Page 213 - Essentials of physical chemistry

More Kinetics and Some Mechanisms 175

and

K I
[E I]

!
E þ I (E I); K I ¼
[E free ] [I]
[E free ] ¼ [E tot ] [E S] [E I] ¼ [E tot ] [E S] K I [E free ] [I],

so we rearrange to find [E free ]:

([E tot ] [E S])
:
(1 þ K I [I])
[E free ] ¼
Now we proceed with the familiar Michaelis–Menten derivation for the usual steady-state
approximation:

d[E S]
¼ k 1 [E free ] [S] k 1 [E S] k 2 [E S] ffi 0:
dt

Insert the formula for [E free ] into the steady state, so

k 1 [S]([E tot ] [E S])
¼ (k 1 þ k 2 )[E S]:
(1 þ K I [I])

But we need to isolate [E S] and you should follow this step with pencil and paper:

k 1 [S] [E tot ] [E S]k 1 [S] k 1 [S] þ (1 þ K I [I])(k 1 þ k 2 )
þ (k 1 þ k 2 )[E S] ¼ [E S] :
(1 þ K I [I]) ¼ (1 þ K I [I]) (1 þ K I [I])
Cancel the (1 þ K I [I]) denominator on both sides of the equation and we obtain

k 1 [S] [E tot ]
:
(1 þ K I [I])(k 1 þ k 2 ) þ k 1 [S]
[E S] ss ¼
As before V ¼ k 2 [E S] ss and V max ¼ k 2 [E tot ], and these values would be measured using the
Michaelis–Menten measurements without I. Thus, we have

k 2 k 1 [S] [E tot ]
,
V ¼
(1 þ K I [I])(k 1 þ k 2 ) þ k 1 [S]
1 (k 1 þ k 2 )
appear.
=
now use k 1 ¼ and we see K M ¼
(1 k 1 ) k 1
k 2 [S] [E tot ] V max [S]
,
V ¼ ¼
(1 þ K I [I])K M þ [S] (1 þ K I [I])K M þ [S]