2.6.3. Energies of Orbitals

The energy of an electron in a hydrogen atom is determined solely by the principal quantum number. Thus the energy of the orbitals in hydrogen atom increases as follows: 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f.

In hydrogen, the energy levels of the electron are determined by the principal quantum number (n), which characterizes the size and energy of the orbital. As n increases, the energy levels become higher (less negative). For hydrogen, the energy levels can be listed in ascending order: the 1s orbital has the lowest energy, followed by the 2s and 2p orbitals, which are equal in energy, and so on for the 3s, 3p, and 3d orbitals. Higher principal quantum numbers correspond to orbitals that are further from the nucleus and have higher energies.

Imagine a ladder where each step represents an energy level. The bottom step is the most stable and has the least potential energy (like the 1s orbital). As you climb each ladder rung (stepping up to 2s, 2p, etc.), you move away from the ground (the nucleus) and gain energy until eventually, you're at the top of the ladder where the potential energy is the highest. This visual helps understand that the higher you go, the less stable and more energetic your position becomes.

Although the shapes of 2s and 2p orbitals are different, an electron has the same energy when it is in the 2s orbital as when it is present in 2p orbital. The orbitals having the same energy are called degenerate.

Degenerate orbitals are orbitals that have the same energy level despite having different shapes. In the case of the 2s and 2p orbitals, they are both considered to be at the same energy level even though 2s is spherical and 2p has a dumbbell shape. This means that when an electron occupies either the 2s or 2p orbital, it experiences the same energy, which is significant for understanding electron configurations.

Think of a playground with a slide and a swing that are both at the same height from the ground. Though the slide and the swing are shaped differently, they give you the same potential energy when you sit on them at that height. Similarly, the 2s and 2p orbitals are at the same energy level, allowing electrons to occupy either one without additional energy cost.

The energy of an electron in a multi-electron atom, unlike that of the hydrogen atom, depends not only on its principal quantum number (shell), but also on its azimuthal quantum number (subshell).

In multi-electron systems, electrons are subjected to more complex interactions due to the presence of multiple nuclei and electron repulsion. Although the principal quantum number (n) indicates the energy level, the azimuthal quantum number (l) introduces further differentiation in energy levels among orbitals. For instance, in the same principal shell, the energy order is typically s < p < d < f due to variances in how effective the shielding from the nucleus is among the electrons in different orbital types.

Consider a set of stairs where each floor represents an energy level determined mainly by how far you are from the ground (n). However, if you also think about the type of flooring (l) like hardwood, carpet, or tile, you realize that each type affects how comfortable it feels to stand on. Similarly, the atomic orbitals have varied energies based on both their distance from the nucleus and their type, meaning that s orbitals are 'lower' or 'more comfortable' than p orbitals at the same level.