The centrifugal barrier and intermolecular potential

In reality molecules are not hard spheres. The collision partners are subject to attractive and repulsive forces, these forces have a considerable effect upon the trajectory of a pair of colliding particles and the energetics of the collision process. This is best illustrated by the fact that a collision event may still occur between two particles where the intemuclear separation at the point of closest approach is greater than the sum of the atomic radii of the collision partners. The contrast of this situation to the situation of the non-interacting hard sphere model of atomic collisions is shown in Figure 4.7.

The relative motion of two reactant species can be specified in terms of a relative position vector R of magnitude R, aligned at an angle \j/ with respect to a fixed direction as shown in Figure 4.7. The presence of the intermolecular forces directly affects both the separation R of the two reactants and the orientation angle \\f. The magnitude of the interspecies separation with time R(t), describes the approach of the reactants and the subsequent separation of the reaction products. However, to fully describe the reaction dynamics and relative position of the species, it is also necessary to consider the angle of the relative orientation \\f of the reactants with respect to a fixed direction. The necessity for this is that the relative rotation of the reactants, that is the change in the orientation angle \j/ with time, results in a net reduction in the energy available for reaction. The moment of inertia of the reactants is given by / = hence, the smaller the interspecies separation, the lower is I and by virtue of the conservation of angular momentum (/co) the higher will be the angular velocity O). Hence, by

reducing R and thereby increasing (O the net effect, as demonstrated in Equation 4.23, is to increase the rotational kinetic energy Er

Er =

dr

(4.23) As the two reactants converge, there is therefore an effective reduction in the amount of kinetic energy available to overcome the potential barrier V(R). We see the relative rotation itself acting as a barrier to reaction which prevents the two reactant species from getting as close as would be possible if

CO had zero value. Hence, the total collision energy E j is the sum of the relative kinetic energy and the

potential energy as given in Equation 4.24:

Et= T + V ( R ) = ^

V dry

2

(4.24)

O

V'

F igu re 4.7 The trajectory in the centre-of-mass frame of a collision between tw o ‘real’ particles at impact parameter b,

relative velocity v and reduced mass p. The relative separation R (t) can be view ed as the path taken by particle o f mass p, and initial velocity v, when deflected by a potential V(R) centred on the stationary target at

O.

Note, the kinetic energy is expressed as two terms in Equations 4.23 and 4.24. This is explained as arising because the kinetic energy is defined in terms of the relative position vector R(t).

Consequently, as the colliding particles approach one another, R(t) changes with time and the kinetic energy term therefore requires two terms. Firstly, a term that corresponds to the magnitude of the relative separation of the colliding species, and secondly, a term corresponding to the changing direction ij/ of the relative position vector.

From Appendix 4, consideration of the angular momentum L of the relative motion gives a relationship between the angular velocity co and the impact parameter b:

L - |Xvè = /co = |Li/?^(0 (4.25) Hence, from Equation 4.25 we get:

(0 =

vb

(4.26) If the collision partners are at initial separations large enough so that the potential energy V(R)has zero value, the total energy of the collision E j may be described purely in terms of the initial relative velocity V and the reduced mass p,:

^ (4.27)

Et ” 2

Therefore, by substituting the expressions for co given in Equation 4.26 into Equation 4.24 we get:

Ej = T + V{ R) = ^ dt

2 , 2

p

2 R^ + V(R)

Equating the two expressions for Et in Equations 4.27 and 4.28 gives:

El — Ej +

dt + V(R)

(4.28)

(4.29) Thus, the energy of rotation acts to reduce the kinetic energy available to overcome the intermolecular potential. A convenient representation of the effective kinetic energy, which is sometimes known as the

line o f centres energy, is therefore obtained by rearranging Equation 4.24 to give:

R^) 2 1 dr + y ( ^ )

(4.30) Hence, the motion of the approaching particles may be viewed as the motion of a particle of mass p, in a potential V(R) with an effective kinetic energy .

An alternative perspective combines the energy of rotation (centrifugal energy) and the potential energy, such that the centrifugal energy acts as a repulsive contribution to an effective potential Veff giving:

V ^ { R ) = V( R) + E , ^ (4-31)

The energy of rotation therefore contributes to Veff, which acts as a repulsive potential to prevent the close convergence of the collision partners and is hence termed the centrifugal barrier to reaction [23].

dr

(4.32) From Equation 4.31 it is clear that as the colliding partners approach each other, the effective potential increases. As the total collision energy is conserved, the increase in Vgff necessarily leads to a fall in the kinetic energy term in Equation 5.32. Ultimately, the colliding particles reach their smallest

separation Rq, often known as the distance o f closest approach or turning point. At Rq the kinetic energy term is zero. Hence, the total collision energy is, at this point, equal to the effective potential, thus enabling Rq to be determined from the implicit equation in Veff [23].

