Which of the following sets of quantum numbers represents the highest energy of an atom?

For multi-electron atoms, the relative energy of subshells is predicted by the $$n + \ell$$ rule.

1. Arrange subshells in increasing order of $$n + \ell$$ (principal quantum number $$n$$ plus azimuthal quantum number $$\ell$$).
2. If two subshells have the same $$n + \ell$$ value, the subshell with the larger $$n$$ lies higher in energy.

Compute $$n + \ell$$ for each option:

Option A $$n = 3,\; \ell = 2$$ ⟹ $$n + \ell = 3 + 2 = 5$$

Option B $$n = 4,\; \ell = 2$$ ⟹ $$n + \ell = 4 + 2 = 6$$

Option C $$n = 4,\; \ell = 0$$ ⟹ $$n + \ell = 4 + 0 = 4$$

Option D $$n = 3,\; \ell = 0$$ ⟹ $$n + \ell = 3 + 0 = 3$$

• The largest $$n + \ell$$ value is $$6$$ for Option B.
• No other option ties with this value, so no further comparison is needed.

Hence the subshell with the highest energy among the given choices is described by Option B.

Option B which is: $$n = 4, \ell = 2, m = 1, s = +1/2$$