Law of Radioactivity

Q. The half life of ${}_{92}^{238}U$ undergoing $\alpha$ decay is $4.5\times {10}^9$ years. What is the activity of 4 $g$ sample of ${}_{92}^{238}U$ ?

1. $4.9\times {10}^4$ per second
2. $3.6\times {10}^4$ per second
3. $1.2\times {10}^4$ per second
4. $5.6\times {10}^4$ per second

Q. Half-life of $B{i}^{210}$ is $5$ days. If we start with $50,000$ atoms of this isotope, the number of atoms left after $10$ days is

1. $5,000$
2. $25,000$
3. $12,500$
4. $20,000$

Q. Concentration of a radioactive sample decreases exponentially with time.

1. True
2. False

Q. The rate at which a particular decay process occurs in a radioactive sample, is proportional to the number of radioactive nuclei present. If N is the number of radioactive nuclei present at some instant, the rate of change of N is $\frac{dN}{dt}=-\lambda N$.
Consider radioactive decay of A to B which may further decay either to X or to Y. ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$ are decay constant for A to B decay, B to X decay and B to Y decay respectively. If at $t=0$ number of nuclei of A, B, X and Y are $N_0,N_0$, zero and zero respectively and $N_1,N_2,N_3,N_4$ are number of nuclei A, B, X and Y at any instant.

Rate of accumulation of at any instant will be

1. $N_1{\lambda }_1+N_2{\lambda }_2+N_3{\lambda }_3$
2. $N_1{\lambda }_1-N_3{\lambda }_2-N_4{\lambda }_3$
3. $N_1{\lambda }_1-N_2{\lambda }_2-N_2{\lambda }_3$
4. $N_1{\lambda }_1+N_2{\lambda }_2-N_2{\lambda }_3$

Q. The nuclei of two radioactive isotopes of same substance $A^{236}$ and $A^{234}$ are present in the ratio 4 : 1 in an ore obtained from Mars. Their half lives are $30\text{min}$ and $60\text{min}$ respectively. Both isotopes are alpha emitters and the activity of the isotope with half life $30\text{min}$ is one Rutherford. Calculate after how much time (in min) their activities will become identical.

Q. Nuclei of a radioactive element is produced at constant rate $\alpha$ and they decay with decay constant $\lambda$. At $t=0$, the number of nuclei is 0. The number of active nuclei at time $t$ is

1. $\frac{\alpha }{\lambda }(1-e^{-\lambda t})$
2. $\alpha (1-e^{-\lambda t})$
3. $\frac{\alpha }{\lambda }-e^{-\lambda t}$
4. $\alpha e^{-\lambda t}$

Q. The rate at which a particular decay process occurs in a radioactive sample, is proportional to the number of radioactive nuclei present. If N is the number of radioactive nuclei present at some instant, the rate of change of N is $\frac{dN}{dt}=-\lambda N$.

The number of nuclei of B will first increase then after a maximum value, it will decreases, if

1. ${\lambda }_1>{\lambda }_2+{\lambda }_3$
2. ${\lambda }_1={\lambda }_2={\lambda }_3$
3. ${\lambda }_1={\lambda }_2+{\lambda }_3$
4. For any values of ${\lambda }_1,{\lambda }_2$ and ${\lambda }_3$

Q. You have prepared a nice sample of $N_{\circ }$ radioactive nuclei of decay constant $\lambda$, and want to show it off to your friends. When they arrive units of time later, you open the lid and find there are only _______ nuclei left intact.

1. $N_{\circ }{e}^{\lambda t}$
2. $N_{\circ }{e}^{-\lambda t}$
3. $N_{\circ }$
4. $N_{\circ }\lambda t$

Q. Radioactive material '$A$' has decay constant '$8\lambda$' and material '$B$' has decay constant '$\lambda$'. Initially they have same number of nuclei. After what time, the ratio of number of nuclei of material 'B' to that 'A' will be $\frac{1}{e}$?

1. $\frac{1}{9\lambda }$
2. $\frac{1}{\lambda }$
3. $\frac{1}{7\lambda }$
4. $\frac{1}{8\lambda }$

Q. A sample of ${}_{19}{K}^{40}$ disintegrates into two nuclei Ca and Ar with decay constant ${\lambda }_{ca}=4.5\times {10}^{-10}{s}^{-1}$ and ${\lambda }_{Ar}=0.5\times {10}^{-10}{s}^{-1}$ respectively. The time after which $99\%$ of ${}_{19}{K}^{40}$ get decayed is

1. $6.2\times {10}^{9}sec$
2. $9.2\times {10}^{9}sec$
3. $7.2\times {10}^{9}sec$
4. $4.2\times {10}^{9}sec$

Q. The radioactivity of a sample is X at time $t_1$ and Y and a time $t_2$. If the mean life time of the specimen is $\tau$, the number of atoms that have disintigrated in the time interval $t_1-t_2$ is

1. $X{t}_1-Y{t}_2$
2. $X-Y$
3. $\frac{X-Y}{\tau }$
4. $(X-Y)\tau$

Q. At time $t=0$, a container has $N_0$ radioactive atoms with a decay constant $\lambda$. In addition, $C$ numbers of atoms of the same type are being added to the container per unit time. How many atoms of this type are there at $t=T$ ?

1. $\frac{C}{\lambda }{e}^{-\lambda T}-N_0{e}^{-\lambda T}$
2. $\frac{C}{\lambda }{e}^{-\lambda T}+N_0{e}^{-\lambda T}$
3. $\frac{C}{\lambda }(1-e^{-\lambda T})+N_0{e}^{-\lambda T}$
4. $\frac{C}{\lambda }(1+e^{-\lambda T})+N_0{e}^{-\lambda T}$

Q. The rate at which a particular decay process occurs in a radioactive sample, is proportional to the number of radioactive nuclei present. If N is the number of radioactive nuclei present at some instant, the rate of change of N is $\frac{dN}{dt}=-\lambda N$.

At $t=\infty$, which of the following incorrect?

1. $N_2=0$
2. $N_3=\frac{2N_0{\lambda }_2}{{\lambda }_2+{\lambda }_3}$
3. $N_4=\frac{2N_0{\lambda }_2}{{\lambda }_2+{\lambda }_3}$
4. $N_3+N_4+N_1+N_2=2N_0$

Q.

Who discovered radioactivity?

1. Einstein

2. Newton

3. C.V. Raman

4. Becquerel

Q. Radioactivity of an unstable element depends upon the number of nuclei present in it.

1. False
2. True