Question

A freshly prepared radioactive source of half life 2 hours emits radiation of intensity which is 32 times the permissible safe level. The minimum time after which it would be possible to work safely with the source is
A. 6 hours
B. 10 hours
C. 24 hours
D. 128 hours

Answer 1

The half-life of the substance is given as 2 hours. Find the number of half-lives when the substance is decayed 32 times. Multiply by this number of times to find the required time.

Complete step by step solution:
A radioactive substance undergoes continuous disintegration. There are atoms which disintegrate in the beginning. There are atoms which disintegrate at the end. Half life of a radioactive substance is defined as the time during which half of the number of atoms present initially in the sample of the element decay or the time during which number of atoms left undecayed in the sample are half the total number of atoms present initially in the sample.

Here, half-life of the radioactive substance, T = 2 hours.
Amount of radioactive substance left after time,
$t = \dfrac{N}{{{N_0}}} \\\ \Rightarrow t = {\left( {\dfrac{1}{2}} \right)^{t/T}}$
Where ${N_0}$ is the number of atoms present initially, and N is the number of atoms present after time $t$ and $t = n \times T =$ total time of n half lives. It is given that,
$\dfrac{N}{{{N_0}}} = \dfrac{1}{{32}}$
$\Rightarrow\dfrac{1}{{32}}= {\left( {\dfrac{1}{2}} \right)^{t/T}}$

$$\Rightarrow {\left( {\dfrac{1}{2}} \right)^5} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{T}}} \\\ \Rightarrow \dfrac{t}{T} = 5 \\\$$

Where T = 2 hour
$\Rightarrow \,\,t = 2 \times 5 \\\ \therefore t = 10\,hours$
Hence, after 10 hours it would be possible to work safely with the source.

The correct option is B.

Note: The rate of disintegration of a radioactive sample is called activity of the sample, which is directly proportional to the number of atoms left undecayed in the sample. Amount of radioactive substance left after any time t depends on the half life of the substance. The value of t varies exponentially. So, look out for given values and try to find out the relation in terms of half-lives. Use the properties of exponents and logarithms correctly.