1. Write the electronic configuration of potassium

Potassium has atomic number $19$, so it has $19$ electrons.

Its electronic configuration is:

$1s^2\,2s^2\,2p^6\,3s^2\,3p^6\,4s^1$

So the outermost electron is in the $4s$ orbital.

2. Determine the quantum numbers for a $4s$ electron

For the orbital $4s$:

• Principal quantum number: $n = 4$
• For an $s$-orbital, azimuthal quantum number: $l = 0$
• Magnetic quantum number for $l=0$: $m = 0$
• Spin quantum number can be $s = +\frac{1}{2}$ or $-\frac{1}{2}$

Since the option gives $+\frac{1}{2}$, that is acceptable.

Thus the four quantum numbers are:

$n=4,\quad l=0,\quad m=0,\quad s=+\frac{1}{2}$

3. Check the options

• A: $n=3, l=0, m=1$ is impossible because for $l=0$, $m$ must be $0$.
• B: $n=4, l=0, m=0, s=+\frac{1}{2}$ ✅ correct
• C: $n=2, l=0, m=0$ corresponds to $2s$, not the outermost electron.
• D: $n=4, l=2$ corresponds to $4d$, but potassium outermost electron is in $4s$.

Therefore, the correct option is B.