Chapter 5. Collisions and Molecular Dynamics

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data. Although they may lack the theoretical rigour of more fundamental theoretical formulations, these more empirical theories are important since, apart from their historical interest, they often act as pointers for the development of more quantitative theoretical models.

These can be directed towards real systems and can help us avoid using highly sophisticated

calculations to treat the wrong target. To quote the late John W. Tukey, who made major contributions to statistics, both in industrial and academic areas: “An approximate answer to the

right question is worth a great deal more than a precise answer to an approximate question”.

It is worth noting that John Tukey’s initial background was chemistry.

At a first glance, an ab initio theory would seem to be the most satisfactory way to

treat theoretically the chemical kinetics of any system, since this does not need to introduce any experimental data. However, only those systems involving 3–4 atoms in the

gas phase can be solved with chemical accuracy (Ϯ4 kJ/mol) by this route. For the vast

majority of reactions of chemical interest, we need either to introduce major simplifications into the most sophisticated theories so that they can be applied, or to opt for a simpler theory, including empirical parameters, which can be used without a posteriori

changes to estimate the rate constants. These approaches differ because in the first case,

approximations are made when the method is applied, while with the latter approach, the

approximations come within the formulation of the theory. In both cases, we hope that

we can understand and interpret any discrepancies between the observed and the calculated kinetic data. The approach we use frequently depends upon the nature of the chemical system. The biggest problem is how to interpret any disagreement between

experiment and calculation. This could result from experimental error, from wrong

approximations when we apply the theory or incorrect assumptions in the theory itself.

In the latter case, the theory is wrong and must be rejected. We must try to avoid the tendency of attributing any discrepancies between theoretical prediction and experimental

observation to approximations introduced in the application of the theory, rather than to

any weakness in the theory itself. The continued use of any obsolete theory past its “sellby date”, is scientifically incorrect as well as hindering the development of more appropriate models.

As a scientific theory is intended to simulate and help us understand the behaviour of

part of or the whole Universe, it should be kept sufficiently simple to be understandable,

and its proof should be purely operational. The relevant question should not be whether it

is true or false, since this does not have any meaning from a scientific viewpoint. Instead

we should ask: “does it work”?

Even so, the answer to this question may not be completely objective. We can discard a

theory that contains errors of logic, but a similar decision based upon comparison between

theoretical and experimental data depends on how rigorous we want the agreement to be.

Based on these ideas, we will discuss the simple collision theory, which is attractive

because of the relationship it provides between the orientation and the energy of the molecules involved in collisions and their reactivity. However, it is not actually satisfactory as

a theoretical hypothesis for polyatomic systems, since it cannot be used quantitatively for

the calculation of rates of chemical reactions. However, for triatomic systems whose

potential energy surfaces (PESs) are well defined, the calculation of trajectories using this

approach provides an excellent method for calculating the reaction rates. Thus, we will

restrict this type of calculation to simple systems. For polyatomic systems or reactions in



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solution, it is totally impracticable and we will make use of other theories to be discussed

in the following chapter.



5.1



SIMPLE COLLISION THEORY



The simplest model for the rates of chemical reactions assumes that every time there is a

bimolecular collision, there is a reaction. From calculations of the frequency of collisions

in this system, we can then determine the rate constant.

To calculate the number of collisions per unit time, we need a model for the behaviour

of molecules in these systems. The simplest approach involves a system of two gases, A

and B, whose molecules behave as hard spheres, which are characterised by impenetrable

radii rA and rB. A collision between A and B occurs when their centres approach within a

distance dAB, such that

dAB = rA + rB



(5.1)



If we assume that the molecules of B are fixed and that those of A move with an average

–

2 –

velocity v A, each molecule A sweeps a volume dAB v A per unit time which contains sta2

tionary molecules of B. The area p ϭ dAB , is known as the collision cross section. If

there are NB/V molecules of type B per unit volume, the number of collisions of a molecule of type A with the stationary molecules B will be



zAB =



2

dAB vA N B

V



(5.2)



If the total number of molecules of A per unit volume is NA/V, then the total number of collisions of A with B per unit volume is given by



Z AB =



2

dAB vA N B N A

V2



(5.3)



