Question

what is the maximum number of electrons that can have n=4, m_l=-1 in an atom?

Explanation:

Step1: Determine possible \( l \) values for \( n = 4 \)

For a given principal quantum number \( n \), the azimuthal quantum number \( l \) can take values from \( 0 \) to \( n - 1 \). So when \( n = 4 \), \( l = 0, 1, 2, 3 \).

Step2: Analyze \( m_l = - 1 \) for each \( l \)

• For \( l = 0 \), \( m_l \) can only be \( 0 \), so \( m_l=-1 \) is not possible.
• For \( l = 1 \) (p - orbital), \( m_l=- 1,0, + 1 \), so \( m_l = - 1 \) is allowed.
• For \( l = 2 \) (d - orbital), \( m_l=-2,-1,0,+1,+2 \), so \( m_l = - 1 \) is allowed.
• For \( l = 3 \) (f - orbital), \( m_l=-3,-2,-1,0,+1,+2,+3 \), so \( m_l = - 1 \) is allowed.

Step3: Determine number of orbitals with \( m_l=-1 \)

For \( l = 1 \), there is 1 orbital with \( m_l=-1 \); for \( l = 2 \), there is 1 orbital with \( m_l=-1 \); for \( l = 3 \), there is 1 orbital with \( m_l=-1 \). So total number of orbitals with \( m_l=-1 \) and \( n = 4 \) is \( 3 \) (since \( l = 1,2,3 \) each contribute one orbital with \( m_l=-1 \)).

Step4: Calculate maximum electrons per orbital

Each orbital can hold a maximum of 2 electrons (due to Pauli exclusion principle, electrons have opposite spins).

Step5: Calculate total electrons

Total number of electrons \(= 3\times2=6\)

Answer:

6