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Chapter 7: Problem 141

Are the following statements true for the hydrogen atom only, true for all atoms, or not true for any atoms? a. The principal quantum number completely determines the energy of a given electron. b. The angular momentum quantum number, \(\ell\), determines the shapes of the atomic orbitals. c. The magnetic quantum number, \(m_{\ell}\), determines the direction that the atomic orbitals point in space.

Short Answer

Expert verified
a. True for hydrogen atom only. b. True for all atoms. c. True for all atoms.

Step by step solution

01

a. Principal quantum number and electron energy

For hydrogen atom, the energy of a given electron depends only on the principal quantum number (n). This is due to the fact that in a hydrogen atom, there's only one electron and the energy levels are directly connected to the principal quantum number. However, for multi-electron atoms, the energy of an electron depends not only on the principal quantum number (n) but also on the angular momentum quantum number (ℓ). This is because in multi-electron atoms, electron-electron interactions and additional forces come into play, which affects the energy levels. Therefore, the statement "The principal quantum number completely determines the energy of a given electron" is true for hydrogen atom only.
02

b. Angular momentum quantum number and atomic orbital shapes

The angular momentum quantum number (ℓ) determines the shapes of the atomic orbitals, regardless of the type of atom (hydrogen or multi-electron atoms). This is because (ℓ) takes values from 0 to n-1 and describes the type of orbital, such as s, p, d, and f orbitals, which have distinct shapes. Therefore, the statement "The angular momentum quantum number, ℓ, determines the shapes of the atomic orbitals" is true for all atoms.
03

c. Magnetic quantum number and atomic orbital direction

The magnetic quantum number (\(m_{\ell}\)) determines the orientation of the atomic orbitals in space. It takes values from -ℓ to +ℓ, including 0, and corresponds to the number of possible orientations of the orbitals. This property is consistent for all atoms, regardless of whether they are hydrogen or multi-electron atoms. Therefore, the statement "The magnetic quantum number, \(m_{\ell}\), determines the direction that the atomic orbitals point in space" is true for all atoms.

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Most popular questions from this chapter

Three elements have the electron configurations \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6}\), \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\), and \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{1}\). The first ionization energies of these elements (not in the same order) are \(0.419,0.735\), and \(1.527 \mathrm{MJ} / \mathrm{mol}\). The atomic radii are \(1.60,0.98\), and \(2.35 \AA\). Identify the three elements, and match the appropriate values of ionization energy and atomic radius to each configuration.

Arrange the following groups of atoms in order of increasing size. a. \(\mathrm{Te}, \mathrm{S}, \mathrm{Se}\) b. \(\mathrm{K}, \mathrm{Br}, \mathrm{Ni}\) c. \(\mathrm{Ba}, \mathrm{Si}, \mathrm{F}\)

In defining the sizes of orbitals, why must we use an arbitrary value, such as \(90 \%\) of the probability of finding an electron in that region?

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=0, \ell=0, m_{\ell}=0\) b. \(n=2, \ell=1, m_{\ell}=-1, m_{s}=-\frac{1}{2}\) c. \(n=3, m_{s}=+\frac{1}{2}\) d. \(n=2, \ell=2\) e. \(n=1, \ell=0, m_{\ell}=0\)

Rank the elements \(\mathrm{Be}, \mathrm{B}, \mathrm{C}, \mathrm{N}\), and \(\mathrm{O}\) in order of increasing first ionization energy. Explain your reasoning.
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