Example Arrhenius plot
Physical Chemistry
Lecture 6
Mechanisms of chemical reactions
Useful linear form
Slope of plot gives
the activation
energy directly
Many rate constants
seem to obey this
relationship
Chemical kinetics
Understand the nature of reactions



Measure concentration changes with time
Determine effects of various parameters (e.g.
pressure, temperature, relative concentration,
presence of other chemical substances)
Explain observed changes in chemical terms
Requires integration of theory of chemical
action with experimental results


Development of a mechanism of reaction
Distinguish between possible mechanisms by
comparison of prediction to observed results
A exp( Ea / RT )
Arrhenius behavior is
empirical






Useful parameterization of
temperature-dependent rate
constants
A  pre-exponential factor
Ea  activation energy
Often seen in empirical analysis
of kinetic data
Activation energy related to the
likelihood of a reactive event
Modern chemical-reaction
theory does not readily
predict the Arrhenius
equation
Surprising how often
Arrhenius behavior is seen
Seen in analysis because it
is simple
Rate constants of some
reactions do not obey
Arrhenius’s simple equation


Example: decomposition of
diacetylene
Indicates complexity of the
chemical reaction
Elementary reactions
Chemical reactions are often more complex than presented in
the balanced equation

Empirical equation of Arrhenius
(van’t Hoff)
Ea  1 
 
R T 
Non-Arrhenius behavior
Arrhenius theory
k (T ) 
ln k (T )  ln A 

Do not always occur as a single step from initial state to final
state
Order is not necessarily the stoichiometric coefficient
Some reactions do occur in a single step -- elementary
reactions



Generally involve simple mono- or bimolecular interactions
May involve “unusual” species, i.e. species that are not thought of
because they are not very stable
Order in elementary reactions is the stoichiometry number, which
is called the molecularity
H2
H
 2 O  H 2O2
 Br  HBr
v  k[ H 2 ]1[O ]2
v  k[ H ]1[ Br ]1
1
Complex reactions
Complex reactions are
considered to be
comprised of a series of
elementary reactions
The simultaneous
occurrence of these
elementary reactions
leads to an overall
reaction mechanism
Example: reaction of
hydrogen and bromine
to produce HBr
H2
 Br2
Parallel reactions
 2 HBr
Elementary reaction mechanism
Br2
Br  H 2
H  Br2
H  HBr
Br  Br





2 Br
HBr  H
HBr  Br
 Br
H2
Br2



Analytical
expressions for
concentrations are
possible

kI

P

Allows mathematical expressions for
concentrations
At reaction end, both products are
present
kA
A 
B
Example: Reaction goes to
completion; neglect reverse
reaction
B
kB

A
Can always postulate a reverse
reaction to any reaction

[ A(t )]  [ A(0)]e  k At
[ I (t )]


k A [ A(0)]  k At
e  e kI t
kI  k A



k e  k I t  k I e  k At
[ P(t )]  [ A(0)]1  A
kI  k A






kA
e ( k A  k B )t 
k A  kB

 kA
[ B (t )]  [ A(0)]
 kA  kB


kA
e ( k A  k B ) t 
k A  kB

Reversible reactions
In principle, all
reactions are reversible
In practice, for many
reactions equilibrium
lies at one extreme or
the other
Examples of equilibrium
kB
A 
B
kC
A 
C
[ A(t )]  [ A(0)]e  ( k B  kC )t
[ B(t )]  [ A(0)]

kB
1  e ( k B  kC ) t
k B  kC

kC
[C (t )]  [ A(0)]
1  e  ( k B  kC ) t
k B  kC
When reverse reaction is not
inconsequential, it must be
included
Reversible first-order reactions
can be solved mathematically
 kB
[ A(t )]  [ A(0)]
 kA  kB
Parallel first-order reactions
One reactant reacts along two
different pathways
Consider both reactions to be of first
order
Three parallel reactions
Ratios of equilibrium
product concentrations
give ratios of rate
constants
In kinetics, one often neglects
steps considered
inconsequential
kA
A 
I
I
kB
kC

[C ]eq
Reversible reactions
Sequential two-step reaction
Two subsequent
first-order steps
Rate equations for
both steps are
integrable
[ B ]eq
Ratio of final
concentrations
determined by the rate
constants
Example: reactions of
benzyl penicillin under
acidic conditions




Chair – boat interchange
of cyclohexane
Monomer-dimer
conversion
NH2
N
O
H
O
O
H
O
R
NH2
R
 Proflavin
 Carboxylic acid
O
2
R
O
H
2
Simple collision theory of gasphase kinetics
To participate in a
A  B  Products
bimolecular reaction,
molecules must
v  Z AB
approach each other
vmax  Z AB   AB v AB ,ave n*A nB*
closely
8kT
  AB N 0
[ A][ B ]
SCT: gas-phase reaction

rate proportional to
collision frequency
SCT does not generally
 AB  collision cross  section
agree with experimental
rates
8kT
k max   AB N 0
Points out how to think

about theory of
chemical reactions
Experimental SCT and
Arrhenius parameters




p  steric factor
Emin  minimum energy for
reaction
8kT
p  AB N 0


8kT
p  AB N 0

23.85
14.6
0.039
25.42
15.5
0.076
O + O3  2 O2
273 –900
23.21
20.0
0.037
N + NO  N2 + O
300 – 6000
23.28
~0
0.040
CH3 + C6H6  CH4 + C6H5
456 – 600
17.04
38.5
2.7  10-5
BH3 + BH3CO  CO + B2H6
273 – 333
19.34
29.3
PH3 + B2H6  PH3BH3 +BH3
249 – 273
14.97
47.7
7.4  10-6
CO + O2  O + CO2
2400 – 3000
21.97
213.4
4.3  10-3
F2 + ClO2  F + FClO2
227 – 247
16.37
33.5
5.2  10-5
Activated-complex theory
Developed by H. Eyring, M.
Evans, and M. Polanyi
Activated complex is a
precursor to products

exp( Emin / RT ) [ A][ B]



