Topic 2 Experiment and theory

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1 Topic 2 Experiment and theory Combination of experiment and theory to determine rate data and understanding on reactions for application in combustion

2 Transition state theory A + BC ABC AB + C Q - partition function, h is Planck s constant and k B is the Boltzmann constant. indicates the transition state Properties of transition state (and reactants if spectroscopic and thermodynamic data are not available) determined from electronic structure calculations. The formulation given above is the canonical form and gives the rate coefficient as a function of T

Note the change from molecular to molar units and that R = k B N A, wheren A is the Avogadro number.

3 Thermodynamic formulation of TST where K is the equilibrium constant and DE 0 is the difference in zero point energies of the reactants and products. DG is the (molar) Gibbs energy change for the reaction. Note the change from molecular to molar units and that R = k B N A, wheren A is the Avogadro number. Applying this equation to TST: _ is the Gibbs energy of activation. We will discuss later when we talk about variational transition state theory

4 Approach to calculating rate coefficients using TST The approach is most simple for a constrained transition state one where there is a significant maximum on the reaction coordinate. The potential energy surface (PES) is calculated using electronic structure methods and the TS located. The energy of the TS relative to the reactants is determined along with the structure of the TS (which allows the moments of inertia to be calculated) and the vibrational frequencies. k (T) can then be calculated using Transition State Theory, and assuming rigid rotor harmonic oscillator (RRHO) behaviour Problems arise with hindered internal rotors and more complex calculations are needed. Quantum mechanical tunnelling through the energy barrier can be important for H transfer reactions.

5 b

6 Example: OH + CH 3 OCH 3 (DME) CH 2 OCH 3 + H 2 O Calculated data for reactants and TS Units cm-1: OH: Rotational constant, B : 18.59; w: DME: A,B,C: 1.519, 0.355, ; w: 188.7, , , , , , , , , , , , , , , , , , , , TS: A,B,C: 0.368, 0.124, w: 82.81, , , , , , , , , , , , , , , , , , , , , , , Imaginary frequency: Zero point energy of TS, relative to reactants: -0.6 kj mol -1

7 OH + DME k 1 /cm 3 molecule -1 s -1 5E-11 1E Temperature / K This work OH + DME This work OH + d-dme This work OD + d-dme Fit to OH + DME this work Arif et al ref 14 Bonard et al Mellouki et al ref 16 Nelson et al ref 17 Nelson et al (Rel Rate) ref 17 Perry et al ref 18 Tully et al ref 19 Wallington et al ref 20 Cook et al ref 30 Wallington et al ref 62 De More and Bayes 1999 ref 63 1E E-3 2.0E-3 3.3E-3 5.0E-3 1/Temperature(K) Importance of tunnelling; asymmetric Eckart barrier (See Miller, J Am Chem Soc, 1979, 101,6810 NB: presence of van der Waals well before TS see later discussion of Inner and outer TSs. Important in reactions of OH with oxygenates. several TSs -??? effects

8 Microcanonical rate theory Microcanonical rate theory examines the rate at a specific energy; it gives: where W (E ) is the sum of states at the transition state, from energy zero to E and N(E) is the density of states at energy E for the reactant(s). We will use this approach when dealing with master equation calculations. Integrating this expression over a Boltzmann distribution of reactants gives the TST result for k(t).

9 Accuracy of TST calculations Kinetic accuracy Uncertainty of barrier height is typically ~ 4 kj mol -1 ( 1 kcal mol -1 ). Graph shows a plot of exp(e/rt) vs T with E = 4 kj mol-1, which represents The uncertainty factor in k deriving from the uncertainty in the activation energy. Deficiencies in the TS model Rigid rotor harmonic oscillator model Tunnelling model generally use Eckart. Improvements higher level ab initio calculations, variational effects, anharmonicities, multidimensional tunnelling Often tune the barrier height obtained from lower level calculations to experimental data, allowing improved extrapolation of the latter

