How many electrons in an atom can have the principal quantum number n = 6?

• A. 12
• B. 36
• C. 72
• D. 24

Answer: (C)

Verified step by step guidance

1. Recall that the principal quantum number \( n \) defines the 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 \( n = 6 \), \( l \) can be 0, 1, 2, 3, 4, or 5.
3. Each \( l \) corresponds to a subshell with a specific number of magnetic quantum number \( m_l \) values, which range from \( -l \) to \( +l \), giving \( 2l + 1 \) orbitals per subshell.
4. Each orbital can hold 2 electrons (one with spin up and one with spin down). Therefore, the total number of electrons for \( n = 6 \) is calculated by summing over all subshells: 2 \( \times \) \( \sum_{l=0}^{5} (2l + 1) \).
5. Calculate the sum \( \sum_{l=0}^{5} (2l + 1) \), multiply by 2 to account for electron spin, and this will give the maximum number of electrons that can have \( n = 6 \).