The question asks for the maximum number of electrons in an atom that correspond to a specific set of quantum numbers.

Quantum Numbers Provided:

• $n = 4$ (principal quantum number, indicating the energy level)
• $l = 3$ (azimuthal quantum number, indicating the sublevel, here $l = 3$ corresponds to the $f$-orbital)
• $m_l = -2$ (magnetic quantum number, specifying an orbital in the $f$-sublevel)
• $m_s = +\frac{1}{2}$ (spin quantum number, indicating the spin orientation of the electron)

Explanation:

1. Interpretation of the Set of Quantum Numbers:

   • The quantum numbers $n = 4$ and $l = 3$ describe the $4f$-orbital.
   • $m_l = -2$ specifies a particular orbital within the $f$-sublevel.
   • $m_s = +\frac{1}{2}$ describes one electron within that orbital with a specific spin.

2. Electron Capacity:

   • Each orbital (defined by $m_l$) can hold a maximum of 2 electrons, one with $m_s = +\frac{1}{2}$ and one with $m_s = -\frac{1}{2}$.
   • Since $m_s = +\frac{1}{2}$ is already specified, only one electron fits this description.

Answer:

The maximum number of electrons with the given quantum numbers is: $\boxed{1}$ (Option E)

Let me know if you'd like a deeper explanation or have further questions!

Here are 5 follow-up questions to deepen your understanding:

1. What do the quantum numbers $n, l, m_l,$ and $m_s$ represent in atomic structure?
2. How does the value of $l$ determine the type of orbital (e.g., $s, p, d, f$)?
3. What is the maximum number of electrons in an $f$-sublevel?
4. How does the Pauli Exclusion Principle relate to quantum numbers?
5. How are quantum numbers used to describe electron configurations?

Tip: Remember, the spin quantum number $m_s$ ensures that no two electrons in the same orbital can have identical sets of quantum numbers!