10 k (cm 3 mol -1 s -1 ) Effects of incorporating higher level methods into TST Barker et al.: Chem Phys Letters, 2010, 499, OH + H 2. High level ab intio calculation (CFOUR) + anharmonicity and coupling between different modes + multidimensional tunnelling. Klippenstein: OH + OH O + H 2 O Green: calculation from moderate level ab initio. (Barrier = 2.8 kcal mol -1 ) Barrier constrained to experimental data (1.2 kcal mol -1 ) Black: High level ab initio (0.7 kcal mol - 1 ) Red: High level constrained to experimental data (1.0 kcal mol -1 ) Bedjanian et al. (1999) Bahng & Macdonald (2007) Woolridge et al. (1994) Hong et al. (2010) Sutherland et al. (1991) Lifshitz and Michael (1991) Moderate-level ab initio Data-constrained model High-level ab initio High-level ab initio +0.3 kcal T(K)

11 More complex system Barker et al. J Phys Chem Letters, 3, 1549 (2012): OH + CO H + CO2 High level ab initio calculations, semi classical transition state theory. Incorporate tunnelling. Plot shows results from first principles calculation no tuning of the potential energy surface. Reaction is pressure dependent. Plot shows high and low pressure limits.

12 Radical + radical reactions e.g. CH 3 + H, CH 3 + CH 3 No barrier on surface. Transition states determined variationally

13 W W

14 Correlation of reactant and product modes for CH 3 +H Vibrational frequencies / cm -1 CH 3 CH 4

15 Correlation of reactant and transition state modes of motion A molecule has 3N degrees of freedom where N is the number of atoms. An atom has 3 degrees of translational freedom A linear molecule has 3 translational, 2 rotational and 3N-5 vibrational degrees of freedom A non-linear molecule has 3 translational, 3 rotational and 3N-6 vibrational degrees of freedom. CH 3 + H CH 4 T R V 6 0 8(+reaction coord) 3T 3T 1R 1R 2T 2R *** 1T Reaction coordinate 2R 2V(deformation) *** 1V(umbrella) + 2V(deformation) 3V(deformation) *** 3V(C-H stretch) 3V(deformation)

16 Methyl and Ethane frequencies CH 3 C 2 H 6

17 CH 3 + CH 3 Slagle et al.j. Phys. Chem. 1988, 92, Laser flash photolysis + photionization mass spectrometry (PIMS) at low pressures and absorption spectroscopy (AS) at high pressures. Reaction is second order in radical so absolute, not relative, concentration needed. Use absorption cross section for AS and calibration against loss of precursor for PIMS.

18 Rate coefficient vs pressure

19 Theory Wagner and Wardlaw, J. Phys. Chem. 1988, 92, Applied flexible transition state theory to calculate microcanonical rate constants, k(e,j) with a RRKM model: Two adjustable parameters linked to (i) efficiency of collisional energy transfer and (ii)the evolution of the transitional modes. Obtained best fit to experimental data and then fitted to the Troe parameterisation.

21 More recent calculations, based on ab intio surfaces Harding et al Phys. Chem. Chem. Phys., 2007, 9,4055 k depends sensitively on potential energy, V, as radicals approach. V calc depends on the level of theory used. Calculated k (T) varies by > factor of 10 as level of theory is changed. k (T) calc may be more accurate than k (T) expt, because the latter depends on extrapolation

22 High pressure limit rate coefficients for CH 3 +CH 3 C 2 H 6. The color key is as follows: gray, CASPT2; green, CASSCF; purple, B3LYP; blue, MPW1K; orange, MP2. The black symbols represent experimental results

23 Klippenstein p122

24 Experimental determination of k (CH 3 + H)) Laser flash photolysis producing H and CH3 with [H]<<[CH3] H by resonance fluorescence, CH 3 by absorption. Need absolute [CH 3 ], since k (H) = k[ch 3 ] Brouard et al. J. Phys. Chem. 1989, 93,

25 Experimental results for CH 3 + H CH 3 absorption analysed via Where D(t)=DI/I 0 H fluorescence analysed via Where k 2 refers to CH 3 + CH 3 and k 3 to other 1 st order loss processes for Plot show rate coefficents vs p at 300, 400, 500, 600 K.

