Page 106 - Mechanism and Theory in Organic Chemistry

Interpretation of Rate Constants 95

T is the temperature in degrees Kelvin. The units of A, called the pre-exponential
factor, are the same as those of k,,,: for a first-order rate constant, time-'; for a
second-order rate constant, 1 mole-' time-'. We use the notation k,,, to empha-
size that the equation applies to the observed rate constant, which may or may
not be simply related to the microscopic k's characterizing the individual steps
of a reaction sequence.
If we write Equation 2.50 in the form of Equation 2.51, we see at once a
resemblance to the familiar relation 2.52 between the equilibrium constant of a
C)
-E, = RTln -
-AGO = RTln K (2.52)
reaction and its free-energy change. Hence it is natural to interpret E, as an
energy. This energy is called the Arrhenius activation energy, or simply activation
energy, and may be crudely interpreted as the height of an energy barrier over
which reactants must pass on their way to products. Yet because k,,, will not in
general correspond to the microscopic constant for a single step, the origin of the
activation energy on the molecular level is not well defined. To obtain a more
precise idea of the dynamic behavior of molecules during a reaction, we turn to
the transition-state theory.

Transition State Theory36
The transition state theory is confined to consideration of single elementary re-
action steps, and is meaningful only when applied to a single microscopic rate
constant. The theory postulates that when two molecules come together in a
collision that leads to products (or when a single molecule in a unimolecular step
follows the motions that cause the chemical change), they pass through a con-
figuration of maximum potential energy called the transition state. In order to
understand this concept fully, we must first digress to consider some ideas about
potential energy surfaces.

Potential energy surfaces Because each of the Natoms in a molecule can
move in three mutually perpendicular and therefore independent directions, a
molecule has a total of 3N degrees of freedom. But since we think of a molecule
as a unit, it is useful to divide these degrees of freedom into three categories. If
the atoms were fixed relative to each other, the position of the rigid molecule in
space would be defined by specifying six quantities: the three cartesian coordi-
nates of its center of mass and three rotational angles to indicate its orientation
in space. Hence there remain 3N - 6 degrees of freedom which are internal
vibrational motions of the atoms with respect to each other. (A linear molecule

Transition state theory is discussed in standard texts on physical chemistry, kinetics, and physical
organic chemistry. See, for example, (a) W. J. Moore, Physical Chemistry, 3rd ed., Prentice-Hall,
Englewood Cliffs, N.J., 1962, p. 296; (b) S. W. Benson, Thermochemical Kinetics, Wiley, New York,
1968; (c) K. J. Laidler, Chemical Kinetics, 2nd ed., McGraw-Hill, New York, 1965; (d) K. B. Wiberg,
Physical Organic Chemistry, Wiley, New York, 1964; (e) L. P. Hammett, Physical Organic Chemistry,
2nd ed., McGraw-Hill, New York, 1970. For a different approach to chemical dynamics, see
(f) D. L. Bunker, Accts. Chem. Res., 7, 195 (1974).