1 SF Chemical Kinetics. Lecture 5 . Microscopic theory of chemical reaction kinetics. Microscopic theories of chemical reaction kinetics. • A basic aim is to calculate the rate constant for a chemical reaction from first principles using fundamental physics. • Any microscopic level theory of chemical reaction kinetics must result in the derivation of an expression for the rate constant that is consistent with the empirical Arrhenius equation. • A microscopic model should furthermore provide a reasonable interpretation of the pre-exponential factor A and the activation energy E A in the Arrhenius equation. • We will examine two microscopic models for chemical reactions : – The collision theory. – The activated complex theory. • The main emphasis will be on gas phase bimolecular reactions since reactions in the gas phase are the most simple reaction types.
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1
SF Chemical Kinetics
Lecture 5
Microscopic theory of chemical reaction kinetics
Microscopic theories of chemical reaction kinetics
bull A basic aim is to calculate the rate constant for a chemical reaction from first principles using fundamental physics
bull Any microscopic level theory of chemical reaction kinetics must result in the derivation of an expression for the rate constant that is consistent with the empirical Arrhenius equation
bull A microscopic model should furthermore provide a reasonable interpretation of the pre-exponential factor A and the activation energy EA in the Arrhenius equation
bull We will examine two microscopic models for chemical reactions ndash The collision theory
ndash The activated complex theory
bull The main emphasis will be on gas phase bimolecular reactions since reactions in the gas phase are the most simple reaction types
2
References for Microscopic Theory of Reaction Rates
bull Effect of temperature on reaction ratendash Burrows et al Chemistry3 Section 87 pp383-389
bull Collision Theory Activated Complex Theoryndash Burrows et al Chemistry3 Section 88 pp390-395
ndash Atkins de Paula Physical Chemistry 9th edition Chapter 22 Reaction Dynamics Section 221 pp832-838
ndash Atkins de Paula Physical Chemistry 9th edition Chapter 22 Section224-225 pp 843-850
Collision theory of bimolecular gas phase reactions
bull We focus attention on gas phase reactions and assume that chemical reactivity is due to collisions between molecules
bull The theoretical approach is based on the kinetic theory of gasesbull Molecules are assumed to be hard structureless spheres Hence themodel neglects the discrete chemical structure of an individualmolecule This assumption is unrealistic
bull We also assume that no interaction between molecules until contactbull Molecular spheres maintain size and shape on collision Hence thecentres cannot come closer than a distance d given by the sum ofthe molecular radii
bull The reaction rate will depend on two factors
bull the number of collisions per unit time (the collision frequency)bull the fraction of collisions having an energy greater than a certain
threshold energy E
3
Simple collision theory quantitative aspects
Products)()( kgBgA
Hard sphere reactantsMolecular structure anddetails of internal motionsuch as vibrations and rotationsignored
Two basic requirements dictate a collision event
bull One must have an AB encounter over a sufficiently short distance to allowreaction to occur
bull Colliding molecules must have sufficient energy of the correct type to overcome the energy barrier for reaction A threshold energy E is required
Two basic quantities are evaluated using the Kinetic Theory of gases the collision frequency and the fraction of collisions that activate moleculesfor reactionTo evaluate the collision frequency we need a mathematical way to definewhether or not a collision occurs
Ar
Brd
Area = s
A
B
The collision cross section for two molecules can be regarded to be the area within which the center of theprojectile molecule A must enter around the target molecule Bin order for a collision to occur
22
BA rr ds
The collision cross section sdefines when a collision occurs
Effective collisiondiameter
BA rr d
4
Gas molecules exhibita spread or distributionof speeds
Tk
mv
Tk
mvvF
BB 2exp
24)(
223
2
)(vF
v
Maxwell-Boltzmann velocity distribution function
bull The velocity distribution curvehas a very characteristic shape
bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediatespeeds
bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed
bull The shape of the Gaussian distribution curve changes as the temperature is raised
bull The maximum of the curve shifts tohigher speeds with increasing
temperature and the curve becomesbroader as the temperatureincreases
bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature
JC Maxwell1831-1879
The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB
rBAAB vnnZ 2d
