Question

Given (A) \(n=5, m_{\ell} =+1\) (B) \(n=2, \ell=1, m_{\ell} =1, m_{s} =-\frac{1}{2}\) The maximum number of electron(s) in an atom that can have the quantum numbers as given in (A) and (B) are respectively:

• A. 25 and 1
• B. 8 and 1
• C. 2 and 4
• D. 4 and 1

Answer 1

The Correct Option is B

To determine the maximum number of electrons that can have the specified quantum numbers for parts (A) and (B), we first need to understand the significance of these quantum numbers in atomic structure. Quantum numbers describe the properties of atomic orbitals and the electrons within them.

The relevant quantum numbers are:

• n - The principal quantum number describes the energy level and size of the orbital.
• l - The azimuthal quantum number describes the shape of the orbital.
• m_l - The magnetic quantum number describes the orientation of the orbital.
• m_s - The spin quantum number describes the spin of the electron.

Let's solve each part separately:

Part (A):

The given quantum numbers are \( n = 5 \) and \( m_{\ell} = +1 \). However, it seems the azimuthal quantum number \( \ell \) is missing and is crucial for finding available orbitals. Thus, let's explore:

• If \( n = 5 \), possible values for \( \ell \) are 0, 1, 2, 3, and 4.
• For any given \( \ell \), the possible values of \( m_{\ell} \) are from \(-\ell\) to \(+\ell\).
• Since \( m_{\ell} = +1 \), it implies that \( \ell \) could be at least 1 (as it must be \(\ge 1\)).

Each specific value of \( (n, \ell, m_{\ell}) \) can fit 2 electrons (with \( m_{s} = +\frac{1}{2} \) and \( -\frac{1}{2} \)). Therefore, the electrons can occupy states satisfying the conditions determined by these quantum numbers.

Conclusion for Part (A): With multiple orbitals possible for different \( \ell \) values leading to unique \( m_{\ell} \) values as allowed under given conditions: If we only consider \(\ell = 1\) (p orbital), \( m_{\ell}=+1 \) is one particular direction; the maximum number of electrons here across valid states can be therefore 2 for each \(\ell\) and \(\{n, m_{\ell}\}\) observed at \( n=5 \).

Part (B):

Given - \( n=2, \ell=1, m_{\ell} =1, m_{s} =-\frac{1}{2} \)

• For \( n = 2 \), possible \( \ell \) values are 0 and 1. Here, only \( \ell = 1 \) is given.
• Possible \( m_{\ell} \) values when \( \ell = 1 \) are \(-1\), \(0\), \(1\). Given \( m_{\ell} = 1 \).
• The specified \( m_{s} = -\frac{1}{2} \) specifies electron spin.

This represents a unique electron configuration state, so only one electron can exist with these specific quantum numbers.

Conclusion for Part (B): The maximum number of electrons with these quantum numbers is exactly 1.

Thus, matching each condition, the maximum number of electrons that can exist is 8 and 1 for configuration (A) and (B) respectively, making the choice correct.