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Combustion Chemistry, Part 2

Combustion Chemistry, Part 2
Summary of Part 1: Overview, Chem Basics • Fuels are both Boon and Bane – Oil: cheaptoproduce convenient highdensity energy carrier – Current system not sustainable, esp. greenhouse – So we need: 1) higher efficiency 2) new energy carrier • Maybe an Alternative Fuel • If so, need to understand how it behaves… • Fuel Chemistry is Tricky – NOT Arrhenius single step – Details matter to understand why fuels behave differently • Many clever (tricky) methods for experimentally measuring rates…but you cannot measure everything • …Most of the rest of the lectures focus on understanding fuel chemistry theoretically / computationally Combustion Chemistry, Part 2 2 Plan of attack • Start with Thermodynamics/Thermochemistry – Including equilibrium Statistical Mechanics • Next simple Rate Theory – conventional Transition State Theory – highPlimit (thermalized) • Then Fancier Rate Theory – variational TST – Pdependence of rate coefficients • Then Mechanisms combining many species, reactions 3 Thermodynamics/Thermochemistry All Kinetics is Leading Toward Equilibrium. So good to start by figuring out where we are going (later we can worry about how fast we will get there…) Part of Thermo is about phase behaviour (e.g. volatility, miscibility) “Thermochemistry” is about reactions. st 1 Law gives energy density, final temperature nd 2 Law related to detailed balance (and so reverse rate coefficients), final composition at equilibrium 4 Fuel volatility must match fuel injector, and certain volatility ranges are particularly hazardous to store. Gasoline boils 150 C, jet 200 C, diesel 300 C Large molecules pyrolyze at lower T than b.p., and are solids at room temperature. 5 Volatility, Solvation • Phase behavior of mixtures is complicated subject in itself – How liquid fuel evaporates in engine – Formation of aerosols in atmosphere (particulate pollution) – Distillation is main separation process in refinery – Some liquids are inmiscible, so can have several liquid phases • Don’t want this to occur in fuel tank or fuel injector • Strongly affects mobility in environment • Volatility (Partial Pressure of species in gas phase) depends on solvation of the molecule in the liquid phase – Nonbonded interactions (Enthalpic interactions) – Entropy of molecule in liquid (mostly hardsphere) • Boiling Point is just when sum of all the partial pressures equals atmospheric pressure. Purecompound boiling point not directly related to Henry’s Law constant when species is in solution 6 Thermodynamics • We need thermodynamic data to: – Determine the heat release in a combustion process (need enthalpies and heat capacities) – Calculate the equilibrium constant for a reaction – this allows us to relate the rate coefficients for forward and reverse reactions (need enthalpies, entropies (and hence Gibbs energies, and heat capacities). • This lecture considers: – Classical thermodynamics and statistical mechanics – relationships for thermodynamic quantities – Sources of thermodynamic data – Measurement of enthalpies of formation for radicals – Active Thermochemical Tables – Representation of thermodynamic data for combustion models 7 Various thermodynamic relations are needed to determine heat release and the relations between forward and reverse rate coefficients A statement of Hess’s Law n is the i stoichiometric coefficient o For ideal gas or ideal solution, a =P /P or i i o a = C /C . Must use same standard state i i o used when computing DG . Solids have a =1. i For surface sites use fractional occupation. 8 Tabulated thermodynamic quantities. 