nj = number density of molecule j (units m-3)
Mean relative velocityUnits m2s-1
Mean relative velocity evaluated via kinetic theory
Tk
mv
Tk
mvvF
dvvFvv
BB 2exp
24)(
223
2
0
Average velocity of a gas molecule
Maxwell-Boltzmann velocityDistribution function
m
Tkv B
8
Mass ofmolecule
Some maths
MB distribution of velocitiesenables us to statistically estimate the spread ofmolecular velocities in a gas
BA rr d
5
We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across
22
BAr vvv
j
Bj
m
Tkv
8
Tkv B
r
8
Reduced mass
BA
BA
mm
mm
Hence the collision numberbetween unlike moleculescan be evaluated
rBAAB vnnZ 2d
BA
BBAAB
nZn
TknnZ
21
8
s
Collision frequencyfactor
For collisions between like molecules vvr 2The number of collisions per unittime between a single A molecule and other AMolecules
21
82
A
BAA
m
TknZ
s
Total number of collisionsbetween like molecules
21
2 8
2
2
2
A
BA
AAAA
m
Tkn
nZZ
s
We divide by 2 to ensureThat each AA encounterIs not counted twice
Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue e
Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value
TkF
B
exp
eee
E
6
The simple collision theory expression for the reaction rate R between unlikemolecules
TknZn
dt
dnR
B
BAA
expe
21
8
sTk
Z B
The rate expression for abimolecular reaction between A and B A
A B
dcR kc c
dt
A
A BA B
A A
A AA
E N
n nc c
N N
dn dcN
dt dt
e
We introduce molar variables
Hence the SCT rate expression
expA
A A B
dc ER ZN c c
dt RT
The bimolecular rate constant forcollisions between unlike molecules
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
Avogadro constant
The rate constant for bimolecular collisionsbetween like molecules
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
sBoth of these
expressions aresimilar to the Arrhenius equation
CollisionFrequencyfactor
7
We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor
RT
EAk A
obs exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AB encounters
AA encounters
s B
A
ABobs
kNA
ATAzA
8
m
kNA
ATAzA
BA
AAobs
s
82
bull SCT predictsthat the pre-exponentialfactor should depend on temperature
Pre-exponentialfactor
bull The threshold energy and theactivation energy can also becompared 2
ln
RT
E
dT
kd A 2
2ln
RT
RTE
dT
kd
2
RTEEA
Arrhenius
SCT
EEA bull Activation energy exhibits
a weak T dependence
SCT a summary
bull The major problem with SCT is that the threshold energy E is very difficult to evaluate from first principles
bull The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT
bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors
bull We can incorporate P into the SCTexpression for the rate constant
bull For many gas phase reactionsP is considerably less than unity
bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of the chemical nature of the reactants and products
bull What has gone wrong The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction
bull We need a better theory that takes molecular structure into account The activated complex theory does just that
calcAAP exp
RT
EPzk
RT
EPzk
AA
AB
exp
exp
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
2
References for Microscopic Theory of Reaction Rates
bull Effect of temperature on reaction ratendash Burrows et al Chemistry3 Section 87 pp383-389
bull Collision Theory Activated Complex Theoryndash Burrows et al Chemistry3 Section 88 pp390-395
ndash Atkins de Paula Physical Chemistry 9th edition Chapter 22 Reaction Dynamics Section 221 pp832-838
ndash Atkins de Paula Physical Chemistry 9th edition Chapter 22 Section224-225 pp 843-850
Collision theory of bimolecular gas phase reactions
bull We focus attention on gas phase reactions and assume that chemical reactivity is due to collisions between molecules
bull The theoretical approach is based on the kinetic theory of gasesbull Molecules are assumed to be hard structureless spheres Hence themodel neglects the discrete chemical structure of an individualmolecule This assumption is unrealistic
bull We also assume that no interaction between molecules until contactbull Molecular spheres maintain size and shape on collision Hence thecentres cannot come closer than a distance d given by the sum ofthe molecular radii
bull The reaction rate will depend on two factors
bull the number of collisions per unit time (the collision frequency)bull the fraction of collisions having an energy greater than a certain
threshold energy E
3
Simple collision theory quantitative aspects
Products)()( kgBgA
Hard sphere reactantsMolecular structure anddetails of internal motionsuch as vibrations and rotationsignored
Two basic requirements dictate a collision event
bull One must have an AB encounter over a sufficiently short distance to allowreaction to occur
bull Colliding molecules must have sufficient energy of the correct type to overcome the energy barrier for reaction A threshold energy E is required
