An orbital is occupied by an electron with the quantum numbers \(n=4, l=1 .\) How many orbitals of this type are found in a multi-electron atom? (a) \(4 p, 3\) (b) \(4 s, 1\) (c) \(4 d, 5\) (d) \(4 p, 6\)

The principal quantum number, denoted by the symbol \( n \), plays a crucial role in the field of quantum chemistry. It is integral to understanding the arrangement of electrons within an atom. Think of \( n \) as the address for an electron's energy level or shell; it signifies the distance of an electron's orbital from the nucleus and, consequently, its energy.

When we say an electron has a principal quantum number \( n = 4 \), we are indicating that this electron is located in the fourth energy level from the nucleus. The energy of the electron, as well as the size of its orbital, increases with higher values of \( n \). The principal quantum number can be any positive integer—that is, \( n = 1, 2, 3, 4, ... \), with each step up reflecting a move to a higher and more distant energy level around the nucleus.

Why is this important? Well, an electron's energy level is tied to its chemical properties. When atoms interact with each other, the principal quantum number helps us understand these interactions since electrons lower in energy (closer to the nucleus) are generally less reactive compared to those at higher levels. Understanding the concept of the principal quantum number is vital for predicting the behavior of atoms in chemical bonding and reactions.

The azimuthal quantum number, also known as the angular momentum quantum number and represented by \( l \), gives us more than just the shape of the electron cloud or orbital in an atom; it tells a story about the subshells and the type of chemical bonding an atom can participate in. The value of \( l \) is dependent on the principal quantum number, \( n \), as it can range from 0 up to \( n - 1 \).

Each value of \( l \) corresponds to a different subshell type, which is labeled using specific letters to indicate its shape: \( l = 0 \) is an s subshell (spherical shape), \( l = 1 \) is a p subshell (dumbbell shape), \( l = 2 \) corresponds to the d subshell (clover shape), and \( l = 3 \) is for the f subshell (complex shape). These subshells house the atomic orbitals, where electrons are most likely to be found.

Understanding the Azimuthal Quantum Number's Role

For instance, if an electron possesses \( n = 4 \) and \( l = 1 \), as in our exercise, the electron is in a '4p' orbital. Recognising this allows chemists to predict the electron's angular momentum, identifying how it will behave under the influence of magnetic fields—a phenomenon that underpins technologies like MRI scanning. Moreover, the azimuthal quantum number partially governs the energy differences between subshells, directly influencing the energy that electrons absorb or emit when moving from one subshell to another.

Atomic orbitals are the 'neighborhoods' of electrons within an atom defined by quantum numbers. They are not paths that electrons travel, but rather regions in space where there is a high probability of finding an electron. These orbitals are shaped by the principal and azimuthal quantum numbers, \( n \) and \( l \), giving each its unique geometrical form that can be spherically symmetrical, dumbbell-shaped, or even more complex.

When considering the question of how many orbitals of a certain type are found in a multi-electron atom, we look at these quantum numbers for guidance. The azimuthal quantum number, in particular, helps us define the number of orbitals in a subshell by the formula \( 2l+1 \). So, a p subshell (\( l=1 \)) will have \( 2(1)+1=3 \) orbitals, regardless of the principal quantum number's value.

Distribution of Electrons in Orbitals

According to the Pauli exclusion principle, each orbital can hold a maximum of two electrons with opposite spins. With this knowledge, we can determine that a 4p subshell will contain three orbitals and can hold up to six electrons. This distribution of electrons across orbitals in different subshells is instrumental in understanding the electronic structure of elements, predicting how they will interact in chemical reactions, and explaining the periodic table's structure. Addressing the exercise presented, by knowing the number and type of orbitals, students can visualize the electron distribution and recognize that for \( n=4 \) and \( l=1 \), there are three '4p' orbitals in a multi-electron atom, corresponding to answer (a).