26 CH 3 + H. Parameterisation and determination of k Parameterisation using Troe method (see earlier) Also determined k using master equation method coupled with inverse Laplace transform (see later)

27 Klippenstein p99 Positive T dependence for k, : contrast ve dependence for CH 3 +CH 3

28 Su and Michael, Proc Comb Inst 2002, 29, 1219 Shock tube study of the thermal decomposition of C 2 D 5 I/CH 3 I/Kr mixtures which generated D atoms and CH 3 radicals. [H] and [D] were monitored by ARAS. A rate constant of cm 3 molecule -1 s -1 was measured for the reaction CH 3 + D CH 2 D + H. This rate constant was converted to the high pressure limit for CH 3 + H CH 4 using the theoretical ratio of 1.6 determined theoretically by Klippenstein, Georgieskii, and Harding. ( Proc. Comb. Inst. 2002, 29, 1229.)

29 Klippenstein p106 Gorin model (D M Golden) Excluded angles of approach

31 1/Weff = 1/Winner + 1/Wouter

32 OH + C2H4

33 E. E. Greenwald, S. W. North, Y. Georgievskii and S. J. Klippenstein, J. Phys. Chem. A, 2005, 109,

34 Fits to available data slight adjustment of inner TS energy

35 Master equation model for pressure dependent reactions

36 Modelling dissociation and association reactions master equation analysis Collisional energy transfer between grains Dissociation and association A+B Energy grains - Bundles of energy levels AB Set up rate equation for concentration in each grain. Express as matrix equation: dc/dt = Mc Time dependent grain concentrations depend on initial concentrations and on eigenvalues and eigenvectors of M Eigenvalue of smallest magnitude is the negative of the dissociation rate coefficient.

37 Solution: Master equation for dissociation

38 H.O. Pritchard The quantum theory of unimolecular reactions (CUP 1982) Log(-l i ) for dissociation vs pressure at fixed T Numerically smallest eigenvalue gives the rate coefficient fig shows limits k at high p and k 0 [M] at low p. Relaxation eigenvalues are well separated and the system is well behaved a rate coefficient can be assigned.

39 Association reaction

40 Structure of M for an isomerisation reaction

41 More complex reactions. A+B C D A + B E+F Competition between relaxation and reaction E+F C D Similar approach to dissociation problem Numerically smallest eigenvalues related to phenomenological rate constants (chemically significant eigenvalues)

42 Inverse Laplace transform and association rate coefficients Inverting this relationship allows k(e) to be determined from the high pressure limiting rate constant. Most effectively performed using the association rate constant. Davies et al. Chem Phys Letters, 1986, 126,

function, K(b) is the equilibrium constant and N(E) is the rovibronic density of

43 Microcanonical dissociation rate constants from inverse Laplace transform of canonical association rate constant b = 1/RT Q(b) is the rovibronic partition function, K(b) is the equilibrium constant and N(E) is the rovibronic density of states of the association complex. N p is the convoluted densities of states of the reactant species A + B C

44 OH + C 2 H 4 at low T OH + C 2 H 4 HOC 2 H 4

45 Comparison of master equation fits with theory of Greenwald et al.

46 High temperature, OH + C 2 H 4 Hanson group (Vasu et al. J. Phys. Chem. A 2010, 114, ) OH radicals were produced by shock-heating t-butyl hydroperoxide, Me 3 COOH, and monitored by laser absorption near nm K, 2.3 atm 1201 K

47 OH + C 2 H 4 (Vasu et al.)

48 Uncertainty contributions in the determination of OH + C 2 H 4 Contributions from uncertainty in rate coefficients: CH 3 COCH 3 + OH (fractional uncertainty 0.3) 0.1% CH 3 + OH 3 CH 2 + H 2 O (factor 2) 3.9% CH 3 + OH CH 3 OH (factor of 2) 1.5% CH 3 + CH 3 C 2 H= (Factor of 2) 1.5% Also contributions from temperature, fitting process, OH abosorption coefficient, mixture concentration, wavemeter reading Overall uncertainty: 22.8%

4 kcal mol -1 ) by reference to experimental data.

