Page 110 - Mechanism and Theory in Organic Chemistry

Interpretation of Rate Constants 99
The situation can be visualized if only one vibrational degree of freedom
besides the reaction coordinate is included. Then we have the three-dimensional
potential energy surface of Figure 2.8, two valleys meeting over a mountain pass.40
If we climb along the reaction coordinate out of one valley over the pass into the
other, we go over an energy maximum along the reaction coordinate, but the
surface rises in the perpendicular direction and we are therefore following a
potential energy minimum with respect to the motion perpendicular to the
reaction coordinate.
The second point about the surface is that it shows only the potential energy.
The total energy of the molecular system is the sum of its kinetic and potential
energies. The molecules exchange kinetic energy by collisions, and are distributed
over a range of total energies, with many at low energies and fewer at higher
energies. It is tempting to think of a reaction as following the path of a pack horse
up out of the left-hand valley and over the pass into the right-hand one. This
model is quite inappropriate; a much better way to think of the situation is to
imagine many birds flying in the valleys at various levels, the levels representing
the various possible total energies. The individual birds can go up or down by
receiving or giving up some energy to their surroundings, but the vertical
distribution of birds is in equilibrium and remains unchanged. The birds are
flying around at random, and those that are high enough may in their wanderings
happen to sail over the pass and join the population in the other valley. The rate
of passage of the birds from one side to the other depends on the height of the pass
and on the vertical distribution of the birds. In the molecular system the vertical
distribution is determined by the temperat~re.~~

Thermodynamics of the Transition State
In developing the transition state theory, we shall take advantage of the fact that
most of the motions in a reacting molecular system are ordinary vibrations,
rotations, and translations. Only the one normal mode corresponding to the
reaction coordinate is doing something peculiar by coming apart to form new
molecules. We shall postulate therefore that the molecules going over the barrier
are in equilibrium with all the other reactant molecules, just as in our bird analogy
we said that the birds that can get over the pass are just those that happen to be
high enough up and headed in the right direction.
We assume that in Reaction 2.53 there are at any instant some molecules

40 This surface could never be the complete one in a molecular system, as it would require that
3N - 6 + 1 = 3 (nonlinear) or 3N - 5 + 1 = 3 (linear), neither of which have solutions for N
an integer. A three-dimensional reaction coordinate diagram like Figure 2.8 is thus always only a
projection of a surface of higher dimensionality, just as the two-dimensional one is.
41 We must also remember that, although we tend to think of the atoms as classical particles, their
motions are actually determined by the rules of quantum mechanics. If we tried to follow in detail
the motions of the atoms in a molecule crossing the barrier with just enough total energy to get over,
we would come up against the uncertainty principle just as we do in trying to follow electron
motions, and would be unable to say just how the atoms got from one place to the other. For further
discussion see W. F. Sheehan, J. Chem. Educ., 47, 254 (1970).