In an atom, total number of electrons having quantum numbers n = 4, $|m_l|$ = 1 and $m_s$ = $-\frac{1}{2}$ is ________.
Correct Answer: 6
Approach Solution - 1
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.
Approach Solution -2
We are given: n = 4, |mₗ| = 1, and mₛ = –½.
We need to find how many electrons in an atom can have these values of quantum numbers.
Step 2: Identify possible values of quantum numbers.
For n = 4, possible subshells are: l = 0 (s), 1 (p), 2 (d), and 3 (f).
Given |mₗ| = 1, the possible values of mₗ are +1 and –1.
Hence, we must consider orbitals that have mₗ = +1 or –1.
Step 3: Check possible subshells.
- For l = 1 (p orbital): mₗ = –1, 0, +1 → includes ±1 (2 orbitals).
- For l = 2 (d orbital): mₗ = –2, –1, 0, +1, +2 → includes ±1 (2 orbitals).
- For l = 3 (f orbital): mₗ = –3, –2, –1, 0, +1, +2, +3 → includes ±1 (2 orbitals).
Step 4: Count total orbitals with given |mₗ| = 1.
From all subshells (p, d, f), total orbitals having mₗ = +1 or –1 = 2 + 2 + 2 = 6 orbitals.
Step 5: Spin condition.
Since mₛ = –½, each orbital will contain one electron with that spin.
Hence, total number of electrons = 6.
Step 6: Final Answer.
The total number of electrons having n = 4, |mₗ| = 1, and mₛ = –½ is 6.
Final Answer: 6
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