Structure of Atom Chapter-Wise Test 10

Correct answer Carries: 4.

Wrong Answer Carries: -1.

The number of spectral lines produced when an electron falls from \( n = 9 \) to \( n = 6 \) in a hydrogen atom is:

Number of spectral lines = \( \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2} = \frac{(9 - 6)(9 - 6 + 1)}{2} = 6 \). Transitions: 9→6, 9→7, 9→8, 8→6, 8→7, 7→6.

A. 4
B. 6
C. 3
D. 5
2

The energy required to excite an electron from \( n = 3 \) to \( n = 6 \) in a hydrogen atom is \( 1.8164 \times 10^{-19} \, \text{J} \). What is its frequency? (\( h = 6.626 \times 10^{-34} \, \text{J s} \))

\( v = \frac{\Delta E}{h} = \frac{1.8164 \times 10^{-19}}{6.626 \times 10^{-34}} = 2.741 \times 10^{14} \, \text{Hz} \).

\( 4.112 \times 10^{14} \, \text{Hz} \)
\( 2.741 \times 10^{14} \, \text{Hz} \)
\( 1.371 \times 10^{14} \, \text{Hz} \)
\( 5.482 \times 10^{14} \, \text{Hz} \)
2

The energy of an electron in the first orbit of \( \text{He}^+ \) is \( -54.4 \, \text{eV} \). What is the wavelength of the photon emitted when the electron transitions from \( n = 4 \) to \( n = 2 \) in \( \text{He}^+ \)? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( c = 3.0 \times 10^8 \, \text{m s}^{-1} \), \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \))

For \( \text{He}^+ \) (Z = 2), \( E_n = -54.4 / n^2 \). \( E_2 = -54.4 / 4 = -13.6 \, \text{eV} \), \( E_4 = -54.4 / 16 = -3.4 \, \text{eV} \). \( \Delta E = -3.4 - (-13.6) = 10.2 \, \text{eV} = 1.632 \times 10^{-18} \, \text{J} \). \( \lambda = \frac{hc}{\Delta E} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{1.632 \times 10^{-18}} = 1.217 \times 10^{-7} \, \text{m} = 121.7 \, \text{nm} \).

\( 121.7 \, \text{nm} \)
\( 486.1 \, \text{nm} \)
\( 30.4 \, \text{nm} \)
\( 164.0 \, \text{nm} \)
1

The energy of an electron in the first orbit of a hydrogen atom is \( -2.18 \times 10^{-18} \, \text{J} \). What is the wavelength of the photon emitted when an electron in \( \text{Be}^{3+} \) transitions from \( n = 6 \) to \( n = 3 \)? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( c = 3.0 \times 10^8 \, \text{m s}^{-1} \))

For \( \text{Be}^{3+} \) (Z = 4), \( E_n = -Z^2 \times 2.18 \times 10^{-18} / n^2 \). \( E_3 = -16 \times 2.18 \times 10^{-18} / 9 = -3.8711 \times 10^{-18} \, \text{J} \), \( E_6 = -16 \times 2.18 \times 10^{-18} / 36 = -9.6778 \times 10^{-19} \, \text{J} \). \( \Delta E = -9.6778 \times 10^{-19} - (-3.8711 \times 10^{-18}) = 2.9033 \times 10^{-18} \, \text{J} \). \( \lambda = \frac{hc}{\Delta E} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{2.9033 \times 10^{-18}} = 6.845 \times 10^{-8} \, \text{m} = 68.45 \, \text{nm} \).

\( 68.45 \, \text{nm} \)
\( 136.9 \, \text{nm} \)
\( 34.23 \, \text{nm} \)
\( 91.27 \, \text{nm} \)
1

The ionization energy of \( \text{Li}^{2+} \) is \( 122.4 \, \text{eV} \). What is the energy required to excite an electron from \( n = 3 \) to \( n = 6 \)? (\( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \))

For \( \text{Li}^{2+} \) (Z = 3), \( E_n = -122.4 / n^2 \). \( E_3 = -122.4 / 9 = -13.6 \, \text{eV} \), \( E_6 = -122.4 / 36 = -3.4 \, \text{eV} \). \( \Delta E = -3.4 - (-13.6) = 10.2 \, \text{eV} = 1.632 \times 10^{-18} \, \text{J} \).

