How many electrons in an atom can have \(n=4, l=2, m=-2\) and \(s=+\frac{1}{2} ?\) (a) 1 (b) 2 (c) 5 (d) 10

In chemistry, the Pauli exclusion principle is a fundamental rule that is pivotal in defining the arrangement of electrons in an atom. It was formulated by Wolfgang Pauli in 1925, and it states that no two electrons in an atom can have the same set of four quantum numbers. These quantum numbers are addresses that locate the position of an electron in an atom and describe its energy, angular momentum, orientation in space, and spin.

Specifically, the principle dictates that if you have two electrons in the same orbital, they must have opposite spins — identified by their spin quantum numbers as +1/2 or -1/2. Because of this, any orbital can hold a maximum of two electrons. To better visualize this, imagine an orbital as a room with only two beds — one for each electron — and each bed can only be occupied by one electron with a unique spin. This concept helps predict the electronic structure of atoms, and it explains why electron configurations are built up in a certain way — an arrangement that minimizes the repulsion between electrons and results in the most stable electron cloud.

The azimuthal quantum number, also known as the angular momentum quantum number, is symbolized by 'l' and it provides essential information about the shape of an electron's orbital. The values of 'l' are integers that range from 0 to one less than the principal quantum number 'n' (i.e., from 0 to n-1). For each value of 'n', there are n different values of 'l'.

Here is how the values of the azimuthal quantum number correspond to different subshells:

• For l=0, the subshell is an 's' orbital, which is spherical in shape.
• For l=1, it indicates a 'p' orbital, with a dumbbell shape.
• For l=2, the subshell is a 'd' orbital, which generally has a more complex, cloverleaf shape.
• For l=3, it corresponds to an 'f' orbital, with an even more intricate shape.

Each of these subshells has a different number of orbitals and can hold different numbers of electrons, contributing to the complexity of the electron configuration within an atom.

Electron configuration is the arrangement of electrons around the nucleus of an atom in its orbitals. It provides a visual representation of where electrons are likely to be found within an atom. Understanding electron configuration is vital for elucidating chemical properties and bonding behaviors of elements.

The arrangement follows a set of rules known as 'Aufbau principle', 'Hund's rule' and the Pauli exclusion principle. According to the Aufbau principle, electrons fill the lowest energy orbitals first before moving to higher energy levels. Hund's rule states that electrons will occupy each orbital singly before any orbital is doubly occupied, and as per the Pauli exclusion principle, each electron in an atom must have a unique set of quantum numbers.

This structured fill-up process results in a specific sequence for electron arrangement, often noted by the energy levels and the subshells, such as 1s² 2s² 2p⁶, and so on. This notation indicates the number of electrons in each subshell and grants insight into the chemical reactivity and bonding potential of an atom. By knowing the electron configuration, one can predict how an atom will interact with others, its magnetic properties, and the types of chemical bonds it will form.