Page 415 - Elements of Chemical Reaction Engineering Ebook

386 Nonelementary Reaction Kinetics Chap. 7

We need to replace The rate of disappearance of the substrate, -rs, is
unbound enzyme
concentration (E) in -r, = kl(E)(S) - k2(E.S) (7-78)
the rate law
The net rate of formation of the enzyme-substrate complex is

rE.s k,(E)(S) - k2(E*S) - k3(W)(E*S) (7-79)
We note from the reaction sequence that the enzyme is not consumed by
the reaction. The total concentration of the enzyme in the system, (Et), is con-
stant and equal to the sum of the concentrations of the free or unbonded
enzyme E and the enzyme-substrate complex E * S:
Total enzyme
concentration, (E,) = (E) + (Ems) (7-80)
bound + free
Rearranging Equation (7-80), the enzyme concentration becomes
(E) = (EA - (E.S) (7-81)

Substituting Equation (7-81) into Equation (7-79) and using the PSSH for the
enzyme complex gives

rE.s = 0 = kl[(E,) - (E.S)](S) - k,(E.S) - k3(E.S)(W) (7-82j
Solving for (E * S) yields

(7-83)

Next, substituting Equation (7-8 1) into Equation (7-78) yields

-r, = kl[(E,) - (E*S)](S) - k2(E*S) (7-84)
Subtracting Equation (7-82) from Equation (7-84), we get
-r, = k3(W)(E-S) (7-85)
Substituting for (E-S) gives us

The final form of (7-86)
the rate law
Note: Throughout, E, 5 (E,) = total concentration of enzyme with typical units
(kmol/m3).

7.4.2 Michaelis-Menten Equation

Because the reaction of urea and urease is carried out in aqueous solu-
tion, water is, of course, in excess, and the concentration of water is therefore
considered constant. Let

k; = k3(W) and