(a) How many subshells are there for principal quantum number \(n=5\) ? (b) Identify the subshells in the form \(5 \mathrm{~s}\), etc. (c) How many orbitals are there in the shell with \(n=5\) ?

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It introduces a different perspective compared to classical mechanics, where energy and matter exhibit characteristics of both particles and waves.

In the realm of quantum mechanics, the idea of quantization is key. Energy levels are discrete, meaning electrons can only inhabit certain energy levels around an atom. These levels are defined by quantum numbers, which describe values of conserved quantities in the dynamics of the quantum system. The most significant of these is the principal quantum number, denoted by 'n'. It determines the energy level of an electron and broadly its distance from the nucleus.

In the provided exercise, understanding the role of quantum mechanics offers students a foundational concept that helps them make sense of how subshells and orbitals are structured within an atom.

Electron subshells are divisions of electron shells of an atom, where each shell corresponds to a principal quantum number 'n'. These subshells are designated by letters such as 's', 'p', 'd', 'f', and so on, which correspond to angular momentum quantum numbers of 0, 1, 2, 3, respectively.

For the principal quantum number 'n=5', as in the exercise, there are 5 subshells labeled as 5s, 5p, 5d, 5f, and 5g. These labels indicate not only the energy level (denoted by the number 5) but also the shape and complexity of the region in space where there is a high probability of finding an electron. By understanding that each subshell can contain a different number of orbitals, students can better grasp the arrangement of electrons in an atom and why certain subshells have more capacity to hold electrons than others.

For example, the s-subshell can only contain 1 orbital, while the p-subshell can contain 3, showing a simple but fundamental hierarchical structure within atoms based on these quantum properties.

Atomic orbitals represent regions in an atom where electrons are likely to be found, defined by a set of quantum numbers including the principal, angular momentum, and magnetic quantum numbers. Each orbital can hold a maximum of two electrons with opposite spins.

Using the 'n=5' shell from our exercise, we can calculate that there are a total of 25 orbitals. This is derived by adding the number of orbitals per type of subshell. Subshells have the following number of orbitals: s has 1, p has 3, d has 5, f has 7, and g has 9. The variety and shapes of these orbitals - spherical for an s orbital, dumbbell-shaped for p orbitals, and more complex forms for d and f orbitals - play a critical role in chemical bonding and reactions.

The understanding of atomic orbitals is essential for students as it helps explain why atoms exhibit certain electronic configurations and how they engage in chemical bonding, further illustrating the interconnectedness of quantum mechanics, subshell structure, and chemical properties of elements.