Atoms and Nuclei

A nuclear reactor starts producing a radioactive nuclide X from t = 0, at a constant rate of α per second. Each decay of X produces energy E0, which is utilized to heat a liquid of mass m and specific heat s. Assuming no heat loss from the ...

Consider a hydrogen atom with $v_k, r_k,$ and $K_k$ denoting the velocity, orbital radius and kinetic energy of the electron in the $k^{{th}}$ orbit, respectively. The electron undergoes a transition from the $n^{{th}}$ orbit, emitting radi...

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 th...

A particle of mass $m$ is moving in a circular orbit under the influence of the central force $F(r)=-k r$, corresponding to the potential energy $V(r)=k r^2 / 2$, where $k$ is a positive force constant and $r$ is the radial distance from th...

In a radioactive decay process, the activity is defined as $A=-\frac{d N}{d t}$, 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 a...

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.

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

The minimum kinetic energy needed by an alpha particle to cause the nuclear reaction ${ }_{7}^{16} \mathrm{~N}+$ ${ }_{2}^{4} \mathrm{He} \rightarrow{ }_{1}^{1} \mathrm{H}+{ }_{8}^{19} \mathrm{O}$ in a laboratory frame is $n$ (in $M e V$. A...

A heavy nucleus N, at rest, undergoes fission N \(\to\) P + Q, where P and Q are two lighter nuclei. Let \(\delta\) = MN \(-\) MP \(-\) MQ, where MP, MQ and MN are the masses of P, Q and N, respectively. EP and EQ are the kinetic energies o...

Which of the following statement(s) is(are) correct about the spectrum of the hydrogen atom?

A heavy nucleus Q of half-life 20 minutes undergoes alpha-decay with probability of 60% and beta-decay with probability of 40%. Initially, the number of Q nuclei is 1000. The number of alpha-decays of Q in the first one hour is

In an X-ray tube, electrons emitted from a filament (cathode) carrying current I hit a target (anode) at a distance d from the cathode. The target is kept at a potential V higher than the cathode resulting in emission of continuous and char...

A particle of mass m moves in circular orbits with potential energy V(r) = Fr, where F is a positive
constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the
radius of the particle’s orbit is ...

Suppose a \(_{88}^{226}Ra\) nucleus at rest and in ground state undergoes \(\alpha\)-decay to a \(_{86}^{222}Rn\) nucleus in its excited state. The kinetic energy of the emitted \(\alpha\) particle is found to be 4.44 MeV. $$_{86}^{222}Rn...

A free hydrogen atom after absorbing a photon of wavelength \(\lambda\)a gets excited from the state n = 1 to the state n = 4. Immediately after that the electron jumps to n = m state by emitting a photon of wavelength \(\lambda\)e. Let t...

In a radioactive sample, \({}_{19}^{40}K\) nuclei either decay into stable \({}_{20}^{40}Ca\) nuclei with decay constant 4.5 \(\times\) 10-10 per year or into stable \({}_{18}^{40}Ar\) nuclei with decay constant 0.5 \(\times\) 10-10 per...

Consider a hydrogen-like ionized atom with atomic number \(Z\) with a single electron. In the emission spectrum of this atom, the photon emitted in the \(n=2\) to \(n=1\) transition has energy \(74.8eV\) higher than the photon emitted in th...

In a radioactive decay chain, \({}_{90}^{232}Th\) nucleus decays to \({}_{82}^{212}Pb\) nucleus. Let \({N_\alpha }\) and \({N_\beta }\) be the number of \(\alpha\) and \({\beta ^ - }\) particles, respectively, emitted in this decay process...

An electron in a hydrogen atom undergoes a transition from an orbit with quantum number \({n_i}\) to another with quantum number \({n_f}\). \({V_i}\) and \({V_f}\) are respectively the initial and final potential energies of the electron. I...

\({}^{131}{\rm I}\) is an isotope of Iodine that \(B\) decays to an isotope of Xenon with a half-life of \(8\) days. A small amount of a serum labelled with \({}^{131}{\rm I}\) is injected into the blood of a person. The activity of the am...

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by \(E = {3 \over 5}{{Z(Z - 1){e^2}} \over {4\pi {\varepsilon _0}R}}\)The measured masses of the neutron, \(_1^1H\), $$_7^{15}N$...

