Chapter 5 - Collision Theory.pdf

Collision theory
 The simplest quantitative account of reaction rates is in terms of
collision theory, which can be used only for the discussion of
reactions between simple species in the gas phase.
■ Collision theory:
 We shall consider the bimolecular elementary reaction;
A + B → P v = k2[A][B]
where P denotes products, and aim to calculate the second-order
rate constant k2.
We expect the rate v to be proportional to the rate of collisions, and
therefore to the mean speed of the molecules, c ∝ (T/M)1/2 where M
is the molar mass of the molecules, their collision cross-section, σ,
and the number densities NA and NB of A and B.
v ∝σ(T/M)1/2NANB ∝ σ (T/M)1/2[A][B]

Collision theory
However, a collision will be successful only if the kinetic energy
exceeds a minimum value, the activation energy, Ea, of the reaction.
This requirement suggests that the rate constant should also be
proportional to a Boltzmann factor of the form e−Ea/RT. So we can
anticipate, by writing the reaction rate in the form;
k2 ∝σ (T/M)1/2e−Ea /RT
Not every collision will lead to reaction even if the energy
requirement is satisfied, because the reactants may need to collide
in a certain relative orientation. This ‘steric requirement’ suggests
that a further factor, P, should be introduced, and that
k2 ∝σ P (T/M)1/2e−Ea /RT
This expression has the form predicted by collision theory.
k2 ∝ steric requirement × encounter rate × minimum energy
requirement

Collision theory
■ (a) Collision rates in gases:
We have anticipated that the reaction rate, and hence k2, depends
on the frequency with which molecules collide. The collision
density, ZAB, is the number of (A,B) collisions in a region of the
sample in an interval of time divided by the volume of the region and
the duration of the interval.
Where σ is the collision cross-section and μ is the reduced mass,
Similarly, the collision density for like molecules at a molar
concentration [A] is

Collision theory
■ The collision cross-section for two molecules can be regarded
to be the area within which the projectile molecule (A) must enter
around the target molecule (B) in order for a collision to occur. If the
diameters of the two molecules are dA and dB, the radius of the target
area is d = 1/2(dA + dB) and the cross-section is πd2.

Collision theory
■ The energy requirement:
According to collision theory, the rate of change in the molar
concentration of A molecules is the product of the collision density
and the probability that a collision occurs with sufficient energy.
Collision density can be incorporated by writing the collision cross-
section as a function of the kinetic energy of approach of the two
colliding species, and setting the cross-section, σ(ε), equal to zero if
the kinetic energy of approach is below a certain threshold value, εa.
Later, we shall identify NAεa as Ea, the (molar) activation energy of
the reaction. Then, for a collision with a specific relative speed of
approach vrel (not, at this stage, a mean value),

Collision theory
■ The steric requirement:
We can accommodate the disagreement between experiment and
theory by introducing a steric factor, P, and expressing the reactive
cross-section, σ *, as a multiple of the collision cross-section, σ * =
Pσ. Then the rate constant becomes
The collision cross-section is the target
area that results in simple deflection of
the projectile molecule; the reaction
cross-section is the corresponding area
for chemical change to occur on
collision.

Collision theory
■ Diffusion-controlled reactions:
Encounters between reactants in solution occur in a very different
manner from encounters in gases. Reactant molecules have to jostle
their way through the solvent, so their encounter frequency is
considerably less than in a gas. However, because a molecule also
migrates only slowly away from a location, two reactant molecules
that encounter each other stay near each other for much longer than
in a gas. This lingering of one molecule near another on account of
the hindering presence of solvent molecules is called the cage effect.
Such an encounter pair may accumulate enough energy to react
even though it does not have enough energy to do so when it first
forms.

