Identify a correct set of quantum numbers that represents the...

The energy of an electron in a multi-electron atom is determined by the $(n + l)$ rule. A higher value of $(n + l)$ corresponds to a higher energy level. If two orbitals have the same $(n + l)$ value, the one with the higher $n$ value has higher energy.

Calculating $(n + l)$ for the given options:

For $n = 3, l = 0$: $n + l = 3 + 0 = 3$

For $n = 3, l = 1$: $n + l = 3 + 1 = 4$

For $n = 3, l = 2$: $n + l = 3 + 2 = 5$

For $n = 4, l = 0$: $n + l = 4 + 0 = 4$

The maximum value of $(n + l)$ is $5$, which corresponds to the $3d$ orbital. Thus, the set of quantum numbers $n = 3, l = 2, m_l = +1, s = +1/2$ represents the highest energy.

Answer: $n = 3 , l = 2 , m_l = +1 , s = + 1/2$