Consider the diffusion situation described in section 9.5. Instead of having pure diffusion, we put a catalyst into the fluid that forces a first-order chemical reaction \(a \rightarrow b\) to occur. Resolve the transient problem for the concentration profile assuming dilute solutions, a constant surface concentration of \(a\), and impermeable container walls.

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9.5 TRANSIENT SYSTEMS The addition of time dependence to any situation is conceptually no more difficult than adding another spatial dimension. Practically, the solutions become more difficult to obtain analytically and numerically. Two flavors of time dependence can be added. If we add a first-order time deriva- tive to Laplace's equation, we introduce an exponentially decaying function of time into the solu- tion that leads to a dissipative process. Adding a second-order time derivative introduces a periodic time function into the solution and leads to a wavelike oscillatory process. Adding both leads to a damped traveling wave solution. The preferred method of obtaining analytic solutions to time-dependent transport problems involves using Green's functions [6]. The economy of using Green's functions stems from the fact that multidimensional solutions can be built up from a product of single-dimensional, time- dependent, Green's functions. To date there is no preferred method of solving these equations numerically. Several techniques can be found in Finlayson [11] and White [12], but each equation has its own peculiarities and the numerical method used depends upon which method can yield a solution, the accuracy desired, and the patience of the person waiting for the computer to run the program. This is particularly true for nonlinear equations.