Question

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. In each case, change the incorrect value (or values) to make the set valid. (a) \(n=4, \ell=2, m_{\ell}=0, m_{s}=0\) (b) \(n=3, \ell=1, m_{\ell}=-3, m_{s}=-1 / 2\) (c) \(n=3, \ell=3, m_{\ell}=-1, m_{s}=+1 / 2\)

Short answer

Change the incorrect quantum values as follows: (a) \(m_s=\pm 1/2\); (b) \(m_{\ell}=-1, 0, \text{or } 1\); (c) \(\ell=0, 1, \text{or } 2\).

Step-by-step solution

Review Quantum Numbers and Valid Ranges

Quantum numbers describe the position and spin of an electron in an atom. They include the principal quantum number \(n\), which can be any positive integer; the azimuthal quantum number \(\ell\), which can range from 0 to \(n-1\); the magnetic quantum number \(m_{\ell}\), which can range from \(-\ell\) to \(+\ell\); and the spin quantum number \(m_{s}\), which can be \(+1/2\) or \(-1/2\).

Analyze Set (a)

For the set \(n=4, \ell=2, m_{\ell}=0, m_{s}=0\), the issue lies with the spin quantum number \(m_{s}=0\). The spin quantum number must be either \(+1/2\) or \(-1/2\). Change \(m_{s}\) to either of these values to make the set valid.

Analyze Set (b)

For the set \(n=3, \ell=1, m_{\ell}=-3, m_{s}=-1/2\), the magnetic quantum number \(m_{\ell}\) is incorrect. It must lie in the range \(-\ell \leq m_{\ell} \leq \ell\), meaning \(m_{\ell}\) should be between \(-1\) and \(+1\). Change \(m_{\ell}\) to any value within this range, such as \(-1\).

Analyze Set (c)

For the set \(n=3, \ell=3, m_{\ell}=-1, m_{s}=+1/2\), the issue is with the azimuthal quantum number \(\ell=3\). Since \(\ell\) must be less than \(n\), it should range from 0 to 2. Change \(\ell\) to a valid number such as 2 for consistency.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by \(n\), is a fundamental aspect of quantum mechanics and provides crucial information about the energy level in which an electron resides within an atom. It's like the local neighborhood of an electron's home, determining the overall size and energy of that orbit. The principal quantum number can take any positive integer value: \(n = 1, 2, 3, \ldots\).

• Higher \(n\) values: Correspond to electrons located further from the nucleus, hence they possess higher energies.
• Orbital size: The size of the orbital increases with \(n\), signifying larger probability regions where an electron might be found.

Understanding the principal quantum number helps in visualizing how electrons populate the energy levels of an atom, thereby dictating the atom's chemical properties. Each energy level can hold multiple electrons, but the specific capacity is influenced by the other quantum numbers. Knowing the value of \(n\) allows us to explore the possible shapes and orientations of electron clouds within an atom.

Magnetic Quantum Number

The magnetic quantum number, expressed as \(m_{\ell}\), provides insight into the orientation of an electron's cloud in three-dimensional space. It's crucial for identifying how an electron's orbital, or the "room" it occupies in its "house," is oriented relative to external magnetic fields.

• Possible values: The range of \(m_{\ell}\) is determined by the azimuthal quantum number \(\ell\). Specifically, \(m_{\ell}\) can be any integer from \(-\ell\) to \(+\ell\).
• Example: If \(\ell = 2\), then \(m_{\ell}\) can be \(-2, -1, 0, +1, \) or \(+2\).
• Significance: Different \(m_{\ell}\) values represent different orientations of the orbitals. This becomes particularly important in the presence of magnetic fields, where the directionality of orbitals can affect how electrons interact with external forces.

The magnetic quantum number is particularly meaningful in visualizing the complex structures of atoms in fields such as chemistry and physics. It assists in predicting the magnetic properties and spectral lines of atoms when exposed to external fields.

Spin Quantum Number

The spin quantum number, represented as \(m_{s}\), is a unique quantum number that describes an intrinsic property of electrons, analogous to a tiny bar magnet spinning around its axis. Unlike other quantum numbers, the spin quantum number does not depend on any orbital characteristics but rather the nature of the particle itself.

• Possible values: The spin of an electron can only be \(+1/2\) or \(-1/2\). This is commonly referred to as "spin up" or "spin down."
• Fundamental importance: The inclusion of spin has profound implications in quantum mechanics, contributing to the overall quantum state of an electron in an atom.
• Pauli Exclusion Principle: According to this principle, no two electrons in the same atom can occupy the same quantum state simultaneously. Thus, spin quantum numbers allow for two electrons to exist within the same orbital as long as they have opposite spins.

The spin quantum number is essential for understanding the magnetic properties of materials and plays a pivotal role in applications ranging from quantum computing to magnetism in everyday devices. It helps explain phenomena like atomic bonding and electron configurations, thus being foundational to modern atomic theory.