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Chapter 30: Problem 57

Answer the following questions using the quantum mechanical model of the atom. (a) For a given principal quantum number \(n\), what values of the angular momentum quantum number \(\ell\) are possible? (b) For a given angular momentum, what values of the magnetic quantum number \(m_{\ell}\) are possible? Which of the following subshell configurations are not allowed? For those that are not allowed, give the reason why. (a) \(3 \mathrm{~s}^{1}\), (b) \(2 \mathrm{~d}^{2}\) (c) \(3 \mathrm{~s}^{4}\) (d) \(4 \mathrm{p}^{8}\) (e) \(5 \mathrm{f}^{12}\)

Short Answer

Expert verified
Invalid configurations are (b) 2d虏, (c) 3s鈦�, and (d) 4p鈦�; reasons: \(d\) doesn't exist for \(n=2\), \(s\) can hold up to 2, and \(p\) can hold up to 6.

Step by step solution

01

Determine Possible Values of Angular Momentum Quantum Number (\(\ell\))

For a given principal quantum number \( n \), the angular momentum quantum number \( \ell \) can take on integer values from 0 up to \( n-1 \). Thus, for \( n = 3 \), \( \ell \) can be 0, 1, or 2.
02

Determine Possible Values of Magnetic Quantum Number (\(m_\ell\))

For a given \( \ell \), the magnetic quantum number \( m_\ell \) can take integer values ranging from \( -\ell \) to \( +\ell \). For example, if \( \ell = 1 \), then \( m_\ell \) can be -1, 0, or 1.
03

Analyze Each Configuration

For each subshell configuration, check its validity based on the allowed number of electrons.- (a) \(3s^1\): Valid, as the \(s\) subshell can hold up to 2 electrons.- (b) \(2d^2\): Invalid, as there is no \(d\) subshell for \(n=2\) (\(\ell\) can only be 0 or 1).- (c) \(3s^4\): Invalid, as the \(s\) subshell can only hold up to 2 electrons.- (d) \(4p^8\): Invalid, as the \(p\) subshell can only hold up to 6 electrons.- (e) \(5f^{12}\): Valid, as the \(f\) subshell can hold up to 14 electrons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Numbers
In the quantum mechanical model of the atom, quantum numbers act as the atom's address system. These numbers are crucial for defining the state of an electron precisely. There are four main quantum numbers:

  • Principal Quantum Number ( "): Denotes the main energy level or shell.
  • Angular Momentum Quantum Number ( "): Describes the shape of the orbital.
  • Magnetic Quantum Number ( ): Specifies the orientation of the orbital.
  • Spin Quantum Number: Indicates the two fundamental spin states of the electron.
These numbers work together to give a complete description of an electron's position and energy within an atom.
Principal Quantum Number
The principal quantum number, represented by \( n \), tells us the primary energy level or shell of an electron. It is always a positive integer (1, 2, 3, and so on). As \( n \) increases, the electron's energy and its average distance from the nucleus also increase. This number is key in determining the overall size of the electron cloud.

For example, if \( n = 3 \), the electron is in the third energy level, which is further from the nucleus. This indicates more energy and a larger orbital size.
Angular Momentum Quantum Number
The angular momentum quantum number, symbolized by \( \ell \), defines the shape of an orbital within a given energy level. Its possible values range from 0 to \( n-1 \).

  • \( \ell = 0 \): s orbital (spherical shape)
  • \( \ell = 1 \): p orbital (dumbbell shape)
  • \( \ell = 2 \): d orbital (cloverleaf shape)
  • \( \ell = 3 \): f orbital (complex shape)
These shapes affect how electrons are distributed around the nucleus, influencing chemical bonding and reactivity.
Magnetic Quantum Number
The magnetic quantum number, denoted as \( m_\ell \), specifies the orientation of an orbital in space. For each value of \( \ell \), \( m_\ell \) can range from \( -\ell \) to \( +\ell \), including zero. This quantum number determines the directionality of an orbital.

For instance, if \( \ell = 1 \) (p orbital), \( m_\ell \) can be -1, 0, or 1, allowing three orientations. Each orientation corresponds to a different spatial direction of the orbital, essential for understanding how atoms form bonds in molecules.

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Most popular questions from this chapter

Using the Bohr model, determine the ratio of the energy of the \(n\) th orbit of a triply ionized beryllium atom \(\left(\mathrm{Be}^{3+}, Z=4\right)\) to the energy of the \(n\) th orbit of a hydrogen atom \((\mathrm{H})\).

Concept Simulation 30.1 at reviews the concepts on which the solution to this problem depends. The electron in a hydrogen atom is in the first excited state, when the electron acquires an additional \(2.86 \mathrm{eV}\) of energy. What is the quantum number \(n\) of the state into which the electron moves?

Problem (a) What is the ionization energy of a hydrogen atom that is in the \(n=4\) state? (b) Determine the ratio of the ionization energy for \(n=4\) to that of the ground state.

Write down the fourteen sets of the four quantum numbers that correspond to the electrons in a completely filled \(4 \mathrm{f}\) subshell.

Suppose that the molybdenum \((Z=42)\) target in an X-ray tube is replaced by a silver \((Z=47)\) target. Do (a) the cutoff wavelength \(\lambda_{0}\) and (b) the wavelength of the \(K_{\alpha}\) X-ray photon increase, decrease, or remain the same? Assume that the voltage across the tube is constant and is sufficient to produce characteristic X-rays from both targets. Provide a reason for each answer. The voltage across the X-ray tube is \(35.0 \mathrm{kV}\). Determine (a) the cutoff wavelength \(\lambda_{0}\) and (b) the wavelengths of the \(K_{\alpha}\) X-ray photons emitted by the molybdenum and silver targets. Verify that your answers are consistent with your answers to the Concept Questions.
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