Two basic quantities are evaluated using the Kinetic Theory of gases the collision frequency and the fraction of collisions that activate moleculesfor reactionTo evaluate the collision frequency we need a mathematical way to definewhether or not a collision occurs
Ar
Brd
Area = s
A
B
The collision cross section for two molecules can be regarded to be the area within which the center of theprojectile molecule A must enter around the target molecule Bin order for a collision to occur
22
BA rr ds
The collision cross section sdefines when a collision occurs
Effective collisiondiameter
BA rr d
4
Gas molecules exhibita spread or distributionof speeds
Tk
mv
Tk
mvvF
BB 2exp
24)(
223
2
)(vF
v
Maxwell-Boltzmann velocity distribution function
bull The velocity distribution curvehas a very characteristic shape
bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediatespeeds
bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed
bull The shape of the Gaussian distribution curve changes as the temperature is raised
bull The maximum of the curve shifts tohigher speeds with increasing
temperature and the curve becomesbroader as the temperatureincreases
bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature
JC Maxwell1831-1879
The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB
rBAAB vnnZ 2d
nj = number density of molecule j (units m-3)
Mean relative velocityUnits m2s-1
Mean relative velocity evaluated via kinetic theory
Tk
mv
Tk
mvvF
dvvFvv
BB 2exp
24)(
223
2
0
Average velocity of a gas molecule
Maxwell-Boltzmann velocityDistribution function
m
Tkv B
8
Mass ofmolecule
Some maths
MB distribution of velocitiesenables us to statistically estimate the spread ofmolecular velocities in a gas
BA rr d
5
We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across
22
BAr vvv
j
Bj
m
Tkv
8
Tkv B
r
8
Reduced mass
BA
BA
mm
mm
Hence the collision numberbetween unlike moleculescan be evaluated
rBAAB vnnZ 2d
BA
BBAAB
nZn
TknnZ
21
8
s
Collision frequencyfactor
For collisions between like molecules vvr 2The number of collisions per unittime between a single A molecule and other AMolecules
21
82
A
BAA
m
TknZ
s
Total number of collisionsbetween like molecules
21
2 8
2
2
2
A
BA
AAAA
m
Tkn
nZZ
s
We divide by 2 to ensureThat each AA encounterIs not counted twice
Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue e
Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value
TkF
B
exp
eee
E
6
The simple collision theory expression for the reaction rate R between unlikemolecules
TknZn
dt
dnR
B
BAA
expe
21
8
sTk
Z B
The rate expression for abimolecular reaction between A and B A
A B
dcR kc c
dt
A
A BA B
A A
A AA
E N
n nc c
N N
dn dcN
dt dt
e
We introduce molar variables
Hence the SCT rate expression
expA
A A B
dc ER ZN c c
dt RT
The bimolecular rate constant forcollisions between unlike molecules
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
Avogadro constant
The rate constant for bimolecular collisionsbetween like molecules
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
sBoth of these
expressions aresimilar to the Arrhenius equation
CollisionFrequencyfactor
7
We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor
RT
EAk A
obs exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AB encounters
AA encounters
s B
A
ABobs
kNA
ATAzA
8
m
kNA
ATAzA
BA
AAobs
s
82
bull SCT predictsthat the pre-exponentialfactor should depend on temperature
Pre-exponentialfactor
bull The threshold energy and theactivation energy can also becompared 2
ln
RT
E
dT
kd A 2
2ln
RT
RTE
dT
kd
2
RTEEA
Arrhenius
SCT
EEA bull Activation energy exhibits
a weak T dependence
SCT a summary
bull The major problem with SCT is that the threshold energy E is very difficult to evaluate from first principles
bull The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT
bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors
bull We can incorporate P into the SCTexpression for the rate constant
bull For many gas phase reactionsP is considerably less than unity
bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of the chemical nature of the reactants and products
bull What has gone wrong The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction
bull We need a better theory that takes molecular structure into account The activated complex theory does just that
calcAAP exp
RT
EPzk
RT
EPzk
AA
AB
exp
exp
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
3
Simple collision theory quantitative aspects
Products)()( kgBgA
Hard sphere reactantsMolecular structure anddetails of internal motionsuch as vibrations and rotationsignored
Two basic requirements dictate a collision event
bull One must have an AB encounter over a sufficiently short distance to allowreaction to occur
bull Colliding molecules must have sufficient energy of the correct type to overcome the energy barrier for reaction A threshold energy E is required
