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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,
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.
“Thermochemistry” is about reactions.
1 Law gives energy density, final temperature
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
• 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
Various thermodynamic relations are needed to
determine heat release and the relations between
forward and reverse rate coefficients
A statement of
n is the
For ideal gas or ideal solution, a =P /P or
a = C /C . Must use same standard state
used when computing DG . Solids have a =1.
For surface sites use fractional occupation.
8 Tabulated thermodynamic quantities.
1. Standard enthalpy of formation
Standard enthalpy change of formation, D H
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)
D H = 74.8 kJ mol
The standard enthalpies of formation of C(s)
and H (g) are both zero
Computing K (T), G(T), H(T), S(T)
𝑜 0 𝑜 ′
𝐻 𝑇 =𝑈 + =𝐻 (𝑇 ) + 𝐶 𝑇 𝑑𝑇′
0 0 ′
𝑆 𝑇 =𝑆 (T ) + (𝐶 𝑇 /𝑇′ )𝑑𝑇′
0 0 0
𝐺 𝑇 = 𝐻 (𝑇 )−𝑇 𝑆 (𝑇 )
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
nonidealities are conventionally hidden in the activities as
Fugacities or “activity coefficients”.
C (T) is expressed in several different formats causing some confusion:
NIST/Benson tabulated C (T ) or several different polynomialtype expansions:
Shomate, Wilhoit (beware typos in original paper), two different NASA formats.
C (T) can also be expressed by statistical mechanics formulas.
𝑃𝑉11 Now maintained by Elke Goos.
12 LHV and UHV
• Fuels are classified by their Heating Value, i.e. their heat of
• Two variants are commonly used: Lower Heating Value and Upper
• LHV assumes all the H O formed is in gas phase, this is realistic
for engines where the H O leaves in the exhaust. Note in a real
engine the H O in the exhaust would be hot, but the LHV
calculation usually assumes room temperature steam.
• UHV assumes all the H O formed is in liquid phase. This is
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
Given H ’s and C (T)’s
and assuming stoichiometry,
you can solve for T
At true combustion T,
equilibrium concentrations of
Species other than CO2 H2O
14 At low P, high T, CO+O2 equilibrium important
so Adiabatic Flame T varies a bit with P
15 Standard entropy
Based on the 3 law of
The entropy of any perfectly
crystalline material at T = 0 is
Standard molar entropy, S
The entropy of 1 mol of a
NB – calculation using
substance in its standard state
statistical mechanics – next
based on the 3 law
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
number of quantum states with energy E .
• Q contains enough information to compute all the
normal thermochemical quantities. For example
Helmholtz Free Energy = U – TS = G PV = k T ln Q
U(T,V) = k T ∂(ln Q)/∂T
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
(ZPE) = ½ h S n separately. Then E = h S n n and
i vib i i
q = P ( 1 – exp (hn /k T))
vib i B
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
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
q 8p Ik T/sh
For nonlinear molecules there are 3 distinct moments of inertia Ia,Ib,Ic
q p q
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
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
ln Q = N ln q
Where q = qvib qtrans qrot qelec using the modified q
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
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
are significantly different, but usually q stays about the same as in gas
21 Thermodynamic and spectroscopic data from NIST
• E.g. Methane, gas phase. Selected thermodynamic data, ir
spectra, vibrational and electronic energy levels
Quantity Value Units Method Reference
DH 74.87 kJ mol Review Chase
DH 74.6±0.3 kJ mol Review Manion
S 188.66±0.42 J mol K N/A Colwell
22 Beware Internal Rotors Floppy Motions
• The normal vibrational partition function formulas are for
• 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
∂ V/ ∂x ∂y
• From the second derivatives, one can do normalmode analysis to
compute all the smallamplitudelimit vibrational frequencies n
• 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
• 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
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:
– 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
• Recommended value by IUPAC: D H (298.15 K) =
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
Litorja and Ruscic, J Chem Phys, 1997, 107, 9852
• Measure the photionization threshold for CH and the
appearance potential of CH from CH photexcitation.
Obtain the dissociation energy of CH H:
CH CH + e R1
CH CH + H + e R2
R2R1: CH CH + H
29 Computed data (Quantum Chemistry)
• Geometries, vibrational frequencies, entropies, energies, means
for comparing data CH
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.
• Laser flash photolyisis, H
• 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
• Solve rate equations – gives
biexponential decay of H, k
and k and hence K from
analysis. Vary T, enthalpy of
reactions from second or
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
• Berkowitz et al provide an extensive dataset for RH bond energies
using radical kinetics, gasphase acidity cycles, and photoionization
32 Thermodynamic databases
• Active, internally consistent thermodynamic
– 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
A Grid ServiceBased Active Thermochemical Table
Framework von Laszewski et al.
Part of a
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
Ruscic et al., J. Phys. Chem. A
2006, 110, 65926601
38 39 Comparison with Howard data
40 Example of current accuracy in ATcT
CH + H CH + H
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
atomization energies (TAE0) for CH and CH and the ATcT enthalpy
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
42 NEAT protocol
43 Incorporation of thermodynamics data into rate
• 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)
• Linked to ATcT and used in Chemkin.
44 Burcat database. Entry for CH
• 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
• First 7 entries are a17 for 1000 – 6000 K. 2 set are
a1a7 for 200 – 1000 K. Temp ranges specified in line
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
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.
48 Programs to estimate thermo with Group Additivity
• 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
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
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
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
• How can we use computed V(R) to compute rate coefficients k(T)
54 How does one compute a rate coefficient using
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.
Q = ∫dp dq exp(βH) (1Q(q ))/h
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.
• 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
• 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
• 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.
2) Compute V(R) and the 2 derivatives ∂ V/∂R ∂R at those two special
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
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
• 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
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)
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….
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
• X axis is E
• The Yield is a
Chen, Green, Moore
J. Chem. Phys. (1988)
Following on earlier expts
by Bitto et al.
increasing with E
Now everyone believes k(E) = N (E)/hr (E)
• 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
= 1/hQreac ∫N (E)exp(E/k T)dE integrate by parts
= 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
is rarely important
at combustion T.
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