Summarizing Remarks λ λ λ. : equilibrium geometry

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1 112 Summrizing Remrks Summrizing Remrks The theory underlying chemicl processes, in prticulr chemicl equilibrium is mture science. The bsis of the edifice is Quntum Mechnics! For prticulr volume nd number of toms (or moleculr species) one cn solve Ĥψ = Eψ λ λ λ The eqution is esy to write down in full generlity. Solving it is hrd (no surprise s it contins the full richness of chemicl phenomen) n chemistry, the Born- Oppenheimer or Potentil Energy surfce plys crucil role: t llows us to tlk bout moleculr structure. For electronic ground sttes, the PES usully hs quite deep wells. Ech minimum corresponds to stble isomer, nd we cn meningfully ssign M : Mss of isomer R e : equilibrium geometry ω : hrmonic vibrtionl frequencies (usully in cm- 1) i E : energy t the bottom of well el E zp = i 1 2 ω i This prt of Quntum Chemistry is very well developed, nd over the pst 40 yers, user- friendly computer progrms re redily vilble to do clcultions for ) Molecules in gs phse: robust b) Solid stte, periodic systems: vilble, firly ccurte Summrizing Remrks 112

2 113 c) Solution Stte: Use continuum solvtion models, perhps incorporting some solvent molecules explicitly. Use clssicl simultion techniques. From the info from Quntum Mechnics for single molecules, one cn obtin mcroscopic properties of equilibrium thermodynmics. This works very well for (idel) gses nd lso solids. For rel cses, one cn mke corrections (ctivity coefficients). Solutions re most difficult. One needs to run clssicl simultions. t cn be very expensive to tret the molecules quntum mechniclly then. Most often one uses force field methods nd clssicl mechnics to describe tomic motion. f we focus on gses (s done in this clss), the link between Quntum Mechnics nd thermodynmics is provided by Sttisticl Mechnics. N q Q N molecules (Boltzmnn pproximtion) N! q = q molecule = q t q R q v q el q zp q n Trnsltionl, rottionl, vibrtionl (without zeropoint), electronic, zeropoint, nucler. Using del gs, hrmonic oscilltor, rigid rotor pproximtions, this cn be ccurtely obtined using info from Quntum Chemistry (Gussin clcultions, see file Mtlb Gussin on website) Connection to thermo, in cnonicl ensemble ) A = k B T lnq (,, ) ATV N (Helmholtz free energy) All properties from thermodynmics b) Chemicl equilibrium constnts (using pressure rther thn volume) K eq ( q ) υi = for rection υ = 0 Access to thermochemistry from first principle theory A second spect of interest is chemicl kinetics. This subject is not quite s robust. The microscopic theory cn be bsed on the time- dependent Schrödinger eqution (Quntum Mechnics). This is quite complicted, nd only few body systems hve been studied. t cn give Quntum stte to Quntum stte rte constnts in scttering events. These re often quite detiled spectroscopic studies, nd/or involving extensive computtions (quntum dynmics). Most of Prcticl chemicl kinetics is rooted in trnsition stte theory, nd then Quntum Chemistry plys mjor role gin. Besides minim on the PES, one cn locte trnsition sttes, i.e. extrem with one imginry frequency. The prtition function t the trnsition stte (except for specil tretment of the rection coordinte, which yields the universl fctor k T B ) cn be h obtined using the stndrd recipe of quntum chemistry, exctly the sme s for minim. Using trnsition stte theory, the rte constnt to go from R TS P is given by Summrizing Remrks 113

3 114 = k B T h q ( TS ) ( R) q The fctor kt h is universl, but lso the most suspicious prt of the derivtion. t is correct only by order of mgnitude. Using the usul formuls for the prtition function one cn obtin E / RT k A T e f ( ) =Δ +Δ, the ctivtion brrier, is independent of temperture. ( ) Here, E Eel Ezp AT typiclly P would tke the form T (power lw). The precise form is hrd to clculte, nd is even hrd to mesure. The ctivtion energy is most importnt. n textbooks, one often finds E depends (slightly) on T. This depends on (rbitrry) definitions. much prefer the picture we get from Quntum Mechnics: energies do not depend on T. Brriers on PES do not depend on T. Put AT. t cptures the ll remining temperture dependence in the pre- exponentil fctor ( ) effects of rottionl, trnsltionl nd the remining vibrtionl motion. n ddition there re contributions from kinetic theory nd collision theory, which did not discuss in clss. These things re discussed in the textbook by Reid nd Engel. One further note: we hve seen how the prtition functions for the nucler spin degree of freedom nd moleculr rottion hve to be treted together due to the puli principle for nuclei (either bosons or fermions). t leds to the symmetry fctor in the rottionl prtition function. t is n intricte quntum mechnicl effect. Luckily the high temperture pproximtion is ccurte for the rottionl prtition function nd the finl result is very simple. Now, using Quntum Mechnics, Quntum Chemistry, Sttisticl Mechnics nd Trnsition stte theory, we hve microscopic theory to clculte forwrd nd bckwrd rtes for elementry rections. This is precisely the info one needs to set up rte equtions for so- clled micro- kinetics model. To model kinetics one lists set of elementry rections r = A + bb cc + dd k b A b c B kb C d D The forwrd nd bckwrd rte constnts re relted by thermodynmics k b = K eq = C eq A eq c D eq B eq d b Summrizing Remrks 114

