Bohr's Atomic Model and Energy Levels

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LEVELJEE Main

An electron in a hydrogen like atom is in an excited state. It has a total energy of $- 3.4 eV$ . Calculate (a) the kinetic energy, (b) the de-Broglie wavelength of the electron.

JEE Advanced 2018

LEVELJEE Main

Consider a hydrogen-like ionised 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 $74.8 eV$ higher than the photon emitted in the $n = 3$ to $n = 2$ transition. The ionisation energy of the hydrogen atom is $13.6 eV$ . The value of $Z$ is ............ .

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 2014

LEVELJEE Main

If $λ_{Cu}$ is the wavelength of $K_{α}$ , X-ray line of copper (atomic number 29) and $λ_{MO}$ is the wavelength of the $K_{α}$ , X-ray line of molybdenum (atomic number 42), then the ratio $λ_{Cu} / λ_{Mo}$ is close to

As per Bohr model, the minimum energy (in eV) required to remove an electron from the ground state of doubly ionized Li atom ( $Z = 3$ ) is

$K_{α}$ wavelength emitted by an atom of atomic number $Z = 11$ is $λ$ . Find the atomic number for an atom that emits $K_{α}$ radiation with wavelength $4 λ$

(A)

Z = 6

(B)

Z = 4

(C)

Z = 11

(D)

Z = 44

JEE Advanced 2015

LEVELJEE Main

Consider a hydrogen atom with its electron in the $n^{th}$ orbit. 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 ( $h c = 1242 eV nm$ )

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 2016

LEVELJEE Main

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

* Multiple Correct Options

(A)

Relative change in the radii of two consecutive orbitals does not depend on

Z

(B)

Relative change in the radii of two consecutive orbitals varies as

1/ n

(C)

Relative change in the energy of two consecutive orbitals varies as

1/ n^{3}

(D)

Relative change in the angular momenta of two consecutive orbitals varies as

1/ n

LEVELJEE Main

Assertion If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic X-rays do not change. Reason When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

(A)

If Assertion is true, Reason is true; Reason is the correct explanation for Assertion.

(B)

If Assertion is true, Reason is true; Reason is not a correct explanation for Assertion.

(C)

If Assertion is true; Reason is false.

(D)

If Assertion is false; Reason is true.

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 Main

The largest wavelength in the ultraviolet region of the hydrogen spectrum is 122 nm. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is

In the Bohr model of the hydrogen atoms

* Multiple Correct Options

(A)

the radius of the

n^{th}

orbit is proportional

n^{2}

(B)

the total energy of the electron in the

n^{th}

orbit is inversely proportional to

n

(C)

the angular momentum of the electron in an orbit is an integral multiple of

h / π

(D)

the magnitude of the potential energy of the electron in any orbit is greater than its kinetic energy.

JEE Advanced 2016

LEVELJEE Main

A hydrogen atom in its ground state is irradiated by light of wavelength $970 \overset{˚}{A}$ . Taking $h c / e = 1.237 \times 1 0^{- 6} eVm$ and the ground state energy of hydrogen atom as $- 13.6 eV$ , the number of lines present in the emission spectrum is

LEVELJEE Main

Which one of the following statements is wrong in the context of X-rays generated from an X-ray tube ?

(A)

Wavelength of characteristic X-rays decreases when the atomic number of the target increases

(B)

Cut-off wavelength of the continuous X-rays depends on the atomic number of the target

(C)

Intensity of the characteristic X-rays depends on the electrical power given to the X-ray tube

(D)

Cut-off wavelength of the continuous X-rays depends on the energy of the electrons in the X-ray tube

LEVELJEE Main

The electron in a hydrogen atom makes a transition from an excited state to the ground state. Which of the following statement is true ?

(A)

Its kinetic energy increases and its potential and total energy decreases

(B)

Its kinetic energy decreases, potential energy increases and its total energy remains the same

(C)

Its kinetic and total energy decreases and its potential energy increases

(D)

Its kinetic, potential and total energy decreases

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 $- 13.6 eV$ .

LEVELJEE Main

The wavelength of the first spectral line in the Balmer series of hydrogen atom is 6561 \AA. The wavelength of the second spectral line in the Balmer series of singly ionized helium atom is

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})

LEVELJEE Main

A hydrogen atom and a $Li^{2 +}$ ion are both in the second excited state. If $l_{H}$ and $l_{L i}$ are their respective electronic angular momenta, and $E_{H}$ and $E_{L i}$ their respective energies, then

(A)

l_{H} > l_{L i}

and

∣ E_{H} ∣ > ∣ E_{L i} ∣

(B)

l_{H} = l_{L i}

and

∣ E_{H} ∣ < ∣ E_{L i} ∣

(C)

l_{H} = l_{L i}

and

∣ E_{H} ∣ > ∣ E_{L i} ∣

(D)

l_{H} < l_{L i}

and

∣ E_{H} ∣ < ∣ E_{L i} ∣

LEVELJEE Main

The $K_{α}$ X-ray emission line of tungsten occurs at $λ = 0.021 nm$ . The energy difference between $K$ and $L$ levels in this atom is about

A photon collides with a stationary hydrogen atom in ground state inelastically. Energy of the colliding photon is 10.2 eV. After a time interval of the order of micro second another photon collides with same hydrogen atom inelastically with an energy of 15 eV. What will be observed by the detector ?

(A)

2 photons of energy 10.2 eV

(B)

2 photons of energy 1.4 eV

(C)

One photon of energy 10.2 eV and an electron of energy 1.4 eV

(D)

One photon of energy 10.2 eV and another photon of energy 1.4 eV

JEE Advanced 2013

LEVELJEE Advanced

The radius of the orbit of an electron in a Hydrogen-like atom is $4.5 a_{0}$ where $a_{0}$ is the Bohr radius. Its orbital angular momentum is $\frac{3 h}{2 π}$ . It is given that $h$ is Planck constant and $R$ is Rydberg constant. The possible wavelength(s), when the atom de-excites, is (are)

* Multiple Correct Options

(A)

\frac{9}{32 R}

(B)

\frac{9}{16 R}

(C)

\frac{9}{5 R}

(D)

\frac{4}{3 R}

JEE Advanced 2000

LEVELJEE Main

Electrons with energy $80 keV$ are incident on the tungsten target of an X-ray tube. K-shell electrons of tungsten have $72.5 keV$ energy. X-rays emitted by the tube contain only

(A)

a continuous X-ray spectrum (Bremsstrahlung) with a minimum wavelength of

\approx 0.155 \overset{˚}{A}

(B)

a continuous X-ray spectrum (Bremsstrahlung) with all wavelengths

(C)

the characteristic X-ray spectrum of tungsten

(D)

a continuous X-ray spectrum (Bremsstrahlung) with a minimum wavelength of

\approx 0.155 \overset{˚}{A}

and the characteristic X-ray spectrum of tungsten

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } As per Bohr model, the minimum energy (in eV) required to remove an electron from the ground state of doubly ionized Li atom (Z=3) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Kα​ wavelength emitted by an atom of atomic number Z=11 is λ. Find the atomic number for an atom that emits Kα​ radiation with wavelength 4λ

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

Assertion If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic X-rays do not change. Reason When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

The largest wavelength in the ultraviolet region of the hydrogen spectrum is 122 nm. The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is

In the Bohr model of the hydrogen atoms

Which one of the following statements is wrong in the context of X-rays generated from an X-ray tube ?

The electron in a hydrogen atom makes a transition from an excited state to the ground state. Which of the following statement is true ?

The wavelength of the first spectral line in the Balmer series of hydrogen atom is 6561 \AA. The wavelength of the second spectral line in the Balmer series of singly ionized helium atom is

.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; } The Kα​ X-ray emission line of tungsten occurs at λ=0.021 nm. The energy difference between K and L levels in this atom is about

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Electrons with energy 80 keV are incident on the tungsten target of an X-ray tube. K-shell electrons of tungsten have 72.5 keV energy. X-rays emitted by the tube contain only

As per Bohr model, the minimum energy (in eV) required to remove an electron from the ground state of doubly ionized Li atom ( $Z = 3$ ) is

$K_{α}$ wavelength emitted by an atom of atomic number $Z = 11$ is $λ$ . Find the atomic number for an atom that emits $K_{α}$ radiation with wavelength $4 λ$

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

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

The $K_{α}$ X-ray emission line of tungsten occurs at $λ = 0.021 nm$ . The energy difference between $K$ and $L$ levels in this atom is about

Electrons with energy $80 keV$ are incident on the tungsten target of an X-ray tube. K-shell electrons of tungsten have $72.5 keV$ energy. X-rays emitted by the tube contain only