Correct Answer - Option 4 : 300 days
Concept:
Rate of decay and decay constant (disintegration constant):
- In radioactive decay, the no. of atoms decaying at any time 't' is proportional to the no. of such atoms 'N' present.
- This is the characteristic of first-order reactions.
- The rate of the decay constant or disintegration constant is given by:
\(\frac{{dN}}{{dt}} = - \lambda N\)
Where, N = no. of atoms present, λ = Disintegration constant (unit = time-1)
Half-life period:
- It is the time required for the decay of one-half of the amount of species.
- Represented as t½.
Relationship between half-life period and decay constant:
\(\lambda = \frac{{0.693}}{{{t_½}}}\)
Calculations:
Given: λ = 0.00231 per day
To find: t½ =?
We know,
\(\lambda = \frac{{0.693}}{{{t_½}}}\) ⇒ \(0.00231 = \frac{{0.693}}{{{t_½}}}\)
⇒ t½ = 300 days
Hence,
The disintegration constant λ of a radioactive element is 0.00231 per day. Its half-life is 300 days.
- The half-life period depends only on the decay constant and independent of the amount of radioactive substance.
- The smaller the half-life of a radionuclide, the greater is its instability.
- The Law of radioactive decay is expressed as
\(\lambda = \frac{{2.303}}{t} \times \log \left( {\frac{{{N_0}}}{N}} \right)\)