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 s...

A hydrogen atom in its ground state is irradiated by light of wavelength 970\(\mathop A\limits^o\).Taking hc = 1.237 \(\times\) 10\(-\)6 eVm and the ground state energy of hydrogen atom as \(-\) 13.6 eV, the number of lines present in the...

The isotope \(_5^{12}B\) having a mass 12.014 u undergoes \(\beta\)-decay to \(_6^{12}C\). \(_6^{12}C\) has an excited state of the nucleus (\(_6^{12}C\)*) at 4.041 MeV above its ground state. If \(_5^{12}B\) decays to \(_6^{12}C\)*, the m...

Highly excited states for hydrogen-like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principle quantum number n, where n >> 1. Which of the following statement(s) is(are) true?

A fission reaction is given by \(_{92}^{236}U \to _{54}^{140}Xe + _{38}^{94}Sr + x + y\), where x and y are two particles. Considering \(_{92}^{236}U\) to be at rest, the kinetic energies of the products are denoted by $${K_{Xe}},{K_{Sr}},{...

For a radioactive material, its activity A and rate of change of its activity R are defined as \(A = - {{dN} \over {dt}}\) and \(R = - {{dA} \over {dt}}\), where N(t) is the number of nuclei at time t. Two radioactive source P(mean life $...

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 electri...

Consider a hydrogen atom with its electron in the nth orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is (hc = 1242 eV nm)

Match the nuclear processes given in Column I with the
appropriate option(s) in Column II:

If \(\lambda\)Cu is the wavelength of K\(\alpha\) X-ray line of copper (atomic number 29) and \(\lambda\)Mo is the wavelength of the K\(\alpha\) X-ray line of molybdenum (atomic number 42), then the ratio \(\lambda\)Cu/\(\lambda\)Mo is clos...

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 :

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

The correct statement is

The radius of the orbit of an electron in a hydrogen-like atom is 4.5a0, where a0 is the Bohr radius. Its orbital angular momentum is \({{3h} \over {2\pi }}\). It is given that h is Planck constant and R is Rydberg constant. The possible wa...

A freshly prepared sample of a radioisotope of half-life 1386 s has activity 103 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 t...

If the anti-neutrino had a mass of 3 eV/c2 (where c is the speed of light) instead of zero mass, what should be the range of the kinetic energy, K, of the electron?

What is the maximum energy of the anti-neutrino?

The wavelength of the first spectral line in the Balmer series of hydrogen atom is 6561 \(\mathop A\limits^o\). The wavelength of the second spectral line in the Balmer series of singly-ionized helium atom is

In a CO molecule, the distance between C (mass = 12 amu) and O (mass = 16 amu), where 1 amu \(= {5 \over 3} \times {10^{ - 27}}\) kg, is close to :

It is found that the excitation frequency from ground to the first excited state of rotation for the CO molecule is close to \({4 \over \pi } \times {10^{11}}\) Hz. Then, the moment of inertia of CO molecule about its centre of mass is clos...

A diatomic molecule has moment of inertia I. By Bohr's quantization condition, its rotational energy in the nth level (n = 0 is not allowed) is

To determine the half-life of a radioactive element, a student plots a graph of \(\ln \left| {{{dN(t)} \over {dt}}} \right|\) versus t. Here, \({{dN(t)} \over {dt}}\) is the rate of radioactive decay at time t. If the number of radioactive ...

Results of calculations for four different designs of a fusion reactor using D-D reaction are given below. Which of these is most promising based on Lawson criterion?

Assume that two deuteron nuclei in the core of fusion reactor at temperature T are moving towards each other, each with kinetic energy 1.5 kT, when the separation between them is large enough to neglect Coulomb potential energy. Also neglec...

In the core of nuclear fusion reactor, the gas becomes plasma because of

The speed of the particle, that can take discrete values, is proportional to

A radioactive sample S1 having activity of 5 \(\mu\)Ci has twice the number of nuclei as another sample S2 which has an activity of 10 \(\mu\)Ci. The half lives of S1 and S2 can be :

The ratio of the kinetic energy of the \(n=2\) electron for the H atom to that of He\(^+\) ion is