Two basic quantities are evaluated using the Kinetic Theory of gases the collision frequency and the fraction of collisions that activate moleculesfor reactionTo evaluate the collision frequency we need a mathematical way to definewhether or not a collision occurs
Ar
Brd
Area = s
A
B
The collision cross section for two molecules can be regarded to be the area within which the center of theprojectile molecule A must enter around the target molecule Bin order for a collision to occur
22
BA rr ds
The collision cross section sdefines when a collision occurs
Effective collisiondiameter
BA rr d
4
Gas molecules exhibita spread or distributionof speeds
Tk
mv
Tk
mvvF
BB 2exp
24)(
223
2
)(vF
v
Maxwell-Boltzmann velocity distribution function
bull The velocity distribution curvehas a very characteristic shape
bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediatespeeds
bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed
bull The shape of the Gaussian distribution curve changes as the temperature is raised
bull The maximum of the curve shifts tohigher speeds with increasing
temperature and the curve becomesbroader as the temperatureincreases
bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature
JC Maxwell1831-1879
The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB
rBAAB vnnZ 2d
nj = number density of molecule j (units m-3)
Mean relative velocityUnits m2s-1
Mean relative velocity evaluated via kinetic theory
Tk
mv
Tk
mvvF
dvvFvv
BB 2exp
24)(
223
2
0
Average velocity of a gas molecule
Maxwell-Boltzmann velocityDistribution function
m
Tkv B
8
Mass ofmolecule
Some maths
MB distribution of velocitiesenables us to statistically estimate the spread ofmolecular velocities in a gas
BA rr d
5
We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across
22
BAr vvv
j
Bj
m
Tkv
8
Tkv B
r
8
Reduced mass
BA
BA
mm
mm
Hence the collision numberbetween unlike moleculescan be evaluated
rBAAB vnnZ 2d
BA
BBAAB
nZn
TknnZ
21
8
s
Collision frequencyfactor
For collisions between like molecules vvr 2The number of collisions per unittime between a single A molecule and other AMolecules
21
82
A
BAA
m
TknZ
s
Total number of collisionsbetween like molecules
21
2 8
2
2
2
A
BA
AAAA
m
Tkn
nZZ
s
We divide by 2 to ensureThat each AA encounterIs not counted twice
Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue e
Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value
TkF
B
exp
eee
E
6
The simple collision theory expression for the reaction rate R between unlikemolecules
TknZn
dt
dnR
B
BAA
expe
21
8
sTk
Z B
The rate expression for abimolecular reaction between A and B A
A B
dcR kc c
dt
A
A BA B
A A
A AA
E N
n nc c
N N
dn dcN
dt dt
e
We introduce molar variables
Hence the SCT rate expression
expA
A A B
dc ER ZN c c
dt RT
The bimolecular rate constant forcollisions between unlike molecules
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
Avogadro constant
The rate constant for bimolecular collisionsbetween like molecules
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
sBoth of these
expressions aresimilar to the Arrhenius equation
CollisionFrequencyfactor
7
We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor
RT
EAk A
obs exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AB encounters
AA encounters
s B
A
ABobs
kNA
ATAzA
8
m
kNA
ATAzA
BA
AAobs
s
82
bull SCT predictsthat the pre-exponentialfactor should depend on temperature
Pre-exponentialfactor
bull The threshold energy and theactivation energy can also becompared 2
ln
RT
E
dT
kd A 2
2ln
RT
RTE
dT
kd
2
RTEEA
Arrhenius
SCT
EEA bull Activation energy exhibits
a weak T dependence
SCT a summary
bull The major problem with SCT is that the threshold energy E is very difficult to evaluate from first principles
bull The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT
bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors
bull We can incorporate P into the SCTexpression for the rate constant
bull For many gas phase reactionsP is considerably less than unity
bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of the chemical nature of the reactants and products
bull What has gone wrong The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction
bull We need a better theory that takes molecular structure into account The activated complex theory does just that
calcAAP exp
RT
EPzk
RT
EPzk
AA
AB
exp
exp
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
4
Gas molecules exhibita spread or distributionof speeds
Tk
mv
Tk
mvvF
BB 2exp
24)(
223
2
)(vF
v
Maxwell-Boltzmann velocity distribution function
bull The velocity distribution curvehas a very characteristic shape
bull A small fraction of moleculesmove with very low speeds asmall fraction move with very high speeds and the vast majorityof molecules move at intermediatespeeds
bull The bell shaped curve is called aGaussian curve and the molecularspeeds in an ideal gas sample areGaussian distributed