Collision theory
■ Diffusion-controlled reactions:
The complicated overall process can be divided into simpler parts by
setting up a simple kinetic scheme. We suppose that the rate of
formation of an encounter pair AB is first-order in each of the
reactants A and B:
A + B → AB v = kd[A][B]
kd (where the d signifies diffusion) is determined by the diffusional
characteristics of A and B. The encounter pair can break up without
reaction or it can go on to form products P. If we suppose that both
processes are pseudofirst-order reactions (with the solvent perhaps
playing a role), then we can write
AB → A + B v = kd′[AB] and
AB → P v = ka[AB]

Collision theory
The concentration of AB can now be found from the equation for the
net rate of change of concentration of AB:
The rate of formation of products is therefore;
Two limits can now be distinguished. If the rate of separation of the
unreacted encounter pair is much slower than the rate at which it
forms products, then k′
d << ka and the effective rate constant is
In this diffusion-controlled limit, the rate of reaction is governed by
the rate at which the reactant molecules diffuse through the solvent.

Collision theory
An activation-controlled reaction arises when a substantial activation
energy is involved in the reaction AB → P.
Then ka << kd′
where K is the equilibrium constant for A + B AB. In this limit, the
reaction proceeds at the rate at which energy accumulates in the
encounter pair from the surrounding solvent.
■ The rate of a diffusion-controlled reaction is calculated by
considering the rate at which the reactants diffuse together. the rate
constant for a reaction in which the two reactant molecules react if
they come within a distance R* of one another is
kd = 4πR*DNA
where D is the sum of the diffusion coefficients the two reactant
species in the solution.

Transition state theory
 We know that, an activated complex forms between reactants as
they collide and begin to assume the nuclear and electronic
configurations characteristic of products.
 We also saw that the change in potential energy associated with
formation of the activated complex accounts for the activation
energy of the reaction.
 We now consider a more detailed calculation of rate constants
using transition state theory (also widely referred to as activated
complex theory).
 Transition state theory is an attempt to identify the principal
features governing the size of a rate constant in terms of a model
of the events that take place during the reaction.

■ Transition state theory pictures a reaction between A and B as
proceeding through the formation of an activated complex, C‡ , in a
rapid pre-equilibrium.
When we express the partial pressures, pJ, in terms of the molar
concentrations, [J], by using pJ = RT[J], the concentration of
activated complex is related to the (dimensionless) equilibrium
constant by
The activated complex falls apart by unimolecular decay into
products, P, with a rate constant k‡:
The Eyring equation

The Eyring equation
It follows that,
Our task is to calculate the unimolecular rate constant k‡ and the
equilibrium constant K‡.

The Eyring equation
(a) The rate of decay of the activated complex
An activated complex can form products if it passes through the
transition state, the arrangement the atoms must achieve in order
to convert to products.
If its vibration-like motion along the reaction coordinate occurs with a
frequency ν, then the frequency with which the cluster of atoms
forming the complex approaches the transition state is also ν.
Therefore, we suppose that the rate of passage of the complex
through the transition state is proportional to the vibrational
frequency along the reaction coordinate, and write
Where κ is the transmission coefficient. In the absence of
information to the contrary, κ is assumed to be about 1.

The Eyring equation
(b) The concentration of the activated complex:
Equilibrium constant K‡ is given by;
The are the standard molar partition functions and the units of NA
and the are mol−1, so K‡ is dimensionless.
According to the partition function, a vibration of the activated
complex C‡ tips it through the transition state. The partition function
for this vibration is;
where ν is its frequency (the same frequency that determines k‡).

The Eyring equation
If the exponential may be expanded and the partition
function reduces to;
We can therefore write;
Where denotes the partition function for all the other modes of the
complex. The constant K‡ is therefore;
with K‡ a kind of equilibrium constant, but with one vibrational mode
of C‡ discarded.

The Eyring equation
(c) The rate constant:
We can now combine all the parts of the calculation into
At this stage the unknown frequencies ν cancel and, after writing;
We obtain the Eyring equation:
in terms of the partition functions of A, B, and C‡,
so in principle we now have an explicit expression for calculating the
second-order rate constant for a bimolecular reaction in terms of the
molecular parameters for the reactants and the activated complex and
the quantity κ.

