For the orbital angular momentum of an electron in a 2s-orbital.

The question asks for the orbital angular momentum of an electron in a 2s-orbital.

The question asks to identify the valid set of quantum numbers among the given options. The rules for valid quantum numbers are:

• $n$ must be a positive integer ($n \ge 1$).
• $l$ can range from 0 to $n-1$.
• $m$ can range from $-l$ to $+l$, including 0.
• $s$ can be $+\frac{1}{2}$ or $-\frac{1}{2}$.

Let's check each option:

• $n=3$: Valid (positive integer).
• $l=2$: Valid, as $0 \le l \le n-1 \Rightarrow 0 \le 2 \le 3-1 \Rightarrow 0 \le 2 \le 2$.
• $m=-2$: Valid, as $-l \le m \le +l \Rightarrow -2 \le -2 \le +2$.
• $s=+\frac{1}{2}$: Valid. This set of quantum numbers is valid.

• $n=3$: Valid.
• $l=2$: Valid.
• $m=-1$: Valid.
• $s=0$: Invalid. The spin quantum number $s$ must be either $+\frac{1}{2}$ or $-\frac{1}{2}$.

• $n=2$: Valid.
• $l=2$: Invalid. For $n=2$, $l$ can only be 0 or 1 ($0 \le l \le n-1 \Rightarrow 0 \le l \le 2-1 \Rightarrow 0 \le l \le 1$).

This option is identical to option (3).

• $n=2$: Valid.
• $l=2$: Invalid. For $n=2$, $l$ can only be 0 or 1.

Only option (1) represents a valid set of quantum numbers.

The correct option is (1).