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 p _A.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 P_A(x, t).

(a) Write (do not derive) a differential equation that could be used to determine p_A(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 ≈ O)?

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Water (Quiescent)