Thermodynamic aspects
The statistical thermodynamic version of transition state theory rapidly
runs into difficulties because only in some cases is anything known
about the structure of the activated complex.
However, the concepts that it introduces, principally that of an
equilibrium between the reactants and the activated complex, have
motivated a more general, empirical approach in which the activation
process is expressed in terms of thermodynamic functions.
(a) Activation parameters
If we accept that is an equilibrium constant, we can express it in
terms of a Gibbs energy of activation, Δ‡G, through the definition

Then the rate constant becomes;
---------- (1)
Because G = H − TS, the Gibbs energy of activation can be divided
into an entropy of activation, Δ‡S, and an enthalpy of activation,
Δ‡H, by writing; Δ‡G = Δ‡H − TΔ‡S ----------- (2)
When eqn (2) is used in eqn (1) and κ is absorbed into the entropy
term, we obtain;
The formal definition of activation energy, then
gives; Ea=Δ‡H + 2RT, so

It follows that the Arrhenius factor A can be identified as;
The entropy of activation is negative because two reactant species
come together to form one species.
we can identify that additional reduction in entropy, Δ‡Ssteric, as the
origin of the steric factor of collision theory, and write
Thus, the more complex the steric requirements of the encounter, the
more negative the value of Δ‡Ssteric, and the smaller the value of P.
Gibbs energies, enthalpies, entropies, volumes, and heat capacities
of activation are widely used to report experimental reaction rates,
especially for organic reactions in solution.

■ The thermodynamic version of transition state theory simplifies
the discussion of reactions in solution.
■ The statistical thermodynamic theory is very complicated to apply
because the solvent plays a role in the activated complex. In the
thermodynamic approach we combine the rate law
with the thermodynamic equilibrium constant.
Then,
If ko
2 is the rate constant when the activity coefficients are 1 (that is,
ko
2 = k‡K), we can write;
Reactions between ions

At low concentrations the activity coefficients can be expressed in
terms of the ionic strength, I, of the solution by using the Debye–
Hückel limiting law in the form;
with A = 0.509 in aqueous solution at 298 K. Then
-------- (1)
The charge numbers of A and B are zA and zB, so the charge number
of the activated complex is zA + zB; the zJ are positive for cations and
negative for anions.
Equation (1) expresses the kinetic salt effect, the variation of the
rate constant of a reaction between ions with the ionic strength of the
solution.

If the reactant ions have the same sign (as in a reaction between
cations or between anions), then increasing the ionic strength by the
addition of inert ions increases the rate constant.

The dynamics of molecular collisions
■ Reactive collisions:
Molecular beams allow us to study collisions between molecules in
preselected energy states, and can be used to determine the states
of the products of a reactive collision.
Detailed experimental information about the intimate processes that
occur during reactive encounters comes from molecular beams,
especially crossed molecular beams.
It is possible to study the dependence
of the success of collisions on these
variables and to study how they affect
the properties of the out coming
product molecules.

The dynamics of molecular collisions
Chemiluminescence:
■ One method for examining the energy distribution in the products
is infrared chemiluminescence, in which vibrationally excited
molecules emit infrared radiation as they return to their ground
states.
■ By studying the intensities of the infrared emission spectrum, the
populations of the vibrational states may be determined
Laser-induced fluorescence:
■ Another method makes use of laser-induced fluorescence. In this
technique, a laser is used to excite a product molecule from a
specific vibration-rotation level; the intensity of the fluorescence from
the upper state is monitored and interpreted in terms of the
population of the initial vibration-rotation state.

(a) The rate of decay of the activated complex
Not every oscillation along the reaction coordinate takes the complex through the
transition state and centrifugal effect of rotations might also be an important contribution
to the break up of the complex, and in some cases the complex might be rotating too
slowly, or rotating rapidly but about the wrong axis.

Chapter 5 - Collision Theory.pdf

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Chapter 5 - Collision Theory.pdf