(4.33)

4.3.5 The reaction cross-section

The reaction cross-section o^j may be defined in terms of the reaction rate constant such that:

W=VQrxn (4.34)

This relationship can be used to give a qualitative measure of the likelihood of reaction by considering a beam of the incident reactant A of number density Aa passing through a reaction cell containing the collision partner B of number density Nq. The loss of the reactant species A in the beam is given as the rate of reduction in beam intensity /a with distance travelled in the cell

= Kv) Aa Ab= / A( ^ )AB a rx n

where /a (%) is the intensity of the beam of reactant species A at position x.

In a simplified model it is possible to estimate the reaction cross-section by assuming that every collision results in reaction. This is clearly a gross over-simplification as there are many other possible outcomes of a collision such as non-reactive scattering and electron-transfer to name but two. Consequently, Qc > anm-

It is possible to define the likelihood of a collision event leading to reaction as a function of the impact parameter b. We define the opacity function as the fraction of collisions with impact parameter b that result in reaction. Hence, 0< P ( b ) < \ .

As stated in section 4.3.4, the centrifugal barrier effectively acts to keep the reactant molecules apart, preventing reaction. Hence, ignoring possible steiic effects, the reaction is favoured for small impact parameters. In other words, P(b)-^ 0 for high values of b. It is therefore possible to define a limiting condition, that is a reaction will not take place if the impact parameter exceeds a certain magnitude Collisions involving reactants having impact parameters greater than brmx will not lead to the reaction taking place. Hence, the opacity function P(b) has zero value if the impact parameter exceeds b^^.

The inclusion of the opacity function into the equation for the collision cross-section yields the reaction cross-section Qrxn, thus:

=2nbP{b)db (4.36)

^ rxn = 2ti J bP{b)db (4.37) 0

where the opacity function P(b) is given by;

f ( 6 ) = 0 , ( 6 > 6 _ , )

The above derivation of the reaction cross-section Gnm may be treated quantum mechanically. With the reaction probability defined for each value of the quantum number /, this gives:

1=0

where is the de Broglie wave number [23] and P(l) is the probability of reaction for a given value of 1. In the case where many I values provide a contribution to the reaction it is possible to use the semi-classical correspondence bks ~ (/ + 0.5) to convert the summation over I to an integral over b.

This treatment results in Equation 4.37.

4.3.6 Collision energy-dependence and the interm olecular potential

As mentioned in section 4.3.4 the collision energy must be of sufficient magnitude to overcome the centrifugal barrier to allow a reaction to proceed. Central to deterrnining the kinetic energy available for reaction is the role of the intermolecular potential V{R) between the reactants. A prototypical form of the intermolecular potential is given by:

V( R) = - ^ (4.40)

As previously defined, the effective potential is given by:

Hence substituting Equation 4.40 into 4.41 gives:

(4.41)

(4.42) Simple differentiation of Equation 4.42 above yields /?max,

(4.43) The parameter is simply the value of R that corresponds to the maximum of the effective potential

Veff. This gives a criterion for reaction, namely that the reactants have non-zero kinetic energy at /? = /?max. That is, using Equations 4.29 and 4.32:

R-

Ej — V{ R) - Ej

> 0

As can be seen from Equation 4.44, increasing the impact parameter for a given value of Ej

means of defining another parameter, that is the largest value of the impact parameter, 6max for which the molecules may overcome the effective potential and undergo reaction. This results in the condition for reaction reverting to the form of Equation 4.45.

E ^ - V { R ) - E ^ =

0

Hence, having obtained an expression for b^ax the reaction cross-section may be obtained using Equation 4.46, which is derived fi-om Equation 4.37 above.

a ^ = 2 n p ] b - ^ = n P b l^

A general form of the intermolecular potential is given by Levine and Bernstein [23] as Equation 4.40. Incorporating Equation 4.40 into Equation 4.43 gives /^max for values of 5 > 2, in Equation 4.40.

2

r l. = s C

2E jb'

( s - 2 ) (4.47)

Hence, using the above Equations, the reaction cross-section Gnm is given by: