Page 105 - Mechanism and Theory in Organic Chemistry

If, in the two-step mechanism in Equations 2.37-2.38, it is not justifiable to
assume that B is consumed as fast as formed, [B] will increase and then decrease;
the rate of disappearance of A will not equal the rate of appearance of C, and the
stationary-state approximation is not valid. This situation requires a more general
approach.35

Preliminary equilibrium In a second common limiting case ofthe
two-step mechanism, the second step is slow. Then ordinarily the reverse of the
first step will be important, so we need to use Equations 2.44-2.45. With the first
step and its reverse much faster than the second step, nearly all B formed returns
to A. Now there is an equilibrium always maintained between A and B and the
second step is rate-determining. We can therefore write an equilibrium constant
for Equation 2.44,

Then since

we can at once write the rate equation 2.48 for rate of formation of C in terms of
starting material A.

The mechanism thus predicts first-order behavior, with an observed rate constant

If the equilibrium constant K can be measured independently, k, can be recovered.
A judicious combination of the stationary-state, preliminary-equilibrium,
and rate-determining step concepts will often yield the rate equation for more
complex reaction schemes. An example is given in Problem 6.

2.6 INTERPRETATION OF RATE CONSTANTS
The utility of rate constants for understanding reaction mechanisms depends
largely on interpreting them in terms of energies. Energy information is ordinarily
obtained from rate data by either of two methods, one empirical and the other
more theoretical.

The Arrhenius Equation
The temperature dependence of observed rate constants follows the Arrhenius
equation (2.50) with good accuracy for most reactions. A and E, are parameters
-
determined experimentally, R is the gas constant, 1.986 cal OK-I mole-l, and

35 See note 34.