a straight line is btained when ln [], r, as here, ln (p/pº) is pltted against t; the slpe gives -k. 2006/03/4 22

23 Exercise 2 In the experiment f N 2 O 5 decmpsitin in liquid brmine, the cncentratin f N 2 O 5 varied with time as fllws: t/s [N 2 O 5 ]/(ml L - ) Cnfirm that the reactin is first-rder in N 2 O 5 and determine the rate cnstant with prper unit. ns: The plt f ln [N 2 O 5 ]/[N 2 O 5 ] versus t is a straight line with a slpe f -2.x0-3 s /03/4 23

24 First-rder Rate Equatin The half-lift t ½ f a reactin is the time required fr half f the reactant t disappear. Fr st rder reactin P: k t /2 ln [ ] ln 2 [ ] 2 t/2 t/2 In terms f relaxatin time ln 2 t /2 τ ln 2 k Fr st rder reactin, relaxatin time /k. [ ] [ ] - / τ [ ] [ ] k t e, r ln ln - t τ 2006/03/4 24

25 Experimental Kinetics and Gas Reactins Fig. 8. Fr a first-rder reactin, ne-half f the reactant disappears in t ½ independent f the initial cncentratin. 2006/03/4 25

26 Experimental Kinetics and Gas Reactins Fig. 8.a fr a first-rder reactin, the relaxatin time is independent f the initial cncentratin. []/e []/e 2 []/e /03/4 26

$t [ ] [ ] ln t [ ] [ ] x f as the fractin f$

27 First-rder ismerizatin n ismerizatin f cyclprpane, the rate cnstant is fund by: k ln t [ ] [ ] ln t [ ] [ ] x f as the fractin f ismerizatin x ln lg t [ ] t ( f ) 2006/03/4 27

28 Secnd-rder Rate Equatin The rate equatin fr a secnd-rder reactin P [ ] d υ - dt k[ ] 2 d[ ] [ ] r - 2 kdt If the cncentratin f is [] at t and [] 2 at t 2, [] [] 2 d[ ] t2 -k dt, r - [ ] 2 t [ ] [ ] - - If the initial cncentratin f [] [] at t 0 - [ ] [ ] kt, ( t - t ) plt f /[] versus t is linear with a slpe f k, the rate cnstant has a unit f L ml - s - 2 and t /2 -k k 2 [ ] 2006/03/4 28

29 Secnd-rder reactin plt f /[] versus t is linear with a slpe f k. - [ ] [ ] kt, [ ] [ ] [ ] kt k t eff 2006/03/4 29

30 Secnd-rder Rate Equatin The rate equatin fr a secnd-rder reactin a P [ ] d υ - a dt k[ ] 2 d[ ] [ ] If the cncentratin f is [] at t and [] 2 at t 2, If the initial cncentratin f is [] at t 0, - [ ] [ ] [ ] [ ] 2 - akt, akdt k dt The half-life t ½ is inversely prprtinal t the initial cncentratin. r ak - ( ) t 2 -t and t /2 2 ak [ ] 2006/03/4 30

31 Experimental Kinetics and Gas Reactins Fig. 8.2 Fr a secnd-rder reactin + B Prducts, with [] [B], the half-life t ½ is inversely prprtinal t [] ; that is, t ½ /k[], thus, when [] is reduced by a factr f 2, the half-life dubles. 2006/03/4 3

32 Secnd-rder Rate Equatin The rate equatin fr a secnd-rder reactin + B P T integrate this equatin, it is cnvenient if we define a prgress variable x [] -[] t [B] -[B] t, here the initial cncentratins at t0 are [] fr and [B] fr B, als [] [B]. Since dx dt [ ] d υ - dt k kt k[ ][ B] ([ ] )([] ) - x - B ([] [ ]) B ln - x [ ] [ ] x ( t) x ( 0) - x ([ ] - x )([] B - x ) - ln dx [] B [] B 2006/03/ x x dx x dx x dx x B - x - B B - x - B - x ([ ] )[] ( ) 0 ([ ] [ ] )[] ( ) 0 ([ ] [ ] )[ ( ] ) k t 0 dt

