Question

Benzoyl peroxide, the substance most widely used against acne, has a half-life of 9.8 × 103 days when refrigerated. How long will it take to lose 5% of its potency (95% remaining)? Assume that this is a first-order reaction. Give your answer in scientific notation.

Answer 1

Answer:

$7.3\times 10^2\ days$

Explanation:

Given that:

Half life = $9.8\times 10^3$ days

$t_{1/2}=\frac{\ln2}{k}$

Where, k is rate constant

So,  

$k=\frac{\ln2}{t_{1/2}}$

$k=\frac{\ln2}{9.8\times 10^3}\ days^{-1}$

The rate constant, k = 0.00007 days⁻¹

Using integrated rate law for first order kinetics as:

$[A_t]=[A_0]e^{-kt}$

Where,  

$[A_t]$ is the concentration at time t

$[A_0]$ is the initial concentration

Given:

5 % is lost which means that 0.05 of $[A_0]$ is decomposed. So,

$\frac {[A_t]}{[A_0]}$ = 1 - 0.05 = 0.95

t = ?

$\frac {[A_t]}{[A_0]}=e^{-k\times t}$

$0.95=e^{-0.00007\times t}$

t = 732.76 days = $7.3\times 10^2\ days$