1. Standard enthalpy of formation o Standard enthalpy change of formation, D H f The standard enthalpy change when 1 mol of a substance is formed from its elements in their reference states, at a stated temperature (usually 298 K). The reference state is the most stable state at that temperature, and at a pressure of 1 bar. e.g. C(s) + 2H (g)  CH (g) 2 4 o 1 D H = 74.8 kJ mol f The standard enthalpies of formation of C(s) and H (g) are both zero 2 9 Computing K (T), G(T), H(T), S(T) eq 𝑇 𝑜 0 𝑜 ′ 𝐻 𝑇 =𝑈 + =𝐻 (𝑇 ) + 𝐶 𝑇 𝑑𝑇′ 𝑜 𝑝 𝑇 𝑇 0 0 ′ o 𝑆 𝑇 =𝑆 (T ) + (𝐶 𝑇 /𝑇′ )𝑑𝑇′ 𝑜 𝑝 𝑇 0 0 0 𝐺 𝑇 = 𝐻 (𝑇 )−𝑇 𝑆 (𝑇 ) 0 𝐾 𝑇 =exp (−𝐺 𝑇 ) None of these depend on Pressure (they are for standard state) Same K works at all pressures. For nonideal gases, the eq nonidealities are conventionally hidden in the activities as Fugacities or “activity coefficients”. C (T) is expressed in several different formats causing some confusion: p NIST/Benson tabulated C (T ) or several different polynomialtype expansions: p i Shomate, Wilhoit (beware typos in original paper), two different NASA formats. C (T) can also be expressed by statistical mechanics formulas. p 10 𝑒𝑞 𝑃𝑉11 Now maintained by Elke Goos. 12 LHV and UHV • Fuels are classified by their Heating Value, i.e. their heat of combustion. • Two variants are commonly used: Lower Heating Value and Upper Heating Value. • LHV assumes all the H O formed is in gas phase, this is realistic 2 for engines where the H O leaves in the exhaust. Note in a real 2 engine the H O in the exhaust would be hot, but the LHV 2 calculation usually assumes room temperature steam. • UHV assumes all the H O formed is in liquid phase. This is 2 realistic for bomb calorimetry experiments, where the final temperature is usually pretty low. So UHV is easier to measure. But it can be a big overestimate of the true heat delivered by the fuel in an engine. 13 Adiabatic Flame Temperatures Compute by noting H (T ) = H (T ) out out in in for an adiabatic process o Given H ’s and C (T)’s p and assuming stoichiometry, you can solve for T out At true combustion T, equilibrium concentrations of Species other than CO2 H2O are significant… 14 At low P, high T, CO+O2 equilibrium important so Adiabatic Flame T varies a bit with P 15 Standard entropy Standard entropy rd Based on the 3 law of Thermodynamics: C /T p The entropy of any perfectly crystalline material at T = 0 is zero  Standard molar entropy, S m 0 T The entropy of 1 mol of a NB – calculation using substance in its standard state statistical mechanics – next rd based on the 3 law slide Sometimes entropies of formation are used, but this makes no difference to entropies of reaction provided consistency is maintained 16 Statistical Mechanics Basics • There is a quantity Q called the “partition function” Q = S g exp(E /k T) i i B where Ei are the possible energies of the molecule (quantum mechanics only allows certain quantized energies), and g is the i number of quantum states with energy E . i • Q contains enough information to compute all the normal thermochemical quantities. For example Helmholtz Free Energy = U – TS = G PV = k T ln Q B 2 U(T,V) = k T ∂(ln Q)/∂T B 17 Quantized Energies Partition Functions We usually approximate each of the vibrations in a molecule as a harmonic oscillator. (This is not always an accurate approximation, but it really simplifies the math) The quantized energy of a harmonic oscillator with characteristic frequency n are: I recommend you choose the zero of energy to be the lowest state (all the vibrations have n =0), and handle the zeropointenergy i (ZPE) = ½ h S n separately. Then E = h S n n and i vib i i 1 q = P ( 1 – exp (hn /k T)) vib i B 18 3 Translational Partition Function (Particle in a Cube, V=L ) 19 Rotational Partition Functions For a Linear Molecule, assumed to be Rigid with Moment of Inertia “I”: Each J state has (2J+1) M states, so g = 2J+1 Making a similar approximation as for translation, and considering the effects of symmetry (with symmetry number s), we obtain 2 2 q  8p Ik T/sh rot B For nonlinear molecules there are 3 distinct moments of inertia Ia,Ib,Ic ½ 3/2 q  p q rotnonlinear rot 3/2 where I is replaced by sqrt(I I I ) a b c 20 Total partition function Above we gave partition functions for certain motions of a single molecule in a volume V. For N identical noninteracting molecules in a volume V (e.g. an ideal gas): N Q = (q q q q ) / N vib trans rot elec ln Q N ln (q q q q ) – N ln N vib trans rot elec Conventionally people replace the V in q with V/N = RT/P and just write trans ln Q = N ln q Where q = qvib qtrans qrot qelec using the modified q trans Note for most stable molecules there is only one accessible electronic state, so q 1. For most radicals q 2. For molecules with lowlying nondegenerate elec elec electronic states one should evaluate the partition function exactly. If the molecules are interacting weakly (e.g. nonideal gas), one can correct the expression above for Q using the Equation of State. For liquids q and q rot trans are significantly different, but usually q stays about the same as in gas vib phase. 