As indicated above, we have assumed that the molecules of B are stationary to obtain the

above expression. In practice, Figure 5.1 shows that, for each pair of molecules, A and B,

involved in a collisional trajectory, we can define a relative velocity vAB, which is related

to their velocities, vA and vB, according to



(



2

2

vAB = vA + vB − 2 vA vB cos



)



1



2



(5.4)



The value of cos can vary between Ϫ1 and 1. As all values of between 0 and 360° are

equally probable, the positive and negative values of cos will cancel out, and the mean

value will be zero. Thus we obtain

2

2

2

vAB = vA + vB



(5.5)



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Figure 5.1 Relative velocities for a collisional trajectory between molecules A and B.



Assuming that the molecular velocities are described by a Maxwell–Boltzmann distribution, the mean velocity of the molecules of a gas A is given by eq. (1.11). Consequently,

the relative mean molecular velocity of molecules of types A and B will be



vAB =



8 k BT



(5.6)



where the reduced mass is ϭ mAmB/(mA ϩ mB). Introducing this value for the relative

velocity in eq. (5.3), we obtain



Z AB =



2

dAB vAB N B N A

V2



(5.7)



If we consider a gas that contains only molecules of type A, the total number of collisions

would be



Z AA =



2

2

1 4rA vAA N A

2

V2



(5.8)



where the factor ½ arises because we cannot count the same molecule twice. Since from

eq. (5.5), the relative mean velocity of the molecules of a gas can be related to the mean

–

–

velocity of the molecules, v AAϭ √2 v A, eq. (5.8) can be rewritten as

Z AA =



2

2

2 2 rA vA N A

V2



(5.9)



The expressions for the number of collisions per unit time, also known as the collision densities, contain two factors involving the number of molecules per unit volume. These can be

expressed in terms of pressures or molar concentrations, giving the rate constant for a bimolecular reaction between molecules of A and B in the experimentally more meaningful form



kAB =



v



R AB



2

= dAB



8 k BT



(5.10)



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We can test this collision theory approximation by comparing the value calculated for the

elementary reaction

NO2 + CO ⎯⎯ NO + CO2

→



(5.I)



at 870 K with the experimental value, kexp ϭ 1.5ϫ102 dm3 molϪ1 secϪ1. From the van der

Waals radii of N and O atoms, the reactant bond lengths, and assuming that these behave

as hard spheres, we can estimate that dAB ϭ 4ϫ10Ϫ10 m. The reduced mass

can be calculated from the relative molecular masses of nitrogen dioxide and carbon

monoxide

=



46 × 28

1.661 ×10 −27 = 2.89 ×10 −26 kg

46 + 28



(5.11)



where the units conversion factor in the unified atomic mass unit, from which we obtain the

relative mean velocity, using the relationship 1 J ϭ 1 kg m2 secϪ2,

vNO2 ,CO =



8 × 1.38 × 10 −23 × 870

= 1.03 × 103 m sec −1

3.14 × 2.89 × 10 −26



(5.12)



such that we calculate the rate constant to be kAB ϭ 5.17ϫ10Ϫ16 m3 moleculesϪ1 secϪ1 ϭ

3.11ϫ1011 dm3 molϪ1secϪ1. We can see from the comparison between the experimental

and calculated values that the latter are a factor of at least 109 greater than the former.

In this simple collision theory, there are two implicit approximations which lead to calculated values that are considerably greater than the experimental ones: (i) not all the collisions occur with a favourable orientation for reaction; (ii) we need a certain amount of

energy for the conversion of the reactants into products, and not all the collisions have

enough energy to produce this chemical transformation.

The first of these approximations can be corrected for by introducing a geometric factor, g Յ 1. This can be calculated using the molecular geometries of the reactants and products. In the example of reaction (5.I), it is necessary that in the collision the NO bond is

extended and its oxygen atom directly approaches the carbon atom of CO for the reaction

to occur (Figure 5.2). When the N᎐O and C᎐O bonds are collinear, the chemical transformation results from the stretching of these bonds, since these stretching vibrations are of

high frequency and energy. In this model, the geometric factor g is given by the product

of the orientation factor for the CO molecule, ½, and that of the NO2 molecule, which, as

indicated in Figure 5.2, will be approximately 2/17. Therefore, for this reaction, g ϭ 1/17.