bull The shape of the Gaussian distribution curve changes as the temperature is raised
bull The maximum of the curve shifts tohigher speeds with increasing
temperature and the curve becomesbroader as the temperatureincreases
bull A greater proportion of thegas molecules have high speeds at high temperature than at low temperature
JC Maxwell1831-1879
The collision frequency is computed via the kinetictheory of gasesWe define a collision number (units m-3s-1) ZAB
rBAAB vnnZ 2d
nj = number density of molecule j (units m-3)
Mean relative velocityUnits m2s-1
Mean relative velocity evaluated via kinetic theory
Tk
mv
Tk
mvvF
dvvFvv
BB 2exp
24)(
223
2
0
Average velocity of a gas molecule
Maxwell-Boltzmann velocityDistribution function
m
Tkv B
8
Mass ofmolecule
Some maths
MB distribution of velocitiesenables us to statistically estimate the spread ofmolecular velocities in a gas
BA rr d
5
We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across
22
BAr vvv
j
Bj
m
Tkv
8
Tkv B
r
8
Reduced mass
BA
BA
mm
mm
Hence the collision numberbetween unlike moleculescan be evaluated
rBAAB vnnZ 2d
BA
BBAAB
nZn
TknnZ
21
8
s
Collision frequencyfactor
For collisions between like molecules vvr 2The number of collisions per unittime between a single A molecule and other AMolecules
21
82
A
BAA
m
TknZ
s
Total number of collisionsbetween like molecules
21
2 8
2
2
2
A
BA
AAAA
m
Tkn
nZZ
s
We divide by 2 to ensureThat each AA encounterIs not counted twice
Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue e
Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value
TkF
B
exp
eee
E
6
The simple collision theory expression for the reaction rate R between unlikemolecules
TknZn
dt
dnR
B
BAA
expe
21
8
sTk
Z B
The rate expression for abimolecular reaction between A and B A
A B
dcR kc c
dt
A
A BA B
A A
A AA
E N
n nc c
N N
dn dcN
dt dt
e
We introduce molar variables
Hence the SCT rate expression
expA
A A B
dc ER ZN c c
dt RT
The bimolecular rate constant forcollisions between unlike molecules
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
Avogadro constant
The rate constant for bimolecular collisionsbetween like molecules
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
sBoth of these
expressions aresimilar to the Arrhenius equation
CollisionFrequencyfactor
7
We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor
RT
EAk A
obs exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AB encounters
AA encounters
s B
A
ABobs
kNA
ATAzA
8
m
kNA
ATAzA
BA
AAobs
s
82
bull SCT predictsthat the pre-exponentialfactor should depend on temperature
Pre-exponentialfactor
bull The threshold energy and theactivation energy can also becompared 2
ln
RT
E
dT
kd A 2
2ln
RT
RTE
dT
kd
2
RTEEA
Arrhenius
SCT
EEA bull Activation energy exhibits
a weak T dependence
SCT a summary
bull The major problem with SCT is that the threshold energy E is very difficult to evaluate from first principles
bull The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT
bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors
bull We can incorporate P into the SCTexpression for the rate constant
bull For many gas phase reactionsP is considerably less than unity
bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of the chemical nature of the reactants and products
bull What has gone wrong The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction
bull We need a better theory that takes molecular structure into account The activated complex theory does just that
calcAAP exp
RT
EPzk
RT
EPzk
AA
AB
exp
exp
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
5
We now relate the average velocity to the meanrelative velocityIf A and B are different molecules thenThe average relative velocity is given byThe expression across
22
BAr vvv
j
Bj
m
Tkv
8
Tkv B
r
8
Reduced mass
BA
BA
mm
mm
Hence the collision numberbetween unlike moleculescan be evaluated
rBAAB vnnZ 2d
BA
BBAAB
nZn
TknnZ
21
8
s
Collision frequencyfactor
For collisions between like molecules vvr 2The number of collisions per unittime between a single A molecule and other AMolecules
21
82
A
BAA
m
TknZ
s
Total number of collisionsbetween like molecules
21
2 8
2
2
2
A
BA
AAAA
m
Tkn
nZZ
s
We divide by 2 to ensureThat each AA encounterIs not counted twice
Fraction of moleculeswith kinetic energy greater Than some minimum Thresholdvalue e
Molecular collision iseffective only iftranslational energyof reactants is greater than somethreshold value
TkF
B
exp
eee
E
6
The simple collision theory expression for the reaction rate R between unlikemolecules
TknZn
dt
dnR
B
BAA
expe
21
8
sTk
Z B
The rate expression for abimolecular reaction between A and B A
A B
dcR kc c
dt
A
A BA B
A A
A AA
E N
n nc c
N N
dn dcN
dt dt
e
We introduce molar variables
Hence the SCT rate expression
expA
A A B
dc ER ZN c c
dt RT
The bimolecular rate constant forcollisions between unlike molecules
