The e quantum numbers of four electrons (e1 to e4) are given below : $$ \begin{array}{ccccccccc}n & l & m & s & & n & l & m & s \\ 3 & 0 & 0 & +1 / 2 & e 2 & 4 & 0 & 0 & 1 / 2 \\ 3 & 2 & 2 & -1 / 2 & e 4 & 3 & 1 & -1 & 1 / 2\end{array} $$ \(\begin{array}{r}\text { The quar } \\ e 1 & n \\ e 3 & 3\end{array}\) The correct order of decreasing energy of these electrons is: (a) \(e 4>e 3>e 2>e 1\) (b) \(e 2>e 3>e 4>e 1\) (c) \(e 3>e 2>e 4>e 1\) (d) \(e 1>e 4>e 2>e 3\)

The principal quantum number, denoted as 'n', serves as one of the critical components in determining the properties of electrons within an atom. It is integral in defining the overall size and energy level of an electron's orbit, with larger values of 'n' indicating orbits that are farther from the nucleus. This quantum number takes on positive integer values - 1, 2, 3, and so on.

An electron with a higher principal quantum number is positioned in a higher energy level or shell, which correlates to a greater potential energy due to the increased distance from the positively charged nucleus. In our exercise, the principal quantum number was key to distinguishing between the energy levels of different electrons and establishing which occupied the higher energy state. Remember, the higher the value of 'n', the higher the energy level of the electron.

The azimuthal quantum number, designated by the letter 'l', indicates the shape of the electron's orbital and influences the energy level within a given principal quantum number. Each value of 'l' corresponds to a specific type of orbital, divisible into subshells labeled as 's', 'p', 'd', and 'f'. The value of 'l' can range from 0 to 'n' - 1, where for 's' orbitals, 'l' is 0, for 'p' orbitals, 'l' is 1, and so on.

Orbitals with higher azimuthal quantum numbers have more complex shapes and generally higher energy within the same principal quantum number. Therefore, in scenarios where electrons inhabit the same shell (same 'n'), those with higher 'l' values are at a higher energy state. Our exercise demonstrated this concept, where we compared the energy of electrons based on both 'n' and 'l' when 'n' was the same for multiple electrons.

Orbital energy levels are the distinct energy states that electrons can occupy in an atom. These levels are determined by the principal and azimuthal quantum numbers. Within each energy level, there can be one or more sublevels corresponding to the azimuthal quantum number.

Additionally, factors such as the electron's interactions with other electrons, nuclear charge, and the specific distribution of electrons can influence orbital energies. For example, in multi-electron atoms, electrons can repel each other, altering the energy of an orbital compared to a hydrogen-like (single electron) system. In the exercise, ordering the electrons by decreasing energy required a comprehensive analysis of these orbital energy levels based on the quantum numbers.

Electron configuration refers to the arrangement of electrons in an atom's orbitals, following the Pauli exclusion principle, Hund's rule, and the Aufbau principle. The Pauli exclusion principle states that no two electrons can have the same set of four quantum numbers; therefore, each electron in an orbital must have opposing spins. Hund's rule suggests that electrons will fill degenerate orbitals (orbitals with the same energy) singly before they pair up. The Aufbau principle guides the order in which orbitals are filled, starting from the lowest energy level to higher ones.

The information about each electron's quantum numbers in our exercise helped determine their respective positions in the atom's electron configuration. Correctly identifying the configuration is vital for understanding chemical properties and reaction behaviors. In summary, electron configuration is the roadmap for 'housing' electrons in an atom's 'neighborhood' of orbitals.