Problem 34 - Chemistry and Chemical Reactivity

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) \(n=3, \ell=3, m_{\ell}=0\) (c) \(n=6, \ell=5, m_{\ell}=-1\) (b) \(n=2, \ell=1, m_{\ell}=0\) (d) \(n=4, \ell=3, m_{\ell}=-4\)

Step-by-Step Solution

Answer

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

Step 1: Analyze Set (a)

Quantum numbers must follow specific rules. The quantum number \( n \) is the principal quantum number and can be any positive integer. The azimuthal quantum number \( \ell \) must be an integer from 0 to \( n-1 \). Here, \( n=3 \) and \( \ell=3 \), which is not valid because \( \ell \) should be less than \( n \).

Step 2: Analyze Set (b)

For set (b), \( n=2 \) and \( \ell=1 \). These numbers follow the rule \( 0 \leq \ell < n \). The magnetic quantum number \( m_{\ell} \) must be in the range \(-\ell \leq m_{\ell} \leq \ell\). Here, \( m_{\ell}=0 \), which is valid since \( -1 \leq 0 \leq 1 \).

Step 3: Analyze Set (c)

For set (c), \( n=6 \) and \( \ell=5 \). This satisfies the condition \( 0 \leq \ell < n \). Additionally, \( m_{\ell} \) is \(-1\), which fits in the range \(-5 \leq -1 \leq 5\). Thus, this set is valid.

Step 4: Analyze Set (d)

Here, \( n=4 \) and \( \ell=3 \) meet the condition. However, \( m_{\ell} = -4 \), which must satisfy \(-\ell \leq m_{\ell} \leq \ell\). The range should be \(-3 \leq m_{\ell} \leq 3\), making this set invalid.

Key Concepts

Understanding the Principal Quantum Number

The principal quantum number, denoted by \( n \), is a critical component in the quantum mechanical model of an atom. It primarily signifies the size and energy level of an electron's orbital. The principal quantum number can take any positive integer value (1, 2, 3,...), which determines how far the orbital is from the nucleus. This distance influences the potential energy of electrons in that orbital.
For instance, when \( n = 1 \), the electron is closer to the nucleus and usually found in the lowest energy state, known as the ground state. As \( n \) increases, the electron's orbit becomes larger and it occupies higher energy levels.
A larger \( n \) implies a greater number of allowed azimuthal quantum numbers (\( \ell \)). Specifically, each value of \( n \) can have \( \ell \) ranging from 0 to \( n-1 \). For example, if \( n = 3 \), \( \ell \) can be 0, 1, or 2 but not 3.

Exploring the Azimuthal Quantum Number

The azimuthal quantum number or angular momentum quantum number, denoted by \( \ell \), describes the shape of an electron's orbital. The values of \( \ell \) are restricted by the principal quantum number, such that \( \ell \) ranges from 0 to \( n-1 \).
Each value corresponds to a specific orbital shape:

• \( \ell=0 \): The orbitals are spherically shaped (s-orbitals).
• \( \ell=1 \): The orbitals are shaped like dumbbells (p-orbitals).
• \( \ell=2 \): These orbitals resemble cloverleaves (d-orbitals).
• \( \ell=3 \): These are more complex (f-orbitals).

Thus, \( \ell \) not only affects the energy of orbitals but also their spatial orientation and complexity. Furthermore, \( \ell \) limits the possible values of the magnetic quantum number (\( m_\ell \)), which ranges from \(-\ell \) to \( \ell \). A set of quantum numbers becomes invalid if \( \ell \) is not within the specified range by \( n \). For example, \( \ell=3 \) when \( n=3 \) is invalid.

Decoding the Magnetic Quantum Number

The magnetic quantum number, denoted by \( m_\ell \), determines the orientation of the orbital within a magnetic field. Its values depend on the azimuthal quantum number \( \ell \).
The range for \( m_\ell \) is from \(-\ell \) to \( \ell \), including zero. For instance, if \( \ell = 2 \), \( m_\ell \) could be \(-2, -1, 0, 1, \) or \( 2 \). This means five possible orientations are available for the d-orbital of an electron.
These orientations determine how electrons are distributed in an atom's sublevels and how they interact with external magnetic fields. Incorrect values of \( m_\ell \) result in invalid quantum numbers. For example, if \( \ell = 3 \), \( m_\ell \) should lie within the set \{-3, -2, -1, 0, 1, 2, 3\}. Using \( m_\ell = -4 \) is invalid because it lies outside of this permissible range.