Question

In an atom, total number of electrons having quantum numbers n = 4, $|m_l|$ = 1 and $m_s$ = $-\frac{1}{2}$ is ________.

Answer 1

Correct Answer: 6

The quantum numbers are defined as follows:
\(n = 4\): Principal quantum number.
\(|m_l| = 1\): Absolute value of the magnetic quantum number.
\(m_s = -\frac{1}{2}\): Spin quantum number.
Step 1: Determine the possible values of \(l\) and \(m_l\)
For \(n = 4\), the possible values of the azimuthal quantum number \(l\) are:
\[l = 0, 1, 2, 3.\]
For each \(l\), the possible values of \(m_l\) are as follows:
\(l = 0\): \(m_l = 0\).
\(l = 1\): \(m_l = -1, 0, +1\).
\(l = 2\): \(m_l = -2, -1, 0, +1, +2\).
\(l = 3\): \(m_l = -3, -2, -1, 0, +1, +2, +3\).
From the given condition \(|m_l| = 1\), the possible values of \(m_l\) are:
\[m_l = -1 \text{ or } +1.\]
Step 2: Count the orbitals corresponding to \(n = 4\) and \(|m_l| = 1\)
For \(l = 1\): \(m_l = \pm1\) (2 orbitals).
For \(l = 2\): \(m_l = \pm1\) (2 orbitals).
For \(l = 3\): \(m_l = \pm1\) (2 orbitals).
The total number of orbitals with \(|m_l| = 1\) is:
\[2 + 2 + 2 = 6 \, \text{orbitals}.\]
Step 3: Assign electrons with \(m_s = -\frac{1}{2}\)
Each orbital can hold one electron with \(m_s = -\frac{1}{2}\). Thus, the total number of electrons is:
\[6 \, \text{electrons}.\]
Final Answer: 6.