Given (A) n = 5, $$m_l$$ = +1 (B) n = 2, $$l$$ = 1, $$m_l$$ = -1, $$m_s$$ = -1/2. The maximum number of electron(s) in an atom that can have the quantum numbers as given in (A) and (B) are respectively:

To solve this problem, we need to determine the maximum number of electrons in an atom that can have the quantum numbers specified in parts (A) and (B). Quantum numbers describe the properties of atomic orbitals and the electrons within them. The principal quantum number $$n$$ indicates the shell, the azimuthal quantum number $$l$$ indicates the subshell (where $$l$$ ranges from 0 to $$n-1$$), the magnetic quantum number $$m_l$$ indicates the orbital (where $$m_l$$ ranges from $$-l$$ to $$+l$$), and the spin quantum number $$m_s$$ indicates the electron spin (either $$+\frac{1}{2}$$ or $$-\frac{1}{2}$$).

Starting with part (A): given $$n = 5$$ and $$m_l = +1$$. We are not given $$l$$ or $$m_s$$, so we must consider all possible values of $$l$$ for $$n = 5$$. Since $$n = 5$$, $$l$$ can be 0, 1, 2, 3, or 4.

For each $$l$$, we check if $$m_l = +1$$ is possible. The value of $$m_l$$ must satisfy $$-l \leq m_l \leq +l$$:

• If $$l = 0$$, $$m_l$$ can only be 0. So $$m_l = +1$$ is not possible.
• If $$l = 1$$, $$m_l$$ can be $$-1, 0, +1$$. So $$m_l = +1$$ is possible.
• If $$l = 2$$, $$m_l$$ can be $$-2, -1, 0, +1, +2$$. So $$m_l = +1$$ is possible.
• If $$l = 3$$, $$m_l$$ can be $$-3, -2, -1, 0, +1, +2, +3$$. So $$m_l = +1$$ is possible.
• If $$l = 4$$, $$m_l$$ can be $$-4, -3, -2, -1, 0, +1, +2, +3, +4$$. So $$m_l = +1$$ is possible.

Thus, $$m_l = +1$$ is possible for $$l = 1, 2, 3, 4$$. For each of these $$l$$ values, there is exactly one orbital with $$m_l = +1$$. Each orbital can hold up to 2 electrons (one with $$m_s = +\frac{1}{2}$$ and one with $$m_s = -\frac{1}{2}$$). Therefore:

• For $$l = 1$$, the orbital with $$m_l = +1$$ can hold 2 electrons.
• For $$l = 2$$, the orbital with $$m_l = +1$$ can hold 2 electrons.
• For $$l = 3$$, the orbital with $$m_l = +1$$ can hold 2 electrons.
• For $$l = 4$$, the orbital with $$m_l = +1$$ can hold 2 electrons.

Adding these together: $$2 + 2 + 2 + 2 = 8$$ electrons. Hence, the maximum number of electrons with $$n = 5$$ and $$m_l = +1$$ is 8.

Now for part (B): given $$n = 2$$, $$l = 1$$, $$m_l = -1$$, and $$m_s = -\frac{1}{2}$$. Here, all four quantum numbers are specified. This uniquely identifies a specific orbital (defined by $$n$$, $$l$$, and $$m_l$$) and a specific spin state ($$m_s$$).

For $$n = 2$$ and $$l = 1$$, the possible $$m_l$$ values are $$-1, 0, +1$$. The orbital with $$m_l = -1$$ is one specific orbital. Each orbital can hold a maximum of 2 electrons, but they must have different spins: one with $$m_s = +\frac{1}{2}$$ and one with $$m_s = -\frac{1}{2}$$. Since we are specifying $$m_s = -\frac{1}{2}$$, there can be only one electron in this atom that has exactly these quantum numbers: $$n = 2$$, $$l = 1$$, $$m_l = -1$$, and $$m_s = -\frac{1}{2}$$.

Therefore, the maximum number of electrons for part (B) is 1.

Combining the results, the maximum number of electrons for (A) is 8 and for (B) is 1. Comparing with the options:

• A. 25 and 1
• B. 8 and 1
• C. 2 and 4
• D. 4 and 1

Hence, the correct answer is Option B.