University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2014 Lecture 20: Transition State Theory. ERD: 25.14

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1 Univrsity of Wasinton Dpartmnt of Cmistry Cmistry 453 Wintr Quartr 04 Lctur 0: Transition Stat Tory. ERD: 5.4. Transition Stat Tory Transition Stat Tory (TST) or ctivatd Complx Tory (CT) is a raction mcanism oriinally dvlopd to dscrib as pas collision ractions. It as t nral raction scm: k + ( ) (0.) Wr t ractants and collid to first form a transition stat or activatd complx wic is dsinatd (). T ida bind TST is tat t transition stat is an unstabl, sort-livd complx. In t cas of a simpl diatomic collision t transition stat () consists of t - pair joind by a vry wak bond. T collidin molculs and ar assumd in quilibrium wit t transition stat wr C [ ] [ ][ ] (0.) Tis quilibrium notation owvr involvs t transition stat wic actually xists at a nry maximum and is tus transitory in natur, as its nam implis. For t diatomic raction mcanism t rat is ivn by rat k k C [ ][ ] [ ][ ] (0.3) To valuat t rat constant k, w can apply statistical mtods to q qv C V V (0.4) q q V V wr t partition functions of,, and ar q, q, q, rspctivly. T tratmnt of q rquirs som xplanation. It is assumd tat wn and form t transition complx t complx acquirs translation, rotational, vibrational and lctronic motions, wic all must b rflctd in t partition function, i..

2 q q q q q trans rot vib lc ( π ) / π D / 3 / k T m 8 I (0.5) σ ( π m ) 8π I D0 / 3 / k T σ In 0.5 t mass of t transition stat in t translational partition function is just m m + m. In t rotational partition function mm wr t rducd mass µ, I µ ( R ) is t m + m momnt of inrtia of t transition complx and. T trm is t dnracy of t round lctronic stat of t transition complx and D D is t dissociation nry of t 0 transition complx. T vibrational motion is tratd in t followin way. W assum as and com totr to form t transition complx tat translational motion alon t raction coordinat is vntually convrtd in part to a vibrational motion of t bond also dirctd alon t raction coordinat. W us t notation q, transq, rotq, vib D0 / (0.6) wr is a rducd vibrational partition q, vib / k T function. Now t vibrational of t bond in t transition complx is assumd to occur at a vry low frquncy suc tat q, vib (0.7) / k T ν ( / ) In otr words t bond vibration is calculatd in t i tmpratur limit. Now w also assum tat t transition complx is convrtd to product witin a vibrational priod so tat k ν. Usin tis xprssion and quation 0.7 w obtain q, transq, rot D0 / k C k V ν V (0.8) q, transq, rot V D0 / Tis is calld t Eyrin quation. To quation 0.8 is addd ad oc a constant κ <. Tis is calld t transmission cofficint and xprsss t fact tat not all collisions rsult in t formation of t

3 transition complx. Normally 0.5 < κ <, but κ can b quit small for atomic collisions. T final form for t Eyrin quation is κ V (0.9) q, transq, rot D0 / wr is absnt t dr of vibrational frdom tratd in t i tmpratur limit and ivin ris to t factor of. Raction rats ar normally masurd as pr mol quantitis so quation 0.9 is multiplid by vaadro s numbr RT, m Nκ V κ V (0.0). Exampls of Diatomic Transition Stat Calculations For t collision of two atoms w av V π ( m + m ) k T 8π µ ( ) ( R ), trans, rot 3 q q V ( π ( m) ) q 3 V ( π ( m) ) q 3 Tn w obtain for t rat constant κ ( π ( m ) ) 8 ( ) 0 / + m R k T D k T π µ ( π ( m) ) ( π ( m) ) 8π k T κ ( R ) µ D0 / (0.) (0.) If D0 is rportd in units of Jouls pr mol w must writ: 8π D0 / RT κ ( R ) (0.3) µ C. Mor Complicatd ractions: Trnary Complxs T simplst xampl of a trnary transition complx is t isotop xcan H + D HD+ H (0.4) T transition complx as linar form H H D wr t bonds li alon t raction coordinat. T rat constant as t form: q HHD D0 / κ (0.5) D H

4 H wr D0 D0 D0. T partition functions for q D and q H ar obtaind as usual for atomic and diatomic spcis. T transition complx partition function q HHD is tratd as follows. T transition complx is linar so it as a sinl momnt of inrtia and rotations ar calculatd by t sam procdur usd for a linar triatomic lik CO xcpt tat σ for HHD. T vibrational partition function is tratd as follows. Not for a linar triatomic tr ar four vibrational mods in t partition function corrspondin to symmtric and asymmtric strtcs and two quivalnt bndin mods H H D H H D H H D asymmtric symmtric Doubly d nrat strtc strtc bnd T asymmtric strtc contributs to t raction coordinat, is trfor tratd in t i tmpratur limit and yilds t k T/ trm in t Eyrin quation. T otr tr vibrational mods rmain in t transition complx partition function ( ( ) ) π m 3 H + md 8π IHHD qhhd 3 j/ j ν (0.6).C. Gibbs Enry of ctivation Rturnin to t nral xprssion for t raction rat κ C, w dfin formally t Gibbs nry of activation G RTln (0.7) Usin quation 8.5 w now dfin t kintic constant as: G / RT, m κ (0.8) Usin t corrspondin rlationsip btwn t Gibbs nry, ntalpy and ntropy... G H T S w furtr obtain: G / RT S / R H / RT κ κ (0.9) Now for an idal as G U + ( PV) T S U + RT n T S RTln ln U (0.0) T RT W can now apply quation 0.0 to t quation κ C to obtain ln ln C U ln κ + + (0.) T T T T RT H U + PV U + RT n. For t raction lso for idal ass ( ) +, n-. Trfor C

5 ln U RT H + RT H + RT + + T T RT RT RT RT T rrnius/van t Hoff quation is for comparison ln Ea T RT W conclud E H + RT for n-. Mor nrally a ( ) Ea H n RT (0.) (0.3) + (0.4) Usin 0.4 w writ out t rrnius rat law for n-: S / R ( Ea RT) / RT S / R Ea / RT κ κ (0.5)