Atomic Structure | Physics JEE previous year questions with Solutions

Q 21 :

Consider the Bohr model of hydrogen like atoms, the ratio of the radius of $5^{th}$ orbit of the electron in ${Li}^{2 +}$ and ${He}^{+}$ is [2025]

$\frac{3}{2}$
$\frac{4}{9}$
$\frac{9}{4}$
$\frac{2}{3}$

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(4)

$r = a_{0} \cdot \frac{n^{2}}{Z}$

For ${Li}^{2 +}$ : $r_{5} = a_{0} \cdot \frac{25}{3}$

for ${He}^{+}$ : $r_{5} = a_{0} \cdot \frac{25}{2}$

$∴ \frac{r_{{Li}^{2 +}}}{r_{{He}^{+}}} = \frac{2}{3}$

Q 22 :

Given below are two statements:

Statement (I) : The dimensions of Planck's constant and angular momentum are same.

Statement (II) : In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.

In the light of the above statements, choose the most appropriate answer from the options given below: [2025]

Both Statement I and Statement II are correct.
Statement I is incorrect but Statement II is correct.
Statement I is correct but Statement II is incorrect.
Both Statement I and Statement II are incorrect.

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(3)

E = hf

$M L^{2} T^{- 2} = [h] \times [T^{- 1}]$

$[h] = [M L^{2} T^{- 1}]$

$L = [M V R] = [M L^{2} T^{- 1}]$

$L = \frac{n h}{2 π}$

L is integral multiple of $\frac{h}{2 π}$

Q 23 :

In a hydrogen like ion, the energy difference between the $2^{nd}$ excitation energy state and ground is 108.8 eV. The atomic number of the ion is [2025]

4
2
1
3

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4

(4)

$△ E = 13.6 Z^{2} [\frac{1}{n_{1}^{2}} - \frac{1}{n_{2}^{2}}]$

$13.6 Z^{2} [\frac{1}{1} - \frac{1}{9}] = 108.8$

$\frac{13.6 \times 8}{9} (Z^{2}) = 108.8 \Rightarrow Z = 3$

Q 24 :

An electron in the hydrogen atom initially in the fourth excited state makes a transition to n energy state by emitting a photon of energy 2.86 eV. The integer value of n will be ________. [2025]

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(2)

$E = 13.6 (\frac{1}{n_{1}^{2}} - \frac{1}{n_{1}^{2}})$

$2.86 = 13.6 (\frac{1}{n^{2}} - \frac{1}{5^{2}})$

$\frac{1}{n^{2}} = 0.21 + \frac{1}{25}$

$n^{2} = 4 \Rightarrow n = 2$

Q 25 :

A photon is emitted in transition from $n = 4$ to $n = 1$ level in hydrogen atom. The corresponding wavelength for this transition is (given, $h = 4 \times 10^{- 15}$ eV s) [2023]

941 nm
974 nm
99.3 nm
94.1 nm

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(4)

$\frac{h c}{λ} = + 13.6 eV [\frac{1}{1} - \frac{1}{4^{2}}]$

$\Rightarrow \frac{4 \times 10^{- 15} \times 3 \times 10^{- 8}}{λ} = 13.6 [\frac{15}{16}]$

$So, λ = 94.1 nm$

Q 26 :

Speed of an electron in Bohr's $7^{th}$ orbit for Hydrogen atom is $3.6 \times 10^{6} m/s.$ The corresponding speed of the electron in $3^{rd}$ orbit, in m/s is [2023]

$1.8 \times 10^{6}$
$7.5 \times 10^{6}$
$3.6 \times 10^{6}$
$8.4 \times 10^{6}$

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2

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(4)

$V_{n} \propto \frac{Z}{n}$

$z = 1$

$∴$ $V_{n} \propto \frac{1}{n}$

$∴$ $\frac{V_{3}}{V_{7}} = \frac{7}{3}$

$∴$ $V_{3} = \frac{7}{3} V_{7} = \frac{7}{3} \times 3.6 \times 10^{6} m/s = 8.4 \times 10^{6} m/s$

Q 27 :

The radius of electron’s second stationary orbit in Bohr’s atom is R. The radius of 3rd orbit will be [2023]

$\frac{R}{3}$
$2.25 R$
$3 R$
$9 R$

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(2)

$r \propto \frac{n^{2}}{Z}$

$\frac{r_{2 nd}}{r_{3 rd}} = {(\frac{n_{2}}{n_{3}})}^{2}$

$\Rightarrow \frac{R}{r_{3 rd}} = {(\frac{2}{3})}^{2}$

$\Rightarrow r_{3 rd} = \frac{9}{4} R = 2.25 R$

Q 28 :

A small particle of mass $m$ moves in such a way that its potential energy $U = \frac{1}{2} m ω^{2} r^{2}$ where $ω$ is constant and $r$ is the distance of the particle from origin. Assuming Bohr's quantization of momentum and circular orbit, the radius of $n^{th}$ orbit will be proportional to. [2023]

$\sqrt{n}$
$n$
$n^{2}$
$\frac{1}{n}$

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(1)

$U = \frac{1}{2} m ω^{2} r^{2}$

$F = - \frac{d U}{d r} = - m ω^{2} r$

$Now m ω^{2} r = \frac{m v^{2}}{r} \Rightarrow v = ω r \dots (i)$

$and m v r = \frac{n h}{2 π} \dots (i i)$

$From (i) and (ii),$

$m ω r^{2} = \frac{n h}{2 π} \Rightarrow r \propto \sqrt{n}$

Q 29 :

A light of energy 12.75 eV is incident on a hydrogen atom in its ground state. The atom absorbs the radiation and reaches to one of its excited states. The angular momentum of the atom in the excited state is $\frac{x}{π} \times 10^{- 17} eV s$ . The value of $x$ is _________

(use $h = 4.14 \times 10^{- 15} eVs, c = 3 \times 10^{8} {ms}^{- 1}$ ) [2023]

0%

(828)

$In the ground state energy = - 13.6 eV$

$So energy \frac{- 13.6 eV}{n^{2}} = - 13.6 + 12.75$

$\frac{- 13.6 eV}{n^{2}} = - 0.85$

$n = \sqrt{16} = 4$

$Angular momentum = \frac{n h}{2 π} = \frac{4 h}{2 π} = \frac{2 h}{π}$

$Angular momentum = \frac{2}{π} \times 4.14 \times 10^{- 15}$ $= \frac{828 \times 10^{- 17}}{π} eVs$

Q 30 :

Experimentally it is found that 12.8 eV energy is required to separate a hydrogen atom into a proton and an electron. So the orbital radius of the electron in a hydrogen atom is $\frac{9}{x} \times 10^{- 10} m$ . The value of the $x$ is _________.

$(1 eV = 1.6 \times 10^{- 19} J, \frac{1}{4 π ε_{0}} = 9 \times 10^{9} {Nm}^{2} / C^{2}$ and electronic charge $= 1.6 \times 10^{- 19} J C)$ [2023]

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(16)

$Binding energy of system = \frac{k e^{2}}{2 r} joule and \frac{k e^{2}}{2 r} = 12.8 eV$

$\frac{9 \times 10^{9} \times {(1.6 \times 10^{- 19})}^{2}}{2 r} = 12.8 \times 1.6 \times 10^{- 19}$

$\Rightarrow r = \frac{9 \times 10^{9} \times 1.6 \times 10^{- 19}}{12.8 \times 2}$

$\Rightarrow r = \frac{9 \times 10^{- 10}}{16}$

Topic Question Set

Q 21 :

Consider the Bohr model of hydrogen like atoms, the ratio of the radius of $5^{th}$ orbit of the electron in ${Li}^{2 +}$ and ${He}^{+}$ is [2025]

Q 23 :

In a hydrogen like ion, the energy difference between the $2^{nd}$ excitation energy state and ground is 108.8 eV. The atomic number of the ion is [2025]

Q 24 :

An electron in the hydrogen atom initially in the fourth excited state makes a transition to n energy state by emitting a photon of energy 2.86 eV. The integer value of n will be ________. [2025]

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Q 25 :

A photon is emitted in transition from $n = 4$ to $n = 1$ level in hydrogen atom. The corresponding wavelength for this transition is (given, $h = 4 \times 10^{- 15}$ eV s) [2023]

Q 26 :

Speed of an electron in Bohr's $7^{th}$ orbit for Hydrogen atom is $3.6 \times 10^{6} m/s.$ The corresponding speed of the electron in $3^{rd}$ orbit, in m/s is [2023]

Q 27 :

The radius of electron’s second stationary orbit in Bohr’s atom is R. The radius of 3rd orbit will be [2023]

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