Question

A CSI investigator finds a tube of benzoyl peroxide, the substance most widely used against acne, in the refrigerator of a murder suspect. He claims that he used this tube two weeks ago which worked to clear up his acne; today’s date is April 21, 2008. The tube has a date stamp (production date) of January 8, 1995. Any less than 95% active ingredient has no medicinal effect. Benzoyl peroxide has a half-life of 9800 days. Assume first order kinetics. Is he lying?

Answer 1

Ans:

For a first order reaction, rate constant (k) is related to half-life (t_1/2) through the equation

k = 0.693/t_1/2 = 0.693/9800 = 7.0714 x 10^-5 days^-1

Also, rate constant (k) can be connected to the concentration of the reactant (benzoyl peroxide) through the equation,

Where, k = rate constant of the first order reaction

            t = time of the reaction

            [R]_o = Initial concentration of benzoyl peroxide

            [R]_t = Concentration of benzoyl peroxide at time t

Time, ‘t’ = {Number of days between January 8, 1995 and April 21, 2008 – 2 weeks} =

{Number of days between January 8, 1995 and April 21, 2008 – 14 days} = 4838 days

Let’s take the initial concentration of benzoyl peroxide in the tube (on the production date) as 100% = [R]_o

So, equation 1 takes the form;

7.0714 x 10^-5 = (2.303/4838) log₁₀ (100/[R]_t)

[R]_t = 71.0310 %

Two weeks before April 21, 2008, the tube consisted only 71.0310% benzoyl peroxide (active ingredient). Since anything less than 95% active ingredient has no medicinal effect, we can draw the conclusion that the person is lying.