33 Calculus T slve the integral b are cnstants, use the methd f partial fractins. First, ( a -x )( b-x ) kt ( a -x )( b x ) and integrate the expressin n the right. x dx ( a -x )( b-x ) b a x () t dx x ( 0) - a x x dx b a x 0 a b x 0 0 b x dx x where a and [ ( x ) ( x )] x ln a + ln b 0 ln b a b a ( b x ) a ( a x )b 2006/03/4 33

34 Secnd-rder Rate Equatin The integrated rate law fr 2 nd rder reactin a + bb P: υ [ ] d d[b] - - a dt b dt [ ][ B] If the initial reactants are nt in stichimetric prprtins, i.e. k b[] a[b] kt [ ] -a[ B] b ln [ ][ B] [ ] [ B] If the stichimetric numbers f and B are unity (ab), then [ ] -[ B] [ ][ B] [ ] [ B] [] B -[ ] k t ln plt f ln ([]/[B]) versus t is linear. ln [ B] [ B] [ ] [ ] 2006/03/4 34

35 Experimental Kinetics and Gas Reactins 5 The variatin with time f the cncentratin f a reactant in a secnd-rder reactin. The grey line is the crrespnding decay in a first-rder reactin with the same initial rate. Fr this illustratin, k large 3k small. 2006/03/4 35

36 Zer-rder Rate Equatin The reactin is zer rder if the rate is independent f the cncentratin f the reactant. e.g. catalysis, phtchemical reactins The rate equatin fr a zer-rder reactin Prducts d [ ] υ - k r -d [ ] kdt dt If the initial cncentratin f is [] at t 0, [ ] [ ] The zer-rder rate cnstant has a unit f ml L - s - r ml m -3 s - and the half-life t ½ is t - /2 [ ] 2k kt 2006/03/4 36

37 General -rder Rate Equatin Fr a reactin with -rder in single reactant, P, the general rate equatin is [ ] d υ - k dt [ ] α d[ ] [ ] α r - kdt If the cncentratin f is [] at t and [] 2 at t 2, d [ ] t2 k dt, r - [ ] t ( -) α [ ] α- [ ] α [] 2 - k - 2 [] α 2 Let the initial cncentratin f is [] at t 0 - t - α- [ ] [ ] α ( α -) k, ( t -t ) ( 2α- -) ( ) k[ ] α- α- [ ] and t, /2 α- plt f ln[] versus ln(t ½ ) is linear with a slpe f -( -), the unit f rate cnstant depends n the rder f reactin. 2006/03/4 37

38 General -rder Rate Equatin Fr a reactin with -rder in single reactant, a P, the general rate equatin is [ ] d υ - k a dt [ ] α d[ ] [ ] α If the cncentratin f is [] at t and [] 2 at t 2, Let the initial cncentratin f is [] at t 0 r d [ ] t2 k d, r - [ ] t α t - [ ] α- [ ] α - akdt k dt [] 2 - k - 2 [] α 2 - [ ] [ ] ( α -) k t a( -) kt α - α- α α- ( 2 -) ( ) k[ ] α- α- [ ] and t/2, α- a 2006/03/4 38 ( t -t )

39 Table III Integrated rate laws rder Reactin Rate law kinetic t ½ 0 P k [] / 2k kt x fr 0 x [] P k[] ln2/ k kt ln {[] / ([] -x)} 2 P k[] 2 / k[] kt ln {x / [] ([] -x) ( n- 2 -) n 2 P k [] n ( ) [ ] n- n- k kt - ( -) - - ([ ] ) [ ] n n n x 2006/03/4 39

40 Table III Integrated rate laws rder Reactin Rate law kinetic 2 + B P k[][b] kt ([] [ ]) B - ln [ ] [ ] - x - ln [] B [] B - x 2 +2B P k[][b] kt ([] B -2[ ] ) ln [ ] [ ] -x -ln [] B [] B -2x 2 P (autcatalysis) k [][P] kt ([ ] + [ P] ) ln [ ] [ ] -x -ln [] P [] P + x 2006/03/4 40

