Chapter 14, Problem 14.48P

The presence of CO₂ in solution is essential to the growth of aquatic plant life, with CO₂ used as a reactant in the photosynthesis. Consider a stagnant body of water in which the concentration of CO₂(p_A) is everywhere zero. At time t = 0, the water is exposed to a source of CO₂, which maintains the surface (x = 0) concentration at a fixed value $p_{A.0}$ . For time t > 0, CO₂ will begin to accumulate in the water, hut the accumulation is inhibited by CO₂ consumption due to photosynthesis. The time rate at which this consumption occurs per unit volume is equal to the product of a reaction rate constant k₁ and the local CO, concentration $p_A\left(x,t\right)$ .
(a) Write (do not derive) a differential equation that could be used to determine $p_A\left(x,t\right)$ in the water. What does each term in the equation represent physically’?
(b) Write appropriate boundary conditions that could be used to obtain a particular solution, assuming a “deep” body of water. What would be the form of this solution for the special case of negligible CO₂ consumption $\left(k\approx 0\right)$ ?