For the gas-phase equilibrium the uncatalyzed rate constants at are (forward…

CHEMISTRY · Multi-part · AP · Chemistry · Kinetics · Kinetics

Problem

For the gas-phase equilibrium

\mathrm{SO_2Cl_2\,(g)} \rightleftharpoons \mathrm{SO_2\,(g)} + \mathrm{Cl_2\,(g)}

the uncatalyzed rate constants at

350\,\text{K}

are

k_f = 1.5 \times 10^{-5}\,\text{s}^{-1}

(forward decomposition)

k_r = 3.0 \times 10^{-5}\,\text{s}^{-1}

(reverse recombination) A solid catalyst is introduced that increases *both* rate constants by the same factor of

4.0

. Assume that each elementary step is first order in the species that decomposes or recombines and that the system initially contains only

\mathrm{SO_2Cl_2}

at a pressure of

0.800\,\text{atm}

. (a) Write an expression for the equilibrium constant

K_p

in terms of the given rate constants and calculate its numerical value at

350\,\text{K}

before the catalyst is added. (b) Determine the numerical value of

K_p

after the catalyst is added. Show your reasoning. (c) Derive an expression for the characteristic time

\tau

required for the system to reach

63\%

of its final approach to equilibrium for a reversible first-order reaction of this type. Your expression should be in terms of

k_f

and

k_r

. (d) Calculate the ratio

\tau_{\text{catalyzed}}/\tau_{\text{uncatalyzed}}

using the data provided, and interpret what this ratio implies about the time needed to reach equilibrium.