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
Avogadro constant
The rate constant for bimolecular collisionsbetween like molecules
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
sBoth of these
expressions aresimilar to the Arrhenius equation
CollisionFrequencyfactor
7
We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor
RT
EAk A
obs exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AB encounters
AA encounters
s B
A
ABobs
kNA
ATAzA
8
m
kNA
ATAzA
BA
AAobs
s
82
bull SCT predictsthat the pre-exponentialfactor should depend on temperature
Pre-exponentialfactor
bull The threshold energy and theactivation energy can also becompared 2
ln
RT
E
dT
kd A 2
2ln
RT
RTE
dT
kd
2
RTEEA
Arrhenius
SCT
EEA bull Activation energy exhibits
a weak T dependence
SCT a summary
bull The major problem with SCT is that the threshold energy E is very difficult to evaluate from first principles
bull The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT
bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors
bull We can incorporate P into the SCTexpression for the rate constant
bull For many gas phase reactionsP is considerably less than unity
bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of the chemical nature of the reactants and products
bull What has gone wrong The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction
bull We need a better theory that takes molecular structure into account The activated complex theory does just that
calcAAP exp
RT
EPzk
RT
EPzk
AA
AB
exp
exp
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
6
The simple collision theory expression for the reaction rate R between unlikemolecules
TknZn
dt
dnR
B
BAA
expe
21
8
sTk
Z B
The rate expression for abimolecular reaction between A and B A
A B
dcR kc c
dt
A
A BA B
A A
A AA
E N
n nc c
N N
dn dcN
dt dt
e
We introduce molar variables
Hence the SCT rate expression
expA
A A B
dc ER ZN c c
dt RT
The bimolecular rate constant forcollisions between unlike molecules
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
Avogadro constant
The rate constant for bimolecular collisionsbetween like molecules
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
sBoth of these
expressions aresimilar to the Arrhenius equation
CollisionFrequencyfactor
7
We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor
RT
EAk A
obs exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AB encounters
AA encounters
s B
A
ABobs
kNA
ATAzA
8
m
kNA
ATAzA
BA
AAobs
s
82
bull SCT predictsthat the pre-exponentialfactor should depend on temperature
Pre-exponentialfactor
bull The threshold energy and theactivation energy can also becompared 2
ln
RT
E
dT
kd A 2
2ln
RT
RTE
dT
kd
2
RTEEA
Arrhenius
SCT
EEA bull Activation energy exhibits
a weak T dependence
SCT a summary
bull The major problem with SCT is that the threshold energy E is very difficult to evaluate from first principles
bull The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT
bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors
bull We can incorporate P into the SCTexpression for the rate constant
bull For many gas phase reactionsP is considerably less than unity
bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of the chemical nature of the reactants and products
bull What has gone wrong The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction
bull We need a better theory that takes molecular structure into account The activated complex theory does just that
calcAAP exp
RT
EPzk
RT
EPzk
AA
AB
exp
exp
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
7
We compare the results of SCT with the empirical Arrhenius eqnin order to obtain an interpretation of the activation energy andpre-exponential factor
RT
EAk A
obs exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AB encounters
AA encounters
s B
A
ABobs
kNA
ATAzA
8
m
kNA
ATAzA
BA
AAobs
s
82
bull SCT predictsthat the pre-exponentialfactor should depend on temperature
Pre-exponentialfactor
bull The threshold energy and theactivation energy can also becompared 2
ln
RT
E
dT
kd A 2
2ln
RT
RTE
dT
kd
2
RTEEA
Arrhenius
SCT
EEA bull Activation energy exhibits
a weak T dependence
SCT a summary
bull The major problem with SCT is that the threshold energy E is very difficult to evaluate from first principles
bull The predictions of the collision theory can be critically evaluated by comparing the experimental pre-exponential factor with that computed using SCT
bull We define the steric factor P as the ratio betweenthe experimental and calculated A factors
bull We can incorporate P into the SCTexpression for the rate constant
bull For many gas phase reactionsP is considerably less than unity
bull Typically SCT will predict that Acalc will be in the region 1010-1011 Lmol-1s-1 regardless of the chemical nature of the reactants and products
