All About Atoms Problems

This section provides 100 problems to test your understanding of atomic physics, including calculations of Bohr model energy levels, radii, spectral wavelengths, quantum numbers, electron configurations, and spectral line energies, as well as applications like atomic spectroscopy in spacecraft navigation. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering atomic physics, a key topic for JEE/NEET success.

Numerical Problems

1. An electron in a hydrogen atom is in the $n=2$ state. Calculate its energy in eV.

   • (a) -3.39 eV
   • (b) -3.40 eV
   • (c) -3.41 eV
   • (d) -3.42 eV

2. A hydrogen atom undergoes a transition from $n=3$ to $n=2$. Calculate the wavelength of the emitted photon in nm.

   • (a) 655 nm
   • (b) 656 nm
   • (c) 657 nm
   • (d) 658 nm

3. Calculate the radius of the $n=4$ orbit in a hydrogen atom in Å.

   • (a) 8.45 Å
   • (b) 8.46 Å
   • (c) 8.47 Å
   • (d) 8.48 Å

4. A hydrogen atom transitions from $n=4$ to $n=1$. Calculate the energy of the emitted photon in eV.

   • (a) 12.74 eV
   • (b) 12.75 eV
   • (c) 12.76 eV
   • (d) 12.77 eV

5. Find the shortest wavelength in the Balmer series for hydrogen in nm.

   • (a) 364.5 nm
   • (b) 364.6 nm
   • (c) 364.7 nm
   • (d) 364.8 nm

6. An alpha particle ($q_1=2e$) scatters off a gold nucleus ($Z=79$) with impact parameter $b=2\times {10}^{-14}\text{m}$ and velocity $v=1\times {10}^7\text{m/s}$. Calculate the scattering angle $\theta$ in degrees.

   • (a) 11.5°
   • (b) 11.6°
   • (c) 11.7°
   • (d) 11.8°

7. A hydrogen atom electron transitions from $n=5$ to $n=2$. Calculate the wavelength in nm.

   • (a) 434.0 nm
   • (b) 434.1 nm
   • (c) 434.2 nm
   • (d) 434.3 nm

8. Calculate the energy difference between $n=3$ and $n=1$ in a hydrogen atom in eV.

   • (a) 12.08 eV
   • (b) 12.09 eV
   • (c) 12.10 eV
   • (d) 12.11 eV

9. Find the radius of the $n=2$ orbit in a hydrogen atom in Å.

   • (a) 2.11 Å
   • (b) 2.12 Å
   • (c) 2.13 Å
   • (d) 2.14 Å

10. A hydrogen atom emits a photon with $\lambda =121.6\text{nm}$. Calculate the energy of the photon in eV.

    • (a) 10.19 eV
    • (b) 10.20 eV
    • (c) 10.21 eV
    • (d) 10.22 eV

11. Calculate the frequency of a photon emitted during a transition from $n=3$ to $n=1$ in a hydrogen atom in Hz.

    • (a) 2.92 $\times {10}^{15}$
    • (b) 2.93 $\times {10}^{15}$
    • (c) 2.94 $\times {10}^{15}$
    • (d) 2.95 $\times {10}^{15}$

12. A hydrogen atom electron is in the $n=5$ state. Calculate its energy in eV.

    • (a) -0.543 eV
    • (b) -0.544 eV
    • (c) -0.545 eV
    • (d) -0.546 eV

13. Find the longest wavelength in the Lyman series for hydrogen in nm.

    • (a) 121.5 nm
    • (b) 121.6 nm
    • (c) 121.7 nm
    • (d) 121.8 nm

14. Calculate the radius of the $n=3$ orbit in a hydrogen atom in Å.

    • (a) 4.75 Å
    • (b) 4.76 Å
    • (c) 4.77 Å
    • (d) 4.78 Å

15. A hydrogen atom transitions from $n=4$ to $n=3$. Calculate the wavelength in nm.

    • (a) 1875 nm
    • (b) 1876 nm
    • (c) 1877 nm
    • (d) 1878 nm

16. An electron in a hydrogen atom is in the $n=1$ state. Calculate its energy in Joules.

    • (a) -2.17 $\times {10}^{-18}$
    • (b) -2.18 $\times {10}^{-18}$
    • (c) -2.19 $\times {10}^{-18}$
    • (d) -2.20 $\times {10}^{-18}$

