JEE Physics Atoms MCQs Set 18

Physics Objective Questions and Answers: Atoms

Access targeted multiple-choice questions for Atoms designed to align with the latest JEE academic syllabus for JEE Physics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.

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Multiple Choice Questions

Question 1: Highly excited states for hydrogen-like atoms (Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where \( n\gg1 \). Consider: (i) the relative change in the radii of two consecutive orbitals does not depend on Z; (ii) the relative change in the radii of two consecutive orbitals varies as 1/n; (iii) the relative change in the energy of two consecutive orbitals varies as \( 1/n^3 \); (iv) the relative change in the angular momenta of two consecutive orbitals varies as 1/n. The correct statements are
(a) (i), (ii) and (iv)
(b) (i) and (iii)
(c) (ii), (iii) and (iv)
(d) (i), (iii) and (iv)
Show Answer & Explanation

Answer: (a) (i), (ii) and (iv)

Explanation:
1. \( r_n\propto\dfrac{n^2}Z \): \( \dfrac{\Delta r}r\approx\dfrac2n \), independent of Z and varying as 1/n. Angular momentum \( \propto n \): \( \dfrac{\Delta L}L=\dfrac1n \).
2. \( E\propto\dfrac{Z^2}{n^2} \): \( \dfrac{\Delta E}E\approx\dfrac2n \), which varies as 1/n, not \( 1/n^3 \), so (iii) is false.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 2: The electron in a hydrogen atom makes a transition \( n_1\to n_2 \), where \( n_1 \) and \( n_2 \) are the principal quantum numbers of the two states. Assume Bohr's model. The time period of the electron in the initial state is eight times that in the final state. The possible values of \( n_1 \) and \( n_2 \) are
(a) \( n_1=8,\ n_2=2 \)
(b) \( n_1=8,\ n_2=1 \)
(c) \( n_1=4,\ n_2=2 \)
(d) \( n_1=6,\ n_2=2 \)
Show Answer & Explanation

Answer: (c) \( n_1=4,\ n_2=2 \)

Explanation:
1. Time period \( T\propto n^3 \).
2. \( \dfrac{T_1}{T_2}=\left(\dfrac{n_1}{n_2}\right)^3=8\Rightarrow n_1=2n_2 \). Only \( n_1=4,n_2=2 \) satisfies this among the options.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 3: In the Bohr model of the hydrogen atom, consider: (i) the radius of the nth orbit is proportional to \( n^2 \); (ii) the total energy of the electron in the nth orbit is inversely proportional to n; (iii) the angular momentum of the electron in an orbit is an integral multiple of \( h/\pi \); (iv) the magnitude of the potential energy of the electron in any orbit is greater than its kinetic energy. The correct statements are
(a) (ii) and (iii)
(b) (i) and (ii)
(c) (iii) and (iv)
(d) (i) and (iv)
Show Answer & Explanation

Answer: (d) (i) and (iv)

Explanation:
1. \( r_n\propto n^2 \) (i). \( |U|=2K \), so the potential energy magnitude exceeds the kinetic energy (iv).
2. \( E_n\propto\dfrac1{n^2} \) (ii false). Angular momentum is an integral multiple of \( \dfrac h{2\pi} \), not \( \dfrac h\pi \) (iii false).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 4: The recoil speed of a hydrogen atom after it emits a photon in going from the n = 5 state to the n = 1 state is (take the mass of the hydrogen atom \( 1.67\times10^{-27} \) kg)
(a) 41.7 m/s
(b) 4.17 m/s
(c) 0.417 m/s
(d) 417 m/s
Show Answer & Explanation

Answer: (b) 4.17 m/s

Explanation:
1. Photon energy \( E=13.6\left(1-\tfrac1{25}\right)=13.06 \) eV; momentum \( p=\dfrac Ec \).
2. \( v=\dfrac{E}{mc}=\dfrac{13.06\times1.6\times10^{-19}}{1.67\times10^{-27}\times3\times10^8}=4.17 \) m/s.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 5: In the Bohr model of the hydrogen atom, the ratio of the kinetic energy to the total energy of the electron in a quantum state n is
(a) 1
(b) -1/2
(c) -1
(d) 2
Show Answer & Explanation

Answer: (c) -1

Explanation:
1. Kinetic energy \( K=\dfrac{13.6}{n^2} \) and total energy \( E=-\dfrac{13.6}{n^2} \).
2. \( \dfrac KE=-1 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 6: A hydrogen atom in its ground state is irradiated by light of wavelength 970 \(Å\). Taking \( hc/e=1.237\times10^{-6} \) eV m and the ground state energy as -13.6 eV, the number of lines present in the emission spectrum is
(a) 6
(b) 3
(c) 10
(d) 4
Show Answer & Explanation

