How long will it take for the caffeine to decay?

Question

The half liter of caffeine in your bloodstream is about 5 hours. How long will it take for the caffeine to decay to 15% of the original amount?

it will take ____ hours

Answer 1

In an insane world where this half-liter is pure caffeine, first of all, you won't be having a good time. But we'll figure it out anyway.

We need to get from 500 mL of caffeine to 15% of that, so calculate:

500*0.15 = 75 mL. That's our goal.

We'll use our decay equations to solve this, the first of which is:

t_1/2 = 0.693/k

We know our half life is 5 hours, so we'll use that to solve for our decay constant, k. We must know this before moving on. Solving for k, you should get 0.1386.

Now let's use our other decay equation to find how long it will take:

ln(N/N_o) = -kt

where N is our desired amount (75 mL), N_o is our initial 500 mL, k is the decay constant calculated above, and t is time. Since we calculated k using hours, our answer should also have units of hours. Plugging in, we get

ln(75/500) = -(0.1386)t

solve for t and you should get that this poor sap has to survive t = 13.69 hours until only 75 mL remains in the bloodstream.