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 (pA) 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 pA,0. For time t > 0, CO2 will begin to accumulate in the water, but 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 CO2 concentration pA(x, t).
(a) Write (do not derive) a differential equation that could be used to determine pA(x, t) 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 (k1 = 0)?
(b) The DVD will gradually become more opaque over time as the reaction proceeds. The ability to read the DVD will depend on how well the laser light can penetrate through the thickness of the DVD. Therefore, it is important to know the volume-averaged molar concentration of product, Cprod, as a function of time. Write an expression for Cprod in terms of the oxygen molar concentration, assuming that every mole of oxygen that reacts with the polymer results in p moles of product.