21 Thermodynamic and spectroscopic data from NIST • http://webbook.nist.gov/chemistry/ • E.g. Methane, gas phase. Selected thermodynamic data, ir spectra, vibrational and electronic energy levels Quantity Value Units Method Reference 1  DH 74.87 kJ mol Review Chase f 1998 1  DH 74.6±0.3 kJ mol Review Manion f 1 1  S 188.66±0.42 J mol K N/A Colwell 1963 22 Beware Internal Rotors Floppy Motions • The normal vibrational partition function formulas are for harmonic oscillators. • Some types of vibrational motions (torsions/internal rotations, umbrella vibrations, pseudorotations) are NOT harmonic. – E.g. rotations about CC bonds – Puckering of 5membered rings like cyclopentane • Many of the entropy values in standard tables are derived using approximate formulas to account for internal rotation. Who knows what formulas they used to estimate other floppy motions. They can be significantly in error • If you care about the numbers, read the footnotes in the tables to see how the numbers were computed. Just because it is in a table does not mean it is Truth. • Always read the error bars 23 Often impossible to measure all the vibrational frequencies… so use quantum chemistry to fill in the gaps • For many quantum chemistry methods, people have implemented software that efficiently computes the second derivatives of the potential energy surface 2 ∂ V/ ∂x ∂y m n • From the second derivatives, one can do normalmode analysis to compute all the smallamplitudelimit vibrational frequencies n i • It is hard (essentially impossible) to compute V exactly for multielectron molecules. However, there are many good approximate methods: e.g. B3LYP, CCSD, CASPT2, MRCI, HF – After the slash is the name of the basis set used when expanding the molecular orbitals: e.g. 631G, TZ2P, etc. • Currently, most people use Density Functional Theory approximations to compute the second derivatives of V, e.g. M08, M06, B3LYP, … 24 Computational Chemistry Comparison and Benchmark DataBase http://cccbdb.nist.gov/ • The CCCBDB contains links to experimental and computational thermochemical data for a selected set of 1272 gasphase atoms and molecules. Tools for comparing experimental and computational idealgas thermochemical properties • Species in the CCCBDB – Mostly atoms with atomic number less than than 18 (Argon). A few have Se or Br. – Six or fewer heavy atoms and twenty or fewer total atoms. Exception: Versions 8 and higher have a few substituted benzenes with more than six heavy atoms. Versions 12 and higher have brominecontaining molecules. • Specific experimental properties 1. Atomization energies 2. Vibrational frequencies 3. Bond lengths 4. Bond angles 5. Rotational constants 6. Experimental barriers to internal rotation 25 Methane 26 Enthalpies of formation of radicals • Enthalpies of formation of stable compounds, such as hydrocarbons, are determined from measurements of enthalpies of combustion, using Hess’s Law. • This approach is not feasible for radicals. An IUPAC evaluation of thermodynamic data for radicals can be found in Ruscic et al J Phys Chem Ref Data, 2005, 34, 573. • Example: CH . Determined by: 3 – Kinetics, e.g. J Am Chem Soc, 1990, 112, 1347 – Photoionization spectroscopy, e.g. J Chem Phys, 1997, 107, 9852 – Electronic structure calculations, e.g. J Chem Phys, 2001, 114, 6014 o • Recommended value by IUPAC: D H (298.15 K) = f 1 146.7 ± 0.3 kJ mol 27 Kinetics and thermodynamics of alkyl radicals Seetula et al. J Am Chem Soc, 1990, 112, 1347 • Measured k(T) for R + HI, using laser flash photolysis / photoionization mass spectrometry, and combined with existing data for reverse reaction (I + RH) to determine equilibrium constant. Enthalpy of reaction determined by second and third law methods 28 Photoionization spectrum of CH 3 Litorja and Ruscic, J Chem Phys, 1997, 107, 9852 • Measure the photionization threshold for CH and the 3 appearance potential of CH from CH photexcitation. 