Structure of Atom Chapter-Wise Test 1

Correct answer Carries: 4.

Wrong Answer Carries: -1.

The de Broglie wavelength of an electron is \( 5.0 \times 10^{-10} \, \text{m} \) and its kinetic energy is \( 3.026 \times 10^{-19} \, \text{J} \). What is the uncertainty in its position? (\( h = 6.626 \times 10^{-34} \, \text{J s} \))

Momentum \( p = \frac{h}{\lambda} = \frac{6.626 \times 10^{-34}}{5.0 \times 10^{-10}} = 1.3252 \times 10^{-24} \, \text{kg m s}^{-1} \). \( \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \), \( \Delta x \geq \frac{h}{4\pi \Delta p} = \frac{6.626 \times 10^{-34}}{4 \times 3.14 \times 1.3252 \times 10^{-24}} = 3.98 \times 10^{-11} \, \text{m} \).

\( 1.99 \times 10^{-11} \, \text{m} \)
\( 5.27 \times 10^{-11} \, \text{m} \)
\( 3.98 \times 10^{-11} \, \text{m} \)
\( 6.62 \times 10^{-11} \, \text{m} \)
3

An electron in \( \text{He}^+ \) transitions from \( n = 3 \) to \( n = 1 \). What is the wavelength of the emitted photon in nm? (\( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \), \( c = 3.0 \times 10^8 \, \text{m s}^{-1} \))

For \( \text{He}^+ \) (Z = 2), wavenumber \( \bar{v} = Z^2 R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) = 4 \times 1.097 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = 4.389 \times 10^7 \times \frac{8}{9} \approx 3.901 \times 10^7 \, \text{m}^{-1} \).

Wavelength \( \lambda = \frac{c}{\nu} = \frac{3.0 \times 10^8}{c \cdot \bar{v}} = \frac{1}{\bar{v}} = \frac{1}{3.901 \times 10^7} \approx 2.563 \times 10^{-8} \, \text{m} = 25.63 \, \text{nm} \).

A. \( 25.63 \, \text{nm} \)
B. \( 30.38 \, \text{nm} \)
C. \( 102.5 \, \text{nm} \)
D. \( 91.13 \, \text{nm} \)
2

The wavelength of the fifth line in the Pfund series of a hydrogen atom is: (\( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \))

Pfund series: \( n_1 = 5 \), fifth line is \( n_2 = 10 \). \( \bar{v} = 1.097 \times 10^7 (1/25 - 1/100) = 1.097 \times 10^7 \times 3/100 = 3.291 \times 10^5 \, \text{m}^{-1} \). \( \lambda = 1 / \bar{v} = 3.039 \times 10^{-6} \, \text{m} = 3039 \, \text{nm} \).

\( 2279 \, \text{nm} \)
\( 3741 \, \text{nm} \)
\( 3039 \, \text{nm} \)
\( 4654 \, \text{nm} \)
3

What is the frequency of light emitted when an electron in \( \text{He}^+ \) transitions from \( n = 2 \) to \( n = 1 \)? (\( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \), \( c = 3.0 \times 10^8 \, \text{m s}^{-1} \))

Wavenumber \( \bar{v} = Z^2 R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) = 4 \times 1.097 \times 10^7 \left( \frac{1}{1^2} - \frac{1}{2^2} \right) = 3.291 \times 10^7 \, \text{m}^{-1} \). Frequency \( v = c \bar{v} = 3.0 \times 10^8 \times 3.291 \times 10^7 = 9.873 \times 10^{15} \, \text{Hz} \).

\( 4.936 \times 10^{15} \, \text{Hz} \)
\( 6.582 \times 10^{15} \, \text{Hz} \)
\( 9.873 \times 10^{15} \, \text{Hz} \)
\( 3.291 \times 10^{15} \, \text{Hz} \)
3

The ratio of the velocities of an electron in the second orbit of \( \text{H} \) to the third orbit of \( \text{Li}^{2+} \) is: (\( v_1 \) for H = \( 2.19 \times 10^6 \, \text{m s}^{-1} \))

For \( \text{H} \) (Z = 1), \( v_2 = 2.19 \times 10^6 / 2 = 1.095 \times 10^6 \, \text{m s}^{-1} \). For \( \text{Li}^{2+} \) (Z = 3), \( v_3 = 3 \times 2.19 \times 10^6 / 3 = 2.19 \times 10^6 \, \text{m s}^{-1} \). Ratio = \( 1.095 \times 10^6 / 2.19 \times 10^6 = 0.5 \).

