( ) ( ) ( ) + ( ) Ä ( ) Langmuir Adsorption Isotherms. dt k : rates of ad/desorption, N : totally number of adsorption sites.

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1 Langmuir Adsorption sotherms... 1 Summarizing Remarks... Langmuir Adsorption sotherms Assumptions: o Adsorp at most one monolayer o Surace is uniorm o No interaction between adsorbates All o these assumptions are somewhat dubious a ( ) + ( ) Ä ( ) A g M surace AM surace k kd dθ kpn a A ( 1 θ) kn d θ ka, k : rates o ad/desorption, N : totally number o adsorption sites d P : partial pressure o A A dθ At uilibrium 0 kpn ( 1 θ) ( ) a A d a A d a a knθ kp k + kp θ kp a A KPA θ K ka / k (uilibrium constant) d kd + kapa 1+ KPA Or KP θ A 1 θ Describes ractional coverage as a unction o partial pressure Vads Other expression or θ, θ (volume o adsorbed gas/ull monolayer coverage) VM Vads KP V 1 + KP Divide by KVadsVM P M 1+ KP Vads KPVM ( ) KVM P Vads VM Langmuir Adsorption sotherms and Summarizing Remarks 1

2 plot 1 P vs 1 V ads K can also be measured as unction o T. This gives access to ln K ΔH ads (thermo) T RT Δ Hads Langmuir sotherm with dissociation a molecule A dissociates when adsorbing on the surace, we have A + M Ä AM dθ kp a A( N( 1 θ) ) kd ( Nθ) 0 at uilibrium ( empty sides) ( ( 1 θ) ) ( θ) a A d kp N k N θ kp a A 1 θ ( ) 1/ θ kp a A 1 θ Summarizing Remarks 1/ ( kp a A) ( kp) 1/ θ 1/ P dependence rather than P 1 + a A The theory underlying chemical process, in particular chemical uilibrium is a mature science. The basis o the ediice is Quantum Mechanics! For a particular volume and number o atoms (or molecular species) one can solve Ĥψ Eψ λ λ λ Summarizing Remarks and Summarizing Remarks

3 The uation is easy to write down in ull generality. Solving it is hard (no surprise as it contains the ull richness o chemical phenomena) n chemistry, the Born- Oppenheimer or Potential Energy surace plays a crucial role: t allows us to talk about molecular structure. For General electronic states, the PES usually has quite deep wells. Each minimum corresponds to a stable isomer, and we can meaningully assign M : Mass o isomer ω : harmonic vibrational ruencies i E : energy at the bottom o wall el 1 Ezp h ωi i R r e : uilibrium geometry This part o Quantum Chemistry is very well developed, over the past 40 years, user riendly computer programs are available to do calculations or a) Molecules in gas phase: robust b) Solid state, periodic: available c) Solution State: Use continuum solvation models, perhaps incorporating some solvent molecules explicitly From the ino rom Quantum Mechanics, one can obtain macroscopic properties o uilibrium thermodynamics. This works very well or (ideal) gases and also solids. For real cases, one can make corrections (activity coeicients). Solutions are most diicult. One needs to run classical simulations. we ocus on gases, the link between Quantum Mechanics and thermodynamics is provided by Statistical Mechanics. N q Q N molecules (Boltzmann approximation) N! molecule q q qq q% q q q t R v el zp n Translational, rotational, vibrational (without zeropoint), electronic, zeropoint, nuclear Using deal gas, harmonic oscillator, rigid rotor approximations, this can be accurately obtained using ino rom Quantum Chemistry (see ile Matlab Gaussian on website) Summarizing Remarks and Summarizing Remarks 3