3+ 4 Obtain the dissociation energy of CH H: 3 + CH  CH + e R1 3 3 + CH  CH + H + e R2 4 3 R2R1: CH  CH + H 4 3 29 Computed data (Quantum Chemistry) • Geometries, vibrational frequencies, entropies, energies, means for comparing data CH 3 30 Direct measurement of equilibrium constant for reactions involving radicals: H + C H C H 2 4 2 5 • Brouard et al. J. Phys. Chem. 1986, 90,445450 • Laser flash photolyisis, H atom resonance fluorescence. • Reactions involved: H + C H  C H k 2 4 2 5 1 C H  H + C H k 2 5 2 4 1 H  diffusive loss k d • Solve rate equations – gives biexponential decay of H, k 1 and k and hence K from 1 1 analysis. Vary T, enthalpy of reactions from second or third law. 31 RH bond energies: Extensive tabulation and review Berkowitz et al. 1994, 98, 2744 • The bond enthalpy change at 298 K is the enthalpy change for the reaction RH  R + H: • The bond energy (change) or dissociation energy at zero K is: • Bond energies can be converted to bond enthalpy changes using the relation U = H + pV = H +RT, so that, for RH  R + H, DU = DH +RT. At zero K, the dissociation energy is equal to the bond enthalpy change. • Berkowitz et al provide an extensive dataset for RH bond energies using radical kinetics, gasphase acidity cycles, and photoionization mass spectrometry 32 Thermodynamic databases • Active, internally consistent thermodynamic databases: – ATcT Active thermochemical tables. Uses and network approach. Ruscic et al. J. Phys. Chem. A 2004, 108, 99799997. – NEAT . Network of atom based thermochemistry. Csaszar and Furtenbacher: Chemistry – A european journal, 2010,16, 4826 33 A Grid ServiceBased Active Thermochemical Table Framework von Laszewski et al. 34 An Part of a thermo chemical network showing the basic ideas 35 36 Evidence for a Lower Enthalpy of Formation of Hydroxyl Radical Ruscic et al. J Phys Chem, 2001, 105, 1 37 Accurate Enthalpy of Formation of Hydroperoxyl Radical, HO2 Ruscic et al., J. Phys. Chem. A 2006, 110, 65926601 38 39 Comparison with Howard data 40 Example of current accuracy in ATcT 3 CH + H  CH + H 2 2 B. Ruscic, private communication of unpublished ATcT datum from ver. 1.112 of ATcT TN (2012) • the enthalpy of the reaction is equivalent to the difference in bond dissociation energies of H and CH , and noting that D (CH ) = 2 2 0 2 TAE0(CH )TAE0(CH), the recently published ATcT total 2 atomization energies (TAE0) for CH and CH and the ATcT enthalpy 2 of formation for H produce a quite accurate 0 K enthalpy of the reaction of 3.36 ± 0.08 kcal/mol (14.04 ± 0.35 kJ/mol). The latest ATcT value is nearly identical, 3.38 ± 0.04 kcal/mol (14.15 ± 0.18 kJ/mol), though it has further gained in accuracy due to additional refinements of the ATcT TN • Propagating the uncertainty in the equilibrium constant: • Determine at 1000 K for combustion applications and at 10 K for applications in interstellar chemistry. 41 From a Network of Computed Reaction Enthalpies to AtomBased Thermochemistry (NEAT) A. G. Csaszar and T. Furtenbacher, Chemistry – A european journal, 2010,16, 4826 Abstract: A simple and fast, weighted, linear least squares refinement protocol and code is presented for inverting the information contained in a network of quantum chemically computed 0 K reaction enthalpies. This inversion yields internally consistent 0 K enthalpies of formation for the species of the network. 42 NEAT protocol 43 Incorporation of thermodynamics data into rate calculations: Burcat: http://garfield.chem.elte.hu/Burcat/burcat.html • Provides data in NASA polynomial form, with 7 parameters that are related to necessary thermodynamic functions of state via: Cp/R = a1 + a2 T + a3 T2 + a4 T3 + a5 T4 H/RT = a1 + a2 T /2 + a3 T2 /3 + a4 T3 /4 + a5 T4 /5 + a6/T S/R = a1 lnT + a2 T + a3 T2 /2 + a4 T3 /3 + a5 T4 /4 + a7 Where H(T) = DH (298) + H(T) H(298) f • Linked to ATcT and used in Chemkin. 