bull What has gone wrong The SCT assumption of hard sphere collision neglects the important fact that molecules possess an internal structureIt also neglects the fact that the relative orientation of the colliding molecules will be important in determining whether a collision will lead to reaction
bull We need a better theory that takes molecular structure into account The activated complex theory does just that
calcAAP exp
RT
EPzk
RT
EPzk
AA
AB
exp
exp
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
8
RT
EPzk
RT
EPzk
AA
AB
exp
exp
RT
Ez
RT
ETkNk
AB
BA
exp
exp
821
s
AB encounters
RT
Ez
RT
E
m
TkNk
AA
BA
exp
exp2
21
s
AA encounters
Steric factor(Orientation requirement)
Energy criterionTransportproperty
Weaknessesbull No way to compute P from molecularparameters
bull No way to compute E from first principlesTheory not quantitative or predictiveStrengthsbullQualitatively consistent with observation (Arrhenius equation)bull Provides plausible connection between microscopic molecular properties andmacroscopic reaction rates
bull Provides useful guide to upper limits for rateconstant k
Summary of SCT
Henry Eyring1901-1981
Developed (in 1935) theTransition State Theory(TST)orActivated Complex Theory(ACT)of Chemical Kinetics
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
9
Potential energy surface
bull Can be constructed from experimental measurements or from Molecular Orbital calculations semi-empirical methodshelliphellip
Various trajectories through the potential energy surface
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
10
CABABCBCA
Potential energy hypersurface for chemical reaction between atom and diatomic molecule
AE
AE
0U
Reading reaction progress on PE hypersurface
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
11
Transition state
Ea
En
erg
y
Reaction coordinate
Activated complex
Transition state theory (TST) or activated complex theory (ACT)
bull In a reaction step as the reactant molecules A and B come together they distort and begin to share exchange or discard atoms
bull They form a loose structure ABDagger of high potential energy called theactivated complex that is poised to pass on to products or collapse back to reactants C + D
bull The peak energy occurs at thetransition state The energy difference from the ground state is the activation energy Ea of the reaction step
bull The potential energy falls as the atoms rearrange in the cluster and finally reaches the value for the products
bull Note that the reverse reaction step also has an activation energy in this case higher than for the forward step
Ea
A + BC + D
Transition state theory
bull The theory attempts to explain the size of the rate constant kr and its temperature dependence from the actual progress of the reaction (reaction coordinate)
bull The progress along the reaction coordinate can be considered in terms of the approach and then reaction of an H atom to an F2 molecule
bull When far apart the potential energy is the sum of the values for H andF2
bull When close enough their orbitals start to overlapndash A bond starts to form between H and the closer F atom H FF
ndash The FF bond starts to lengthen
bull As H becomes closer still the H F bond becomes shorter and stronger and the FF bond becomes longer and weakerndash The atoms enter the region of the activated complex
bull When the three atoms reach the point of maximum potential energy (the transition state) a further infinitesimal compression of the HFbond and stretch of the FF bond takes the complex through the transition state
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
12
Thermodynamic approach
bull Suppose that the activated complex ABDagger is in equilibrium with the reactants with an equilibrium constant designated KDagger and decomposes to products with rate constant kDagger
KDagger kDagger
A + B activated complex ABDagger products where KDagger
bull Therefore rate of formation of products = kDagger [ABDagger ] = kDagger KDagger [A][B]
bull Compare this expression to the rate law rate of formation of products = kr [A][B]
bull Hence the rate constant kr = kDagger KDagger
bull The Gibbs energy for the process is given by ΔDagger G = minusRTln (KDagger ) and so KDagger = exp(minus ΔDagger GRT)
bull Hence rate constant kr = kDagger exp (minus(ΔDagger H minus TΔDagger S)RT)
bull Hence kr = kDagger exp(ΔDagger SR) exp(minus ΔDagger HRT)
bull This expression has the same form as the Arrhenius expression ndash The activation energy Ea relates to ΔDagger H
ndash Pre-exponential factor A = kDagger exp(ΔDagger SR)
ndash The steric factor P can be related to the change in disorder at the transition state
[A][B]
][AB=
Statistical thermodynamic approach
bull Suppose that a very loose vibration-like motion of the activated complex ABDagger with frequency v along the reaction coordinate tips it through the transition state ndash The reaction rate is depends on the frequency of that motion Rate = v [ABDagger]
