How many electron(s) in an atom can have \(n=3, l=2 ?\) (a) 1 (b) 2 (c) 5 (d) 10

The principal quantum number, denoted by the symbol \( n \), is one of the most fundamental concepts in quantum mechanics when it comes to the atomic structure. It indicates the main energy level occupied by an electron within an atom. As an analogy, think of \( n \) as the floor number in a hotel; the higher the floor, the higher the energy level of the electron.

The principal quantum number takes positive integer values (\( n=1, 2, 3, \ldots \)). The energy of an electron generally increases with an increase in \( n \). In practical terms, if an electron has \( n=3 \), like in our exercise, it's in the third energy level of the atom. This number is crucial because it partly determines the electron's energy and the size of the orbital in which the electron can be found.

The azimuthal quantum number, also known as the angular momentum quantum number and represented by \( l \), defines the shape of an electron's orbital, and to a certain extent, the energy of the orbital in a given energy level. It can take on any integer value from 0 to \( n-1 \) for each principal quantum number. For example, if \( n=3 \), then \( l \) could be 0, 1, or 2.

These \( l \) values correspond to different subshells--s (0), p (1), d (2), f (3), and so on--which tells us about the shape of the electron cloud. When we say \( l=2 \), it signifies the d subshell, known for its more complex, cloverleaf-like shapes. Each type of subshell has a distinct electron capacity that governs how many electrons it can host.

The magnetic quantum number, symbolized as \( m_l \), provides information about the orientation of the orbital in space relative to the other orbitals. It is determined by the azimuthal quantum number \( l \) and can take values ranging from \( -l \) to \( +l \), which includes zero. So, if \( l=2 \), the possible \( m_l \) values are -2, -1, 0, 1, 2.

Each \( m_l \) value corresponds to a different orientation of the orbital, which means an electron can be found in different spatial directions around the nucleus. Having multiple orientations is like having different seats facing different directions, all within the same section of a stadium. These orientations play a key role in how atoms bond or interact with magnetic fields.

The spin quantum number, denoted by \( m_s \), represents the intrinsic spin of the electron, which is a form of angular momentum. Electrons can spin in one of two possible states: spin-up (+1/2) or spin-down (-1/2).

Spin is a fundamental property of electrons, just like mass or charge, and the two possible spin states contribute to the magnetic properties of an atom. For each orbital orientation \( m_l \), there can be two electrons, one with each spin state. The exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers, so the spin is crucial to distinguish between two electrons occupying the same orbital space.

Atomic subshells are a division of electron shells (principal quantum levels) based on the azimuthal quantum number \( l \). Each electron shell has one or more subshells, and they are labeled s, p, d, f (in ascending order of energy within the same principal quantum level).

• The s subshell (\( l=0 \)) has 1 orbital and can hold a maximum of 2 electrons.
• The p subshell (\( l=1 \)) has 3 orbitals and can hold a maximum of 6 electrons.
• The d subshell (\( l=2 \)) has 5 orbitals and can hold a maximum of 10 electrons, as highlighted in our exercise with \( n=3, l=2 \).
• The f subshell (\( l=3 \)) has 7 orbitals and can hold a maximum of 14 electrons.

Identifying the subshell is critical for understanding the electron configuration of an atom and for predicting chemical properties and bonding behavior. Each subshell has a unique shape and spatial orientation, contributing significantly to the atom's overall electron distribution and chemical characteristics.