The Classic A — B — C Consecutive First-Order Reaction

The next example is a classic problem in both nuclear chemistry as well as chemical engineering. (By the way, a student who complained thathe would never see this problem in real life was sitting in a seminar the very next day when another student was presenting the results of his PhD research showing a time-dependent series of NMR peaks. In the data, a certain peak (A) decreased to form a second peak (B) and that peak reached a maximum but then decreased to form a final peak (C). The PhD candidate then proceeded to use this solution to analyze the kinetics of his data ) The idea is obvious for nuclear processes because nuclear decay follows successive step-by-step transformations from one isotope to 

Let Ao represent the initial gallons of water in tank A, which is 1000 gal here but might be a different number in another situation. Let Aa be the amount of solution (gallons) in tank A at any later time with the similar meaning for Ab and Ac. We can save effort since we know tank A wUl drain in a first-order way with ti/2 = 3 days, so we can immediately write Aa = Ao e .  

The next step is the tricky part because as soon as some water drains into tank B from tank A, some of it will start to drain into tank C, although at a slower rate. Thus, we write 

This is really an amazing mathematical trick because we should see that the two terms in Nb would be part of a derivative of a product if only there was a factor of e , so why not multiply the whole equation by that factor Now we can integrate both sides of the equation  

That gives us a formula for Nb at any time t 0. When will the contents of tank B reach a maximum We need to set 

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