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Chapter 6: Problem 33

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. (a) \(n=2, \ell=2, m_{\ell}=0\) (b) \(n=3, \ell=0, m_{\ell}=-2\) (c) \(n=6, \ell=0, m_{\ell}=1\)

Short Answer

Expert verified
All sets violate quantum number constraints. a) \(\ell < n\); b) \(m_{\ell} = 0\) required; c) \(m_{\ell} = 0\) required.

Step by step solution

01

Understanding Quantum Number Restrictions

To solve this problem, we must understand the restrictions on quantum numbers: 1. Principal quantum number ( "): "): any positive integer 2. Azimuthal quantum number ( : any positive integer 3. "): any integer that satisfies "): must satisfy : possible values of m_{ "): must satisfy m: possible values of "): must be one of the integers from _): integer values that satisfy _): integer values that satisfy m: integer values from m_{ })_): integer values from _): integer values from ").
02

Evaluating Case (a): \(n=2, \ell=2, m_{\ell}=0\)

In this case, the principal quantum number is \(n=2\). The azimuthal quantum number \(\ell\) must satisfy \(0 \leq \ell < n\). Since \(\ell=2\) for \(n=2\), this violates the condition \(\ell < n\). Therefore, this set of quantum numbers is not possible.
03

Evaluating Case (b): \(n=3, \ell=0, m_{\ell}=-2\)

Here, for \(n=3\) and \(\ell=0\), the magnetic quantum number \(m_{\ell}\) can only take the value 0 because \(m_{\ell}\) can range from \(-\ell\) to \(+\ell\). Since \(m_{\ell}=-2\) is not within this range, this set of quantum numbers is not possible.
04

Evaluating Case (c): \(n=6, \ell=0, m_{\ell}=1\)

Similarly, with \(n=6\) and \(\ell=0\), \(m_{\ell}\) must also be 0 for the same reason as in case (b). Since \(m_{\ell}=1\) does not meet this requirement, this set of quantum numbers is not possible.

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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 as \( n \), plays a fundamental role in discerning the energy level of an electron within an atom. It is always a positive integer (e.g., 1, 2, 3, ...). This number defines the major electron shell, or orbit, around the nucleus and determines the overall size and energy of an electron's orbit.

  • Lower values of \( n \) are closer to the nucleus, indicating electrons with lower energy.
  • Higher values of \( n \) are further away, suggesting electrons with higher energy.
For instance, in the example where \( n = 2 \), the electron could reside in the second energy level of the atom. The number directly affects the potential number of sublevels within each major level, which are defined by the azimuthal quantum number.
Azimuthal Quantum Number
The azimuthal quantum number, \( \ell \), is crucial for determining the shape of an electron's orbit within a particular shell. This quantum number can take on integer values starting from 0 up to \( n-1 \). Thus, it reflects the sublevels or subshells possible for a given principal quantum number.

  • \( \ell = 0 \) correlates with an 's' orbital, which is spherical.
  • \( \ell = 1 \) corresponds to 'p' orbitals, which have a dumbbell shape.
  • \( \ell = 2 \) signifies 'd' orbitals, having a more complex clover shape.
For example, if \( n = 3 \), permissible \( \ell \) values are 0, 1, and 2. Importantly, if \( \ell \) exceeds the principal quantum number, like in the case of \( \ell = 2 \) and \( n = 2 \), such configurations are impossible, as the azimuthal quantum number must always be less than the principal quantum number.
Magnetic Quantum Number
The magnetic quantum number, represented as \( m_\ell \), describes the orientation of an electron's orbital with respect to an external magnetic field. This quantum number ranges from \(-\ell\) to \(+\ell\), encompassing all integers in between. Thus, it indicates how many orbitals exist within a given subshell.

  • A subshell characterized by \( \ell = 0 \) (an 's' orbital) will have only one orientation, so \( m_\ell \) can only be 0.
  • For \( \ell = 1 \) (a 'p' orbital), \( m_\ell \) includes -1, 0, and +1, indicating three possible orientations.
  • For \( \ell = 2 \) (a 'd' orbital), \( m_\ell \) ranges from -2 to +2.
In the scenarios provided, if \( \ell = 0 \), the only valid \( m_\ell \) value is 0. Any deviation, such as \( m_\ell = -2 \) or \( m_\ell = 1 \), becomes impossible since they do not align with the allowed range for the given \( \ell \).

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Most popular questions from this chapter

Calculate the wavelength and frequency of light emitted when an electron changes from \(n=4\) to \(n=3\) in the H atom. In what region of the spectrum is this radiation found?

A beam of electrons \(\left(m=9.11 \times 10^{-31} \mathrm{kg} / \text { electron }\right)\) has an average speed of \(1.3 \times 10^{8} \mathrm{m} / \mathrm{s} .\) What is the wavelength of electrons having this average speed?

If sufficient energy is absorbed by an atom, an electron can be lost by the atom and a positive ion formed. The amount of energy required is called the ionization energy. In the \(\mathrm{H}\) atom, the ionization energy is that required to change the electron from \(n=1\) to \(n=\) infinity. Calculate the ionization energy for the He \(^{+}\) ion. Is the ionization energy of the He \(^{+}\) more or less than that of H? (Bohr's theory applies to He \(^{+}\) because it, like the \(\mathrm{H}\) atom, has a single electron. The electron energy, however, is now given by \(E=-Z^{2} R h c / n^{2},\) where \(Z\) is the atomic number of helium.)

Green light has a wavelength of \(5.0 \times 10^{2} \mathrm{nm} .\) What is the energy, in joules, of one photon of green light? What is the energy, in joules, of 1.0 mol of photons of green light?

If energy is absorbed by a hydrogen atom in its ground state, the atom is excited to a higher energy state. For example, the excitation of an electron from \(n=1\) to \(n=3\) requires radiation with a wavelength of \(102.6 \mathrm{nm}\) Which of the following transitions would require radiation of longer wavelength than this? (a) \(n=2\) to \(n=4\) (b) \(n=1\) to \(n=4\) (c) \(n=1\) to \(n=5\) (d) \(n=3\) to \(n=5\)
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