Question: Q. 3. (i) Derive the relation between the decay constant and half life of a radioactive substance.
(ii) A radioactive element reduces to $25 %$ of its initial mass in $\mathbf{1 0 0 0}$ years. Find its half life.
U] [Foreign III 2017]
$$ \begin{aligned} & \text {
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Solution:
Ans. (i) } \ & \text { When } \quad t=T_{1 / 2} \ & \Rightarrow \quad N(t)=\frac{N_{0}}{2} \ & \therefore \quad \frac{N_{0}}{2}=N_{0} e^{-\lambda} T_{1 / 2} \ & \Rightarrow \quad \frac{1}{2}=e^{-\lambda} T_{1 / 2} \ & \Rightarrow \quad-\lambda T_{1 / 2}=-\ln 2 \ & \Rightarrow \mathrm{T}{1 / 2}=\frac{\ln 2}{\lambda}=\frac{0.693}{\lambda} \ & \frac{N}{N{0}}=\left(\frac{1}{2}\right)^{\mathrm{n}} \ & n=\frac{t}{T_{1 / 2}} \ & \frac{N}{N_{0}}=\frac{1}{4}=\left(\frac{1}{2}\right)^{\mathrm{n}} \ & \left(\frac{1}{2}\right)^{\mathrm{n}}=\left(\frac{1}{2}\right)^{2} \end{aligned} $$
$\therefore$ Number of half lives $=2$
$$ \begin{array}{rlrl} \Rightarrow & \frac{1000}{T_{1 / 2}} & =2 \ \Rightarrow \quad T_{1 / 2} & =\frac{1000}{2} \ & =500 \text { years } \end{array} $$
Alternatively :
$$ 1000 \text { years }=2 \text { half lives } $$
$$ \therefore \quad \text { Half life }=500 \text { years } $$
[CBSE Marking Scheme 2017]
Commonly Made Error
- Some students use the wrong formula do wrong substitution and hence get wrong result.
- Many students get confused between substance left behind and the mass disintegrated.