Predicts a temperaturedependent Arrhenius
activation energy
Predicts a temperaturedependent Arrhenius preexponential factor
[ A  B ]
Can be thought of as a
“decomposition”
[ A  B]
Fast equilibrium between
reactants and activated
complex
Ae  Ea / RT
k SCT (T ) 
p
8kT

e  Emin / RT
Arrhenius behavior
Ea
 ln k

T
RT 2
By comparison of derivatives
RT
Ea  Emin 
2
A

p  AB N 0
8kT
e
B  [ A  B ]
A 
Gives products by a firstorder process
Relation of SCT parameters to
Arrhenius parameters

Exists in small amounts
Not seen, like a reactive
intermediate
Can transform in two ways
 Revert to reactants
 Form products
exp( Emin / RT )
k Arr (T ) 
2.9  10-4
(a) The range of validity is expressed in kelvins.
(b) A is in dm3 mol-1 s-1.
(c) Ea is in kJ mol-1.
Like van der Waals’s
improvement of the idealgas law
Many experimental data
reported as Arrhenius
behavior
Comparison with SCT
necessary to connect
theory and experiment
p
0.088
200 – 500

k
Ea
39.3
1000 - 1700
v  Z AB p exp( Emin / RT )
Empirically add two
factors to account for
these features
ln A
24.61
H + HCl  H2 + Cl
SCT neglects two features
Collision must be sufficiently
energetic to cause reaction
Molecules must have proper
orientation to allow reaction
Range
300 – 750
H + HBr  H2 + Br
“Correcting” simple collision
theory

Reaction
H + D2  HD + H


A 
B
 Products
Activated-complex
mathematics
Reaction velocity
proportional to activatedcomplex concentration
Quasi-equilibrium between
reactants and activated
complex
Statistical mechanics defines
disappearance rate constant
of activated complex, f


Defines the enthalpy and
entropy of activation
A means to parameterize
the rate constant
v 
v 

f C

K
a
a A aB
 C
C
C AC B
fK 
C AC B
C
 S  
 H  
RT
 exp 
C AC B
exp
hN 0C 
 R 
 RT 
3
Eyring’s equation
Activated-complex biomolecular rate constant, k2, in
terms of the parameters of the activated complex
Temperature and the Arrhenius
model
Why does the Arrhenius model work so well in many
cases?

 S  
 H  
RT
 exp 

k2 
exp

hN 0C
 R 
 RT 
Not Arrhenius-like behavior

 OH
 CH3

H2 O
From slope and
intercept



H = 12.0 kJ/mol
S = -33.2 J/K-mol
Around room temperature, RT/2 ~ 1.2 kJ/mole
RT/2 varies slowly with temperature compared to the
exponential function
Ea tends to be much larger than RT/2 for many reactions
Emin and Ea both appear to be approximately constant
Computer modeling of
potential-energy surfaces
Potential energy of
configuration of molecules
controls interaction
Model time-dependent
approach of molecules with
(classical or quantum)
simulation
Example:
Evaluate by plotting
ln(k/T) versus 1/T
Example


Can convert between Arrhenius parameters and Eyring
parameters
Evaluation of Eyring
parameters
CH4

Empirical
Not predicted by the simplest theory
Not predicted by activated-complex theory
Partial answer

Difficult to distinguish from Arrhenius behavior under many
circumstances
Another means to parameterize the rate constant


T. Gierczak, R. Talukdar, S. Herndon, G. Vaghjiani, A. Ravishankara,
J. Phys. Chem, 101A, 3125 (1997).
Alternative to
Arrhenius parameters



Relation of Eyring and
Arrhenius parameters
Need to be able to convert between the two
parameterizations of kinetic data
Use differential of Eyring form to show relationships
Phase/Molecularity
Solution
Activation Energy, Ea
H   RT
Pre-exponential factor, A
eRT
S 
exp(
)
hN 0 LC 
R
Gas, unimolecular
H   RT
eRT
S 
exp(
)
hN 0
R
Gas, bimolecular
H   2 RT
e 2 RT
S 
exp(
)
hN 0C 
R
Gas, termolecular
H   3RT
e3 RT
S 
exp(
)
hN 0C  2
R
D + H2  HD + H
Several parameters
Distances
Angles
Computer simulation of gasphase reactions
Reactive encounters
go through the
reactive region
Unreactive
encounter goes to
the reactive region
and return in the
same channel
M. Karplus, R. N. Porter, and R. D. Shamra, J. Chem.
Phys., 43,3258 (1965).
4
Simulation of a reactive event
Follow the time
course of the
approach
Atoms exchange
partners
Time scale is very
short
Repeat many times
and measure
fraction of times
that reactive events
happen
Summary
Theories of simple reactions




Simple collision theory
Modified collision theory
Computer simulation
Activated-complex theory
Parameterization of reaction dynamics



Arrhenius activation energy and pre-exponential
factor
Equilibrium thermodynamic properties of activated
complex from Eyring theory
Simple collision theory parameters
5