4 Connection to thermo, in canonical ensemble a) A ktln Q (,, ) ATV N (Helmholtz ree energy) All properties rom thermodynamics b) Chemical uilibrium constants (using pressure rather than volume) K ( q ) υi or reaction υ 0 Access to thermochemistry rom irst principle theory A second aspect o interest is chemical kinetics. This area is not quite as robust. The microscopic theory can be based on the time- dependent Schrodinger uation (Quantum Mechanics). This is quite complicated, and only ew body systems have been studied. t can give Quantum state to Quantum state rate constants in scattering events Most o Practical chemical kinetics is rooted in transition state theory, and then Quantum Chemistry plays its role again. Besides minima on the PES, one can locate transition states, extrema with one imaginary ruency. The partition unction at the transition state (except or reaction coordinate) can be obtained using the standard recipe, as or minima. Using transition state theory, the rate constant to go rom R TS P is given by k kt h q ( TS ) q ( R) The actor kt h is universal, but also the most suspicious part o the derivation. t is correct only by order o magnitude k ( ) A T e Ea / RT E E E Δ +Δ is independent o temperature. ( ) a el zp AT typically would take the orm (power law). The precise orm is hard to calculate, and ever hard to measure. The activation energy is most important. n textbooks, one oten inds E depends (slightly) on T. This a P at depends on (arbitrary) deinitions. like the picture we get rom Quantum Mechanics: energies do not depend on T. Barriers on PES do not depend on T. Put all remaining temperature dependence in AT ( ). Now, using Quantum Mechanics, Quantum Chemistry, Statistical Mechanics and Transition state theory, we have a microscopic theory to calculate orward and backward rates or elementary reactions. This is precisely the ino one needs to set up rate uations or a so- called micro- kinetics model. Summarizing Remarks and Summarizing Remarks 4

5 To model kinetics one lists a set o elementary reactions k aa + bb Ä cc + dd kb a b c d [ ] [ ] [ ] [ ] r k A B k C D The orward and backward rate constants are related by thermodynamics b c d k C D K a b kb A B At a constant (or particular) T, P k, k, K can all be obtained rom a ratio o partition unctions. This implies K k. kb b n the asymptote o long time, where chemical uilibrium is reached, each o the (net) elementary rates is zero. This is called detailed balance. A kinetic model is deined by a set o elementary reactions and its associated rates and in addition a speciication o initial conditions/concentrations o reagents. Something learned on our way is that reagents themselves might consist o multiple species, in mutual chemical uilibrium. We experimentally cannot start rom arbitrary initial conditions. From the kinetic model + initial conditions, it is usually straight orward to run a numerical simulation to obtain concentration proiles and instantaneous reaction rates. For complicated reactions this is the only way orward. Just like or Quantum Chemistry, one uses a computer to solve the uations. Much o the olklore o chemical kinetics, certainly in textbooks, ruires one to ind simpliied versions o reaction rates. This can certainly provide urther insight on a reaction mechanism. The most useul tools to make approximations are: Pre- uilibrium: This assumes a particular elementary reaction has reached uilibrium and adjusts itsel ast enough to maintain uilibrium throughout [ ] d Steady state: This is invoked by setting 0 or some intermediate species. What this really means is that this concentration adjust itsel very rapidly (instantaneously) to the concentrations o other species. t suices that one o the reactions involving the reaction o intermediate is very ast. think this approximation is easily abused in textbooks. One can veriy the relation obtained by invoking steady state approximation by comparing to numerical simulations Summarizing Remarks and Summarizing Remarks 5

6 An interesting aspect is that the concentration o a species can be very small, even undetectable (eg. H atom radicals) and yet they play crucial role in the reaction mechanism. One may have to deduce such a role rom kinetics proile t is oten stated that a reaction mechanism can never be proven. By monitoring all concentration proiles, one may obtain a pretty solid idea o the important reactions in the mechanism. Something else we discovered in our simulations is that the trend eg. (linear or square root dependence o concentrations) is oten well predicted by approximations. However the exact numerical predictions may be less accurate, i.e. the value o overall eective rate constants in terms o elementary rate constants. A inal note o caution: Sometimes only orward reaction rates are monitored in a reaction mechanism, while the backward rate might not be included (even i it s substantial). This is ishy. The reason oten is that this way one can simpliy on a piece o paper. That is a poor reason. Better to do the simulation using a computer and draw a valid conclusion. Summarizing Remarks and Summarizing Remarks 6