Nuclei

${}^{131}$I is an isotope of Iodine that $\beta$ decays to an isotope of Xenon with a half-life of 8 days. A small amount of a serum labelled with ${}^{131}$I is injected into the blood of a person. The activity of the amount of ${}^{131}$I injected was $2.4 \times 10^5$ Becquerel (Bq). It is known that the injected serum will get distributed uniformly in the blood stream in less than half an hour. After 11.5 hours, 2.5 ml of blood is drawn from the person’s body, and gives an activity of 115 Bq. The total volume of blood in the person’s body, in liters is approximately (you may use $e^x \approx 1 + x$ for $|x| \ll 1$ and $\ln 2 \approx 0.7$).

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

Two deuterons undergo nuclear fusion to form a Helium nucleus. Energy released in this process is : (given binding energy per nucleon for deuteron = 1.1 MeV and for helium = 7.0 MeV)

Imagine that a reactor converts all given mass into energy and that it operates at a power level of $10^9$ watt. The mass of the fuel consumed per hour in the reactor will be : (velocity of light, c is $3\times10^8$ m/s)

The isotope $^{12}_{5}\mathrm B$ having a mass 12.014 u undergoes β-decay to $^{12}_{6}\mathrm C$. $^{12}_{6}\mathrm C$ has an excited state of the nucleus $(^{12}_{6}\mathrm C^*)$ at 4.041 MeV above its ground state. If $^{12}_{5}\mathrm B$ decays to $^{12}_{6}\mathrm C^*$, the maximum kinetic energy of the β-particle in units of MeV is ($1\ \mathrm u=931.5\ \mathrm{MeV}/c^2$, where $c$ is the speed of light in vacuum).

The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by
$$E=\frac35\,\frac{Z(Z-1)e^2}{4\pi\varepsilon_0R}$$
The measured masses of the neutron, $^1_1\mathrm H$, $^{15}_{7}\mathrm N$ and $^{15}_{8}\mathrm O$ are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u, respectively. Given that the radii of both the $^{15}_{7}\mathrm N$ and $^{15}_{8}\mathrm O$ nuclei are same, $1\ \mathrm u=931.5\ \mathrm{MeV}/c^2$ ($c$ is the speed of light) and $e^2/(4\pi\varepsilon_0)=1.44$ MeV fm. Assuming that the difference between the binding energies of $^{15}_{7}\mathrm N$ and $^{15}_{8}\mathrm O$ is purely due to the electrostatic energy, the radius of either of the nuclei is

($1\ \mathrm{fm}=10^{-15}$ m)

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?

Half-lives of two radioactive elements A and B are 20 minutes and 40 minutes, respectively. Initially, the samples have equal number of nuclei. After 80 minutes, the ratio of decayed numbers of A and B nuclei will be :

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life $T$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of $nT$ years, then the value of $n$ is

Match the nuclear processes given in column I with the appropriate option(s) in column II.

For a radioactive material, its activity $A$ and rate of change of its activity $R$ are defined as $A=-\dfrac{dN}{dt}$ and $R=-\dfrac{dA}{dt}$, where $N(t)$ is the number of nuclei at time $t$. Two radioactive sources $P$ (mean life $\tau$) and $Q$ (mean life $2\tau$) have the same activity at $t=0$. Their rates of change of activities at $t=2\tau$ are $R_P$ and $R_Q$, respectively. If $\dfrac{R_P}{R_Q}=\dfrac ne$, then the value of $n$ is

A fission reaction is given by $^{236}_{92}\mathrm U\to{}^{140}_{54}\mathrm{Xe}+{}^{94}_{38}\mathrm{Sr}+x+y$, where $x$ and $y$ are two particles. Considering $^{236}_{92}\mathrm U$ to be at rest, the kinetic energies of the products are denoted by $K_{Xe}$, $K_{Sr}$, $K_x$ (2 MeV) and $K_y$ (2 MeV), respectively. Let the binding energies per nucleon of $^{236}_{92}\mathrm U$, $^{140}_{54}\mathrm{Xe}$ and $^{94}_{38}\mathrm{Sr}$ be 7.5 MeV, 8.5 MeV and 8.5 MeV, respectively. Considering different conservation laws, the correct option(s) is(are)

Let $N_\beta$ be the number of β particles emitted by 1 gram of $^{24}$Na radioactive nuclei (half life = 15 hrs) in 7.5 hours, $N_\beta$ is close to (Avogadro number = $6.023\times10^{23}$/g. mole) :

A radioactive nuclei with decay constant 0.5/s is being produced at a constant rate of 100 nuclei/s. If at $t=0$ there were no nuclei, the time when there are 50 nuclei is :

A piece of bone of an animal from a ruin is found to have $^{14}$C activity of 12 disintegrations per minute per gm of its carbon content. The $^{14}$C activity of a living animal is 16 disintegrations per minute per gm. How long ago nearly did the animal die ? (Given half life of $^{14}$C is $t_{1/2}=5760$ years)

A piece of wood from a recently cut tree shows 20 decays per minute. A wooden piece of same size placed in a museum (obtained from a tree cut many years back) shows 2 decays per minute. If half life of C$^{14}$ is 5730 years, then age of the wooden piece placed in the museum is approximately :

A freshly prepared sample of a radioisotope of half-life 1386 s has activity $10^3$ disintegrations per second. Given that $\ln2=0.693$, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is

The correct statement is

The kinetic energy (in keV) of the alpha particle, when the nucleus $^{210}_{84}\mathrm{Po}$ at rest undergoes alpha decay, is

Match List I of the nuclear processes with List II containing parent nucleus and one of the end products of each process and then select the correct answer using the codes given below the lists: