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  4. Structure of Atom Chapter-Wise Test 14

Structure of Atom Chapter-Wise Test 14

Correct answer Carries: 4.

Wrong Answer Carries: -1.

What is the radius ratio of the third orbit to the first orbit in a hydrogen atom?

Radius \( r_n = r_1 \times n^2 \). For \( n = 3 \), \( r_3 = r_1 \times 9 \). Ratio \( \frac{r_3}{r_1} = 9 \).

4
9
3
6
2

An electron jumps from \( n = 4 \) to \( n = 1 \) in a hydrogen atom. What is the energy of the emitted photon? (\( E_1 = -2.18 \times 10^{-18} \, \text{J} \))

\( \Delta E = E_1 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) = -2.18 \times 10^{-18} \left( \frac{1}{1^2} - \frac{1}{4^2} \right) = -2.18 \times 10^{-18} \times \frac{15}{16} = 2.04375 \times 10^{-18} \, \text{J} \) (emitted energy is positive).

\( 2.04375 \times 10^{-18} \, \text{J} \)
\( 1.635 \times 10^{-18} \, \text{J} \)
\( 2.18 \times 10^{-18} \, \text{J} \)
\( 1.3625 \times 10^{-18} \, \text{J} \)
1

The energy of a photon is given by \( E = h v \), where \( h = 6.626 \times 10^{-34} \, \text{J s} \) and frequency \( v = 4.0 \times 10^{14} \, \text{Hz} \). Calculate the energy in joules.

Energy \( E = h v = 6.626 \times 10^{-34} \times 4.0 \times 10^{14} = 2.6504 \times 10^{-19} \, \text{J} \).

\(2.6504 \times 10^{-19} \, \text{J}\)
\(1.3252 \times 10^{-19} \, \text{J}\)
\(4.9755 \times 10^{-19} \, \text{J}\)
\(6.626 \times 10^{-19} \, \text{J}\)
1

A neutron has a de Broglie wavelength of \( 3.0 \times 10^{-12} \, \text{m} \). What is its kinetic energy? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( m_n = 1.67 \times 10^{-27} \, \text{kg} \))

\( v = \frac{h}{m\lambda} = \frac{6.626 \times 10^{-34}}{1.67 \times 10^{-27} \times 3.0 \times 10^{-12}} = 1.322 \times 10^5 \, \text{m s}^{-1} \). \( KE = \frac{1}{2} m v^2 = \frac{1}{2} \times 1.67 \times 10^{-27} \times (1.322 \times 10^5)^2 = 1.457 \times 10^{-17} \, \text{J} \).

\( 7.285 \times 10^{-18} \, \text{J} \)
\( 2.914 \times 10^{-17} \, \text{J} \)
\( 9.872 \times 10^{-18} \, \text{J} \)
\( 1.457 \times 10^{-17} \, \text{J} \)
4

An electron in \( \text{Li}^{2+} \) has a de Broglie wavelength of \( 2.21 \times 10^{-10} \, \text{m} \) in the second orbit. What is its kinetic energy? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( m_e = 9.1 \times 10^{-31} \, \text{kg} \))

\( \lambda = \frac{h}{mv} \), \( v = \frac{h}{m\lambda} = \frac{6.626 \times 10^{-34}}{9.1 \times 10^{-31} \times 2.21 \times 10^{-10}} = 3.294 \times 10^6 \, \text{m s}^{-1} \). \( KE = \frac{1}{2} m v^2 = \frac{1}{2} \times 9.1 \times 10^{-31} \times (3.294 \times 10^6)^2 = 4.936 \times 10^{-18} \, \text{J} \).

\( 2.18 \times 10^{-18} \, \text{J} \)
\( 9.872 \times 10^{-18} \, \text{J} \)
\( 1.234 \times 10^{-18} \, \text{J} \)
\( 4.936 \times 10^{-18} \, \text{J} \)
4

The angular momentum of an electron in the sixth orbit of a hydrogen atom is \( 6.33 \times 10^{-34} \, \text{J s} \). What is its energy? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( E_1 = -2.18 \times 10^{-18} \, \text{J} \))

\( L = \frac{nh}{2\pi} \), \( n = \frac{6.33 \times 10^{-34} \times 2 \times 3.14}{6.626 \times 10^{-34}} = 6 \). \( E_6 = -2.18 \times 10^{-18} / 36 = -6.056 \times 10^{-20} \, \text{J} \).

\( -2.42 \times 10^{-19} \, \text{J} \)
\( -1.36 \times 10^{-19} \, \text{J} \)
\( -5.45 \times 10^{-19} \, \text{J} \)
\( -6.056 \times 10^{-20} \, \text{J} \)
4

The kinetic energy of a photoelectron is \( 1.5 \times 10^{-19} \, \text{J} \) when light of wavelength \( 400 \, \text{nm} \) is used. What is the work function? (\( h = 6.626 \times 10^{-34} \, \text{J s} \), \( c = 3.0 \times 10^8 \, \text{m s}^{-1} \))

Photon energy \( E = \frac{hc}{\lambda} = \frac{6.626 \times 10^{-34} \times 3.0 \times 10^8}{400 \times 10^{-9}} = 4.9695 \times 10^{-19} \, \text{J} \). \( W_0 = E - KE = 4.9695 \times 10^{-19} - 1.5 \times 10^{-19} = 3.4695 \times 10^{-19} \, \text{J} \).

\( 1.5 \times 10^{-19} \, \text{J} \)
\( 4.9695 \times 10^{-19} \, \text{J} \)
\( 2.6504 \times 10^{-19} \, \text{J} \)
\( 3.4695 \times 10^{-19} \, \text{J} \)
4

An alpha particle and a proton have the same kinetic energy. What is the ratio of their de Broglie wavelengths? (\( m_p = 1.67 \times 10^{-27} \, \text{kg} \), \( m_{\alpha} = 6.64 \times 10^{-27} \, \text{kg} \))

\( \lambda = \frac{h}{\sqrt{2mKE}} \). For equal KE, \( \lambda_{\alpha} / \lambda_p = \sqrt{m_p / m_{\alpha}} = \sqrt{1.67 \times 10^{-27} / 6.64 \times 10^{-27}} = \sqrt{0.2515} \approx 0.501 \).

\( 2.0 \)
\( 0.501 \)
\( 4.0 \)
\( 0.25 \)
2

How many possible values of \( m_s \) are there for an electron with \( n = 4, l = 2, m_l = 1 \)?

For a specific orbital (\( n, l, m_l \)), \( m_s = +1/2 \) or \( -1/2 \). Total = 2.

1
4
2
5
3

The radius of the first orbit of \( \text{Li}^{2+} \) is \( 1.763 \times 10^{-11} \, \text{m} \). What is the kinetic energy of an electron in the third orbit? (\( v_1 \) for H = \( 2.19 \times 10^6 \, \text{m s}^{-1} \), \( m_e = 9.1 \times 10^{-31} \, \text{kg} \))

For \( \text{Li}^{2+} \) (Z = 3), \( r_1 = 5.29 \times 10^{-11} / 3 = 1.763 \times 10^{-11} \, \text{m} \). \( v_n = \frac{Z v_1}{n} \), \( v_3 = \frac{3 \times 2.19 \times 10^6}{3} = 2.19 \times 10^6 \, \text{m s}^{-1} \). \( KE = \frac{1}{2} m v^2 = \frac{1}{2} \times 9.1 \times 10^{-31} \times (2.19 \times 10^6)^2 = 2.18 \times 10^{-18} \, \text{J} \).

\( 2.18 \times 10^{-18} \, \text{J} \)
\( 4.36 \times 10^{-18} \, \text{J} \)
\( 1.09 \times 10^{-18} \, \text{J} \)
\( 9.7 \times 10^{-19} \, \text{J} \)
1

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