Question

Explain briefly why each of the following is not a possible set of quantum numbers for an electron in an atom. (a) \(n=2, \ell=2, m_{\ell}=0\) (b) \(n=3, \ell=0, m_{\ell}=-2\) (c) \(n=6, \ell=0, m_{\ell}=1\)

Short answer

All sets violate quantum number constraints. a) \(\ell < n\); b) \(m_{\ell} = 0\) required; c) \(m_{\ell} = 0\) required.

Step-by-step solution

Understanding Quantum Number Restrictions

To solve this problem, we must understand the restrictions on quantum numbers: 1. Principal quantum number ( "): "): any positive integer 2. Azimuthal quantum number ( : any positive integer 3. "): any integer that satisfies "): must satisfy : possible values of m_{ "): must satisfy m: possible values of "): must be one of the integers from _): integer values that satisfy _): integer values that satisfy m: integer values from m_{ })_): integer values from _): integer values from ").

Evaluating Case (a): \(n=2, \ell=2, m_{\ell}=0\)

In this case, the principal quantum number is \(n=2\). The azimuthal quantum number \(\ell\) must satisfy \(0 \leq \ell < n\). Since \(\ell=2\) for \(n=2\), this violates the condition \(\ell < n\). Therefore, this set of quantum numbers is not possible.

Evaluating Case (b): \(n=3, \ell=0, m_{\ell}=-2\)

Here, for \(n=3\) and \(\ell=0\), the magnetic quantum number \(m_{\ell}\) can only take the value 0 because \(m_{\ell}\) can range from \(-\ell\) to \(+\ell\). Since \(m_{\ell}=-2\) is not within this range, this set of quantum numbers is not possible.

Evaluating Case (c): \(n=6, \ell=0, m_{\ell}=1\)

Similarly, with \(n=6\) and \(\ell=0\), \(m_{\ell}\) must also be 0 for the same reason as in case (b). Since \(m_{\ell}=1\) does not meet this requirement, this set of quantum numbers is not possible.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as \( n \), plays a fundamental role in discerning the energy level of an electron within an atom. It is always a positive integer (e.g., 1, 2, 3, ...). This number defines the major electron shell, or orbit, around the nucleus and determines the overall size and energy of an electron's orbit.

• Lower values of \( n \) are closer to the nucleus, indicating electrons with lower energy.
• Higher values of \( n \) are further away, suggesting electrons with higher energy.

For instance, in the example where \( n = 2 \), the electron could reside in the second energy level of the atom. The number directly affects the potential number of sublevels within each major level, which are defined by the azimuthal quantum number.

Azimuthal Quantum Number

The azimuthal quantum number, \( \ell \), is crucial for determining the shape of an electron's orbit within a particular shell. This quantum number can take on integer values starting from 0 up to \( n-1 \). Thus, it reflects the sublevels or subshells possible for a given principal quantum number.

• \( \ell = 0 \) correlates with an 's' orbital, which is spherical.
• \( \ell = 1 \) corresponds to 'p' orbitals, which have a dumbbell shape.
• \( \ell = 2 \) signifies 'd' orbitals, having a more complex clover shape.

For example, if \( n = 3 \), permissible \( \ell \) values are 0, 1, and 2. Importantly, if \( \ell \) exceeds the principal quantum number, like in the case of \( \ell = 2 \) and \( n = 2 \), such configurations are impossible, as the azimuthal quantum number must always be less than the principal quantum number.

Magnetic Quantum Number

The magnetic quantum number, represented as \( m_\ell \), describes the orientation of an electron's orbital with respect to an external magnetic field. This quantum number ranges from \(-\ell\) to \(+\ell\), encompassing all integers in between. Thus, it indicates how many orbitals exist within a given subshell.

• A subshell characterized by \( \ell = 0 \) (an 's' orbital) will have only one orientation, so \( m_\ell \) can only be 0.
• For \( \ell = 1 \) (a 'p' orbital), \( m_\ell \) includes -1, 0, and +1, indicating three possible orientations.
• For \( \ell = 2 \) (a 'd' orbital), \( m_\ell \) ranges from -2 to +2.

In the scenarios provided, if \( \ell = 0 \), the only valid \( m_\ell \) value is 0. Any deviation, such as \( m_\ell = -2 \) or \( m_\ell = 1 \), becomes impossible since they do not align with the allowed range for the given \( \ell \).