The geometric factor may become much less than unity with increasing complexity of

the reaction in the gas phase. In addition, with reactions in solution, we need to consider

effects of solvent cage, which allow many collisions to occur during an encounter between

the two reactants, such that the effect of the geometric factor becomes more difficult to calculate. However, as we will see, in these systems, too, it is less relevant.

The energetic factor is of fundamental importance in the determination of reaction

probabilities and, hence, rate constants. Once more, we start from the Maxwell–Boltzmann

distribution (eq. 1.10) to calculate this factor. In practice, in collision theory, we normally



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Figure 5.2 Sketch of the origin of the geometric factor in a collision. The favourable orientations

for the attack of the carbon atom of CO are shaded.



use the distribution function of molecular velocities in two dimensions rather than in

three.

⎛ − mv 2 ⎞

dN

m

=

exp ⎜

vdv

N 0 k BT

⎝ 2 k BT ⎟

⎠



(5.13)



Although there is no strong theoretical basis for this decrease in dimensionality, it simplifies the system, and we can rationalise it by considering that at the instant of collision

between the two molecules, the velocity vectors have a common point such that they lie

within a plane. Thus, the components of the velocities within the two dimensions that define

this plane are sufficient to describe an effective collision. The distribution of molecular

energies, based on the above equation, is

dN (

N0



)=



⎛ − ⎞

1

exp ⎜

d

k BT

⎝ k BT ⎟

⎠



(5.14)



which can be integrated to obtain the fraction of collisions with energy equal to or greater

than the critical value, c, for the reaction to occur.



∫



ε > εc



dN (

N0



)=∞



1



⎛ − ⎞



εc



B



B



∫ k T exp ⎜ k T ⎟ d

⎝

⎠

⎛− ⎞

= exp ⎜ c ⎟

⎝ k BT ⎠



(5.15)



If we include this energetic term and the geometric factor g in the expression for the rate

constant, we obtain the final expression for the rate constant given by the collision theory



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that considers only effective collisions.

2

kcol = g dAB



8 k BT



⎛− ⎞

exp ⎜ c ⎟

⎝ k BT ⎠



(5.16)



This expression bears an obvious resemblance to that of the Arrhenius law. In fact the preexponential factor in the latter expression can be identified as

2

A = g dAB



8 RT



(5.17)



The activation energy, expressed in molar terms, is given by



Ea =



d ln ( kcol )

d (1 RT )



= Ec + 1 RT

2



(5.18)



The expressions (5.16)–(5.18) were first applied to bimolecular gas phase reactions by

M. Trautz (1916) and W.C. McC. Lewis (1924). The terms in d and in eq. (5.17) generally compensate each other, such that, if we exclude the geometric factor g, in many cases

log A ϭ10.5Ϯ0.5, with A given in dm3 molϪ1 secϪ1. Steric hindrance, caused by the presence of bulky groups in the positions adjacent to the point of attack, is seen mainly through

an increase in the activation energies of reactions and not through a decrease in the frequency of effective collision. Thus, the simple collision theory does not lead to significant

correlations between the changes in structures and the calculated values for A. Some people have refined this theory by defining an entropy of activation for the molecular collisions, leading to

2

A = g dAB



8 RT



(



exp ⌬S ∗ R



)



(5.19)



The introduction of this extra term in the expression for the pre-exponential factor may

explain values much less than those determined by the frequency of collisions. However,

as in most situations it is not easy to calculate this entropy of activation, this new term is

only of limited use.

The numerical value of the collision theory value rate constant, previously calculated for

the NO2 ϩ CO reaction (3.11ϫ1011 dm3 molϪ1 secϪ1) corresponds to the typical values calculated for the pre-exponential factors of gas-phase bimolecular reactions. Dividing this value

by the molar volume of an ideal gas at standard temperature and pressure (22.421 dm3 molϪ1),

we obtain a relaxation rate, which is approximately (p )Ϫ1 ϭ 1010 atmϪ1 secϪ1 at 273 K. This

implies that the mean free time between collisions of molecules of an ideal gas is 0.1 nsec,

and that the corresponding collision frequency is collϪ1

1010 secϪ1. Some caution must

be made in using these values, since molecules interact with a range of velocities and

instantaneous inter-molecular distances, such that there is no single collision frequency, but



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there is a distribution of time intervals between collisions. This distribution can be given in

terms of a Poisson function in n



P (n) = e −



n



(5.20)



n!



where the mean value of n is ϽnϾ ϭ ϭ T/ coll, and T is the number of collisions per unit

time.