41 Table III Integrated rate laws rder Reactin Rate law kinetic 3 +2B P k [][B] 2 kt 2x ( 2[ ] [ B] )[] ( B 2x )[] B ( 2[ ] [ B] ) + 2 ln [ ] [ ] -x -ln [] B [] B 2x 3 P k [] 2 [P] kt ln( + x) ln ( P + x) + ln ( P + x) x ln( + x) x P ( + x )( P + ) /03/4 4

42 P 0 ( x) 2 ( P + x) utcatalysis Reactin, third-rder rxn. υ [ ] d - dt d[p] dt k [ ] 2 [ P] ln( + x) dx + P P+ 2 ( P + )( + x) ln ( P + x) P P+ 2 ln( + x) ln ( P + x) + ln ( P + x) x ln( + x) x P ( + x )( P + ) /03/4 42

43 Initial Reactin Rate,,,.. Initial rates methd invlves the measurement f the rate f reactin at its beginning fr several different initial cncentratins f reactants.,.. (methd f islatin). 2006/03/4 43

44 Methd f Islatin,.The islatin methd is a technique fr examining the influence f individual species n the rate f a chemical reactin and t determine the rder f the reactin with respect t each cmpnent.,,., /03/4 44

45 Exercise 3 Using the methd f initial rates The recmbinatin f idine atms in the gas phase in the presence f argn was investigated 2 I (g) + r(g) I 2 (g) + r(g) and the rder f the reactin was determined by the methd f initial rates: [I] / (0-5 ml L - ) / (ml L - s - ) (a) 8.70x x0-3.39x x0-2 (b) 4.35x0-3.74x x0-2.57x0 - (c) 8.69x x0-2.38x0-3.3x0-2006/03/4 45

46 Initial Reactin Rate 4 The plt f lg v 0 against (a) lg [I ] 0 fr a given [r] 0, and (b) lg [r] 0 fr a given [I ] /03/4 46

47 Exercise 3 The r cncentratin are (a).0 mml L -, (b) 5.0 mml L -, (c) 0 mml L -. Determine the rders f reactin with respect t the I and r atm cncentratins and the rate cnstant. The plts in Fig shw the lgarithm f (a) the initial rate lg f r. nd against lg [I ] fr a given cncentratin (b) lg against lg [r] fr a given cncentratin f I. The slpe f the tw lines are the rders f reactin with respect t I and r. The intercepts with the vertical axis at lg[x] 0 give lg k. nswer: The slpes are 2 and respectively, s the (initial) rate law is υ [] 2 [ ] r The intercept are k9x0 9 ml -2 L 2 s - k I 2006/03/4 47

48 Pseud First Order Reactin Pseud first-rder rate law ( ): reactin is classified as pseud first rder if the cncentratins f all ther reactants are s great that they can be treated as effectively cnstant; then the rate law has absrbed the values f the cncentratin f the ther reactants. The pseud-first-rder rate cnstant k is directly prprtinal t the higher cncentratin f reactant, if the rate is als first rder fr the substance at lw cncentratin, the verall reactin under these cnditins is said t be pseud-first-rder. If [B]»[], υ k[ ][] B ( k [] B )[ ] k' [ ] 2006/03/4 48

49 Experimental Kinetics and Gas Reactins Fig. 8.4 (a) Plts f [] versus t fr zerhalf-, first-, and secnd-rder reactins with [] ml L -, each having a half-life f min. (b) Linear plts fr the zer-, half-, first-, and secnd-rder reactins. During the first part f the reactin, the cncentratin change with time is nt very different fr different rders. 2006/03/4 49

50 2006/03/4 50

Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University 508 YiFu Lou talk 06/

Stochastic Processes (XI) Hanjun Zhang School of Mathematics and Computational Science, Xiangtan University [email protected] 508 YiFu Lou talk 06/04/2010 - Page 1 Outline 508 YiFu Lou talk 06/04/2010