How many electrons can occupy (a) the 4p-orbitals? (b) the \(3 \mathrm{~d}\)-orbitals? (c) the \(1 \mathrm{~s}\)-orbital? (d) the \(4 \mathrm{f}\)-orbitals?

Quantum numbers play a pivotal role in determining the position and energy of electrons within an atom. Each electron in an atom is described by a set of four quantum numbers: the principal quantum number (), the azimuthal (angular momentum) quantum number (), the magnetic quantum number (), and the spin quantum number ().

The principal quantum number, denoted by 'n', specifies the energy level or shell of an electron and ranges from 1 to infinity. It can be thought of as defining the size of the electron cloud. The azimuthal quantum number, represented by 'l', defines the shape of the orbital and has values ranging from 0 to \( n-1 \), where each number corresponds to a different type of orbital (0 for s, 1 for p, 2 for d, and 3 for f). The magnetic quantum number,, can range from \(-l\) to \(+l\), indicating the orientation of the orbital in space. Lastly, the spin quantum number, with values of \(-\frac{1}{2}\) or \(+\frac{1}{2}\), denotes the two possible orientations of an electron's spin.

Understanding these quantum numbers is essential for grasping the intricacies of electron configuration and the electronic structure of atoms.

Atomic orbitals are regions in an atom where there is a high probability of finding electrons. These orbitals are defined by the quantum numbers and are crucial for understanding the electron capacity of orbitals discussed in the exercise. Based on quantum numbers, electrons reside in s, p, d, or f orbitals with specific shapes—spherical for s orbitals, dumbbell-shaped for p orbitals, clover-shaped for d orbitals, and complex shapes for f orbitals.

As highlighted in the exercise solution, the number of orbitals within these types significantly affect electron capacity. S orbitals have one orbital, which can hold up to 2 electrons. In contrast, p, d, and f orbitals have 3, 5, and 7 orbitals, respectively, which means they can each hold 2 electrons, leading to total capacities of 6, 10, and 14 electrons correspondingly. This relationship between the types of orbitals and their electron capacities is fundamental when predicting the arrangement of electrons in atoms.

Electron configuration is the arrangement of electrons in an atom's orbitals. It follows the Pauli exclusion principle, Hund's rule, and the Aufbau principle to minimize the energy of the atom. In electron configurations, we describe the distribution of electrons among the orbitals based on the increasing order of energy levels.

The notation involves writing the principal quantum number followed by the letter representing the orbital type (s, p, d, f) and a superscript number indicating the amount of electrons in those orbitals. For instance, the solution indicates that the 4p-orbitals can hold 6 electrons, which is represented as 4p^6 in electron configuration notation. This notation reflects the quantum mechanical understanding of atomic structure and is essential for predicting chemical properties and behaviors of elements.

Quantum chemistry is a branch of chemistry focused on the application of quantum mechanics to chemical systems. It explains the behavior of electrons in atoms and molecules, encompassing principles like those involving the electron capacity of orbitals.

Understanding quantum chemistry is vital to grasp how electrons are distributed within atoms, how chemical bonds form, and how molecular shapes determine chemical reactions. The understanding of quantum numbers, atomic orbitals, and electron configurations are core concepts in quantum chemistry and form the theoretical underpinnings for how electronic structure influences chemical properties and reactivity. This field intersects with many areas of chemistry and physics, providing foundational knowledge that is applied in various scientific endeavors, from material science to pharmacology.