Question

A freshly-prepared radioactive source of half-life 2 h emits radiation of intensity which is 64 times the permissible safe level. The minimum time after which it would be possible to work safely with this source is
1. 6 h
2. 12 h
3. 24 h
4. 128 h

Answer 1

Correct Answer - Option 2 : 12 h

CONCEPT: 

• If N0 is the initial number of atoms of nuclide present after n number of half-lives the amount of substance left is given by

\(\Rightarrow \frac{N}{N_{0}} = (\frac{1}{2})^{n}\)

Where N = The number of atoms left after n number of half-lives, N0 = The initial number of atoms,n = The number of half-lives  [ n = \(\frac{T}{t_{\frac{1}{2}}}\) Where T = Time up to which the disintegration observed and \(t_{\frac{1}{2}}\) =Half life]

CALCULATION :

Given - t_1/2 = 2 hours, N₀ = 64 [To work safely the number of reacting molecules must decrease by 64 times]

• The amount of substance left after n  number of half-lives is 

\(\Rightarrow N=N_0 \left(\dfrac{1}{2}\right)^n\)

The above equation can be rewritten as

\(⇒ \dfrac{N}{N_0} = \left(\dfrac{1}{2}\right)^n\)

\(⇒ \dfrac{1}{64}=\left(\dfrac{1}{2}\right)^n\)

\(⇒ (\frac{1}{2})^{6} = (\frac{1}{2})^{n}\)

Comparing both sides of the equation n = 6

• The time after which it is safe to work with the source will be  

⇒ Time = n× \(t_{\frac{1}{2}}\)

Substituting the given values in the above equation

⇒ Time  = 6 × 2 = 12 hours 

• The minimum time after which it would be possible to work safely with this source is 12 hours
• Hence, option 2 is the answer