Question

Which of the following sets of quantum numbers is correct for an electron in 4f orbital?

• A. \(n = 3, \, l = 2, \, m_l = -2, \, m_s = +\frac{1}{2}\)
• B. \(n = 4, \, l = 3, \, m_l = +1, \, m_s = +\frac{1}{2}\)
• C. \(n = 4, \, l = 3, \, m_l = +4, \, m_s = +\frac{1}{2}\)
• D. \(n = 4, \, l = 4, \, m_l = -4, \, m_s = -\frac{1}{2}\)

Hint:

For quantum numbers: - \(n\) (principal quantum number) determines the shell. - \(l\) (azimuthal quantum number) determines the subshell. - \(m_l\) determines the orientation of the orbital, and it ranges from \(-l\) to \(+l\). - \(m_s\) represents the spin, with values of \(\pm \frac{1}{2}\).

Answer 1

The Correct Option is B

Step 1: Understanding the given problem.
We are given that the electron is in the 4f orbital.
The 4f orbital corresponds to \(n = 4\) (principal quantum number) and \(l = 3\) (azimuthal quantum number), which indicates it's an \(f\)-type orbital.
Step 2: Determining valid quantum numbers.
\(n = 4\) means the electron is in the fourth shell.
For the 4f orbital, \(l = 3\). The \(l\) quantum number corresponds to the type of orbital:
\(l = 0\) for \(s\)-orbitals
\(l = 1\) for \(p\)-orbitals
\(l = 2\) for \(d\)-orbitals
\(l = 3\) for \(f\)-orbitals
Step 3: Determining the possible values of \(m_l\).
The magnetic quantum number \(m_l\) can take values from \(-l\) to \(+l\), including zero.
For \(l = 3\), the possible values for \(m_l\) are: \(-3, -2, -1, 0, 1, 2, 3\).
Step 4: Determining the possible values of \(m_s\).
The spin quantum number \(m_s\) can only have two values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
Step 5: Analyzing the options.
Option (1) \(n = 3, \, l = 2, \, m_l = -2, \, m_s = +\frac{1}{2}\) is incorrect because \(n = 3\) corresponds to the third shell, not the fourth shell, and it’s a \(d\)-orbital, not \(f\).
Option (2) \(n = 4, \, l = 3, \, m_l = +1, \, m_s = +\frac{1}{2}\) is correct because it satisfies the conditions for a 4f orbital.
Option (3) \(n = 4, \, l = 3, \, m_l = +4, \, m_s = +\frac{1}{2}\) is incorrect because \(m_l\) must be between \(-3\) and \(+3\).
Option (4) \(n = 4, \, l = 4, \, m_l = -4, \, m_s = -\frac{1}{2}\) is incorrect because \(l\) cannot be 4 for an \(f\)-orbital. For \(n = 4\), the highest value of \(l\) is 3.
Step 6: Conclusion.
Thus, the correct answer is option (2).