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Atoms and Nuclei

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JEE Advanced 1991

LEVELJEE Advanced

A nucleus $X$ , initially at rest, undergoes alpha-decay according to the equation. $_{92}^{A} X \to_{Z}^{228} Y + α$ (a) Find the values of $A$ and $Z$ in the above process. (b) The alpha particle produced in the above process is found to move in a circular track of radius $0.11 m$ in a uniform magnetic field of $3 T$ . Find the energy (in MeV) released during the process and the binding energy of the parent nucleus $X$ . Given that $m (Y) = 228.03 u$ ; $m (_{0}^{1} n) = 1.009 u$ ; $m (_{2}^{4} He) = 4.003 u$ ; $m (_{1}^{1} H) = 1.008 u$ .

JEE Advanced 2016

LEVELJEE Advanced

The electrostatic energy of $Z$ protons uniformly distributed throughout a spherical nucleus of radius $R$ is given by $E = \frac{3}{5} \frac{Z ( Z - 1 ) e ^{2}}{4 π ε _{0} R}$ The measured masses of the neutron, $_{1}^{1} H$ , $_{7}^{15} N$ and $_{8}^{15} O$ are $1.008665 u$ , $1.007825 u$ , $15.000109 u$ and $15.003065 u$ , respectively. Given that the radii of both the $_{7}^{15} N$ and $_{8}^{15} O$ nuclei are same, $1 u = 931.5 MeV/c^{2}$ ( $c$ is the speed of light) and $e^{2} / (4 π ε_{0}) = 1.44 MeV fm$ . Assuming that the difference between the binding energies of $_{7}^{15} N$ and $_{8}^{15} O$ is purely due to the electrostatic energy, the radius of either of the nuclei is ( $1 fm = 1 0^{- 15} m$ )

If the wavelength of the $n^{th}$ line of Lyman series is equal to the de-Broglie wavelength of electron in initial orbit of a hydrogen like element ( $Z = 11$ ). Find the value of $n$ .

JEE Advanced 1993

LEVELJEE Advanced

A neutron of kinetic energy 65 eV collides inelastically with a singly ionized helium atom at rest. It is scattered at an angle of 90° with respect of its original direction. (a) Find the allowed values of the energy of the neutron and that of the atom after the collision. (b) If the atom gets de-excited subsequently by emitting radiation, find the frequencies of the emitted radiation. [Given : Mass of He atom = 4 × ( mass of neutrons ) Ionization energy of H atom = 13.6 eV]

JEE Advanced 1999

LEVELJEE Advanced

Photoelectrons are emitted when 400 nm radiation is incident on a surface of work function 1.9 eV. These photoelectrons pass through a region containing $α$ -particles. A maximum energy electron combines with an $α$ -particle to form a $He^{+}$ ion, emitting a single photon in this process. $He^{+}$ ions thus formed are in their fourth excited state. Find the energies in eV of the photons lying in the 2 to 4 eV range, that are likely to be emitted during and after the combination. [Take $h = 4.14 \times 1 0^{- 15}$ eV-s]

JEE Advanced 2010

LEVELJEE Advanced

Comprehension Passage

The key feature of Bohr's theory of spectrum of hydrogen atom is the quantisation of angular momentum when an electron is revolving around a proton. We will extend this to a general rotational motion to find quantised rotational energy of a diatomic molecule assuming it to be rigid. The rule to be applied is Bohr's quantisation condition.

Question 1:

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

(A)

\frac{1}{n ^{2}} (\frac{h ^{2}}{8 π ^{2} I})

(B)

\frac{1}{n} (\frac{h ^{2}}{8 π ^{2} I})

(C)

n (\frac{h ^{2}}{8 π ^{2} I})

(D)

n^{2} (\frac{h ^{2}}{8 π ^{2} I})

Question 2:

It is found that the excitation frequency from ground to the first excited state of rotation for the CO molecule is close to $\frac{4}{π} \times 1 0^{11} Hz$ . Then the moment of inertia of CO molecule about its centre of mass is close to (Take $h = 2 π \times 1 0^{- 34} J-s$ )

(A)

2.76 \times 1 0^{- 46} kg-m^{2}

(B)

1.87 \times 1 0^{- 46} kg-m^{2}

(C)

4.67 \times 1 0^{- 47} kg-m^{2}

(D)

1.17 \times 1 0^{- 47} kg-m^{2}

Question 3:

In a CO molecule, the distance between C (mass $= 12 amu$ ) and O (mass $= 16 amu$ ), where $1 amu = \frac{5}{3} \times 1 0^{- 27} kg$ , is close to

(A)

2.4 \times 1 0^{- 10} m

(B)

1.9 \times 1 0^{- 10} m

(C)

1.3 \times 1 0^{- 10} m

(D)

4.4 \times 1 0^{- 11} m

JEE Advanced 1988

LEVELJEE Advanced

A particle of charge equal to that of an electron $- e$ , and mass $208$ times of the mass of the electron (called a mu-meson) moves in a circular orbit around a nucleus of charge $+ 3 e$ . (Take the mass of the nucleus to be infinite). Assuming that the Bohr model of the atom is applicable to this system, (a) derive an expression for the radius of the $n^{th}$ Bohr orbit. (b) find the value of $n$ for which the radius of the orbit is approximately the same as that of the first Bohr orbit for the hydrogen atom. (c) find the wavelength of the radiation emitted when the mu-meson jumps from the third orbit to the first orbit. (Rydberg's constant = $1.097 \times 1 0^{7} m^{- 1}$ )

JEE Advanced 2015

LEVELJEE Advanced

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 $K_{Xe}$ , $K_{Sr}$ , $K_{x}$ (2 MeV) and $K_{y}$ (2 MeV), respectively. Let the binding energies per nucleon of $_{92}^{236} U$ , $_{54}^{140} Xe$ and $_{38}^{94} Sr$ be 7.5 MeV, 8.5 MeV and 8.5 MeV, respectively. Considering different conservation laws, the correct options is/are

* Multiple Correct Options

(A)

x = n

y = n

K_{Sr} = 129 MeV

K_{Xe} = 86 MeV

(B)

x = p

y = e^{-}

K_{Sr} = 129 MeV

K_{Xe} = 86 MeV

(C)

x = p

y = n

K_{Sr} = 129 MeV

K_{Xe} = 86 MeV

(D)

x = n

y = n

K_{Sr} = 86 MeV

K_{Xe} = 129 MeV

JEE Advanced 1989

LEVELJEE Advanced

A gas of identical hydrogen-like atoms has some atoms in the lowest (ground) energy level $A$ and some atoms in a particular upper (excited) energy level $B$ and there are no atoms in any other energy level. The atoms of the gas make the transition to a higher energy level by absorbing monochromatic light of photon energy $2.7 eV$ . Subsequently, the atoms emit radiation of only six different energy photons. Some of the emitted photons have an energy of $2.7 eV$ , some have more energy and some less than $2.7 eV$ . (a) Find the principal quantum number of the initially excited level $B$ . (b) Find the ionization energy for the gas atoms. (c) Find the maximum and the minimum energies of the emitted photons.

LEVELJEE Advanced

The element curium $_{96}^{248} Cm$ has a mean life of $1 0^{13} s$ . Its primary decay modes are spontaneous fission and $α$ -decay, the former with a probability of 8% and the latter with a probability of 92%, each fission releases 200 MeV of energy. The masses involved in decay are as follows : $_{96}^{248} Cm = 248.072220 u$ , $_{94}^{244} Pu = 244.064100 u$ and $_{2}^{4} He = 4.002603 u$ . Calculate the power output from a sample of $1 0^{20} Cm$ atoms. ( $1 u = 931 MeV / c^{2}$ )

LEVELJEE Advanced

It is proposed to use the nuclear fusion reaction; $_{1}^{2} H +_{1}^{2} H \to_{2}^{4} He$ in a nuclear reactor of $200 MW$ rating. If the energy from the above reaction is used with a $25$ per cent efficiency in the reactor, how many grams of deuterium fuel will be needed per day? (The masses of $_{1}^{2} H$ and $_{2}^{4} He$ are $2.0141$ atomic mass units and $4.0026$ atomic mass units respectively.)

JEE Advanced 2000

LEVELJEE Advanced

A hydrogen like atom of atomic number $Z$ is in an excited state of quantum number $2 n$ . It can emit a maximum energy photon of $204 eV$ . If it makes a transition to quantum state $n$ , a photon of energy $40.8 eV$ is emitted. Find $n$ , $Z$ and the ground state energy (in $eV$ ) of this atom. Also, calculate the minimum energy (in $eV$ ) that can be emitted by this atom during de-excitation. Ground state energy of hydrogen atom is $- 13.6 eV$ .

JEE Advanced 1992

LEVELJEE Advanced

Light from a discharge tube containing hydrogen atoms falls on the surface of a piece of sodium. The kinetic energy of the fastest photoelectrons emitted from sodium is 0.73 eV. The work function for sodium is 1.82 eV. Find (a) the energy of the photons causing the photoelectrons emission. (b) the quantum numbers of the two levels involved in the emission of these photons. (c) the change in the angular momentum of the electron in the hydrogen atom, in the above transition, and (d) the recoil speed of the emitting atom assuming it to be at rest before the transition. (Ionization potential of hydrogen is 13.6 eV.)

JEE Advanced 2002

LEVELJEE Advanced

A hydrogen-like atom (described by the Bohrs model) is observed to emit six wavelengths, originating from all possible transitions between a group of levels. These levels have energies between $-0.85 eV$ and $-0.544 eV$ (including both these values). (a) Find the atomic number of the atom. (b) Calculate the smallest wavelength emitted in these transitions. (Take $h c = 1240 eV-nm$ , ground state energy of hydrogen atom $= - 13.6 eV$ )

JEE Advanced 2001

LEVELJEE Advanced

A radioactive nucleus $X$ decays to a nucleus $Y$ with a decay constant $λ_{x} = 0.1 s^{- 1}$ , $Y$ further decays to a stable nucleus $Z$ with a decay constant $λ_{y} = 1/30 s^{- 1}$ . Initially, there are only $X$ nuclei and their number is $N_{0} = 1 0^{20}$ . Set-up the rate equations for the populations of $X, Y$ and $Z$ . The population of $Y$ nucleus as a function of time is given by $N_{y} (t) = {N_{0} λ_{x} / (λ_{x} - λ_{y})} [exp (- λ_{y} t) - exp (- λ_{x} t)]$ . Find the time at which $N_{Y}$ is maximum and determine the populations $X$ and $Z$ at that instant.

JEE Advanced 2008

LEVELJEE Advanced

Comprehension Passage

In a mixture of H - He

^{+}

gas (He

^{+}

is singly ionized He atom), H atoms and He

^{+}

ions are excited to their respective first excited states. Subsequently, H atoms transfer their total excitation energy to He

^{+}

ions (by collisions). Assume that the Bohr model of atom is exactly valid.

Question 1:

The quantum number $n$ of the state finally populated in He $^{+}$ ions is

The wavelength of light emitted in the visible region by He $^{+}$ ions after collisions with H atoms is

(A)

6.5 \times 1 0^{- 7} m

(B)

5.6 \times 1 0^{- 7} m

(C)

4.8 \times 1 0^{- 7} m

(D)

4.0 \times 1 0^{- 7} m

Question 3:

The ratio of the kinetic energy of the $n = 2$ electron for the H atom to that of He $^{+}$ ion is

Comprehension Passage

The mass of a nucleus

_{Z}^{A} X

is less than the sum of the masses of

(A - Z)

number of neutrons and

Z

number of protons in the nucleus. The energy equivalent to the corresponding mass difference is known as the binding energy of the nucleus. A heavy nucleus of mass

