Consider our derivation of the time$-$dependent response of a tubular reactor to a pulse input. You will now use some of those ideas to develop a model for fertilizer runoff into a river, a serious problem in many parts of the country. Suppose you treat the river as a tubular reactor with dispersion. Fertilizer can react via a first$-$order decomposition, $-$kc$.$ The rate of addition of fertilizer to the river, per unit

length of river, is H$($x$,$t$),$ where x is the distance down the river and t is time. The amount added to the river depends on time $($when it last rained$)$ and position $($if it$$s close to a field which has had fertilizer added recently$).$ For this problem, first define those quantities that you need to specify the system and then assuming that the flow rate in the river is approximately constant do the following:

a$.$ Derive an equation that describes the concentration of fertilizer as a function of position and time.

b$.$ Determine the boundary conditions for this differential equation. For this, assume the river starts from a lake that may be treated as a CSTR$.$ Assume the lake has fertilizer flowing into it at a rate of G$($t$).$ The river ends up flowing into the ocean at x $=$ L$.$