For a given value of the principal quantum number, \(n\), how do the energies of the \(s, p, d,\) and \(f\) subshells vary for (a) hydrogen, (b) a many-electron atom?

Understanding the principal quantum number is fundamental to comprehending the structure of atoms. It is expressed as the variable 'n' which indicates the overall size and energy level of an orbital where an electron can reside. For hydrogen, since there's only a single electron, the principal quantum number alone determines the energy of its electron. The larger the value of 'n', the farther from the nucleus the electron is likely found and the higher the energy level. This is represented mathematically as:
\[\begin{equation}E_n = -\frac{13.6 \, eV}{n^2}\end{equation}\]
In contrast, for atoms with many electrons, the energy associated with a particular quantum number is influenced not only by 'n' but also by electron-electron interactions and the presence of other quantum numbers, creating a more complex energy relationship.

Subshells are characterized by azimuthal quantum numbers, represented as 'l', which take on values ranging from 0 to (n-1). Each value of 'l' corresponds to a specific type of subshell:

• (l=0) is an 's' subshell
• (l=1) corresponds to a 'p' subshell
• (l=2) matches a 'd' subshell
• (l=3) designates an 'f' subshell

In hydrogen, subshell energies are degenerate—their energy is the same for a given principal quantum number. However, in many-electron atoms, electrons in higher 'l' subshells experience greater repulsion and shielding effects, leading to subtle differences in energy levels even within the same principal quantum number.

In the simplest atom, hydrogen, the electron configuration is determined by the principal quantum number. With only one electron, the energy levels are dictated primarily by the value of 'n' and all subshells within the same principal energy level are degenerate, having the same energy. Hence, a 2s orbital has the same energy as a 2p, 2d, or 2f orbital, which illustrates why hydrogen's electron, while most stable in its ground state (1s), can be excited to higher levels that maintain similar energies, regardless of the subshell type.

The electron configuration of many-electron atoms is a more involved process compared to hydrogen. Here, both the principal (n) and azimuthal (l) quantum numbers are at play, leading to a splitting of energy levels within the same principal energy level. Electrons fill from the lowest to the highest energy subshell, following the 'n+l' rule wherein subshells with a lower sum of 'n+l' have lower energy. Also, for subshells with the same 'n+l' value, the subshell with the lower 'n' value is lower in energy. As a result, the energy ordering for a given principal quantum number 'n' would follow as: s < p < d < f. This concept is crucial in predicting the chemical behavior of elements, as it guides the distribution of electrons into various shells and subshells, ultimately influencing the atom's chemistry.