Question

how many electrons can be assigned to orbitals designated by the quantum numbers n = 2, l = 1?

Explanation:

Step1: Determine the number of orbitals

For a given \( l = 1 \), the magnetic quantum number \( m_l \) can take values \( -l,-l + 1,\cdots,l \). So when \( l=1 \), \( m_l=- 1,0,1 \). The number of orbitals is \( 2l + 1=2\times1 + 1=3 \)

Step2: Calculate the number of electrons

Each orbital can hold a maximum of \( 2 \) electrons (due to the spin quantum number \( m_s=\pm\frac{1}{2} \)). So the number of electrons \( N = 2\times(2l + 1) \). Substituting \( l = 1 \), we get \( N=2\times3=6 \)

Answer:

\( 6 \)