Question

An accident happens in the lab of Professor Utonium, and a radioactive element X is released in the form of a gas at around 4:00 am. Element X has a short half-life (25 min), and the lab would be considered safe when the concentration of X drops by a factor of 10. Considering the decomposition of element X is of first-order, what is the earliest time Professor Utonium can come back to do experiments in the lab

Answer 1

Answer:

5:22 am

Explanation:

The gas X decays following a first-order reaction.

The half-life ($t_{1/2}$) is 25 min. We can find the rate constant (k) using the following expression.

$k = \frac{ln2}{t_{1/2}} =\frac{ln2}{25min} = 0.028 min^{-1}$

We can find the concentration of X at a certain time ($[X]$) using the following expression.

$[X] = [X]_0 \times e^{-k \times t}$

where,

$[X]_0$: initial concentration of X

t: time elapsed

$\frac{[X]}{[X]_0}= e^{-k \times t}\\\frac{1/10[X]_0}{[X]_0}= e^{-0.028min^{-1} \times t}\\t=82min$

The earliest time Professor Utonium can come back to do experiments in the lab is:

4:00 + 82 = 5:22 am