Law of Radioactivity

Q.

A radioactive sample disintegrates via two independent decay processes having half-lives$T_{1/2}^1$ and $T_{1/2}^2$ respectively. The effective half-life, $T_{1/2}$ of the nuclei is

1. None of the above

2. $T_{1/2}=T_{1/2}^1+T_{1/2}^2$

3. $T_{1/2}=\frac{T_{1/2}^1+T_{1/2}^2}{T_{1/2}^1-T_{1/2}^2}$

4. $T_{1/2}=\frac{T_{1/2}^1{T}_{1/2}^2}{T_{1/2}^1+T_{1/2}^2}$

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. 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. Two radioactive material $X_1$ and $X_2$ have decay constant $10\lambda$ and $\lambda$ respectively. Initially, they have the same number of nuclei. The ratio of the number of nuclei $X_1$ to that of $X_2$ will be $\frac{1}{e}$ after time,

1. $\frac{1}{100\lambda }$
   
2. $\frac{1}{11\lambda }$
   
3. $\frac{1}{10\lambda }$
   
4. $\frac{1}{9\lambda }$
   

Q. The radioactive sources $A$ and $B$ of half lives of $2\text{hr}$ and $4\text{hr}$ respectively, initially contain the same number of radioactive atoms. At the end of $2\text{hours}$, their rates of disintegration are in the ratio

1. $4:1$
2. $2:1$
3. $\sqrt{2}:1$
4. $1:1$

Q. A radioactive nucleus $A$ with a half-life $T$, decays into a nucleus $B$. At $t=0$, there is no nucleus $B$. At some time $t$, the ratio of the number of $B$ to that of $A$ is $0.3$. Then, $t$ is given by:

1. $t=\frac{T}{\log \left(1.3\right)}$
2. $t=\frac{T}{\log \left(1.3\right)}$
3. $t=T\frac{\log 1.3}{\log 2}$
4. $t=T\log \left(1.3\right)$

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

1. False
2. True

Q. Unit of radioactivity is rutherford. Its value is

1. $3.7\times {10}^{10}$ disintegrations/sec
2. $3.7\times {10}^6$ disintegrations/sec
3. $1.0\times {10}^{10}$ disintegrations/sec
4. $1.0\times {10}^6$ disintegrations/sec

Q. What is the mass of one Curie of$U^{234}$

1. $3.7\times {10}^{10}gm$
   
2. $2.348\times {10}^{23}gm$
   
3. $1.44\times {10}^{-11}gm$
   
4. $6.25\times {10}^{-34}gm$

Q. If $N_t=N_0{e}^{-\lambda t}$, then number of disintegrated atoms between $t_1$ to $t_2$ $(t_2>t_1)$ will be

1. $N_0[e^{\lambda t_2}-e^{\lambda t_1}]$
2. $N_0[e^{-\lambda t_2}-e^{-\lambda t_1}]$
3. $N_0[e^{-\lambda t_1}-e^{-\lambda t_2}]$
4. None

Q. Let's take a hypothetical radioactive nuclide sample of ${}^{80}X$.
It weighs 80 gm. Find the number of active nuclei remaining after a time of 0.5 second. Given its decay constant is 2 $s^{-1}$

1. $6.023\times {10}^{23}/e$
2. $6.023\times {10}^{23}$ x e
3. $6.023\times {10}^{23}$
4. $Noneofthese$

Q. The count rate of radioactive sample falls from $8.0\times {10}^6$ per second to $1.0\times {10}^6$ per second in 12 hours. What will be the count rate after 20 hours from the beginning?

1. $25\times {10}^4$
2. $4\times {10}^4$
3. $25\times {10}^6$
4. $3\times {10}^4$

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. $20,000$