How many electrons in an atom can have the principal quantum number n = 8?

• A. 16
• B. 128
• C. 8
• D. 64

Answer: (B)

Verified step by step guidance

1. Recall that the principal quantum number \( n \) defines the main energy level or shell of an electron in an atom.
2. For a given \( n \), the possible values of the azimuthal quantum number \( l \) range from 0 to \( n-1 \). So for \( n = 8 \), \( l \) can be 0, 1, 2, ..., 7.
3. For each \( l \), the magnetic quantum number \( m_l \) can take values from \( -l \) to \( +l \), giving \( 2l + 1 \) possible values.
4. Each unique combination of \( n \), \( l \), and \( m_l \) corresponds to an orbital, and each orbital can hold 2 electrons (due to the two possible spin quantum numbers \( m_s = +\frac{1}{2} \) or \( -\frac{1}{2} \)).
5. Calculate the total number of electrons for \( n = 8 \) by summing over all \( l \) values: total electrons = \( 2 \times \sum_{l=0}^{7} (2l + 1) \). This sum gives the total number of orbitals times 2 electrons per orbital.