11.7 The Experimental Study of Fast Reactions
519
Exercise 11.23
a. Verify the steps of algebra leading to Eq. (11.7-7).
b. Verify that Eq. (11.7-8), with Eq. (11.7-9), is a solution to Eq. (11.7-7).
c. Write the expressions for ∆[A] and ∆[B].
If we assume that hydrogen ions in water primarily occur as hydronium ions, the
reaction of hydrogen ions and hydroxide ions can be written
H 3 O
+
+ OH
− k 1
k
1
2H 2 O
(11.7-10)
which has the general form
A + B
k 1
k
1
2C
(11.7-11)
Assume that the reaction is second order in both directions. We write
[C] [C] eq + ∆[C]
(11.7-12a)
[A] [A] eq −
1
2
∆[C]
(11.7-12b)
[B] [B] eq −
1
2
∆[C]
(11.7-12c)
When Eqs. (11.7-12) are substituted into the differential equation for the rate of the
reaction and the necessary steps of algebra are carried out with neglect of terms proportional to (∆[C]) 2 , we obtain
∆[C] ∆[C] 0 e
−t/τ
(11.7-13)
where
1
τ
k 1
[A] eq + [B] eq
2
+ 2k
1 [C] eq
(11.7-14)
E X A M P L E 11.12
At 25 ◦ C, the forward rate constant of Eq. (11.7-10) is equal to 1.4 × 10 11 L mol −1 s −1 . At
this temperature, the dissociation equilibrium constant K w equals 1.008 × 10 −14 mol 2 L −2 .
a. Using Eq. (11.5-3), find the value of k
1 .
b. For pure water, find the relaxation time if a T-jump experiment has a final temperature of
25 ◦ C.
Solution
a. We convert K w , for which the H 2 O activity is expressed in terms of the mole fraction, to
the equilibrium constant for the reaction of Eq. (11.7-10) with all concentrations in terms
of molarities: