Lecture 6: Diffusion nd Rection kinetics 1-1-1 Lecture pln: diffusion thermodynmic forces evolution of concentrtion distribution rection rtes nd methods to determine them rection mechnism in terms of the elementry rection steps bsic rection types problems
Diffusion Thermodynmic force If the chemicl potentil depends on position, the mximum nonexpnsion work μ dw = dμ = dx x pt, Compring with dw = Fdx w = Δμ μ μ + Δμ F μ = x pt, Thermodynmic force
Diffusion Thermodynmic force of concentrtion grdient μ = μ + RTln F ln RT c = RT F = x c x, pt, For idel solution pt Fick s lw of diffusion: Prticles flux: J ~ drift velocity ~ F ~ c x
Diffusion The Einstein reltion c sδatc J = D J = = x AΔt sc sc c = D x s D c DF = = c x RT In the cse of chrged prticle (ion) in electric field we know drift speed vs force reltion for ion mobility, so we cn deduce diffusion constnt s = ue = For exmple: for DzeN AE RT μ = 8 51 m / sv D we find urt = zf Einstein reltion 9 D = 11 m / s
Diffusion The Nernst-Einstein eqution z F= zdf Molr conductivity of ions in the solution λ = μ RT F + + + ν Λ m = ( ν z D + z D ) RT The Stokes-Einstein eqution Frictionl force μe= eze f D = μrt zf D zert zekn AT kt = = = fzf fzen f A Using Stokes s lw D = kt 6πη No chrge involved -> pplicble to ll molecules
Diffusion eqution How concentrtion distribution evolves with time due to diffusion c JA J A J J = = t Al Al l J J = D + D = D + D c+ l = Dl x x x x x x c c = D t x c c c c c Diffusion with convection c JA J A J J = = t Al Al l c J J = cv cv = c c+ l v= vl x c c = v due to convection only t x c x c c c = D v t x x
Solution of diffusion eqution c c c = D v t x x nd order differentil eqution: two boundry condition re required for sptil dependence nd single for time dependence Exmple 1: one surfce of continer (x=) is coted with N molecules t time t= cxt (, ) = n A( π Dt) n 8( π Dt) 1/ 3/ e e x /4Dt Exmple : dissolution infinitely smll solid contining N molecules t time t= in 3D solvent crt (,) = r /4Dt
Diffusion probbilities Probbility to find prticle t given slb of thickness dx is proportionl to the concentrtion there: The men distnce trveled by the prticles: p( x) = c( x) AN dx/ N A cxandx ( ) A 1 x /4Dt x xe dx 1/ N ( πdt) Dt = = = π 1/ x 1/ 1/ = ( Dt) Diffusion in unstirred solution, D=5x1-1 m /S
Rndom Wlk Apprently diffusion cn be modeled s rndom wlk, where prticle is jumping distnce λ in time τ. Direction of the jump is chosen rndomly One dimensionl wlk: p 1/ τ x τ /tλ = πt Compring with the solution of diffusion eqution e D λ τ = Einstein-Smoluchowski eqution Connection between microscopic nd mcroscopic prmeters.
RATES OF CHEMICAL REACTIONS
Rtes of chemicl rections A+ B 3C+ D Instntneous rte of consumption of rectnt: d[ R]/ Instntneous rte of formtion of product: From stoichiometry dp [ ]/ dd [ ] 1 [ ] [ ] 1 [ ] = dc = da = db 3 Rte of the rection: 1 dni v = = ν i dξ In cse of heterogeneous rection the rte will be defined s mol/m s
Rection order Rection rte is generlly dependent on temperture, pressure, concentrtion of species, the phses where rection occurs etc. However, n empiricl reltion clled rte lw exists stting tht: b v= k[ A] [ B]... rte constnt, depends only on T nd not on the concentrtion lgebric dependence on the regent concentrtion rised to some power rection order with respect to species A: α overll rection order: α + β... The power is generlly not equl to the stoichiometric coefficients, hs to be determined from the experiment Order of rection: v = k Zero order. M s 1 v = k[ A][ B] First order in A, first order in B, overll second order. M s 1 1 v k A B 1/ = [ ] [ ] Hlforder in A, first order in B, overll tree-hlves order.
