Question

Two radioactive elements X and Y have half-lives 6 min and 15 min respectively. An experiment starts with 8 times as many atoms of X as Y. How long it takes for the number of atoms of X left to equal the number of atoms of Y left?
(a)- 6 min
(b)- 12 min
(c)- 48 min
(d)- 30 min
(e)- 24 min

Answer 1

Given the concentration of the X is 8 times greater than the concentration of Y, so take a ratio of X: Y as 8:1. By equating the ratio of concentration with the half-life of both the elements we can find the time at which the concentration becomes the same.

Complete step-by-step answer: We are given that the half-life time of X is 6 min. We can write:
${{t}_{1/2}}\text{ of X = 6}$
We are given that the half-life time of Y is 15 min. We can write:
${{t}_{1/2}}\text{ of Y = 15}$
Given the concentration of the X is 8 times greater than the concentration of Y, so take a ratio of X: Y as 8:1.
Let us assume that the number of half-lives of X and Y is x and y.
As the time taken will be the same:
$6x=15y$
By equating the ratio of concentration with the half life, we can write:
$\dfrac{8}{{{(2)}^{x}}}=\dfrac{1}{{{(2)}^{y}}}$
This equation can be solved:
$8={{2}^{x-y}}$
As we know that,
$8={{2}^{3}}$
Combining these, we can write:
x-y = 3
From the equation above and the equation $6x=15y$, we can write the value:
y = 2 and x = 5
Now, putting these values in $6x=15y$,we get:
$6\text{ x 5}=15\text{ x 2 = 30 min}$

Therefore, the correct answer is option (d)- 30 min.

Note: When you are solving these types of questions, the concentration of the elements must be taken correctly. When we have to find the time at which the same amount is present, we can equate the half-life of both elements.