44 Burcat database. Entry for CH 3 • CH3 METHYLRAD STATWT=1. SIGMA=6. IA=IB=.2923 IC=.5846 NU=3004,606.4,3161(2), 1396(2) HF298=146.7 +/0.3 KJ HF0=150.0+/0.3 kJ REF= Ruscic et al JPCRD 2003. HF298=146.5+/0.08 kJ REF=ATcT C Max Lst Sq Error Cp 6000 K 0.44. METHYL RADICAL IU0702C 1.H 3. 0. 0.G 200.000 6000.000 B 15.03452 1 • 0.29781206E+01 0.57978520E02 0.19755800E05 0.30729790E09 0.17917416E13 2 • 0.16509513E+05 0.47224799E+01 0.36571797E+01 0.21265979E02 0.54583883E05 3 • 0.66181003E08 0.24657074E11 0.16422716E+05 0.16735354E+01 0.17643935E+05 4 nd • First 7 entries are a17 for 1000 – 6000 K. 2 set are a1a7 for 200 – 1000 K. Temp ranges specified in line 1 45 46 Very helpful to know typical bond energies Many of the most important reactions in combustion are Habstractions of the form X. + HY = XH + Y. The barrier is lower in the exothermic direction, Ea,reverse Ea,forward DHrxn and often the weakest XH bond is the one that reacts the fastest. 47 Group Additivity • Experimentally, for alkanes it is observed that H, S, and Cp all vary linearly with the number of Carbons • One can assign a value to the increments caused by inserting one more CH2 group into the alkane chain. • This approach works for many different Data for nalkanes chemical functional groups: adding the group to the molecule adds a set amount to H, S.W. Benson constructed tables of these S, Cp called a GAV. Group Addivity Values (GAV). Several researchers, • For S, need to add a especially Bozzelli and Green, have added to these symmetry correction to tables using quantum chemistry to fill in gaps in the sum of the GAV. experimental data. 48 Programs to estimate thermo with Group Additivity • http://webbook.nist.gov/chemistry/grpadd/ • THERGAS (Nancy group) • THERM (Bozzelli) • RMG (Green group, MIT) • Several others... All of these programs are based on Benson’s methods described in his textbook “Thermochemical Kinetics” and in several papers by Benson. See also several improvements to Benson’s method by Bozzelli. Group additivity is related to the “functional group” concept of organic chemistry, and to “Linear Free Energy Relationships” (LFER) and “Linear StructureActivity Relationships” (LSAR). 49 Problems with Group Additivity • While the group additivity method is intuitively simple, it has its drawbacks stemming from the need to consider higherorder correction terms for a large number of molecules. Take cyclopentane as an example, the addition of group contributions o yields H = –103 kJ/mol, yet the experimental value is –76 kJ/mol. The difference is caused by the ring strain, which is not accounted for in the group value of C–(C2,H2) obtained from unstrained, straightchain alkane molecules. • Cyclics are the biggest problem for group additivity, but some other species also do not work well, e.g. some halogenated compounds, and some highly branched compounds. Very small molecules are often unique (e.g. CO, OH), so group additivity does not help with those. • Species with different resonance forms can also cause problems, e.g. propargyl CH2CCH can be written with a triple bond or two double bonds, which should be used when determining the groups 50 Equilibrium minimizes Free Energy G Free Energy Minimizer Software is Available. See e.g. EQUIL in CHEMKIN package. 51 Computational Kinetics, Part 1 Rate Theory 52 53 Start with BornOppenheimer approximation • Electrons are light and have high kinetic energies, so they move very fast compared to the nuclei. So expect nuclei to feel timeaveraged force exerted by swarm of electrons. • Electrons are very quantum mechanical (wavelike, Pauliexclusion principle). Described by Schroedinger’s Wave Equation. • Atoms/Nuclei are much heavier, move slowly, act like classical particles (mostly). Treat them with classical mechanics with some corrections. • So solve the electronmotion problem first, assuming the nuclei are stationary at different