5.62 Physical Chemistry II Spring 2008

Transcription

1 MIT OpenCourseWare Physcal Chemstry II Sprng 2008 For nformaton about ctng these materals or our Terms of Use, vst:

2 5.62 Sprng 2008 Lecture 34 Page Transton State Theory. II. E vs. E a. Knetc Isotope Effect. Want to get k TST nto Arrhenus form k TST =! kt h K " but K " = e G RT % RTln K = &G so k TST =! kt h eg /RT =! kt h es /R e /RT because G = TS NOW : = E + &nrt where n = ( molecules n TS) (molecularty of reacton) (molecularty: e.g., unmolecular, bmolecular, etc.) e.g. n = 2 = So: k TST =! kt h es R e e "E RT k TST BT m e "E RT where m = Surprsngly, theory predcts a temperature dependence to the pre-exponental factor. Ths T-dependence s dffcult to observe expermentally unless the rate constant s measured over a wde temperature range (at least a factor of 5) Now: d lnk TST = ) " d ln! k /R h es e %, + ' + lnt + e (E. * & - = T + E RT 2 (TST) Contrast ths to Arrhenus model: d lnk = E a RT 2

3 5.62 Sprng 2008 Lecture 34 Page 2 dlnk E a TST dlnk = RT = 2 T + E RT 2!E a = RT+ E Agan, the expermental E a s larger than E because E a s a dfference between the average energy of molecules n the pot and the average energy of molecules that react, whle E s a mcroscopc quantty, a threshold energy along the PES. Notce that E a s not a barrer along PES. COMPARISON OF TRANSITION STATE TEORY WIT COLLISION TEORY Calculate k TST n the lmt of the assumptons of collson theory (.e. smplfed TST): ) collsons of hard spheres 2) only translatonal degrees of freedom Treat 2 as an atom a hard sphere of mass 2. [No rotaton, no vbraton] Treat 2 F as a datomc molecule 2 + F! 2 F! F + Wth these assumptons 2 + F 2 F k! kt h " (q 2 trans (q trans / N) % / N) & F / N)(q trans 'q rot e (E RT Note: no vbratonal partton functon for 2 F s ncluded because the one vbratonal mode for the pseudo-datomc molecule transton state has become the reacton coordnate. Also, no rotatonal partton functon for 2 s ncluded because we are treatng 2 as an atom. k! kt h (2"(m 2 + m F )kt) 3/2 & % ( % h 3 N ( 8" 2 I kt e *E RT % (2"m 2 kt) 3/2 (2"m F kt) 3/2 ( )h 2 % h 3 N h 3 N '( The reason there s no rotatonal or vbratonal partton functon for 2 s not that we are assumng the hgh-t lmt, but rather that we are treatng 2 as f t were an atom.

4 5.62 Sprng 2008 Lecture 34 Page 3 Now: I 2 = µd 2!F where µ = m 2 m F m 2 + m F 2 F d 2 F hard sphere collson dameter ) k! N 8kT + " % * + + m F &, m 2 m (. F '-. m 2 "d2 /F e /E RT Ths looks dentcal to the collson theory result, and collson theory s not based on thermodynamcs. " k CT = 8kT % '!d 2 AB e (E 0 RT!µ & Calculate value for k TST n lmt of collson theory assumptons (.e. what fracton of collsons are effectve because they have suffcent translatonal energy along the lne of centers?): 2! = "d 2 F = 30 9 m 2 0 Compare to k TST k % e E RT m 3 mol & s k TST = 3.9!0 7 e "E RT m 3 mol s k TST s smaller because t reflects the more restrctve co-lnear sterc requrement. k CT s an upper bound because collson theory treats reactants as spheres wth no favored drecton of approach (but wth an explct requrement on the effectve collson energy).

5 5.62 Sprng 2008 Lecture 34 Page 4 TRANSITION STATE TEORY AND KINETIC ISOTOPE EFFECT Consder the atom (or proton) transfer reacton X + Y X + Y (or X + + Y X + Y + ) where the orgnal X bond s broken and a new Y bond s formed X = A, Y = B, (X + Y) = C A + B C So the key queston s how do we know whether breakng and makng a bond to occurs n the transton state regon of the reacton coordnate? The sze of the D knetc sotope effect tells us whether the transfer occurs at or before/after the transton state. Knetc sotope effect reacton rates are slower f deuterum s substtuted for hydrogen and a hydrogen bond s nvolved n the reacton. Why? Potental energy of nteracton the same for X and DX (Born- Oppenhemer appproxmaton). 2 hν X 2 hν DX Because zero pont energy of DX s smaller than for X Snce! " k & % ( where k = force constant m '

6 5.62 Sprng 2008 Lecture 34 Page 5! X " m & DX % ( because k X = k DX! DX m X ' Intramolecular potentals are the same. The shape of potental curve doesn t change upon sotopc substtuton. " m! DX =! DX % X ' & m X Snce m DX > m X So 2 h! DX < 2 h! X So D DX X 0 > D 0 dssocaton energy to break D X bond s larger than that to break X Knetc Isotope Effect! k k D Calculate ths usng transton state theory k k D = kt h kt h! q / N "(q A / N)(q B / N) %! q D / N "(q A D / N)(q B D / N) % & e'e & e'e D RT RT k! = q! & q A B Dq D k D " q D % q A B " q % ( +E D ) RT &e 'E But V 0 = V 0 D!E + E D =!V 0! 2 h " + 2 h " D + V h " D! 2 h " D k! = q! & q A B Dq D k D " q D % q A B &e ' " q % 2 h! ( ) '( ) D '( ) D + ( ) & RT " % Most of the sotope effect s n the dfference between the zero pont energes of the deuterated vs. hydrogenated reactants. Can use sotope effect to determne whether a hydrogen bond was nvolved n the transton state. Standard dagnostc n knetcs!