17. Find the shortest wavelength in the Paschen series for hydrogen in nm.

    • (a) 820.2 nm
    • (b) 820.3 nm
    • (c) 820.4 nm
    • (d) 820.5 nm

18. A hydrogen atom emits a photon with energy 1.89 eV. Calculate the wavelength in nm.

    • (a) 655 nm
    • (b) 656 nm
    • (c) 657 nm
    • (d) 658 nm

19. Calculate the energy difference between $n=4$ and $n=2$ in a hydrogen atom in eV.

    • (a) 2.54 eV
    • (b) 2.55 eV
    • (c) 2.56 eV
    • (d) 2.57 eV

20. A hydrogen atom electron transitions from $n=6$ to $n=2$. Calculate the wavelength in nm.

    • (a) 410.1 nm
    • (b) 410.2 nm
    • (c) 410.3 nm
    • (d) 410.4 nm

21. Find the radius of the $n=1$ orbit in a hydrogen atom in Å.

    • (a) 0.528 Å
    • (b) 0.529 Å
    • (c) 0.530 Å
    • (d) 0.531 Å

22. A hydrogen atom transitions from $n=5$ to $n=1$. Calculate the energy of the emitted photon in eV.

    • (a) 13.05 eV
    • (b) 13.06 eV
    • (c) 13.07 eV
    • (d) 13.08 eV

23. Calculate the frequency of a photon emitted during a transition from $n=2$ to $n=1$ in a hydrogen atom in Hz.

    • (a) 2.46 $\times {10}^{15}$
    • (b) 2.47 $\times {10}^{15}$
    • (c) 2.48 $\times {10}^{15}$
    • (d) 2.49 $\times {10}^{15}$

24. A hydrogen atom electron is in the $n=3$ state. Calculate its energy in eV.

    • (a) -1.50 eV
    • (b) -1.51 eV
    • (c) -1.52 eV
    • (d) -1.53 eV

25. Find the longest wavelength in the Balmer series for hydrogen in nm.

    • (a) 655 nm
    • (b) 656 nm
    • (c) 657 nm
    • (d) 658 nm

26. A hydrogen atom transitions from $n=3$ to $n=1$. Calculate the wavelength in nm.

    • (a) 102.5 nm
    • (b) 102.6 nm
    • (c) 102.7 nm
    • (d) 102.8 nm

27. Calculate the energy difference between $n=5$ and $n=3$ in a hydrogen atom in eV.

    • (a) 1.09 eV
    • (b) 1.10 eV
    • (c) 1.11 eV
    • (d) 1.12 eV

28. A hydrogen atom emits a photon with $\lambda =486.1\text{nm}$. Calculate the energy of the photon in eV.

    • (a) 2.54 eV
    • (b) 2.55 eV
    • (c) 2.56 eV
    • (d) 2.57 eV

29. Find the radius of the $n=6$ orbit in a hydrogen atom in Å.

    • (a) 19.03 Å
    • (b) 19.04 Å
    • (c) 19.05 Å
    • (d) 19.06 Å

30. A hydrogen atom transitions from $n=6$ to $n=3$. Calculate the wavelength in nm.

    • (a) 1093 nm
    • (b) 1094 nm
    • (c) 1095 nm
    • (d) 1096 nm

31. A spacecraft sensor detects a hydrogen line at $\lambda =656.3\text{nm}$. Calculate the energy of the photon in eV.

    • (a) 1.88 eV
    • (b) 1.89 eV
    • (c) 1.90 eV
    • (d) 1.91 eV

32. A hydrogen atom electron is in the $n=4$ state. Calculate its energy in Joules.

    • (a) -1.36 $\times {10}^{-19}$
    • (b) -1.37 $\times {10}^{-19}$
    • (c) -1.38 $\times {10}^{-19}$
    • (d) -1.39 $\times {10}^{-19}$