The simple collision theory treatment shows the basic elements for a bimolecular rate

constant: we need a collision to occur, a certain critical energy has to be exceeded and the

particles involved in the collision have to have the correct relative orientations.



5.2



COLLISION CROSS SECTION



Instead of assuming hard sphere behaviour, we can refine the collision theory by using

cross sections calculated on the basis of microscopic interaction potentials of species A and

B during their collision, R. Before we develop the problem for reactive collisions, we will

start by treating the simpler situation of an elastic collision between two bodies subjected

to a central force, which only leads to scattering.

The collision of these two bodies can be seen by a system of coordinates based on the

centre of mass (Figure 5.3). The equations of motion will be determined by the conservation of initial energy, E, and of angular momentum, L. As the direction of L is conserved,

the trajectory has to remain in a plane and the equations of motion can be written, in polar

coordinates

L = vb = r 2



dφ

dt

2



2



1 2 1 ⎛ dr ⎞ 1 2 ⎛ dφ ⎞

E=

v =

+

r ⎜ ⎟ + V (r )

2

2 ⎜ dt ⎟ 2

⎝ ⎠

⎝ dt ⎠



(5.21)



2



1 ⎛ dr ⎞

L2

=

+ V (r )

⎜ dt ⎟ +

2 ⎝ ⎠ 2 r2

where v is the relative incident velocity and b is the impact parameter. Solving these equations, we get

dφ =



L2

dt

r2



(5.22)



and

12



⎧2 ⎡

L2 ⎤ ⎫

⎪

⎪

dr = − ⎨ ⎢ E − V ( r ) −

⎥ ⎬ dt

2 r 2 ⎦⎭

⎪

⎪

⎩ ⎣



(5.23)



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Figure 5.3 Classical trajectory for velocity v and impact parameter b, with deflection angle , following a system of coordinates based on the centre of mass. The classical turning point is rc.



or

⎧2 ⎡

L2 ⎤ ⎫

⎪

⎪

dt = − ⎨ ⎢ E − V (r ) −

2 ⎥⎬

2 r ⎦⎭

⎪

⎪

⎩ ⎣



−1 2



dr

(5.24)



Now, we can write

t ⎛ L ⎞

φ ( t ) = ∫ ⎜ 2 ⎟ dt

−∞ ⎝ r ⎠



= −∫



L r2



r



∞



12



⎧2 ⎡

L2 ⎤ ⎫

⎪

⎪

E − V (r ) −

⎬

⎨ ⎢

2 r 2 ⎥⎪

⎪ ⎣

⎦⎭

⎩



dr

(5.25)



and use the relationship L ϭ vb ϭ b(2 E)½ to obtain

φ ( t ) = ±b ∫



dr



r



∞



⎡ b V (r ) ⎤

r ⎢1 − 2 −

⎥

E ⎦

⎣ r

2



12



2



(5.26)



The sign must be chosen to agree with that of the radial velocity: that is positive for the

species leaving and negative for those approaching.

For any collision, there will always be a maximum approach distance rc for which r(t)

has its minimum value. In this classical turning point all the initial kinetic energy is

converted into potential energy. It is convenient to define t ϭ 0 for r(t) ϭ rc and φ(0)ϭ φc.