\( 13.6 \, \text{eV} \)
\( 6.8 \, \text{eV} \)
\( 10.2 \, \text{eV} \)
\( 40.8 \, \text{eV} \)
3

Which of the following transitions in a hydrogen atom emits light of the shortest wavelength?

Shortest wavelength (highest energy) occurs for the largest \( \Delta E \). \( n = 5 \) to \( n = 1 \) has the largest energy difference.

\( n = 3 \) to \( n = 2 \)
\( n = 5 \) to \( n = 1 \)
\( n = 4 \) to \( n = 3 \)
\( n = 2 \) to \( n = 1 \)
2

What is the wavelength of light emitted when an electron in a hydrogen atom falls from \( n = 3 \) to \( n = 1 \)? (\( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \), \( c = 3.0 \times 10^8 \, \text{m s}^{-1} \))

Wavenumber \( \bar{v} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) = 1.097 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = 9.753 \times 10^6 \, \text{m}^{-1} \). \( \lambda = \frac{1}{\bar{v}} = \frac{1}{9.753 \times 10^6} = 1.025 \times 10^{-7} \, \text{m} = 102.5 \, \text{nm} \).

\( 102.5 \, \text{nm} \)
\( 121.6 \, \text{nm} \)
\( 656.3 \, \text{nm} \)
\( 486.1 \, \text{nm} \)
1

The ratio of the angular momentum of an electron in the third orbit of \( \text{H} \) to the second orbit of \( \text{He}^+ \) is: (\( h = 6.626 \times 10^{-34} \, \text{J s} \))

Angular momentum \( L = \frac{nh}{2\pi} \). For \( \text{H} \) (Z = 1), \( n = 3 \), \( L_3 = \frac{3h}{2\pi} \). For \( \text{He}^+ \) (Z = 2), \( n = 2 \), \( L_2 = \frac{2h}{2\pi} \). Ratio = \( \frac{3h / 2\pi}{2h / 2\pi} = \frac{3}{2} = 1.5 \).

\( 1.5 \)
\( 0.67 \)
\( 3.0 \)
\( 2.0 \)
1

What is the angular momentum of an electron in the second Bohr orbit of a hydrogen atom? (\( h = 6.626 \times 10^{-34} \, \text{J s} \))

Angular momentum \( L = \frac{nh}{2\pi} \). For \( n = 2 \), \( L = \frac{2 \times 6.626 \times 10^{-34}}{2 \times 3.14} = 2.11 \times 10^{-34} \, \text{J s} \).

\( 1.055 \times 10^{-34} \, \text{J s} \)
\( 3.313 \times 10^{-34} \, \text{J s} \)
\( 6.626 \times 10^{-34} \, \text{J s} \)
\( 2.11 \times 10^{-34} \, \text{J s} \)
4

The work function of a metal is \( 2.0 \, \text{eV} \). What is the maximum kinetic energy of an electron ejected by light of wavelength \( 400 \, \text{nm} \)? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( c = 3.0 \times 10^8 \, \text{m s}^{-1} \), \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \))

\( E = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{400 \times 10^{-9}} = 4.9695 \times 10^{-19} \, \text{J} = 3.105 \, \text{eV} \). \( W_0 = 2.0 \, \text{eV} \). \( KE_{\text{max}} = 3.105 - 2.0 = 1.105 \, \text{eV} = 1.768 \times 10^{-19} \, \text{J} \).

\( 4.9695 \times 10^{-19} \, \text{J} \)
\( 3.2015 \times 10^{-19} \, \text{J} \)
\( 2.412 \times 10^{-19} \, \text{J} \)
\( 1.768 \times 10^{-19} \, \text{J} \)
4

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