Which elements are predicted to have the following groundstate electron configurations: (a) \([\mathrm{Kr}] 4 \mathrm{~d}^{10} 5 \mathrm{~s}^{2} 5 \mathrm{p}^{4} ;\) (b) \([\mathrm{Ar}] 3 \mathrm{~d}^{3} 4 \mathrm{~s}^{2}\); (c) \([\mathrm{He}] 2 \mathrm{~s}^{2} 2 \mathrm{p}^{2}\); (d) \([\mathrm{Rn}] 7 \mathrm{~s}^{2} 6 \mathrm{~d}^{2}\) ?

The periodic table is a comprehensive chart that organizes all known chemical elements in an informative array. It reflects the atomic structure of elements and their properties. Elements are arranged in order of increasing atomic number, with rows called periods and columns known as groups. The table is divided into blocks corresponding to the outer electron configurations, making it easier to predict the chemical behavior of an element.

For instance, as shown in the exercise's Step 1, the element selenium (Se) was identified through its electron configuration \( [\mathrm{Kr}] 4\mathrm{d}^{10} 5\mathrm{s}^{2} 5\mathrm{p}^{4} \). This configuration follows the order in which orbitals are filled according to the periodic table and reflects the position of selenium in the p-block of elements.

Orbital theory is a fundamental concept in chemistry that describes the behavior of electrons in atoms. It explains how electrons fill available energy levels and sub-levels within an atom. Orbitals are regions around the nucleus where the probability of finding an electron is the highest.

There are different types of orbitals—s, p, d, and f—each with a unique shape and a specific number of orientations. For example, in Step 2 of the solution, scandium (Sc) has electrons in the 3d and 4s orbitals, indicating that after filling the lower energy 4s orbital, electrons begin to populate the higher energy 3d orbitals. This order of filling is dictated by the principles of orbital theory and is key to understanding the structure of the periodic table.

Quantum numbers describe the unique quantum state of an electron in an atom, which is its address within the atom. There are four quantum numbers: the principal quantum number (n), the angular momentum quantum number (l), the magnetic quantum number (m_l), and the spin quantum number (m_s).

The principal quantum number, n, indicates the main energy level or shell. Angular momentum, l, defines the shape of the orbital, and its values are dependent on n. For l, values range from 0 to (n-1), where 0 corresponds to s-orbitals and 1 to p-orbitals, representing the process seen in Step 3 for Carbon with s and p orbitals being filled. The magnetic quantum number, m_l, tells us the orientation of the orbital, varying from -l to +l. Lastly, m_s describes the electron's spin, which can either be +1/2 or -1/2, indicating its two possible spin states. These quantum numbers not only permit the identification of each electron uniquely but also facilitate the proper construction of an atom's electron configuration, as in the recognition of Rutherfordium (Rf) in Step 4.