\( 1.5 \)
\( 0.5 \)
\( 3.0 \)
\( 0.33 \)
2

The radius of the first Bohr orbit of \( \text{He}^+ \) is \( 2.645 \times 10^{-11} \, \text{m} \). What is the radius of the third orbit of \( \text{Li}^{2+} \) if the Bohr radius for hydrogen is \( 5.29 \times 10^{-11} \, \text{m} \)?

For hydrogen-like atoms, \( r_n = \frac{n^2}{Z} r_1 \). For \( \text{He}^+ \) (Z = 2), \( r_1 = \frac{1^2}{2} \times 5.29 \times 10^{-11} = 2.645 \times 10^{-11} \, \text{m} \). For \( \text{Li}^{2+} \) (Z = 3), \( n = 3 \), \( r_3 = \frac{3^2}{3} \times 5.29 \times 10^{-11} = 3 \times 5.29 \times 10^{-11} = 1.587 \times 10^{-10} \, \text{m} \).

\( 1.587 \times 10^{-10} \, \text{m} \)
\( 2.645 \times 10^{-11} \, \text{m} \)
\( 7.935 \times 10^{-11} \, \text{m} \)
\( 4.761 \times 10^{-10} \, \text{m} \)
1

The ratio of the wavelengths of the second lines in the Balmer and Paschen series of a hydrogen atom is: (\( R_H = 1.097 \times 10^7 \, \text{m}^{-1} \))

Balmer (4→2): \( \bar{v} = 1.097 \times 10^7 (1/4 - 1/16) = 2.057 \times 10^6 \, \text{m}^{-1} \), \( \lambda_1 = 4.861 \times 10^{-7} \, \text{m} \). Paschen (5→3): \( \bar{v} = 1.097 \times 10^7 (1/9 - 1/25) = 7.805 \times 10^5 \, \text{m}^{-1} \), \( \lambda_2 = 1.281 \times 10^{-6} \, \text{m} \). Ratio = \( \lambda_1 / \lambda_2 = 4.861 \times 10^{-7} / 1.281 \times 10^{-6} \approx 0.379 \).

\( 2.64 \)
\( 0.379 \)
\( 1.52 \)
\( 0.185 \)
2

The threshold frequency of a metal is \( 5.0 \times 10^{14} \, \text{Hz} \). What is its work function in eV? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( 1 \, \text{eV} = 1.6 \times 10^{-19} \, \text{J} \))

\( W_0 = h v_0 = 6.626 \times 10^{-34} \times 5.0 \times 10^{14} = 3.313 \times 10^{-19} \, \text{J} \). In eV, \( W_0 = \frac{3.313 \times 10^{-19}}{1.6 \times 10^{-19}} = 2.07 \, \text{eV} \).

\( 1.38 \, \text{eV} \)
\( 3.31 \, \text{eV} \)
\( 4.14 \, \text{eV} \)
\( 2.07 \, \text{eV} \)
4

An electron has a de Broglie wavelength of \( 2.0 \times 10^{-10} \, \text{m} \). What is its kinetic energy in joules? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( m_e = 9.1 \times 10^{-31} \, \text{kg} \))

\( \lambda = \frac{h}{\sqrt{2mKE}} \), so \( KE = \frac{h^2}{2m\lambda^2} = \frac{(6.626 \times 10^{-34})^2}{2 \times 9.1 \times 10^{-31} \times (2.0 \times 10^{-10})^2} = 6.02 \times 10^{-19} \, \text{J} \).

\( 6.02 \times 10^{-19} \, \text{J} \)
\( 3.01 \times 10^{-19} \, \text{J} \)
\( 1.50 \times 10^{-19} \, \text{J} \)
\( 9.04 \times 10^{-19} \, \text{J} \)
1

How many electrons in an atom can have the quantum numbers \( n = 6 \) and \( l = 4 \)?

For \( l = 4 \) (g subshell), orbitals = \( 2l + 1 = 9 \). Electrons = \( 9 \times 2 = 18 \).

14
18
10
22
2

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