Question:

State, with explanation, which diagram corresponds to ka  kb .

Assuming 1st order reactions for both of the two steps, it is possible to solve the rateequations analytically,∗ which gives[A]t = [A]0 e−ka t



ka
(−e−ka t + e−kb t )[A]0[B]t =k−kab




1

−ka t−kb t [C]t = 1 +(kb e− ka e) [A]0ka − kb

The mathematical derivation is far beyond the assumed level of this course. The reason
for showing these equations is because the formula for [C]t is useful to demonstrate the∗The rate equations for this scheme are:dA= −ka A,
dt

dB= ka A − kb B,
dt

dC= kb B
dt

The first equation for A(t) is the same as what we have seen previously in Sec 3.1.2. We put the solution
A(t) = A(0)e−ka t in the second equation. This yields an ordinary differential equation for B(t), forwhich the solution method is well established (though we have no space to expand it here). C(t) can beobtained from a relation A(t) + B(t) + C(t) = constant = A(0), which describes the conservation ofgross materials.

24concept of rate-determining step, as will be discussed next.

4.4 Rate-determining process
The above formula for [C]t can be simplified in two limiting cases where one of therate constants is much larger than the other. For example, when ka  kb , we finde−ka t  e−kb t and 1/(ka − kb ) ' 1/ka . Therefore, [C]t can be simplified as[C]t ' (1 − e−kb t )[A]0
This means that the behaviour of [C]t is determined by the rate constant of the slower
step kb .
Question:Show that when ka  kb ,[C]t ' (1 − e−ka t )[A]0In this way, the slower step determines the overall rate of the product formation.
This is called the Rate-determining or Rate-limiting step. The idea is straightforwardly generalized for reactions involving more than two steps, i.e., more than one
intermediates I1 , I2 , · · · , involved between the reactant R and product P:
R → I1 → I2 → · · · → P

5Steady-state approximation

It is seen in the right panel of the previous diagram that the concentration of B does
not change much but stays low and nearly constant, and therefore d[B]/dt ' 0, whenkb  ka , i.e., when this intermediate specie is highly reactive. This suggests a very
useful and powerful method called the steady-state approximation.
Recipe of the Steady-State Approximation
1. Identify reactive intermediates, I1 , I2 , · · · , in the mechanism
2. Write down
3. Setd[Ii ]
from the assumed mechanism.
dtd[Ii ]= 0 and solve for [Ii ]. (The solution will be denoted by [Ii ]ss .)dt

4. Deduce, and simplify if possible, the rate law by using [Ii ]ss .
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5.1 Application
Let us see how the steady-state approximation works for the following scheme
kf ,kbkA *) B →r P

(49)

which has been examined previously by invoking the pre-equilibrium approximation.
The critical assumption there was the reaction from B to P is very slow such that it
does not disturb very much the equilibrium between A and B. We will now relax thisassumption. To this end, let us follow the recipe of the steady-state approximation. Asbefore, we will assume that all the steps kf , kb and kr are 1st order.
1. We identify B as the reactive intermediate.
2. Noting that B is gained by kf and lost by kb and kr steps,d[B]= kf [A] − kb [B] − kr [B]dt
3. Set the above to be equal to zero, and solve for [B].
kf [A] − kb [B] − kr [B] ' 0 ⇒ [B]ss =kf[A]k b + kr

4. The overall reaction rate is that of the product formation described by the kr step.
We write down this equation and then replace [B] by [B]ss .
Rate =

d[P]kr kf= kr [B] ' kr [B]ss =[A]
dtkb + kr(50)Compare this with the result from the pre-equilibrium assumption, Eq (45):Rate (pre-Eq) ' kr K[A] =kr kf[A]kb

The difference is in the denominator of the rate factor. Remember that the fundamental
assumption in the pre-equilibrium case was kr  kb . This means that the denominatorin Eq (50) may be approximated as kb + kr ' kb . We therefore find that the preequilibrium approximation Eq (45) is reproduced as a limiting case of the steady-state
approximation Eq (50). In other words, the steady-state approximation generalizes, andtherefore covers broader applicability than, the pre-equilibrium approximation.

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