Problem 38 - Chemistry and Chemical Reactivity

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) \(n=2, \ell=2, m_{\ell}=0, m_{\mathrm{s}}=+1 / 2\) (b) \(n=2, \ell=1, m_{\ell}=-1, m_{\mathrm{s}}=0\) (c) \(n=3, \ell=1, m_{\ell}=-2, m_{\mathrm{s}}=+1 / 2\)

Step-by-Step Solution

Answer

Set (a) is invalid because \(\ell\) exceeds the limit, (b) has an invalid \(m_{s}\), and (c) has an invalid \(m_{\ell}\).

Step 1: Understanding Quantum Numbers Overview

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. There are four quantum numbers: principal quantum number \(n\), azimuthal (angular momentum) quantum number \(\ell\), magnetic quantum number \(m_{\ell}\), and spin quantum number \(m_{s}\). The possible values are \(n = 1, 2, 3, \ldots\), \(\ell = 0, 1, 2, \ldots, n-1\), \(m_{\ell} = -\ell, -\ell+1, \ldots, \ell\), and \(m_{s} = +1/2\) or \(-1/2\).

Step 2: Checking Set (a): \(n=2, \ell=2, m_{\ell}=0, m_{s}=+1/2\)

For the set (a), with \(n=2\), \(\ell\) should be 0 or 1 (since \(\ell = n-1)\). Thus, \(\ell=2\) is not possible. To correct it, set \(\ell=1\). Now, \(m_{\ell}=0\) is permissible since for \(\ell=1\), \(m_{\ell}\) can be \(-1, 0, \) or \(1\). The \(m_{s}=+1/2\) is valid.

Step 3: Checking Set (b): \(n=2, \ell=1, m_{\ell}=-1, m_{s}=0\)

In set (b), while \(n=2\), \(\ell=1\), and \(m_{\ell}=-1\) are valid, \(m_{s}=0\) is incorrect. The spin quantum number \(m_{s}\) must be \(+1/2\) or \(-1/2\). Change \(m_{s}\) to \(+1/2\) or \(-1/2\).

Step 4: Checking Set (c): \(n=3, \ell=1, m_{\ell}=-2, m_{s}=+1/2\)

For set (c), \(n=3\) allows \(\ell=0, 1, \) or \(2\), so \(\ell=1\) is possible. However, the value \(m_{\ell}=-2\) is not feasible for \(\ell=1\) because \(m_{\ell}\) can only be \(-1, 0,\) or \(1\). Adjust \(m_{\ell}\) to one of these valid numbers, like \(-1\), \(0\), or \(1\). The \(m_{s}=+1/2\) is correct.

Key Concepts

Principal Quantum Number (n)

The principal quantum number, symbolized as \(n\), is a critical component in understanding the structure of an atom. This number primarily dictates the size and energy level of the electron orbit, often termed as the electron shell.
This number can take positive integer values (\(1, 2, 3, \ldots\)), with each increment representing a level further from the nucleus. Higher \(n\) values hint at electrons residing further out, and thus, having higher potential energy.
The principal quantum number is deeply linked to the overall electron configuration and the eventual chemical behavior of an atom. As \(n\) increases, so does the electron capacity of individual shells, following the formula \(2n^2\).
In the scenarios provided:

• Set (a) and (b): The use of \(n=2\) guides the electron configurations fitting this principal level.
• Set (c): Here, \(n=3\) indicates a higher energy state, allowing more complex electron distributions and angular momentum possibilities.

Azimuthal Quantum Number (ℓ)

The azimuthal quantum number, \(\ell\), frequently referred to as the angular momentum quantum number, provides insight into the shape of the electron cloud or subshell. It can take on values ranging from \(0\) to \(n-1\), where \(n\) is the principal quantum number.
This number determines the orbital shape (s, p, d, f, etc.), defined by different values:

• \(\ell = 0\): s-orbital, spherical in shape.
• \(\ell = 1\): p-orbital, dumbbell-shaped.
• \(\ell = 2\): d-orbital, more complex forms.
• \(\ell = 3\): f-orbital, even more intricate.

Addressing the given scenarios:

• In Set (a): \(\ell\) was set to \(2\) for \(n=2\), which is invalid as \(\ell\) must be less than \(n\). Thus, \(\ell\) could only be \(0\) or \(1\). Correctly, \(\ell=1\) is the adjustment made.
• Set (b): Already satisfied the condition \(\ell = 1\) with \(n = 2\).
• Set (c): Though \(\ell=1\) is valid, it's the magnetic quantum number affected here.

Spin Quantum Number (m_s)

The spin quantum number, denoted as \(m_s\), describes the intrinsic spin of the electron within its orbital. It is not about spatial direction but more about the angular momentum inherent to the electron's nature.
There are only two allowed values for \(m_s\):

• \(+1/2\)
• \(-1/2\)

These depict the two fundamental electron orientations: one "spin up" and the other "spin down". This is crucial as it allows each orbital, defined by \(n\), \(\ell\), and \(m_\ell\), to host two electrons with opposing spins.
In the step-by-step scenarios:

• Set (b): The error stems from setting \(m_s\) to \(0\), which is invalid since electrons must spin in one of the two mentioned states. Adjusting to either \(+1/2\) or \(-1/2\) rectifies it.
• Both Set (a) and Set (c) properly employ \(m_s = +1/2\), aligning with given conditions.

Understanding this quantum number helps grasp electron pairing within orbitals and supports predictions on magnetic properties.