geometries R, yielding a potential field V(R) that the nuclei are moving in. • Done with programs such as GAUSSIAN or MOLPRO • Hard problem, so we use basis set expansions approximations like CCSD(T) or DFT • How can we use computed V(R) to compute rate coefficients k(T) 54 How does one compute a rate coefficient using classical mechanics First, assume ergodicity, i.e. all phase space (q,p) is equally likely to be sampled, biased only by Boltzmann weighting and conservation laws. Divide phase space into “reactant” and “product” regions by specifying ± dividing surface s (q)=0. Sample from “reactant” phase space, and see how fast each trajectory moves from reactant to product. (Only count trajectories which spend significant time as “products”; if they immediately bounce back to reactants we ignore them.) The average time it takes to move from reactant to product is related to the rate coefficient. 55 56 ± ± With the Transition State Assumption of No Recrossing, and choosing s = q , a special coordinate with value of zero at dividing surface, the Classical Phase Space Integrals can be rewritten this pretty way: ± ± Where E = lowest V(R) with q =0 (i.e. on the dividing surface) minus lowest V(R) in reactant space. ± 3N Q = ∫dp dq exp(βH) (1Q(q ))/h reac i.e. it is the integral over the “reactant” phase space, and ± ± ± 3N1 Q = ∫dp dq exp(βH(q =0, p =0)) /h ± ± with no integration over q or p . This integral is on dividing surface. 57 Q is the Heaviside function: Q(x) = 1 if x0, zero otherwise. 58 Why use Transition State Theory 59 However, we cannot completely ignore quantum mechanics for atomic motions… • There is an exact quantum mechanical operator corresponding to the classical phase space integrals, with and without the transition state assumption, see W.H. Miller papers. • Exact version is expensive, biggest case done so far is CH + H, by Manthe. 4 • Several different approximations to the exact formula have been proposed, no consensus yet on best way to proceed. • Some people just ignore the quantum mechanics, and do classical calculations, either phase space integrals or molecular dynamics. But neglecting zeropointenergy of vibrations is a big approximation, and there are also issues about rareevent sampling. • Several patches to molecular dynamics try to include zero point energy approximately. You may be interested in the RPMD method (see RPMDRate program) which avoids some of the TS approximations. • If you are willing to make the TS approximations and some other approximations, you get a cheap and convenient recipe for computing rates… 60 Conventional TST Recipe • Conventional (approximate) TST recipe: • Replace the Q’s with the conventional RRHO formulas used for thermo • Use q /(V/N) rather than dimensionless q proportional to V trans trans ± • Correct E with zeropointenergy difference between reactant and TS. • If mirrorimage TS’s: factor of 2 in symmetry numbers • Multiply by a tunneling correction • To do the conventional TST calculation: 1) Find geometry that minimizes V(R) for reactants and TS (usually a saddle point), typically with a DFT calculation. nd 2 2) Compute V(R) and the 2 derivatives ∂ V/∂R ∂R at those two special m n geometries as accurately as you can afford. 3) Plug those numbers into the TST / stat mech formulas and you probably have a pretty good estimate of k(T) 61 Note that this symmetry number includes Symmetrical internal rotors as well as overall Rotations (“external rotors”). Sometimes the Internal and external rotor symmetry numbers are lumped in with q , don’t double count rot 62 A consistency check • Assuming TST formula is correct: ± k = k T/h Q /Q exp((E E )/k T) forward B reac o,TS o,reac B ± k = k T/h Q /Q exp((E E )/k T) reverse B prod o,TS o,prod B So K = k /k = Q /Q exp((E – E )/k T) c forward reverse prod reac o,prod o,reac B Exactly same thing one gets from Stat Mech Thermo…so at least this rate formula is consistent 63 Homework Question • Consider the reaction C6H7 (cyclohexadienyl)  C6H6(Benzene) + H • Gao et al. J.Phys.Chem.A (2009) determined the delta H for this reaction is 322 kJ/mole. Tsang J.Phys.Chem.