Physical Chemistry Third Edition

26.4 The Activated Complex Theory of Bimolecular Chemical Reaction Rates in Dilute Gases 1109

pass over the saddle and the time required for the passage are used to calculate a rate

constant.^6

The Activated Complex Theory (Transition State Theory)

HenryEyring,1900–1981,wasan

Americanphysical chemist who made

contributionsinvariousareasof

theoretical physical chemistry.

Michael Polanyi,1891–1976,wasa

Hungarian-bornchemist who was

originally trainedasaphysician,and

who laterbecameaprofessorof social

sciences.

Rather than calculating trajectories, one can use statistical mechanics in an approxi-

mate form. The oldest such approximate theory was pioneered in the 1930s by Eyring

and Polanyi, and is called theactivated complex theory, or thetransition-state

theory.^7

We illustrate the activated complex theory for the gas-phase bimolecular reaction

of Eq. (26.4-4). We begin with the following assumptions:

1. The activated complex CDF‡can be identified as a distinct chemical species.


2. The concentration of the activated complex can be obtained by assuming that it is
   in chemical equilibrium with the reactants.


3. The rate of the chemical reaction is equal to the concentration of the activated
   complex timesνpassage, the frequency of passage of an activated complex over the
   maximum in the potential energy:

Rateνpassage[CDF‡] (26.4-6)

In most reactions there is clearly too little time for the activated complex to come to

chemical equilibrium with the reactants, so the second assumption is questionable. In

spite of this fact the theory is more successful than one might expect.

The formation of the activated complex corresponds to the chemical equation

CD+F
CDF‡ (26.4-7)

and its equilibrium constant is given by Eq. (26.3-20):

Kc

([CDF‡]/c◦)

([CD]/c◦)([F]/c◦)

e−∆ε

‡

0 /kBT

(z′′CDF/NAvc◦)

(z′′CD/NAvc◦)(z′′F/NAvc◦)

(26.4-8)

where∆ε‡ 0 is the energy required to form one activated complex in its ground state

from one CD molecule and one F atom in their ground states. It is equal to the height

of the maximum in Figure 26.1c above the energy of the reactants, plus the difference

in the zero-point vibrational energies.

a

b

c

g

h

rGH

rHI

Figure 26.2 A Contour Plot for
the Potential Energy of Nuclear
Motion for a Stable Triatomic
Molecule.

The vibrations of the activated complex resemble those of an ordinary triatomic

molecule except for the asymmetric stretch. A stable linear triatomic molecule GHI

would have a potential energy surface as shown by the contours of equal potential

energy in Figure 26.2, with a relative minimum at pointbinstead of a saddle point.

It has four normal modes: two bends, a symmetric stretch, and an asymmetric stretch.

Motion along the path labeledabccorresponds to the asymmetric stretch, and motion

along the path labeledgbhcorresponds to the symmetric stretch. The activated complex

has three vibrational modes: two bends and a symmetric stretch, which is like that of

GHI because the saddle in the potential energy surface corresponds to a minimum

in thegbhdirection. The asymmetric stretch of the activated complex corresponds to

(^6) R. N. Porter,Ann. Rev. Phys. Chem., 25 , 317 (1974).
(^7) H. Eyring,J. Chem. Phys., 3 , 107 (1935); M. Evans and M. Polanyi,Trans. Faraday Soc., 312 , 875
(1935). See K. J. Laidler and M. C. King,J. Phys. Chem., 87 , 2657 (1983) for a history of the theory.