Select the correct quantum numbers for the highlighted electrons in a set of 5d orbitals.

• A. n = 5, l = 3, m_l = –4, m_s = +1/2
• B. n = 4, l = 4, m_l = +1, m_s = +1/2
• C. n = 5, l = 2, m_l = +1, m_s = +1/2
• D. n = 5, l = 5, m_l = –2, m_s = +1/2
• E. n = 5, l = 2, m_l = +5, m_s = +1/2

Answer: (C)

Verified step by step guidance

1. Step 1: Identify the principal quantum number (n) for the 5d orbitals. Since the orbitals are 5d, the principal quantum number is \(n = 5\).
2. Step 2: Determine the azimuthal quantum number (l) for d orbitals. For d orbitals, \(l = 2\) (since s=0, p=1, d=2, f=3, etc.).
3. Step 3: Analyze the magnetic quantum number (\(m_l\)) values shown in the image. The image shows orbitals labeled with \(m_l\) values from \(-2\) to \(+2\), which matches the allowed \(m_l\) values for \(l=2\) (i.e., \(m_l = -2, -1, 0, +1, +2\)).
4. Step 4: Check the spin quantum number (\(m_s\)) for the highlighted electrons. The spin quantum number can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\). The highlighted electrons have spin up, so \(m_s = +\frac{1}{2}\).
5. Step 5: Combine all quantum numbers for the highlighted electron: \(n = 5\), \(l = 2\), \(m_l = +1\), and \(m_s = +\frac{1}{2}\). This matches the correct set of quantum numbers for the highlighted electron in the 5d orbital.