33. Find the shortest wavelength in the Lyman series for hydrogen in nm.

    • (a) 91.1 nm
    • (b) 91.2 nm
    • (c) 91.3 nm
    • (d) 91.4 nm

34. A hydrogen atom transitions from $n=5$ to $n=4$. Calculate the wavelength in nm.

    • (a) 4050 nm
    • (b) 4051 nm
    • (c) 4052 nm
    • (d) 4053 nm

35. Calculate the energy difference between $n=6$ and $n=2$ in a hydrogen atom in eV.

    • (a) 3.01 eV
    • (b) 3.02 eV
    • (c) 3.03 eV
    • (d) 3.04 eV

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Conceptual Problems

36. What did Thomson’s plum pudding model propose about the atom?

    • (a) Electrons orbit a nucleus
    • (b) Electrons are embedded in a positive sphere
    • (c) Atoms have a dense nucleus
    • (d) Atoms are indivisible

37. What did Rutherford’s gold foil experiment discover?

    • (a) Electrons are negatively charged
    • (b) Atoms have a small, dense nucleus
    • (c) Atoms are mostly solid
    • (d) Electrons are in fixed orbits

38. What is the unit of energy $E_n$ in the Bohr model?

    • (a) eV
    • (b) Radian
    • (c) Hertz
    • (d) Watt

39. What happens to an electron in the Bohr model during a transition from $n=3$ to $n=2$?

    • (a) Absorbs a photon
    • (b) Emits a photon
    • (c) Remains in the same orbit
    • (d) Gains energy

40. What does the principal quantum number $n$ determine in the quantum mechanical model?

    • (a) Orbital shape
    • (b) Orbital orientation
    • (c) Energy level
    • (d) Electron spin

41. What is the unit of the Rydberg constant $R$?

    • (a) m${}^{-1}$
    • (b) Joule
    • (c) Hertz
    • (d) eV

42. What does a larger $n$ in the Bohr model indicate?

    • (a) Lower energy
    • (b) Higher energy (less negative)
    • (c) No energy change
    • (d) Zero energy

43. What happens to the radius $r_n$ in the Bohr model as $n$ increases?

    • (a) Decreases
    • (b) Increases as $n^2$
    • (c) Remains the same
    • (d) Becomes zero

44. What does an emission spectrum show?

    • (a) Continuous spectrum
    • (b) Dark lines on a bright background
    • (c) Bright lines on a dark background
    • (d) No lines

45. What is the dimension of the Bohr radius $a_0$?

    • (a) $[\text{L}]$
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

46. What does the azimuthal quantum number $l$ determine?

    • (a) Energy level
    • (b) Orbital shape
    • (c) Orbital orientation
    • (d) Electron spin

47. What is the significance of $\frac{1}{\lambda }=R(\frac{1}{n_1^2}-\frac{1}{n_2^2})$?

    • (a) Energy level
    • (b) Rydberg formula for spectral lines
    • (c) Orbital shape
    • (d) Electron configuration

48. What happens to the energy $E_n$ in the Bohr model as $n$ increases?

    • (a) Becomes more negative
    • (b) Becomes less negative
    • (c) Remains the same
    • (d) Becomes zero

49. What does the Pauli exclusion principle state?

    • (a) Electrons fill orbitals from lowest to highest energy
    • (b) No two electrons can have the same four quantum numbers
    • (c) Electrons pair with opposite spins
    • (d) Orbitals fill to maximize unpaired electrons

50. How does atomic spectroscopy assist in spacecraft navigation?

    • (a) Increases energy
    • (b) Identifies elements in stars via spectral lines
    • (c) Reduces momentum
    • (d) Increases wavelength

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Derivation Problems

51. Derive the energy levels $E_n=-\frac{13.6}{n^2}\text{eV}$ in the Bohr model.

52. Derive the radius $r_n=n^2{a}_0$ in the Bohr model.

53. Derive the Rydberg formula $\frac{1}{\lambda }=R(\frac{1}{n_1^2}-\frac{1}{n_2^2})$.

54. Derive the scattering angle in Rutherford’s model: $\tan \left(\frac{\theta }{2}\right)=\frac{k{q}_1{q}_2}{m{v}^{2}b}$.

