Energy levels and sublevels Knowledge-Based Study

Study notes

1. In atomic structure, electrons occupy specific regions around the nucleus known as principal energy levels or electron shells. These levels are designated by the principal quantum number, represented by the symbol n.

   The value of n is a positive integer: $n = 1, 2, 3, 4, \dots$

   • Energy and Distance: As the value of n increases, the electron's average distance from the nucleus increases, and its potential energy increases. The level closest to the nucleus is $n = 1$, which corresponds to the lowest energy state.
   • Maximum Electron Capacity: The maximum number of electrons that can occupy a given principal energy level is calculated using the formula $2n^2$.

   Using this formula:

   • Level $n = 1$: maximum $2(1)^2 = 2$ electrons
   • Level $n = 2$: maximum $2(2)^2 = 8$ electrons
   • Level $n = 3$: maximum $2(3)^2 = 18$ electrons
   • Level $n = 4$: maximum $2(4)^2 = 32$ electrons

   Except for the first level, principal energy levels are divided into smaller components called sublevels (or subshells). Each sublevel is designated by a letter that describes its characteristic properties: s, p, d, and f.

   The number of sublevels present in any principal energy level is equal to the value of n:

   • Level $n = 1$ contains 1 sublevel: $1s$
   • Level $n = 2$ contains 2 sublevels: $2s, 2p$
   • Level $n = 3$ contains 3 sublevels: $3s, 3p, 3d$
   • Level $n = 4$ contains 4 sublevels: $4s, 4p, 4d, 4f$

   Within any given principal level in a multi-electron atom, the sublevels differ slightly in energy, increasing in the order: $s < p < d < f$.

   Each sublevel is composed of one or more atomic orbitals. An atomic orbital is a three-dimensional region of space where there is a high probability of finding an electron. By the Pauli exclusion principle, an individual orbital can hold a maximum of 2 electrons with opposite spins.

   The number of orbitals and the resulting maximum electron capacity for each sublevel type are structured as follows:

   • s sublevel: consists of 1 orbital, holding a maximum of $1 \times 2 = 2$ electrons.
   • p sublevel: consists of 3 orbitals, holding a maximum of $3 \times 2 = 6$ electrons.
   • d sublevel: consists of 5 orbitals, holding a maximum of $5 \times 2 = 10$ electrons.
   • f sublevel: consists of 7 orbitals, holding a maximum of $7 \times 2 = 14$ electrons.

   Notice that the number of orbitals increases through consecutive odd integers (1, 3, 5, 7), which systematically determines the electron capacity of each subshell.

   In atoms containing more than one electron, electron-electron repulsions cause sublevels within higher principal levels to overlap in energy.

   Key principles governing sublevel energies include:

   • Within the same principal level: Orbitals in an $s$ sublevel have greater electron penetration near the nucleus than $p$ orbitals, shielding the outer subshells and making $s$ lower in energy than $p$, which is lower than $d$, and so on ($ns < np < nd < nf$).
   • Cross-level overlap: As n increases, the energy gap between principal levels decreases. Consequently, the $4s$ sublevel is slightly lower in energy than the $3d$ sublevel in neutral, isolated atoms prior to filling.
   • Sublevel notation: An electron state is written by prefixing the sublevel letter with its principal energy level number, followed by a superscript showing electron count (for example, $2p^4$ designates four electrons residing in the $p$ sublevel of the second energy level).