Physical Chemistry Third Edition

578 13 Chemical Reaction Mechanisms II: Catalysis and Miscellaneous Topics

0 [R] 5 Km

k 2 [E]total

[R]

rinitial

Figure 13.7 The Initial Rate as a Function of Reactant Concentration for the

Michaelis–Menten Mechanism.

where

Km

k′+k 2

k 1

(13.1-46)

The parameterKmis called theMichaelis–Menten constantor theMichaelis constant.

It is equal to a constant only if the temperature is constant. A version of the Michaelis–

Menten equation can be derived with inclusion of a reverse reaction for step 2. (See

Problem 13.11.) The Michaelis–Menten mechanism has also been studied without use

of the steady-state approximation.^14

Since Eq. (13.1-45) is not easily integrated, the method of initial rates is used

to apply it. A number of experiments are carried out with the same concentration

of enzyme but with different concentrations of reactant, and the initial rate is deter-

mined for each experiment. Figure 13.7 shows the initial rate given by Eq. (13.1-45)

as a function of reactant concentration. Note the resemblance of Eq. (13.1-45) to

Eq. (13.1-17), and the resemblance of Figure 13.7 to Figure 13.2. For small values

of [R] the initial rate is proportional to [R] and for large values of [R] it levels off

and approaches the valuek 2 [E]totalasymptotically. The value ofKmcan be deter-

mined by locating the asymptote and equatingKmto the value of [R] at which

the initial rate is equal to one-half of the asymptotic value, as indicated in the fig-

ure. The number of reactant molecules that react per enzyme molecule per second

is called theturnover number. Its maximum value is equal tok 2 and can range up

to 10^6 s−^1.

Exercise 13.10

a.Show that the initial rate approaches the valuek 2 [E]totalfor large values of [R].

b.Show thatKmis equal to the value of [R] at which the initial rate is equal to half of its

asymptotic value.

c.Show that the maximum value of the turnover number is equal tok 2.

(^14) N. Sundaram and P. C. Wankat,J. Phys. Chem., 102 , 717 (1988).