The complete classical trajectory can be determined by integrating eq. (5.26) from infinity to rc and then from rc forward. However, it is unnecessary to follow the trajectory



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with this detail when we are only observing what happens after the collision. In this case

it is sufficient to know the total deflection, Θ. As shown in Figure 5.3, this deflection is

given by

Θ ( E , b) = − 2φc

= − 2b ∫



∞



rc



dr

⎡ b V (r ) ⎤

r ⎢1 − 2 −

⎥

E ⎦

⎣ r



12



2



(5.27)



2



For us to analyse the behaviour of the angle of deflection as a function of the energy and

of the impact parameter, we have to define the potential. In the simplest case of the potential of hard spheres

V (r ) = 0, r > dAB



(5.28)



V (r ) = ∞, r ≤ dAB



it is obvious that independent of the energy of collision rc ϭ dAB. Under these conditions,

the integration of eq. (5.27) leads to

Θ ( E , b ) = − 2b ∫



= − 2b ∫



∞

dAB



∞

dAB



dr

⎡ b2

⎤

r 2 ⎢1 − 2 − 0 ⎥

r

⎣

⎦

dr



1



2



r r 2 − b2

r =∞



⎧1

⎛ b⎞ ⎫

= − 2b ⎨ cos−1 ⎜ ⎟ ⎬

⎝ r ⎠ ⎭r = d

b

⎩

AB



(5.29)



⎡

⎛ b ⎞⎤

⎛ b⎞

= π − 2 ⎢cos−1 ⎜ ⎟ − cos−1 ⎜

⎥

⎝ ∞⎠

⎝ dAB ⎟ ⎦

⎠

⎣

⎛ b ⎞

= − + 2 cos−1 ⎜

⎝ dAB ⎟

⎠

⎛ b ⎞

= 2 cos−1 ⎜

⎝ dAB ⎟

⎠

For impact parameters greater than the hard sphere diameter dAB, eq. (5.29) requires the

reciprocal of the cosine of a number greater than one, which is not possible. For this case,

Θ(b) ϭ 0. For impact parameters less than dAB, the value of Θ(b) varies between 0 and

(Figure 5.4).



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Figure 5.4 Deflection function Θ(b) for the hard sphere potential of diameter dAB.



Figure 5.5 Lennard–Jones potential, showing the minimum at rm ϭ21/6/



LJ



with an energy V(rm)ϭϪ .



A potential energy function that represents the interaction between two bodies in a more

realistic form is the Lennard–Jones potential

⎡⎛ LJ ⎞ 12 ⎛ LJ ⎞ 6 ⎤

V ( r ) = 4 ⎢⎜

⎟ −⎜ r ⎟ ⎥

⎝

⎠ ⎥

⎢⎝ r ⎠

⎣

⎦



(5.30)



shown in Figure 5.5, where the attractive term in r represents the dispersion forces as well

as dipole–dipole and dipole-induced dipole interactions. The purely empirical repulsive

part was chosen from adjustment to a wide range of atomic potentials. The deflection function of this potential can be calculated using eq. (5.27). The arithmetical treatment of this



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Figure 5.6 Deflection function Θ(E,b) for the Lennard–Jones potential. The minima correspond to

the “rainbow” angle discussed in the text.



solution is very complicated. In Figure 5.6, we show only the results in terms of the

reduced impact parameter b* ϭ b/ LJ for a range of reduced energies E* ϭ E/ . This figure shows three regions with a different dispersive behaviour.

For collisions that are almost head-on (b → 0), the deflection function is very similar to

that of collision between hard spheres (Figure 5.4), because the impinging particle passes

through the attractive part of the potential and is repelled by the wall given by rϪ12. This

behaviour of specular reflection can be found for scattering with almost all potentials.

For very large-impact parameters, the dispersion is also similar to that for the hard

sphere, that is, the impinging particle is hardly scattered in its trajectory. This is due to the

rapid decay of the attractive part of the potential to zero. We should note that this limit is

reached more rapidly when the energy of the incident particle increases.

In the intermediate region of b* and E* we find a new behaviour. The deflection angles

can take considerably negative values, even less than Ϫ . This results from the strong

interaction of the incident particle with the attractive part of the Lennard–Jones potential,

as is shown in Figure 5.7.

The above discussion focussed on the trajectory of a single event, specified by an initial energy and an impact parameter. From the practical point of view, although it is possible to accurately define the initial energy in a molecular beam experiment, it is

impossible to isolate a single impact parameter. Thus, the experimental results are presented in terms of collision cross sections, as shown in Figure 5.8. In the case of elastic

scattering, we can distinguish between the total cross section (E) and the differential

cross section d /d . This latter parameter is defined as the intensity of scattering per solid