bull The activated complex can form products if it passes though the transition state ABDagger
bull The equilibrium constant KDagger can be derived from statistical mechanics ndash q is the partition function for each species
ndash ΔE0 (kJ mol-1)is the difference in internal energy between A B and ABDagger at T=0
bull It can be shown that the rate constant kr is given by the Eyring equationndash the contribution from the critical
vibrational motion has been resolved out from quantities KDagger and qABDagger
ndash v cancels out from the equation
ndash k = Boltzmann constant h = Planckrsquos constant
)RT
ΔE-exp(
q=K 0
BA
ABDagger
)RT
ΔE-exp(
q
h
kT = k Hence
Kh
kT= k
0
BA
AB
r
r
Dagger
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
13
Statistical thermodynamic approach
bull Can determine partition functions qA and qB from spectroscopic measurements but transition state has only a transient existence (picoseconds) and so cannot be studied by normal techniques (into the area of femtochemistry)
bull Need to postulate a structure for the activated complex and determine a theoretical value for qDagger ndash Complete calculations are only possible for simple cases eg H + H2 H2 + H
ndash In more complex cases may use mixture of calculated and experimental parameters
bull Potential energy surface 3-D plot of the energy of all possible arrangements of the atoms in an activated complex Defines the easiest route (the col between regions of high energy ) and hence the exact position of the transition state
bull For the simplest case of the reaction of two structureless particles (eg atoms) with no vibrational energy reacting to form a simple diatomic cluster the expression for kr derived from statistical thermodynamicsresembles that derived from collision theory ndash Collision theory workshelliphelliphelliphellipfor lsquosphericalrsquo molecules with no structure
Example of a potential energy surface
bull Hydrogen atom exchange reactionHA + HBHC HAHB + HC
bull Atoms constrained to be in a straight line (collinear) HA HB HC
bull Path C goes up along the valley and over the col (pass or saddle point) between 2 regions (mountains) of higher energy and descends down along the other valley
bull Paths A and B go over much more difficult routes through regions of high energy
bull Can investigate this type of reaction by collision of molecular atomic beams with defined energy statendash Determine which energy states
(translational and vibrational) lead to the most rapid reaction
Diagramwwwoupcoukpowerpointbtatkins
Mol
HBHC
Mol
HAHB
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
14
Advantages of transition state theory
bull Provides a complete description of the nature of the reaction includingndash the changes in structure and the distribution of energy through the transition
statendash the origin of the pre-exponential factor A with units t-1 that derive from
frequency or velocityndash the meaning of the activation energy Ea
bull Rather complex fundamental theory can be expressed in an easily understood pictorial diagram of the transition state - plot of energy vs the reaction coordinate
bull The pre-exponential factor A can be derived a priori from statistical mechanics in simple cases
bull The steric factor P can be understood as related to the change in order of the system and hence the entropy change at the transition state
bull Can be applied to reactions in gases or liquidsbull Allows for the influence of other properties of the system on the
transition state (eg solvent effects)Disadvantage
bull Not easy to estimate fundamental properties of the transition state except for very simple reactionsndash theoretical estimates of A and Ea may be lsquoin the right ball-parkrsquo but still need
experimental values
dT
kdRTEA
ln2
RT
H
R
S
hc
Tkk B
00
0expexp
000
0
VPUH
RTUEA
internal energy of activation
volume of activation
RTRTmRTnVP
V
1
0
0
0
condensed phases
ideal gases reaction
molecularity
m = 2 bimolecular reaction
mRTEH A 0
m = 1 condensed phases unimolecular
gas phase reactions
m = 2 bimolecular gas phase reactions
RT
E
R
S
hc
Tkk AB exp2exp
0
0
bimolecular gas
phase reaction
pre-exponential factor A
Relating ACT parameters and Arrhenius Parameters
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
15
RT
E
R
Sm
ch
Tkk A
m
Bm expexp
0
10
R
Sm
ch
TkA
m
B
0
10exp R
ES A
kln
T
1Pre-exponential factor related to entropyof activation (difference in entropy betweenreactants and activated complex
R
Sm
ch
TkPZA
m
B
0
10exp
collision theory positive1P
negative1
01
0
0
0
S
SP
SP
steric factor TS less ordered than
reactants
TS more ordered
than reactants
S0 explained interms of changes in translational rotational and vibrationaldegrees of freedom on going fromreactants to TS
m = molecularity
Aln
ACT interpretation of Arrhenius Equation
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