At 25°C, the forward rate constant of Eq. (11.7-10) is equal to 1.4×10^11 Lmol^−1 s^−1. At this temperature, the dissociation equilibrium constant Kw 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

Question

At 25°C, the forward rate constant of [latex]Eq. (11.7-10):H_{3}O^{+}+OH^{−} \xrightleftharpoons[k^{′}_{1}]{k_{1}} 2H_{2}O[/latex] is equal to [latex]1.4×10^{11} Lmol^{−1} s^{−1}[/latex]. At this temperature, the dissociation equilibrium constant [latex]K_{w} equals 1.008×10^{−14} mol^{2} L^{−2}.[/latex]

a. Using [latex]Eq. (11.5-3):\frac{d[B]}{dt} =k_{1}[A]-k_{2}[B][/latex], find the value of [latex]k^{′}_{1}.[/latex]

b. For pure water, find the relaxation time if a T-jump experiment has a final temperature of 25°C.

Accepted Answer

a. We convert [latex]K_{w}[/latex], for which the [latex]H_{2}O[/latex] activity is expressed in terms of the mole fraction, to the equilibrium constant for the reaction of [latex]Eq. (11.7-10):H_{3}O^{+}+OH^{−} \xrightleftharpoons[k^{′}_{1}]{k_{1}} 2H_{2}O[/latex] with all concentrations in terms of molarities:

[latex]K_{1}=\frac{k_{1}}{k^{′}_{1}} =\frac{[H_{2}O]^{2}}{[H^{+}][OH^{-}]} =\frac{(55.35 mol L^{−1})^{2}}{1.008×10^{−14} mol^{2} L^{−2}} =3.039×10^{17}[/latex]

[latex]k^{′}_{1} =\frac{k_{1}}{K_{1}} =\frac{1.4×10^{11} L mol^{−1} s^{−1}}{3.039×10^{17}} =4.6×10^{−7} L mol^{−1} s^{−1}[/latex]

b. From [latex]Eq. (11.7-14):\frac{1}{τ} =k_{1}\frac{[A]_{eq}+[B]_{eq}}{2} +2k^{′}_{1}[C]eq[/latex]

[latex]\frac{1}{τ} =\frac{(1.4×10^{11} L mol^{−1} s^{−1})(2×1.004×10^{−7} mol L^{−1})}{2} +2(4.6×10^{−7} L mol^{−1} s^{−1})(55.35 mol L^{−1})[/latex]

[latex]=1.4×10^{4} s^{−1}[/latex]

[latex]τ=7.1×10^{−5} s[/latex]

Similar expressions for the relaxation time can be derived for other rate laws.[latex]^{a}[/latex]

[latex]^{a}[/latex] K. J. Laidler, Chemical Kinetics, 3rd ed., Harper and Row, New York, 1987, p. 38.

Question 11.12: At 25°C, the forward rate constant of Eq. (11.7-10) is equal...

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