Page 129 - Mechanism and Theory in Organic Chemistry

Now we must analyze Qt. It contains the usual translational and rotational
functions, f: and f: ; the electronic contribution f is unity. It is in the vibrational
part that the difference from the ordinary stable molecule appears. Since one
degree of freedom (that corresponding to the reaction coordinate) is no longer a
vibration, there are only 3N - 7 vibrations in Q:, and these contribute in the usual
way according to Equation A1.32. The motion along the reaction coordinate is

much more like a translation than a vibration and its contribution, f,.,., is there-
fore written as a one-dimensional translational function (p. 1 16),

where 6 is an arbitrary interval measured along the reaction coordinate at the
top of the barrier. (We shall see later that the exact value of 6 is immaterial.)
The expression for k, can now be written in terms of a reduced partition
function, Qt, which contains only the translational, rotational, and vibrational
contributions an ordinary molecule would have, and the special contribution
fR.0. :
(2mtk T)li2 &f AEd
k, = kt S - exp --
h Q RT
A
Next we examine kt. At the top of the barrier, in the interval 6, there will
be transition state molecules moving both to the right, in the direction A + B,
and to the left, in the direction A t B. We want to know the average velocity of
motion of those that are moving from left to right. The energy of motion of one
of the transition state molecules is +mtv2, where v is its velocity. Positive v corre-
sponds to motion from left to right, negative from right to left. The average we
shall find with the aid of a velocity distribution function, that is, a function that
for each velocity is proportional to the probability of finding that velocity. When
the velocity is v, the energy of motion is +mu2. The Boltzmann distribution gives
the ratio of the number of transition states with velocity v to the number with zero
velocity as
N" - exp --
+xu2
- -
No kT
A plot of this function against velocity is the required distribution, and the area
under the curve between v and v + dv is, in the limit as dv + 0, proportional to
the probability of finding a transition state with velocity v. The average is found
by summing over all velocities the product of velocity times the probability of
finding that velocity. Since the total probability of finding some velocity should be
unity, the result must be divided by the total area under the curve to correct for
the lack of normalization. The desired average, d is then given by Equation A1.36,