55. Derive the 1s orbital wave function ${\psi }_{1,0,0}(r)\propto e^{-r/a_0}$.

56. Derive the energy of a photon emitted during a transition from $n_2$ to $n_1$ in the Bohr model.

57. Derive the radius $r_1$ for the $n=1$ orbit in the Bohr model.

58. Derive the wavelength of a spectral line for a given transition in hydrogen.

59. Derive the maximum number of electrons in a subshell with quantum number $l$.

60. Derive the energy difference $\mathrm{\Delta }E$ for a transition in the Bohr model.

61. Derive the probability density $|{\psi }_{1,0,0}{|}^2$ for the 1s orbital.

62. Derive the frequency of a photon emitted during a transition in the Bohr model.

63. Derive the electron configuration of an atom using quantum numbers.

64. Derive the shortest wavelength in a spectral series for hydrogen.

65. Derive the scattering angle $\theta$ for a given impact parameter in Rutherford’s model.

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NEET-style Conceptual Problems

66. What is the unit of wavelength $\lambda$ in the Rydberg formula?

    • (a) Meter
    • (b) Radian
    • (c) Hertz
    • (d) Watt

67. What does a transition from $n=3$ to $n=1$ in hydrogen produce?

    • (a) Absorbed photon
    • (b) Emitted photon
    • (c) No photon
    • (d) Continuous spectrum

68. What is the relationship between $E_n$ and $n$ in the Bohr model?

    • (a) $E_n\propto n$
    • (b) $E_n\propto \frac{1}{n^2}$
    • (c) $E_n$ is independent of $n$
    • (d) $E_n\propto n^2$

69. What happens to the radius $r_n$ in the Bohr model if $n$ decreases?

    • (a) Increases
    • (b) Decreases
    • (c) Remains the same
    • (d) Becomes zero

70. What is the dimension of energy $E_n$ in the Bohr model?

    • (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}]$
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

71. What does the magnetic quantum number $m_l$ determine?

    • (a) Energy level
    • (b) Orbital shape
    • (c) Orbital orientation
    • (d) Electron spin

72. What is the role of atomic spectra in lasers?

    • (a) Increases energy
    • (b) Uses stimulated emission for light amplification
    • (c) Reduces momentum
    • (d) Increases wavelength

73. What happens to the energy levels in the Bohr model as $n$ approaches infinity?

    • (a) Become more negative
    • (b) Approach zero
    • (c) Remain the same
    • (d) Become infinite

74. Why did Rutherford’s model fail to explain atomic stability?

    • (a) Due to quantized energy levels
    • (b) Due to electron radiation and energy loss
    • (c) Due to nuclear size
    • (d) Due to electron spin

75. What is the unit of the Bohr radius $a_0$?

    • (a) Å
    • (b) Joule
    • (c) Hertz
    • (d) Watt

76. What does a bright line in an emission spectrum indicate?

    • (a) Electron absorption
    • (b) Electron emission
    • (c) No transition
    • (d) Continuous energy

77. Which quantum number determines the shape of an orbital?

    • (a) $n$
    • (b) $l$
    • (c) $m_l$
    • (d) $m_s$

78. What is the effect of the Pauli exclusion principle on electron configurations?

    • (a) Allows identical quantum numbers
    • (b) Prevents identical quantum numbers
    • (c) Increases energy levels
    • (d) Reduces energy levels

79. What does a pseudo-force do in a non-inertial frame for atomic calculations?

    • (a) Affects perceived energy levels
    • (b) Affects orbital shapes
    • (c) Creates spectra
    • (d) Reduces momentum

80. What is the dimension of $\frac{k{e}^2}{r}$ in the Bohr model?

    • (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}]$
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

81. What is the role of atomic spectra in spacecraft navigation?

    • (a) Increases energy
    • (b) Identifies elements in stars via spectral lines
    • (c) Reduces momentum
    • (d) Increases wavelength

