Question

A radioactive substance has half-life of 5 years .Probability of decay in 10 years is
(A) 100%
(B) 75%
(C) 50 %
(D) 25 %

Answer 1

Hint: The half-life of a radioactive substance is the amount of time which is required for the quantity by weight of the substance to fall into half its initial value. Hence, after one half life, we have 50 % of the substance remaining and after two half-lives, we will have $\dfrac{1}{4}$ of the substance remaining.

Complete step by step solution:
- In radioactivity, the half-life can be described as the interval of time required for one-half of the atomic nuclei of a radioactive sample to change spontaneously into other nuclear species by emitting energy and particles (radioactive decay). Or in other words it’s the time interval required for the number of disintegrations per second of a radioactive material.
- The half-life of the substance is given as 5 years (T) and we are asked to find probability of decay in 10 years (t). The equation for radioactive decay can be given as
\[N={{N}_{o}}{{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{T}}}\]
The ratio $\dfrac{N}{{{N}_{o}}}$ represents the undecayed fraction. Thus we can modify the above relation as
\[\dfrac{N}{{{N}_{o}}}={{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{T}}}\]
 The decayed fraction can be given as
\[1-\dfrac{N}{{{N}_{o}}}=1-{{\left( \dfrac{1}{2} \right)}^{{}^{t}/{}_{T}}}\]
Let’s substitute the values of t and T
\[1-\dfrac{N}{{{N}_{o}}}=1-{{\left( \dfrac{1}{2} \right)}^{{}^{10}/{}_{5}}}=1-{{\left( \dfrac{1}{2} \right)}^{2}}\]
\[\dfrac{{{N}_{o}}-N}{{{N}_{o}}}=\dfrac{3}{4}\]
 Let’s convert this into percentage by multiplying by100
\[\dfrac{{{N}_{o}}-N}{{{N}_{o}}}\times 100=\dfrac{3}{4}\times 100=75\]
Therefore the probability of decay in 10 years is 75%.

Thus the answer is option (B) 75%.

Note: Keep in mind that the process of radioactive decay is a first order process and therefore the time required for the amount of substance to get halved will be a constant. Also, the half-life just describes the decay of distinct entities.