Nuclei's Question 9 | Physics XII - Part 2 | NCERT Exemplar Questions

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MCQ (Multiple Correct Answer)

Samples of two radioactive nuclides $A$ and $B$ are taken $\lambda_A$ and $\lambda_B$ are the disintegration constants of $A$ and $B$ respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?

Initial rate of decay of $A$ is twice the initial rate of decay of $B$ and $\lambda_A=\lambda_B$

Initial rate of decay of $A$ is twice the initial rate of decay of $B$ and $\lambda_A>\lambda_B$

Initial rate of decay of $B$ is twice the initial rate of decay of $A$ and $\lambda_A>\lambda_B$

Initial rate of decay of $B$ is same as the rate of decay of $A$ at $t=2 \mathrm{~h}$ and $\lambda_B<\lambda_A$

MCQ (Multiple Correct Answer)

The variation of decay rate of two radioactive samples $A$ and $B$ with time is shown in figure. Which of the following statements are true?

Decay constant of $A$ is greater than that of $B$, hence $A$ always decays faster than $B$

Decay constant of $B$ is greater than that of $A$ but its decay rate is always smaller than that of $A$

Decay constant of $A$ is greater than that of $B$ but it does not always decay faster than $B$

Decay constant of $B$ is smaller than that of $A$ but still its decay rate becomes equal to that of $A$ at a later instant

Subjective

$\mathrm{He}_2^3$ and $\mathrm{He}_1^3$ nuclei have the same mass number. Do they have the same binding energy?

Explanation

Nuclei $\mathrm{He}_2^3$ and $\mathrm{He}_1^3$ have the same mass number. $\mathrm{He}_2^3$ has two proton and one neutron. $\mathrm{He}_1^3$ has one proton and two neutron. The repulsive force between protons is missing in ${ }_1 \mathrm{He}^3$ so the binding energy of ${ }_1 \mathrm{He}^3$ is greater than that of ${ }_2 \mathrm{He}^3$.

Subjective

Draw a graph showing the variation of decay rate with number of active nuclei.

Explanation

We know that, rate of decay $=\frac{-d N}{d t}=\lambda N$

where decay constant $(\lambda)$ is constant for a given radioactive material. Therefore, graph between $N$ and $\frac{d N}{d t}$ is a straight line as shown in the diagram.

Subjective

Which sample A or B shown in figure has shorter mean-life?

Explanation

From the given figure, we can say that

$$\begin{aligned} & \text { at } t=0,\left(\frac{d N}{d t}\right)_A=\left(\frac{d N}{d t}\right)_B \\ & \Rightarrow \quad\left(N_0\right)_A=\left(N_0\right)_B \end{aligned}$$

Considering any instant $t$ by drawing a line perpendicular to time axis, we find that $\left(\frac{d N}{d t}\right)_A>\left(\frac{d N}{d t}\right)_B$

$$ \begin{array}{l} \Rightarrow & \lambda_A N_A>\lambda_B N_B\quad \text { (rate of decay of } B \text { is slower ) } \\ \because & N_A>N_B \\ \therefore & \lambda_B>\lambda_A \\ \Rightarrow & \tau_A>\tau_B \quad \left[\because \text { Average life } \tau=\frac{1}{\lambda}\right] \end{array}$$

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