49 Product yields contribution from theory Senosiaian et al, J. Phys. Chem. A 2006, 110, Slight tuning of surface (~0.4 kcal mol -1 ) by reference to experimental data. Note formation of vinyl alcohol >800 K, Confirmed by Taatjes et al. Srinivasan et al. Phys. Chem. Chem. Phys., 2007, 9,

50 Master equation Chemically significant eigenvalues

51 More complex reactions. A+B C D A + B E+F Competition between relaxation and reaction E+F Reminder For a Dissociation reaction: eigenvalue of smallest magnitude is the negative of the dissociation rate constant. C D Similar approach to dissociation problem Numerically smallest eigenvalues related to phenomenological rate constants (chemically significant eigenvalues)

52 Chemically significant eigenvalues for isomerisation Two species, two CSEs System is conservative, so l 1 = 0 Reaction system relaxes to equilibrium state. Relaxation rate constant = l 2 = k f + k r Relaxation time = t t = (k f + k r ) -1

53 Effect of sulfur oxides on fuel oxidation Peter Glarborg Hidden interactions Trace species governing combustion and emissions. 31 st Symposium SO 2 + H (+M) HOSO (+M) (R1) HOSO + H SO + H 2 O (R2) SO + O 2 SO 2 + O (R3) How do we provide rate data for reactions of this sort? Are there hidden complexities in a simple association reaction like (R1)?

54 H + SO 2 experiment + theory 300 OH+SO E rel /kjmol -1 l TS TS2 TS1 TS H+SO 2 HSO 2 k vs p at T = 295, 363 and 423 K 50 0 HOSO Potential energy surface for H + SO 2, from electronic structure calcs Approach: Experimental investigation using vuv LIF for H Master equation analysis, constrained to experimental data Blitz et al, J. Phys. Chem. A 2006, 110, Experimental decay trace for H + SO 2 Gives eienvalue and hence k at selected p and T. Eigenvalues from ME analysis. Red TS1 Blue TS2 Green TS T/K

55 Determination of individual phenomenological rate constants from eigenvalues / eigenvectors Total number of eigenvalues is equal to the total number of grains. The 3 eigenvalues of smallest magnitude relate to the phenomenological rate coefficients of the macroscopic chemical system (chemically significant eigenvalues) H + SO 2 HSO 2 (1) H + SO 2 HOSO (2) H + SO 2 OH + SO (3) H SO 2 HOSO (4) HSO 2 OH + SO (5) HOSO OH + SO (6) l 2 /s l 2 k 2 [SO 2 ]+k -2 k 2 k -1 /k 1 k 2 [SO 2 ] T/K k -2 dc/dt = [SO 2 ] >> [H] pseudo first order conditions

56 l log10 (kr1/10-11 cm 3 molecule -1 s -1 ) Experimental characterisation of the TS2 and TS3 regions of the surface TS2 is significant in the K region. Too slow to study experimentally using LFP.?Characterise using flow reactor methods? TS3 is even more difficult to investigate. Use the reverse reaction, OH + SO, via detailed balance Blitz et al. Proc Comb Inst K / T T/K Use to determine rate constants, k(e) for the reverse reactions H + SO 2 SO + OH HOSO SO + OH

57 l Issues Overlap of chemically significant eigenvalues with energy relaxation eigenvalues: can lead to problems in defining rate constants (e.g. Tsang et al, Robertson et al in alkyl radical decomposition HSO 2 H + SO T/K Use of OH + SO to calculate forward ks using detailed balance. Does detailed balance always apply are the forward and reverse rate constants always related through the equilibrium constant? (Miller and Klippenstein, Miller et al.) Important issue in combustion e.g. CHEMKIN generally introduces forward and reverse reactions, linked via thermodynamics.

