Foundations of Chemical Kinetics
Lecture 17:
Unimolecular reactions in the gas phase:
Lindemann-Hinshelwood theory
Marc R. Roussel
Department of Chemistry and Biochemistry
The factorial
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The number n(n − 1)(n − 2) . . . 1 is called the factorial of n.
Notation: n! (read “n factorial”)
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By convention, 0! = 1.
Marc’s notation vs the textbook’s
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Albert Goldbeter once told me that you knew that a student
was taking ownership of their project when they wanted to
change the notation. . .
Quantity
Sum (number) of states
Density of states
Textbook
G
N
Marc
G
g
Density of states for s harmonic oscillators
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In lecture 6, we derived the following expressions for the sum
and density of states of a harmonic oscillator:
G () = (~ω0 )−1
g () = (~ω0 )−1
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Recall: Roughly speaking, the partition function counts the
number of states with energies below kB T .
Therefore,
kB T
Q≈
~ω0
Note: You can also derive this equation from the harmonic
oscillator partition function by assuming that ~ω0 /kB T is
small, as we did in lecture 12.
Density of states for s harmonic oscillators
(continued)
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If we have s distinguishable, independent harmonic oscillators
whose natural frequencies are ωi , the partition function should
therefore be
s
Y
kB T
Qs ≈
~ωi
i=1
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From the definition of the classical partition function, we have
Z ∞
E
Qs =
gs (E ) exp −
dE
kB T
0
Density of states for s harmonic oscillators
(continued)
Z
Qs =
0
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∞
s
Y
E
kB T
gs (E ) exp −
dE ≈
kB T
~ωi
i=1
The problem now is to find the density of states corresponding
to our partition function Qs .
This problem turns out to be solved by taking a mathematical
operation called an inverse Laplace transform of Qs .
The answer is
gs (E ) =
E s−1
Q
(s − 1)! si=1 ~ωi
Hinshelwood theory
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Recall that a collision-theory treatment badly underestimates
the Lindemann rate constant k1 .
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Hinshelwood’s idea was that the energy acquired in a collision
can be stored in any of the bonds in a molecule, and that this
therefore introduces a statistical factor (the degeneracy of the
corresponding total vibrational energy) into the calculation of
the rate constant.
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A classical treatment, assuming that the temperature is
sufficiently high that we can treat the vibrational levels as
continuous, will use the density of states rather than the
degeneracy.
Hinshelwood theory (continued)
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For simplicity, Hinshelwood assumed that the s vibrational
modes of a molecule had a common vibrational frequency ω0 .
Then,
kB T s
Qs ≈
~ω0
E s−1
gs (E ) ≈
(s − 1)! (~ω0 )s
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The probability that a molecule has vibrational energy
between E and E + dE is thus
E
E s−1
E
gs (E )
exp −
dE =
exp −
dE
Qs
kB T
(kB T )s (s − 1)!
kB T
Hinshelwood theory (continued)
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The probability that a molecule has energy greater than Ea is
therefore
Z ∞
E s−1
E
exp −
dE
s
kB T
Ea (kB T ) (s − 1)!
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This integral gives Γ(s, Ea /kB T )/(s − 1)!, where Γ() is the
incomplete gamma function.
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Typically, Ea kB T . In this case, the integral is well
approximated by
1
(s − 1)!
Ea
kB T
s−1
Ea
exp −
kB T
Hinshelwood theory (continued)
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Assuming a collision-limited rate, the rate constant k1 is
therefore
Ea
Act
Ea s−1
exp −
k1 = Act Pr(E > Ea ) =
(s − 1)! kB T
kB T
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Since Ea /kB T 1,
1
(s − 1)!
Ea
kB T
s−1
1,
which explains why collision theory fails so badly for some
unimolecular reactions.
Hinshelwood theory
Summary and comparison to experiment
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Explains why k1 is larger than the collision-limited value: the
vibrational degeneracy allows molecules to store the same
amount of energy in many different ways, introducing a
statistical factor into the theory.
We assume s oscillators with equal vibrational frequencies.
In practice, we treat s as a parameter which we choose to get
the best fit to the data.
Typically we find that s is about half of the normal modes of
the reactant.
Hinshelwood theory fits the pressure dependence of the
observed rate constant better than plain Lindemann theory.
However, there are still deviations at low pressures.
Because of the strongly T -dependent preexponential factor,
Arrhenius plots for k1 (or k∞ ) should be curved.
They are not.