M

can break into two light nuclei of masses

m_{1}

and

m_{2}

only if

(m_{1} + m_{2}) < M

. Also two light nuclei of masses

m_{3}

and

m_{4}

can undergo complete fusion and form a heavy nucleus of mass

M^{'}

only if

(m_{3} + m_{4}) > M^{'}

. The masses of some neutral atoms are given in the table below: $\begin{array}{llll} _{1}^{1}\text{H} & 1.007825\text{u} & _{1}^{2}\text{H} & 2.014102\text{u} \\ _{3}^{6}\text{Li} & 6.01513\text{u} & _{3}^{7}\text{Li} & 7.016004\text{u} \\ _{64}^{152}\text{Gd} & 151.919803\text{u} & _{82}^{206}\text{Pb} & 205.974455\text{u} \\ _{1}^{3}\text{H} & 3.016050\text{u} & _{2}^{4}\text{He} & 4.002603\text{u} \\ _{30}^{70}\text{Zn} & 69.925325\text{u} & _{34}^{82}\text{Se} & 81.916709\text{u} \\ _{84}^{210}\text{Po} & 209.982876\text{u} & & \end{array}$

Question 1:

The correct statement is

(A)

The nucleus

_{3}^{6} Li

can emit an alpha particle.

(B)

The nucleus

_{84}^{210} Po

can emit a proton.

(C)

Deuteron and alpha particle can undergo complete fusion.

(D)

The nuclei

_{30}^{70} Zn

and

_{34}^{82} Se

can undergo complete fusion.

Question 2:

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

A particle of mass $m$ moves in circular orbits with potential energy $V (r) = F r$ , 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 denoted by $R$ and its speed and energy are denoted by $v$ and $E$ , respectively, then for the $n^{th}$ orbit (here $h$ is the Planck's constant)-

* Multiple Correct Options

(A)

R \propto n^{1/3}

and

v \propto n^{2/3}

(B)

R \propto n^{2/3}

and

v \propto n^{1/3}

(C)

E = \frac{3}{2} (\frac{n ^{2} h ^{2} F ^{2}}{4 π ^{2} m})^{1/3}

(D)

E = 2 (\frac{n ^{2} h ^{2} F ^{2}}{4 π ^{2} m})^{1/3}

JEE Main 2022

LEVELJEE Advanced

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

JEE Advanced 2022

LEVELJEE Advanced

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?

* Multiple Correct Options

(A)

E_{b}^{n} - E_{b}^{p}

is proportional to

Z (Z - 1)

where

Z

is the atomic number of the nucleus.

(B)

E_{b}^{n} - E_{b}^{p}

is proportional to

A^{- \frac{1}{3}}

where

A

is the mass number of the nucleus.

(C)

E_{b}^{n} - E_{b}^{p}

is positive.

(D)

E_{b}^{p}

increases if the nucleus undergoes a beta decay emitting a positron.

Micro-Topic Strategy Matrix

Bohr's Atomic Model and Energy Levels

Nucleus and Nuclear Reaction

Radioactivity

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the wavelength of the nth line of Lyman series is equal to the de-Broglie wavelength of electron in initial orbit of a hydrogen like element (Z=11). Find the value of n.

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In a CO molecule, the distance between C (mass =12 amu) and O (mass =16 amu), where 1 amu=35​×10−27 kg, is close to

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The quantum number n of the state finally populated in He+ ions is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The wavelength of light emitted in the visible region by He+ ions after collisions with H atoms is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The ratio of the kinetic energy of the n=2 electron for the H atom to that of He+ ion is

Comprehension Passage

The correct statement is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The kinetic energy (in keV) of the alpha particle, when the nucleus 84210​Po at rest undergoes alpha decay, is

If the wavelength of the $n^{th}$ line of Lyman series is equal to the de-Broglie wavelength of electron in initial orbit of a hydrogen like element ( $Z = 11$ ). Find the value of $n$ .

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

In a CO molecule, the distance between C (mass $= 12 amu$ ) and O (mass $= 16 amu$ ), where $1 amu = \frac{5}{3} \times 1 0^{- 27} kg$ , is close to

The quantum number $n$ of the state finally populated in He $^{+}$ ions is

The wavelength of light emitted in the visible region by He $^{+}$ ions after collisions with H atoms is

The ratio of the kinetic energy of the $n = 2$ electron for the H atom to that of He $^{+}$ ion is

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