58 1,2-Pentyl isomerisation and dissociation

59 1,2 pentyl isomerisation and dissociation Rate coefficients extracted with Bartis Widom analysis, 400 K 600 K

60 Behaviour at high temperatures: overlap of CS and relaxation eigenvalues

61 Binomial expansion of quadratic solution for l 1 Full triangles: ratio of l 1 from the full quadratic solution to l 1 from the ME. Open squares: ratio of l 1 from the full quadratic solution to l 1 from above approximation

62 Binomial expansion of quadratic solution for l 2 Low T full diamonds: ratio of l 2 from full quadratic solution to l 2 from the ME. Open triangles: ratio of l 2 from Eq. full quadratic solution to l 2 from equation above (full expression. ) Open squares: ratio of -l 2 from full quadratic expression to (k 7 + k -7 ).

63 Comparison of the time dependence of the mole fraction of the 1- and 2-pentyl isomers using the summed grain populations from the ME and using the phenomenological rate coefficients from the ME in a biexponential representation: (a) 600 K; (b) 1000 K. a b

64 Conclusions All wells can contribute to all sink channels irrespective of whether they are directly connected to the transition state that leads to a given set of products. Well-skipping is significant and is characterized by non-standard fall-off curves which exhibit a decline in rate coefficient with increasing pressure, indicative of the competition between collisional relaxation and reaction. Product yields are very sensitive to the difference in dissociation energies for 1- and 2-pentyl. The calculations give a difference of only 4 kj mol -1, and ancillary experiments are essential to define the system more accurately. Because of the complexity of the system, the experiments must be interpreted with a master equation analysis.

65 Autoignition chemistry OH RH R Smaller radical (R 1 ) + Alkene (A 1 ) Termination O 2 O 2 HO 2 + Alkene (A 2 ) RO 2 QOOH Products + OH Propagation O 2 O 2 QOOH OH + R'OOH R'O + OH Branching

$Determination of product yields in C 2 H 5 + O 2 Taatjes et al. (J. Phys. Chem. A 104 (2000) 11549 11560) observed the formation of OH and HO 2, determining the fractional yields.$

66 Determination of product yields in C 2 H 5 + O 2 Taatjes et al. (J. Phys. Chem. A 104 (2000) ) observed the formation of OH and HO 2, determining the fractional yields. Used 100% yield of HO 2 from CH 2 OH + O 2 to calibrate the system. HO 2 yield as T and p Two timescales at higher T OH yield is small. Theoretical interpretation and relevance to autoignition chemistry will be discussed later C 2 H 5 + O 2 C 2 H 5 O 2 * C 2 H 5 O 2 * + M C2H 5 O 2 + M C 2 H 5 O 2 * C 2 H 4 + HO 2 C 2 H 5 O 2 + M C 2 H 4 + HO 2

67 C2H5, C3H7+ O2

Master equation analysis: Miller and Klippenstein, Int J Chem Kinet 33: 654 668, 2001 Threeregimes, low, transition, high T.

68 Master equation analysis: Miller and Klippenstein, Int J Chem Kinet 33: , 2001 Threeregimes, low, transition, high T. In transition region, thermal rate constant jumps from one eigenvalue to the other the two eigenvalues are mixed in this region. At high T, ethyl peroxy is stabilised and HO 2 is formed from its dissociation. k is, in practical terms, independent of p. At low T, reaction involves the pressure dependent formation of RO 2 and direct formation of HO 2

69 Cyclohexyl + O2 Fernendes et al. Phys Chem. Chem. Phys., 2009, 11,

70 Time dependence of OH formation

71 Importance of formally direct route to OH

72 Evidence for chain branching at lower T

73 Dimethyl ether: CH 3 OCH 2 + O2 Eskola et al.

74 CH 3 OCH 2 + O 2 : major mechanism CH 3 OCH 2 +O 2 TS1 TS4 TS5 TS2 TS3(c,t) CH 2 O...CH 2 OOH IM1, RO 2 IM2 QOOH OH+ c-ococ 2CH 2 O+OH CBS-QB//mpw1k/avtz + DZPE TS6 IM3, R'O H+R"CHO