82. What happens to the 1s orbital probability density as $r$ increases?

    • (a) Increases
    • (b) Decreases exponentially
    • (c) Remains the same
    • (d) Becomes zero

83. Why does the Bohr model apply only to hydrogen-like atoms?

    • (a) Due to multiple electrons
    • (b) Due to single electron-nucleus interaction
    • (c) Due to orbital shapes
    • (d) Due to electron spin

84. What is the significance of $13.6\text{eV}$ in the Bohr model?

    • (a) Ionization energy of hydrogen
    • (b) Orbital radius
    • (c) Spectral wavelength
    • (d) Electron spin

85. What is the unit of quantum number $n$?

    • (a) Dimensionless
    • (b) Meter
    • (c) Joule
    • (d) kg·m/s

86. What does a high energy difference $\mathrm{\Delta }E$ in a transition indicate?

    • (a) Low frequency photon
    • (b) High frequency photon
    • (c) No photon
    • (d) Constant frequency

87. What is the physical significance of $e^{-r/a_0}$?

    • (a) Energy level
    • (b) Radial decay in 1s orbital
    • (c) Spectral line
    • (d) Electron configuration

88. Why does the quantum mechanical model use orbitals instead of orbits?

    • (a) Due to fixed paths
    • (b) Due to probability distributions
    • (c) Due to nuclear size
    • (d) Due to electron spin

89. What is the dimension of $\frac{1}{\lambda }$ in the Rydberg formula?

    • (a) $[{\text{L}}^{-1}]$
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

90. How does the Aufbau principle determine electron configurations?

    • (a) Electrons fill orbitals randomly
    • (b) Electrons fill from lowest to highest energy
    • (c) Electrons pair with same spins
    • (d) Electrons avoid pairing

91. What is the role of the spin quantum number $m_s$?

    • (a) Determines energy level
    • (b) Determines orbital shape
    • (c) Determines orbital orientation
    • (d) Determines electron spin

92. What does a dark line in an absorption spectrum indicate?

    • (a) Electron emission
    • (b) Electron absorption
    • (c) No transition
    • (d) Continuous energy

93. What is the physical significance of $n\hslash$?

    • (a) Orbital radius
    • (b) Quantized angular momentum
    • (c) Spectral line
    • (d) Electron configuration

94. What is the dimension of $h\nu$ in atomic spectra?

    • (a) $[\text{M}{\text{L}}^2{\text{T}}^{-2}]$
    • (b) $[\text{M}\text{L}{\text{T}}^{-1}]$
    • (c) $[\text{L}{\text{T}}^{-2}]$
    • (d) $[\text{M}{\text{L}}^2{\text{T}}^{-1}]$

95. Why does the quantum mechanical model use four quantum numbers?

    • (a) To describe orbital shapes
    • (b) To uniquely identify each electron
    • (c) To determine energy levels only
    • (d) To determine spectral lines

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NEET-style Numerical Problems

96. An electron in a hydrogen atom is in the $n=3$ state. Calculate its energy in eV.

    • (a) -1.50 eV
    • (b) -1.51 eV
    • (c) -1.52 eV
    • (d) -1.53 eV

97. A hydrogen atom transitions from $n=4$ to $n=2$. Calculate the wavelength in nm.

    • (a) 486.0 nm
    • (b) 486.1 nm
    • (c) 486.2 nm
    • (d) 486.3 nm

98. Calculate the radius of the $n=5$ orbit in a hydrogen atom in Å.

    • (a) 13.22 Å
    • (b) 13.23 Å
    • (c) 13.24 Å
    • (d) 13.25 Å

99. A hydrogen atom emits a photon with $\lambda =102.6\text{nm}$. Calculate the energy in eV.

    • (a) 12.08 eV
    • (b) 12.09 eV
    • (c) 12.10 eV
    • (d) 12.11 eV

100. Find the energy difference between $n=4$ and $n=1$ in a hydrogen atom in eV.
     - (a) 12.74 eV
     - (b) 12.75 eV
     - (c) 12.76 eV
     - (d) 12.77 eV

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