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Chapter 7: Problem 69

Explain why each of the following sets of quantum numbers would not be permissible for an electron, according to the rules for quantum numbers. \(n=1, l=0, m_{l}=0, m_{s}=+1\) \(n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2}\) \(n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2}\) \(n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2}\) \(n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2}\)

Short Answer

Expert verified
Invalid sets due to: incorrect spin \((+1)\), invalid \(l\) for \(n\), invalid \(m_{l}\), non-positive \(n\), invalid \(m_{s}\).

Step by step solution

01

Understand Quantum Numbers

Quantum numbers describe the properties of an electron in an atom. They are: The principal quantum number \(n\), the azimuthal quantum number \(l\), the magnetic quantum number \(m_{l}\), and the spin quantum number \(m_{s}\). The values must follow these rules: \(n\) is a positive integer. \(l\) ranges from 0 to \(n-1\). \(m_{l}\) ranges from \(-l\) to \(+l\). \(m_{s}\) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
02

Analyze First Set

For \(n=1, l=0, m_{l}=0, m_{s}=+1\), none of these numbers break rules for \(n, l,\) or \(m_{l}\). However, \(m_{s}\) must be \(+\frac{1}{2}\) or \(-\frac{1}{2}\), so \(m_{s}=+1\) is not permissible.
03

Analyze Second Set

For \(n=1, l=3, m_{l}=+3, m_{s}=+\frac{1}{2}\), \(l\) must be less than \(n\). Since \(l=3\) is not less than \(n=1\), this set is not permissible.
04

Analyze Third Set

For \(n=3, l=2, m_{l}=+3, m_{s}=-\frac{1}{2}\), \(m_{l}\) has to be between \(-l\) and \(+l\). Here, \(l=2\), so \(m_{l}=+3\) is not permissible.
05

Analyze Fourth Set

For \(n=0, l=1, m_{l}=0, m_{s}=+\frac{1}{2}\), \(n\) must be a positive integer (1, 2, 3,...), so \(n=0\) is not permissible.
06

Analyze Fifth Set

For \(n=2, l=1, m_{l}=-1, m_{s}=+\frac{3}{2}\), \(m_{s}\) must be \(+\frac{1}{2}\) or \(-\frac{1}{2}\). Thus, \(m_{s}=+\frac{3}{2}\) is not permissible.

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Key Concepts

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

Principal Quantum Number
The principal quantum number, denoted by \( n \), is fundamental to understanding the energy levels of an electron in an atom. It represents the main shell or energy level that an electron occupies. Each electron in an atom is placed within one of these shells.

Here's what you need to know about \( n \):
  • \( n \) must be a positive integer (1, 2, 3, ...).
  • The larger the value of \( n \), the higher the electron鈥檚 energy and the farther it is from the nucleus.
  • Each value of \( n \) can hold a specific number of electrons calculated using the formula \( 2n^2 \).
In the fourth set from the exercise, where \( n=0 \), the value is impermissible because \( n \) cannot be zero.
Azimuthal Quantum Number
The azimuthal quantum number is denoted by \( l \). It describes the subshell of an electron and defines the shape of the orbital.

Important aspects of \( l \) include:
  • It can take on any integer value from 0 up to \( n-1 \).
  • \( l=0 \) corresponds to an s orbital, \( l=1 \) to a p orbital, \( l=2 \) to a d orbital, and \( l=3 \) to an f orbital.
  • This number helps determine the number of angular nodes in an orbital, which affects its shape.
For the second set, \( l=3 \) with \( n=1 \), it's not permissible since \( l \) should be less than \( n \).
Magnetic Quantum Number
The magnetic quantum number, symbolized as \( m_{l} \), determines the orientation of the electron's orbital in space. It adds further detail to the description of an electron's position within an atom.

Key facts about \( m_{l} \):
  • \( m_{l} \) can range from \(-l\) to \(+l\), including zero.
  • Each specific \( m_{l} \) value represents one orientation of an orbital in space.
  • For example, if \( l=2 \), then \( m_{l} \) can be \(-2, -1, 0, +1, +2\).
In the third set, \( m_{l} = +3 \) for \( l = 2 \) is not allowed because \( m_{l} \) must fall within the range from \(-2\) to \(+2\).
Spin Quantum Number
The spin quantum number \( m_{s} \) is unique as it describes the electron's intrinsic angular momentum, also known as "spin." It is the fourth quantum number, providing a complete description of the electron's state.

Details about \( m_{s} \):
  • \( m_{s} \) can only be \(+\frac{1}{2}\) or \(-\frac{1}{2}\), representing the two possible spin states of an electron.
  • These values correspond to the "spin-up" or "spin-down" orientation of an electron in an external magnetic field.
  • This quantum number implies that each orbital can hold a maximum of two electrons, each with opposite spins.
The first set from the exercise includes \( m_{s} = +1 \), which is not valid, and the fifth set includes \( m_{s} = +\frac{3}{2} \), both of which violate the spin quantum number restrictions.

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Most popular questions from this chapter

A line of the Lyman series of the hydrogen atom spectrum has the wavelength \(9.50 \times 10^{-8} \mathrm{~m}\). It results from a transition from an upper energy level to \(n=1\). What is the principal quantum number of the upper level?

Which of the following statements about a hydrogen atom is false? a). An electron in the \(n=1\) level of the hydrogen atom is in its ground state. b). On average, an electron in the \(n=3\) level is farther from the nucleus than an electron in the \(n=2\) state. c). The wavelength of light emitted when the electron goes from the \(n=3\) level to the \(n=1\) level is the same as the wavelength of light absorbed when the electron goes from the \(n=1\) level to \(n=3\) level. d). An electron in the \(n=1\) level is higher in energy than an electron in the \(n=4\) level. e). Light of greater frequency is required for a transition from the \(n=1\) level to \(n=3\) level than is required for a transition from the \(n=2\) level to \(n=3\) level.

The energy of a photon is \(2.70 \times 10^{-19} \mathrm{~J}\). What is the wavelength of the corresponding light? What is the color of this light?

A particular microwave oven delivers 750 watts. (A watt is a unit of power, which is the joules of energy delivered, or used, per second.) If the oven uses microwave radiation of wavelength \(12.6 \mathrm{~cm}\), how many photons of this radiation are required to heat \(1.00 \mathrm{~g}\) of water \(1.00^{\circ} \mathrm{C}\) assuming that all of the photons are absorbed?

An atom in its ground state absorbs a photon (photon 1), then quickly emits another photon (photon 2). One of these photons corresponds to ultraviolet radiation, whereas the other one corresponds to red light. Explain what is happening. Which electromagnetic radiation, ultraviolet or red light, is associated with the emitted photon (photon 2 )?
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