How many electrons may enter the orbital denoted by (a) \(2 p\), (b) \(I\) s, (c) \(3 p\), (d) \(3 d\)

Understanding the Pauli's exclusion principle is crucial when learning about the behavior of electrons in atoms. This principle, introduced by Wolfgang Pauli in 1925, states that no two electrons in an atom can have the same set of four quantum numbers. Since quantum numbers are like addresses for electrons, this ensures that each electron has a unique place in an atom.

What are these quantum numbers? They include the principal quantum number (related to the electron's energy level), the orbital angular momentum quantum number (related to the shape of the orbital), the magnetic quantum number (which describes the orientation of the orbital), and the spin quantum number (which indicates the direction of the electron's spin).

According to the Pauli's exclusion principle, an orbital can hold a maximum of two electrons, and those electrons must have opposite spins. Spin is a fundamental property of electrons and can be thought of as either 'up' or 'down'—represented by spin quantum numbers of +1/2 or -1/2. This is why when you see an electron configuration, paired electrons are shown with arrows pointing in opposite directions, symbolically representing their spins.

Electron configuration refers to the distribution of electrons in an atom's orbitals. Knowing which orbitals electrons occupy helps us predict an element's chemical behavior. Electrons fill orbitals starting from the lowest energy level moving towards higher energy levels, a guideline known as the Aufbau principle. You may visualize this as people entering a theater; they tend to fill the front rows before moving to the back.

The order in which electrons fill the orbitals is based on their energy; the '1s' orbital is filled first, followed by '2s', then '2p', and so on. The order gets more complex with larger atoms, but the principles that dictate the electron configuration remain the same. Chemists use a special notation to represent this configuration, like '1s² 2s² 2p⁶' for neon, which tells us that neon has a full '1s' orbital, a full '2s' orbital, and a full '2p' sublevel, reflecting a particularly stable arrangement of electrons.

Each orbital has a specific capacity to hold electrons, determined by its type ('s', 'p', 'd', or 'f') and its energy level (given by the principal quantum number, 'n'). An 's' orbital, being spherical, has room for two electrons, a 'p' orbital has three dumbbell-shaped sub-orbitals and can hold six electrons, and 'd' orbitals, with five complex-shaped sub-orbitals, can accommodate ten electrons.

The maximum number of electrons within a specific orbital relates directly to the Pauli's exclusion principle and the magnetic quantum numbers. For instance, the solution to the exercise provided indicates that the '2p' orbital has three magnetic quantum numbers (-1, 0, +1), each corresponding to a sub-orbital. With two electrons per sub-orbital (each having opposite spins), we get a total of six electrons for the entire 'p' orbital. As for the '3d' orbital, with its five sub-orbitals, it can hold up to ten electrons. Thus, the orbital capacity is a fundamental concept in predicting and writing out electron configurations of elements.

The angular momentum quantum number, represented by 'l', is fundamental to understanding electron configuration and orbital shapes within an atom. This quantum number can have integer values ranging from 0 to n-1, where 'n' is the principal quantum number. For example, if an electron is in the second energy level (n=2), the possible values for 'l' are 0 and 1, corresponding to 's' and 'p' orbitals, respectively.

The value of 'l' indicates the shape of the orbital—an 's' orbital (l=0) is spherical, 'p' orbitals (l=1) are dumbbell-shaped, 'd' orbitals (l=2) are cloverleaf-shaped, and 'f' orbitals (l=3) are even more complex. The angular momentum quantum number also affects how many sub-orbitals are available: one for 's', three for 'p', five for 'd', and seven for 'f'. This number is crucial because it tells us not only the shape of the orbitals but also how many electrons they can potentially hold.