75 Species profiles, 550 K, 1 bar E E E E E E E E E E E CH3OCH2 IM1 IM2 OH E-05 1E-06

76 Master equation: rate constant analysis I 1 = RO 2 I 2 = QOOH R (+O 2 ) I 1 I 2

77 Phenomenological rate ceoefficients from a Bartis Widom analysis 1.00E+10 k/s E E E E E E E E E+01 k1 k2 k3 k4 k5 k6 k7 k8 k9 1.00E T/K

78 Routes to branching: CH 3 OCH 2 + O 2 (+ O 2 ) Anderson and Carter: Molecular Physics 2008, 106,

79 J. Phys. Chem. A 2012, 116, Abstract...Master Equation Solver for Multi-Energy Well Reactions (MESMER), a user-friendly, object-oriented, open-source code designed to facilitate kinetic simulations over multi-well molecular energy topologies where energy transfer with an external bath impacts phenomenological kinetics. MESMER offers users a range of user options specified via keywords.

80 Master Equation Solver for Multi-Energy well Reactions (MESMER) Open source, object oriented code (C++) facilitates kinetic simulations of multi-well systems where energy transfer with a bath gas impacts phenomenological kinetics Provides interface with results of electronic structure calculations to allow set up of multi-well system Outputs choices include chemically significant (and other) eigenvalues, species concentrations vs time, phenomenological rate constants. Allows fitting to experimental data using c 2 minimisation

81 Excited electronic states in combustion Chemistry of methylene (CH 2 ) CH 2 exists as a triplet ( 3 CH 2 ) and a singlet ( 1 CH 2 ), separated in energy by ~ 9 kcal mol -1. The upper state (singlet) is much more reactive. It is involved, e.g., in the production of C 3 H 3, a soot precursor and in the chemistry of Titan. The singlet is deactivated to the triplet on collision with unreactive (and reactive) gases. Our understanding of the mechanism of deactivation in reactive systems is limited, especially at combustion temperatures

82 Signal Laser flash photolysis CH 2 CO at (308 nm). Detection 1 CH 2 by LIF, e.g. line from 3 12 state in a 1 A 1 (0,0,0) at nm. Pressure 1-10 Torr. Rapid initial rotational relaxation of 1 CH 2 (0,0,0). Collision induced intersystem crossing (CIISC) and reactions investigated for rotationally relaxed 1 CH 2 (0,0,0) Methodology time ( s)

83 Energy Gannon et al. J. Chem. Phys. 2010, 132 (2), CH 2 + M - CIISC k rot j x Rate coefficient / cm 3 molecules -1 s He Ne Ar Kr Xe N 2 SF 6 k W SO i 20 l k il l k ll 10 X ~ ( v v ) 1 2 v3 ~ a ( 0 0 0) Temperature (K)

84 Total fluorescence signal / arbitrary units 1 CH 2 + C 2 H 2 C 3 H 4 * C 3 H 3 + H 1 CH 2 + C 2 H 2 3 CH 2 + C 2 H 2 1. Kinetics from decay of CH 2 the overall rate constant decreases as T. 2. If same mechanism for deactivation as for Ar, then reactive channel becomes unimportant at higher T??? 3. Monitor H using VUV LIF; growth shows same kinetics as 1 CH 2 decay 4. Calibrate H signal to determine what fraction of the 1 CH 2 loss occurs by reaction and what fraction by deactivation CH 2 H Time / s

85 CH2 + C2H2

86 1 CH 2 + C 2 H Rate coefficient / cm -3 molecule -1 s x x10-10 Rate coefficients for reaction And deactivation x Rate coefficient / cm 3 molecule -1 s CH 2 + C 3 H 6 C 2 H 2 C 2 H Temperature / K Rate coefficient for deactivation to the triplet now decreases with T. Doesn t fit in with the behaviour found for inert gases. What is the mechanism? 1.2x x Reaction Deactivation Temperature / K Surface crossing Singlet surface Triplet surface

87 What is the yield of H + C 3 H 3 for the reactive channels? Used Mesmer to calculate the yields of H+C3H3 vs C3H4 (just reactive channels) Results above appropriate to calculations for Titan (Faraday Discuss., 2010, 147, )

Topic 4 Thermodynamics

Topic 4 Thermodynamics 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