Answer: (a) 6

Explanation:
1. Photon energy \( =\dfrac{1.237\times10^{-6}}{970\times10^{-10}}=12.75 \) eV, so \( 13.6\left(1-\tfrac1{n^2}\right)=12.75\Rightarrow n=4 \).
2. Number of lines \( =\dfrac{n(n-1)}2=6 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 7: A hydrogen atom has its electron in the nth orbit. An electromagnetic radiation of wavelength 90 nm is used to ionise the atom. If the kinetic energy of the ejected electron is 10.4 eV, then n is (\( hc=1242 \) eV nm)
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer & Explanation

Answer: (b) 2

Explanation:
1. Photon energy \( =\dfrac{1242}{90}=13.8 \) eV.
2. Binding energy \( =13.8-10.4=3.4 \) eV \( =\dfrac{13.6}{n^2}\Rightarrow n=2 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 8: A hydrogen-like atom (described by the Bohr model) is observed to emit six wavelengths, originating from all possible transitions between a group of four levels. These levels have energies between -0.85 eV and -0.544 eV (including both). The atomic number of the atom is
(a) 2
(b) 4
(c) 5
(d) 3
Show Answer & Explanation

Answer: (d) 3

Explanation:
1. Six lines from four consecutive levels: \( -\dfrac{13.6Z^2}{n^2}=-0.85 \) and \( -0.544 \) for levels n and n + 3.
2. \( \dfrac{0.85}{0.544}=\dfrac{(n+3)^2}{n^2}=\dfrac{25}{16}\Rightarrow n=12 \), then \( Z^2=\dfrac{0.85\times144}{13.6}=9 \), so Z = 3.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 9: For the same hydrogen-like atom (Z = 3), whose four levels run from n = 12 to n = 15, the smallest wavelength emitted in these transitions is close to (take \( hc=1240 \) eV nm)
(a) 4052 nm
(b) 405 nm
(c) 1350 nm
(d) 12400 nm
Show Answer & Explanation

Answer: (a) 4052 nm

Explanation:
1. The smallest energy gap gives the longest wavelength; the largest gap between these levels, n = 15 to n = 12, gives the shortest wavelength of the group.
2. \( \Delta E=13.6\times9\left(\dfrac1{144}-\dfrac1{225}\right)=0.306 \) eV, \( \lambda=\dfrac{1240}{0.306}=4052 \) nm.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 10: For the hydrogen-like atom of the previous type (Z = 4, excited state 2n = 4), the minimum energy that can be emitted during de-excitation is
(a) 40.8 eV
(b) 204 eV
(c) 10.58 eV
(d) 3.4 eV
Show Answer & Explanation

Answer: (c) 10.58 eV

Explanation:
1. The smallest photon energy comes from the closest pair of levels: n = 4 to n = 3.
2. \( 13.6\times16\left(\dfrac19-\dfrac1{16}\right)=217.6\times\dfrac7{144}=10.58 \) eV.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 11: An electron in a hydrogen-like atom is in an excited state with total energy -3.4 eV. Its kinetic energy and the de-Broglie wavelength of the electron are (\( h=6.63\times10^{-34} \) J s, \( m=9.1\times10^{-31} \) kg)
(a) \( 6.8\ \text{eV and }4.7\ Å \)
(b) \( 3.4\ \text{eV and }3.3\ Å \)
(c) \( 1.7\ \text{eV and }9.3\ Å \)
(d) \( 3.4\ \text{eV and }6.6\ Å \)
Show Answer & Explanation

Answer: (d) \( 3.4\ \text{eV and }6.6\ Å \)

Explanation:
1. For a bound electron in a Coulomb field \( K=-E=3.4 \) eV.
2. \( \lambda=\dfrac{h}{\sqrt{2mK}}=\dfrac{6.63\times10^{-34}}{\sqrt{2\times9.1\times10^{-31}\times3.4\times1.6\times10^{-19}}}=6.6\times10^{-10} \) m \( =6.6\ Å \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 12: A particle of charge -e and mass 208 times the mass of the electron (a mu-meson) moves in a circular orbit around a nucleus of charge +3e. The mass of the nucleus is infinite and Bohr's model applies. The radius of the nth orbit, in terms of the Bohr radius \( a_0 \) of hydrogen, is
(a) \( \dfrac{n^2a_0}{208} \)
(b) \( \dfrac{n^2a_0}{624} \)
(c) \( \dfrac{n^2a_0}{3} \)
(d) \( 624\,n^2a_0 \)
Show Answer & Explanation

Answer: (b) \( \dfrac{n^2a_0}{624} \)

Explanation:
1. \( r_n=\dfrac{n^2h^2\varepsilon_0}{\pi mZe^2}\propto\dfrac{n^2}{mZ} \).
2. With \( m=208m_e \) and Z = 3: \( r_n=\dfrac{n^2a_0}{208\times3}=\dfrac{n^2a_0}{624} \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 13: For the mu-meson atom (orbiting charge -e, mass 208 \( m_e \), nucleus +3e), the wavelength of the radiation emitted when the mu-meson jumps from the third orbit to the first orbit is (Rydberg constant \( R=1.097\times10^7 \) m\(^{-1}\))
(a) \( 5.48\ Å \)
(b) \( 0.0548\ Å \)
(c) \( 0.548\ Å \)
(d) \( 1.65\ Å \)
Show Answer & Explanation

Answer: (c) \( 0.548\ Å \)

Explanation:
1. \( \dfrac1\lambda=R\times208\times Z^2\left(1-\dfrac19\right)=1.097\times10^7\times208\times9\times\dfrac89 \).
2. \( \dfrac1\lambda=1.097\times10^7\times1664=1.825\times10^{10} \) m\(^{-1}\), so \( \lambda=5.48\times10^{-11} \) m \( =0.548\ Å \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 14: A doubly ionised lithium atom (Li\(^{2+}\)) has its electron excited from the first to the third Bohr orbit. The number of spectral lines observed in the emission spectrum of this excited system is
(a) 3
(b) 2
(c) 4
(d) 6
Show Answer & Explanation

Answer: (a) 3

Explanation:
1. From n = 3 the possible transitions are 3 to 2, 3 to 1 and 2 to 1.
2. Number of lines \( =\dfrac{3\times2}2=3 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 15: The ionisation energy of a hydrogen-like Bohr atom is 4 rydberg (1 rydberg = 13.6 eV). The radius of the first orbit of this atom, in terms of the Bohr radius \( a_0=0.529\ Å \), is
(a) \( 0.529\ Å \)
(b) \( 0.2645\ Å \)
(c) \( 1.058\ Å \)
(d) \( 0.132\ Å \)
Show Answer & Explanation

Answer: (b) \( 0.2645\ Å \)

Explanation:
1. \( 13.6Z^2=4\times13.6\Rightarrow Z=2 \) (a He\(^+\) ion).
2. \( r_1=\dfrac{a_0}Z=\dfrac{0.529}{2}=0.2645\ Å \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 16: A hydrogen atom in its ground state is excited by monochromatic radiation of wavelength 975 \(Å\). The number of different lines possible in the resulting emission spectrum is
(a) 3
(b) 10
(c) 4
(d) 6
Show Answer & Explanation

Answer: (d) 6

Explanation:
1. Photon energy \( =\dfrac{12400}{975}=12.7 \) eV, so \( 13.6\left(1-\tfrac1{n^2}\right)=12.7\Rightarrow n=4 \).
2. Number of lines \( =\dfrac{4\times3}2=6 \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 17: For the same excited hydrogen atom (excited to n = 4 by 975 \(Å\) radiation), the longest wavelength in the emission spectrum is (\( R=1.097\times10^7 \) m\(^{-1}\))
(a) \( 1.875\ \mu\text m \)
(b) \( 0.656\ \mu\text m \)
(c) \( 0.486\ \mu\text m \)
(d) \( 4.05\ \mu\text m \)
Show Answer & Explanation

Answer: (a) \( 1.875\ \mu\text m \)

Explanation:
1. The longest wavelength comes from the smallest energy gap, n = 4 to n = 3.
2. \( \dfrac1\lambda=R\left(\dfrac19-\dfrac1{16}\right)=\dfrac{7R}{144}\Rightarrow\lambda=1.875\times10^{-6} \) m.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 18: For the same hydrogen-like ion (Z = 5), the energy required to excite the electron from the third Bohr orbit to the fourth Bohr orbit is
(a) 47.2 eV
(b) 8.2 eV
(c) 16.53 eV
(d) 31.0 eV
Show Answer & Explanation

Answer: (c) 16.53 eV

Explanation:
1. \( \Delta E=13.6\times25\left(\dfrac19-\dfrac1{16}\right) \).
2. \( =340\times\dfrac7{144}=16.53 \) eV.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 19: For the same ion (Z = 5), the wavelength of the radiation required to remove the electron from the first Bohr orbit to infinity is (\( hc=1240 \) eV nm)
(a) 3.64 \( Å \)
(b) 364 \( Å \)
(c) 1.0 \( Å \)
(d) 36.4 \( Å \)
Show Answer & Explanation

Answer: (d) 36.4 \( Å \)

Explanation:
1. Ionisation energy of the first orbit \( =13.6\times25=340 \) eV.
2. \( \lambda=\dfrac{1240}{340}=3.64 \) nm \( =36.4\ Å \).

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

Question 20: For the same hydrogen-like ion (Z = 5), the radius of the first Bohr orbit is (Bohr radius \( 5.3\times10^{-11} \) m)
(a) \( 1.06\times10^{-11}\ \text m \)
(b) \( 5.3\times10^{-11}\ \text m \)
(c) \( 2.65\times10^{-10}\ \text m \)
(d) \( 2.1\times10^{-11}\ \text m \)
Show Answer & Explanation

Answer: (a) \( 1.06\times10^{-11}\ \text m \)

Explanation:
1. \( r_1=\dfrac{a_0}Z \).
2. \( =\dfrac{5.3\times10^{-11}}{5}=1.06\times10^{-11} \) m.

Teacher's Note:
1. Write the energy or radius formula for a hydrogen-like atom, with the correct Z, before comparing levels.
2. Check the sign and the order of magnitude: energies of bound levels are negative and shrink as n increases.

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