In a laboratory study, a gas-phase reaction of the type A→products is catalyzed by solid nanoparticles of metal M dispersed in the reaction vessel. Only reactant molecules that collide with an active surface site on a nanoparticle can react.
For times short enough that [A] does not change appreciably, the measured rate is
r(t)=k[A]0[Cat]active(t),
where [A]0 is the constant bulk concentration of A and [Cat]active(t) is the molar concentration of active surface sites (expressed per unit volume of the reaction vessel).
The catalyst gradually deactivates because each active site is independently poisoned in a first-order process with deactivation rate constant kd=1.5×10−4s−1 at the reaction temperature.
At t=0s the concentration of active sites is [Cat]0=3.0×10−5M.
(a) Write a differential equation that describes the time dependence of [Cat]active(t) during the deactivation process.
(b) Solve the differential equation from part (a) to obtain an explicit expression for [Cat]active(t) in terms of [Cat]0, kd, and t.
(c) Using your result from part (b), derive an expression for r(t) for times in which [A]≈[A]0 is valid.
(d) Sketch, on appropriately labeled axes, the qualitative shapes of the curves for r(t) with deactivation and for r(t) in the hypothetical case of no deactivation (kd=0). Explain how the differences between the two curves would appear in experimental rate versus time data.