Question

Which of the following represent valid sets of quantum numbers? For a set that is invalid, explain briefly why it is not correct. (a) \(n=3, \ell=3, m_{\ell}=0\) (b) \(n=2, \ell=1, m_{\ell}=0\) (c) \(n=6, \ell=5, m_{\ell}=-1\) (d) \(n=4, \ell=3, m_{\ell}=-4\)

Short answer

Sets (b) and (c) are valid, while sets (a) and (d) are invalid.

Step-by-step solution

Understanding Quantum Numbers

The principal quantum number \(n\) must be a positive integer: \(n = 1, 2, 3, \ldots\). The azimuthal quantum number \(\ell\) can take integer values from 0 to \(n-1\). The magnetic quantum number \(m_\ell\) can take integer values from \(-\ell\) to \(\ell\).

Analyze Set (a)

Set (a) is \(n=3, \ell=3, m_{\ell}=0\). Since \(\ell\) must be less than \(n\), this set is invalid because \(\ell = 3\) is not less than \(n = 3\).

Analyze Set (b)

Set (b) is \(n=2, \ell=1, m_{\ell}=0\). Since \(\ell = 1\) is within the valid range from 0 to \(n-1 = 1\), and \(m_{\ell}=0\) is within the range from \(-1\) to \(1\), this set is valid.

Analyze Set (c)

Set (c) is \(n=6, \ell=5, m_{\ell}=-1\). Here, \(\ell = 5\) is within the range from 0 to \(n-1 = 5\), and \(m_{\ell} = -1\) is within the range from \(-5\) to \(5\). Therefore, this set is valid.

Analyze Set (d)

Set (d) is \(n=4, \ell=3, m_{\ell}=-4\). While \(\ell = 3\) is within the range from 0 to \(n-1 = 3\), \(m_{\ell} = -4\) is not within the range from \(-3\) to \(3\). Thus, this set is invalid.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \(n\), is a crucial concept in understanding the quantum mechanical description of atoms. It primarily indicates the energy level of an electron in an atom. The values of \(n\) can be any positive integer (1, 2, 3, ...). Each value of \(n\) signifies a different electron shell, or layer, around the nucleus. As \(n\) increases, the electron is located further from the nucleus, corresponding to a higher energy level.

Think of \(n\) like the floors in a multi-story building. The floors are numbered starting from 1 at the ground level. The first floor is closer to the building's entrance (the nucleus), and as you go higher, the floors (energy levels) increase in number. Electrons on higher floors (levels) have higher energy but are further away from the entrance. This helps us understand where an electron might reside around an atom and how much energy it possesses.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized by \(\ell\), provides even more detail about an electron's state. It gives information about the shape of the electron's orbital, which is where the electron is likely to be found. Based on the principal quantum number \(n\), the azimuthal quantum number \(\ell\) can take on integer values ranging from 0 up to \(n-1\).

Each value of \(\ell\) corresponds to a different type of orbital:

• \(\ell = 0\) indicates an s orbital (sphere-shaped)
• \(\ell = 1\) indicates a p orbital (dumbbell-shaped)
• \(\ell = 2\) indicates a d orbital (cloverleaf-shaped)
• \(\ell = 3\) indicates an f orbital (complex shapes)

For example, if \(n = 3\), \(\ell\) can be 0, 1, or 2. Each type of orbital can hold a specific number of electrons, determining how electrons are arranged around the nucleus, contributing to the atom's chemical behavior.

Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_{\ell}\), provides insight about the orientation of the electron's orbital in space. Once you've determined \(n\) and \(\ell\), the possible values of \(m_{\ell}\) range from \(-\ell\) to \(+\ell\).

For example, if \(\ell = 2\), \(m_{\ell}\) can be -2, -1, 0, 1, or 2. This means the d orbital (when \(\ell = 2\)) has multiple orientations in space it can take.

• Each orbital orientation defined by \(m_{\ell}\) can hold two electrons with opposite spins.
• The magnetic quantum number allows for a detailed understanding of how an atom's electrons orient themselves in a magnetic field.

Understanding \(m_{\ell}\) is crucial for predicting how atoms interact with magnetic fields and how they will align themselves at the atomic level, which is essential in fields like chemistry and physics.