Question

Write an acceptable value for each of the missing quantum numbers. (a) \(n=3, \ell=?, m_{\ell}=2, m_{s}=+\frac{1}{2}\) (b) \(n=?, \ell=2, m_{\ell}=1, m_{s}=-\frac{1}{2}\) (c) \(n=4, \ell=2, m_{\ell}=0, m_{s}=?\) (d) \(n=?, \ell=0, m_{\ell}=?, m_{s}=?\)

Short answer

(a) \(l=2\), (b) \(n=3, 4, ...etc.\), (c) \(m_{s}= +\frac{1}{2}, -\frac{1}{2}\), (d) \(n=1, 2, ...etc., m_{\ell}=0, m_{s}= +\frac{1}{2}, -\frac{1}{2}\)

Step-by-step solution

Solve for Missing Quantum Number in (a)

For (a) \(n=3, m_{\ell}=2, m_{s}=+\frac{1}{2}\), we need to find \(\ell\). As \(m_{\ell}\) ranges from \(-l\) to \(l\), \(\ell\) must be equal to or greater than \(m_{\ell}\). The only acceptable value that does not exceed \(n-1\) is \(l=2\).

Solve for Missing Quantum Number in (b)

For (b) \(\ell=2, m_{\ell}=1, m_{s}=-\frac{1}{2}\), we need to find \(n\). As \(\ell\) must be smaller than \(n\), the only acceptable value for \(n\) that is larger than \(\ell=2\) is \(n=3\), \(n=4\), and so on.

Solve for Missing Quantum Number in (c)

For (c) \(n=4, \ell=2, m_{\ell}=0\), we need to find \(m_{s}\). \(m_{s}\) only has two possible values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\). So either of them is an acceptable value.

Solve for Missing Quantum Number in (d)

For (d) \(\ell=0\), we need to find \(n, m_{\ell}, m_{s}\). As \(\ell\) should be less than \(n\), the only acceptable value for \(n\) that is larger than \(\ell=0\) is \(n=1\), \(n=2\), and so on. \(m_{\ell}\) ranges from \(-\ell\) to \(\ell\), so the only possible value for \(m_{\ell}\) here is \(m_{\ell}=0\). \(m_{s}\) only has two possible values: \(+\frac{1}{2}\) and \(-\frac{1}{2}\). So either of them is an acceptable value.

Key concepts

Principal Quantum Number (n)

The principal quantum number, represented by the letter \( n \), is one of the four quantum numbers we use to describe electrons in an atom. It defines the primary energy level or shell of an electron. This means \( n \) determines how far an electron is likely to be from the nucleus.
The possible values for \( n \) are positive integers (1, 2, 3, ...). The higher the value of \( n \), the larger the electron’s orbit, which also means higher energy levels. This is because electrons situated further from the nucleus have more energy.

• If \( n = 1 \), the electron is closest to the nucleus.
• If \( n = 2 \), the electron is in the second shell, which is farther out.

This quantum number plays a fundamental role in determining the electron configuration of an atom, influencing the possible values of the other quantum numbers. For instance, the angular momentum quantum number \( \ell \) is always less than \( n \).

Angular Momentum Quantum Number (l)

The angular momentum quantum number, denoted as \( \ell \), describes the shape of the electron's orbital. It is also called the azimuthal quantum number and is crucial for determining the orbital's subshell type, such as s, p, d, or f. The possible values for \( \ell \) range from 0 to \( n-1 \).
If \( n = 3 \), \( \ell \) can be 0, 1, or 2:

• \( \ell = 0 \) refers to an s orbital.
• \( \ell = 1 \) corresponds to a p orbital.
• \( \ell = 2 \) aligns with a d orbital.

In the context of the exercise, for a given \( m_\ell \), \( \ell \) must be equal to or greater than \( | m_\ell | \). For example, if \( m_\ell = 2 \), \( \ell \) must be at least 2, which could further limit or define possible values of \( n \).

Magnetic Quantum Number (m_l)

The magnetic quantum number \( m_\ell \) describes the orientation of the electron orbital in space. It influences the number of orbital positions an electron can occupy within a subshell.
The values for \( m_\ell \) range from \(-\ell\) to \(\ell\), including zero:

• If \( \ell = 1 \), \( m_\ell \) can be \(-1, 0, 1 \).
• If \( \ell = 2 \), \( m_\ell \) can be \(-2, -1, 0, 1, 2 \).

Each value of \( m_\ell \) represents a distinct orientation that an electron orbital can be in. This quantum number is particularly important when considering the magnetic field effects on atom configurations. In terms of electrons, it indicates the precise spatial orientation of the orbital; essentially, which "seat" in the "row" (shell and subshell) the electron is sitting in.

Spin Quantum Number (m_s)

The spin quantum number \( m_s \) describes the intrinsic spin of the electron. Unlike the other quantum numbers that specify the electron’s position or path, \( m_s \) characterizes the electron’s spin direction.
The possible values for \( m_s \) are only \(+\frac{1}{2}\) or \(-\frac{1}{2}\). These values refer to the two opposite directions an electron can spin, commonly referred to as "spin up" and "spin down".

• This spin concept is vital because Pauli's Exclusion Principle states that two electrons in the same orbital cannot have the same set of quantum numbers, including \( m_s \).

So, each orbital can hold two electrons, one with \( m_s = +\frac{1}{2} \) and the other with \( m_s = -\frac{1}{2} \). This property plays a crucial role in determining how electrons pair up in atomic orbitals and fill electron shells.