Nuclei

A nuclear reactor starts producing a radioactive nuclide $X$ from $t=0$, at a constant rate of $\alpha$ per second. Each decay of $X$ produces energy $E_0$, which is utilized to heat a liquid of mass $m$ and specific heat $s$. Assuming no heat loss from the liquid and taking $\lambda$ as the decay constant of $X$, the rate of increase in the temperature of the liquid is:

The binding energy per nucleon of ${}^{209}_{83}Bi$ is ________ MeV.
[Take $m\left({}^{209}_{83}Bi\right)=208.980388$ u, $m_p=1.007825$ u, $m_n=1.008665$ u, 1 u = 931 MeV/c²]

Two nuclei of mass number 3 combine with another nucleus of mass number 4 to yield a nucleus of mass number 10. If the binding energy per nucleon for the mass numbers 3, 4 and 10 are 5.6 MeV, 7.4 MeV and 6.1 MeV, respectively, then in the process, $\Delta Mc^2$ = ________ MeV.

Assuming the experimental mass of $^{12}_{\ 6}\mathrm C$ as 12 u, the mass defect of $^{12}_{\ 6}\mathrm C$ atom is ________ MeV/c².

(Mass of proton = 1.00727 u, mass of neutron = 1.00866 u, 1 u = 931.5 MeV/c² and c is the speed of the light in vacuum).

The energy released when $\dfrac7{17.13}$ kg of $^7_3\mathrm{Li}$ is converted into $^4_2\mathrm{He}$ by proton bombardment is $\alpha\times10^{32}$ eV. The value of $\alpha$ is ________. (Nearest integer)

(Mass of $^7_3\mathrm{Li}$ = 7.0183 u, mass of $^4_2\mathrm{He}$ = 4.004 u, mass of proton = 1.008 u and 1 u = 931 MeV/c² and Avogadro number = $6.0\times10^{23}$)

The energy released if hydrogen atoms are combined to form $^4_2\mathrm{He}$ is ________ MeV.

(Take binding energies per nucleon of $^2_1\mathrm H$ and $^4_2\mathrm{He}$ as 1.1 MeV and 7.2 MeV, respectively)

Two radioactive substances A and B of mass numbers 200 and 212 respectively, shows spontaneous α-decay with same $Q$ value of 1 MeV. The ratio of energies of $\alpha$-rays produced by A and B is ________.

List-I shows various functional dependencies of energy ($E$) on the atomic number ($Z$). Energies associated with certain phenomena are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

A sample initially contains only U-238 isotope of uranium. With time, some of the U-238 radioactively decays into Pb-206 while the rest of it remains undisintegrated.

When the age of the sample is $\mathbf{P}\times10^8$ years, the ratio of mass of Pb-206 to that of U-238 in the sample is found to be 7. The value of P is _____.

[Given: Half-life of U-238 is $4.5\times10^9$ years; $\log_e 2 = 0.693$]

List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose the correct option.

In a radioactive decay process, the activity is defined as $A = -\dfrac{dN}{dt}$, where $N(t)$ is the number of radioactive nuclei at time $t$. Two radioactive sources, $S_1$ and $S_2$ have same activity at time $t = 0$. At a later time, the activities of $S_1$ and $S_2$ are $A_1$ and $A_2$, respectively. When $S_1$ and $S_2$ have just completed their 3rd and 7th half-lives, respectively, the ratio $A_1/A_2$ is _____.

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${}^{16}_{7}\mathrm N + {}^{4}_{2}\mathrm{He} \to {}^{1}_{1}\mathrm H + {}^{19}_{8}\mathrm O$ in a laboratory frame is $n$ (in MeV). Assume that ${}^{16}_{7}\mathrm N$ is at rest in the laboratory frame. The masses of ${}^{16}_{7}\mathrm N$, ${}^{4}_{2}\mathrm{He}$, ${}^{1}_{1}\mathrm H$ and ${}^{19}_{8}\mathrm O$ can be taken to be 16.006 $u$, 4.003 $u$, 1.008 $u$ and 19.003 $u$, respectively, where $1\,u = 930\ \mathrm{MeV}\,c^{-2}$. The value of $n$ is _____.

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_b^p$ and the binding energy of a neutron be $E_b^n$ in the nucleus.

Which of the following statement(s) is(are) correct?

In a radioactive decay chain reaction, ${}^{230}_{90}\mathrm{Th}$ nucleus decays into ${}^{214}_{84}\mathrm{Po}$ nucleus. The ratio of the number of $\alpha$ to number of $\beta^-$ particles emitted in this process is _____.

Nucleus A is having mass number 220 and its binding energy per nucleon is 5.6 MeV. It splits in two fragments 'B' and 'C' of mass numbers 105 and 115. The binding energy of nucleons in 'B' and 'C' is 6.4 MeV per nucleon. The energy Q released per fission will be :

A sample contains $10^{-2}$ kg each of two substances A and B with half lives 4 s and 8 s respectively. The ratio of their atomic weights is 1 : 2. The ratio of the amounts of A and B after 16 s is $\dfrac x{100}$. The value of $x$ is ______.

Which of the following figure represents the variation of $\ln\left(\dfrac R{R_0}\right)$ with $\ln A$ (if $R=$ radius of a nucleus and $A=$ its mass number)

How many alpha and beta particles are emitted when Uranium $\mathrm{{}_{92}U^{238}}$ decays to lead $\mathrm{{}_{82}Pb^{206}}$ ?

A radioactive nucleus can decay by two different processes. Half-life for the first process is 3.0 hours while it is 4.5 hours for the second process. The effective half-life of the nucleus will be :

The Q-value of a nuclear reaction and kinetic energy of the projectile particle, $K_p$ are related as :