Mesuring the rtes of chemicl rections Experimentl mesuring progress of the rection Monitoring pressure in the rection involving gses NO( g) 4 NO ( g) + O ( g) 5 n(1 α) αn 1 αn 3 p= (1 + α) p Absorption t prticulr wvelength (e.g. Br below) H ( g) + Br ( g) HBr( g) Conductnce of the ionic solution ( CH ) CCl( q) + H O( q) ( CH ) COH ( q) + H + ( q) + Cl ( q) 3 3 3 3
Mesuring the Rtes of chemicl rections Flow method concentrtion s function of position concentrtion s function of time Stopped-flow method (down to 1 ms rnge) Flsh photolysis (down to 1 fs = 1-14 s rnge) Chemicl quench flow Freeze quench method
Mesuring the rtes of chemicl rections Determintion of the rte lw Usul technique is the isoltion method, where ll the components except one re present in lrge mounts (therefore their concentrtion is constnt) v= k[ A][ B ] = k [ A] It usully ccompnied by the method of initil rtes, when severl initil concentrtion of A mesured (gin ssuming tht the concentrtions re constnt) v = k [ A ] log v = log k + log[ A ]
Mesuring the rtes of chemicl rections Exmple I( g) + Ar( g) I ( g) + Ar( g) A B α = β =1
Integrted rte lws First order rection. Let s find concentrtion of regent A fter time t da [ ] = ka [ ] A A da [ ] = k [ A ] t Slope -> k [ A] ln = kt [ A ] Hlf-life time required for concentrtion to drop by ½. [ A] = [ A ] e kt 1 [ A ] ln 1/ = ln = ln t1/ = [ A ] k kt Time constnt time required for concentrtion to drop by 1/e: 1 τ = k
Integrted rte lws Second-order rections da [ ] = ka [ ] A A da [ ] [ A ] = k t 1 1 = [ A] [ A ] kt [ A] [ A ] = + kt A 1 [ ] Second order Hlf-life depends on initil concentrtion First order 1 [ A ] 1 [ A ] = t = 1 [ ] [ ] 1/ + kt1/ A k A The concentrtion of the regent drops fster in the 1 st order rection thn in the nd order rection
Rection pproching equilibrium Generlly, most kinetics mesurements re mde fr from equilibrium where reverse rections re not importnt. Close to equilibrium the mount of products is significnt nd reverse rection should be considered. A B v= k[ A] B A v= k'[ B] da = ka [ ] + k [ B] [ A] + [ B] = [ A] da = ka [ ] + k ([ A] [ A]) ( k+ k ) t k + ke [ A] = [ A] k+ k k k [ B] k [ A] eq = [ A], [ B] = [ A], K = = k+ k k+ k A k eq eq [ ] eq k k K k k b Generlly: =... b
The temperture dependence of rection rte The rte constnts of most rection increse with the temperture. Experimentlly for mny rections k follows Arrhenius eqution E ln k = ln A RT A pre-exponentil (frequency) fctor, E ctivtion energy High ctivtion energy mens tht rte constnts depend strongly on the temperture, zero would men rection independent on temperture
The temperture dependence of rection rte Stges of rection: Regents Activtion complex Product Trnsition stte k = Ae E / RT Frction of collision with required energy Activtion energy is the minimum kinetic energy rectnts must hve to form the products. Pre-exponentil is rte of collisions Arrhenius eqution gives the rte of successful collisions. rection coordinte: e.g. chnges of interctomic distnces or ngles
Elementry rections Most rections occur in sequence of steps clled elementry rections. Moleculrity of n elementry rection is the number of molecules coming together to rect (e.g. uni-moleculr, bimoleculr) Uni-moleculr: first order in the rectnt da [ ] A P = k[ A] Bimoleculr: first order in the rectnt da [ ] A + B P = k[ A][ B] Proportionl to collision rte H + Br HBr + Br
Consecutive elementry rections k kb A I P da [ ] = k[ A] di [ ] = k[ A] kb[ I] dp [ ] = kb[ I] Solution for A should be in form: [ A] = [ A] e kt di [ ] k + kb[ I] = k[ A] e [ I] = ( e e )[ A] k k kt kt kt b b [ A] + [ I] + [ P] = [ A] ke [ P] = 1 + ke [ A] kt kt b b kb k
Consecutive elementry rections The (qusi) stedy-stte pproximtion di [ ] = Then di [ ] = k [ A] k [ I] = b nd dp [ ] = k [ I] k [ A] b ( kt ) [ P] 1 e [ A]
Consecutive elementry rections Pre-equilibri k, k k A+ B I b P This condition rises when k >>k b. Then nd K dp [ ] = [ I] k [ A][ B] = k = k [ I] = k K[ A][ B] b b k Second order form with composite rte constnt
Unimoleculr rections As the molecule cquires the energy s result of collision why the rection is still first order? Lindemnn-Hinshelwood mechnism * k * da [ ] A+ A A + A = k[ A] * * k [ ] da * A + A A+ A = k [ A ][ A] * * k da [ ] b * A P = kb[ A ] If the lst step is rte-limiting the overll rection will hve first order kinetics
Lindemnn-Hinshelwood mechnism * k * da [ ] A+ A A + A = k[ A] * * k da [ ] * A + A A+ A = k [ A ][ A] * * k da [ ] b * A P = kb[ A ] da * [ ] * [ ] A = k A k A A k A b * * [ ] [ ][ ] b[ ] k[ A] = k + k [ A] dp * kk b [ A] = k b[ A ] = k + k [ A] b If the rte of dectivtion is much higher tht unimoleculr decy thn: dp kk A kk A = k + k [ A] k b [ ] b [ ] b The Lindemnn-Hinshelwood mechnism cn be tested by reducing the pressure (slowing down the ctivtion step) so the rection will switch to the second order.
The ctivtion energy of the composite rection Let s consider Lindemnn-Hinshelwood mechnism nd pply Arrhenius-like temperture dependence to ech rte constnt k ( E ( )/ )( ( )/ ) RT E b RT Ae Ae ( E ( )/ ) RT Ae = kk AA k = = A b b b { E ( ) + E ( ) E ( )}/ RT e Overll ctivtion energy cn be positive or negtive E ( ) + E ( b) > E ( ) E ( ) + E ( b) < E ( )
Problems P.35. The diffusion coefficient of prticulr RNA molecule is 1.x1-11 m /s. Estimte time required for molecule to diffuse 1 um from nucleus to the cell wll E1.11 The second-order rte constnt for the rection CH 3 COOC H 5 (q) + OH - (q) CH 3 CO - (q) + CH 3 CH OH(q) is.11 dm 3 mol -1 s -1. Wht is the concentrtion of ester fter () 1 s, (b) 1 min when ethyl cette is dded to sodium hydroxide so tht the initil concentrtions re [NOH] =.5 mol dm -3 nd [CH 3 COOC H 5 ] =.1 mol dm -3? E1.17 The effective rte constnt for gseous rection tht hs Lindemnn Hinshelwood mechnism is.5 1-4 s -1 t 1.3 kp nd.1 1-5 s -1 t 1 P. Clculte the rte constnt for the ctivtion step in the mechnism.