4 115 At constnt (or prticulr), functions. This implies K theory, s summrized bove. T P k, k, K cn ll be obtined from rtio of prtition k f b eq f eq =. To get the individul,k b one invokes trnsition stte kb n the symptote of long rection times, when chemicl equilibrium is reched, ech of the (net) elementry rtes is zero. This is clled detiled blnce. A kinetic model is defined by set of elementry rections nd its ssocited rtes nd in ddition specifiction of initil conditions/concentrtions of regents. Something we lerned on our wy is tht regents themselves might consist of multiple species, in mutul chemicl equilibrium. We experimentlly cnnot strt from rbitrry initil conditions. One cn mke things more interesting by using regents equilibrted t different temperture thn the ctul rection temperture. From the kinetic model + initil conditions, it is usully stright forwrd to run numericl simultion to obtin concentrtion profiles nd instntneous rection rtes. For complicted rections this is the only wy forwrd. Just like for Quntum Chemistry, one uses computer to solve the equtions. Much of the folklore of chemicl kinetics, certinly in textbooks, requires one to find simplified versions of rection rtes. This cn certinly provide further insight in rection mechnism. The most useful tools to mke pproximtions re: Pre- equilibrium: This ssumes prticulr elementry rection hs reched equilibrium nd djusts itself fst enough to mintin equilibrium throughout. [ ] d Stedy stte: This is invoked by setting = 0 for some intermedite species. Wht dt this relly mens is tht this concentrtion djusts itself very rpidly (instntneously) to the concentrtions of other more slowly evolving species. t suffices tht one of the rections involving the rection of intermedite is very fst. think this pproximtion is esily bused in textbooks, s it is such powerful wy to mke pproximtions. This doesn t men it is correct. One cn verify the reltion obtined by invoking stedy stte pproximtion by compring to numericl simultions An interesting spect is tht the concentrtion of species cn be very smll, even undetectble (eg. H tom rdicls) nd yet they my ply crucil role in the rection mechnism. One my hve to deduce such role from the kinetic concentrtion profiles. This cn led to very interesting scientific detective stories. Determining rection mechnisms cn be rel fun! t is often stted tht rection mechnism cn never be proven. However,by monitoring ll concentrtion profiles, one my obtin pretty solid ide of ll of the importnt rections in the mechnism. Summrizing Remrks 115

5 116 Something else we discovered in our Mtlb simultions is tht the trend eg. (liner or squre root dependence of concentrtions) is often well predicted by pproximtions. However the exct numericl predictions my be less ccurte, i.e. the vlue of overll effective rte constnts in terms of elementry rte constnts. A finl note of cution: Sometimes only forwrd rection rtes re listed in rection mechnism, while the bckwrd rte might not be included (even if it s substntil). This is suspicious. The reson often is tht this wy one cn simplify mtters on piece of pper. Tht is poor reson. Better to do the simultion using computer nd drw vlid conclusion. n these notes hve occsionlly been criticl of the ssumptions mde in chemicl kinetics. This is no doubt in prt due to my reserch interests. m mostly working on obtining systemtic ccurte solutions to the Schrödinger eqution. The methodologies here idelly pply to ny molecule. There is no need for specific pproximtions, depending on the nture of the molecule. Moreover, in electronic structure the use of computers is mndtory. n chemicl kinetics it ppers one mkes new model for every sitution, nd in text books one mkes (somewht rbitrry) ssumptions for ech cse. A lot of this lck of systemtics would dispper if one embrces the use of computers to solve the resulting differentil equtions. One my still need to nlyse results then to condense things to obtin useful understnding. think it pys to be suspicious of the literture if this involves the derivtion of nlyticl models. The ssumptions mde re often dictted by the desire to find nlyticl solutions, rther thn the vlidity of the ssumptions. This is poor wy to do science, especilly in this ge of powerful computers nd convenient, user friendly softwre. Summrizing Remrks 116