(1986) estimated the entropy of formation of C6H7 is about 375 J/moleK at 550 K. • To a first approximation, the Q’s for the TS and for the reactant are about the same. Making this approximation, and using Gao and Tsang’s numbers, what is k(T) for this reaction at T=550 K What are the units • Does this reaction follow the Arrhenius rate law k(T)=A exp(Ea/RT) Sketch what k(T) looks like on an Arrhenius plot (ln k vs. 1000 K/T). • Compute the reverse rate coefficient, for H atom adding to Benzene at 550 K. What are the units 64 Microcanonical TST: The RRKM rate expression k(E) • The TST equation above assumes a perfect thermal Boltzmann population with a clear temperature T: k(T) • In many combustion reactions, the reactions are so fast that a Boltzmann distribution cannot be established. For those cases it is better to use k(E), and average as necessary over the true energy distribution. • The derivation above repeated for microcanonical case gives ± k(E) = N (E)/hr (E) reac where N(E)=S Q(EEi) and r=dN/dE Probably Wigner knew this formula. Rudy Marcus derived this following on work by Rice, Rampsberger and Kassel, who had derived a different equation called RRK, so the new equation is called RRKM. There was a big controversy about this equation for several decades, and competing formulas for k(E) were proposed. Note that as written N(E) is not smooth, it jumps up when E is high enough to include one more E in the summation. This was never observed…. i 65 …until the 1980’s, when the quantum steps were observed in the reaction CH2CO  CH2 + CO • Y axis is the Yield of a particular state of CH 2 • X axis is E • The Yield is a fraction; numerator k(E) Chen, Green, Moore and denominator J. Chem. Phys. (1988) Following on earlier expts is smoothly by Bitto et al. increasing with E 66 ± Now everyone believes k(E) = N (E)/hr (E) reac • W.H. Miller has derived the exact quantum mechanical operator for N, including tunneling etc. and shown it gives k(E). The exact N gives steps rounded off a bit by tunneling. • A Boltzmann average of k(E) gives k(T) as expected k(T) = 1/Qreac ∫r (E)k(E) exp(E/k T)dE reac B ± = 1/hQreac ∫N (E)exp(E/k T)dE integrate by parts B ± = k T/hQreac ∫r (E)exp(E/k T)dE = k T/h Q /Q exp(bDE) B TS B B reac • Microcanonical detailed balance: r (E)k (E) = r (E)k (E) (= N(E)/h ) reac forward prod reverse 67 One should keep track of J and other conserved quantum numbers in addition to E Currently this is not usually done, but it should be. There is a variant on this algorithm which makes it easy to include nonharmonic modes. Astholz Troe and Wieters (1979). 68 69 See also recent papers by Truhlar on a way to handle coupled torsions. This problem is not completely resolved 70 71 Quantum Mechanically, you don’t need to get over the Barrier You can Tunnel Through the Barrier A way to beat Arrhenius / Boltzmann restriction on reaction rates 72 Tunnelling is very important at low T 73 74 This is another issue which is only partially resolved. Fortunately tunneling is rarely important at combustion T. Extremely important at low T. 75 Summary of Rate Theory • Almost all methods start from BornOppenheimer V(R) • Crucially depend on accuracy of the (approximate) V(R) • Errors in V(R) directly lead to errors in Ea, can give big errors. • Modern methods can achieve pretty accurate reaction barriers (but not always) • Classical Mechanics Rate Theory can be transformed into Quantum Mechanics, but exact quantum mechanics usually too hard to solve • Some simple approximations convert Classical Mechanics into Conventional Transition State Theory, which is much much less expensive to compute. Requires V(R) at only two points. • With reasonable V(R) almost always gives k(T) within order of magnitude of true value. Sometimes better than factor of 2. • Several of the simple approximations in Conventional Transition State Theory are not quite good enough, give small but significant errors. Improved approximations are under development. 76 Some important Qualitative differences between PES’s for different reactions 77