2.14 For each of the following ground-state ions, predict the type of orbital \((1 s, 2 p, 3 d, 4 f\), and so on) from which an electron will need to be removed to form the ions of one greater positive charge: (a) \(\mathrm{Ti}^{2+}\); (b) \(\ln ^{+} ;\)(c) \(\mathrm{Te}^{2-} ;\) (d) \(\mathrm{Ag}^{+} .\)

Understanding electron configuration is essential for predicting how electrons are arranged around an atom and determining which orbital an electron will be removed from or added to. The configuration follows a standard sequence based on the increasing energy levels and sublevels, typically notated in a format like '1s² 2s² 2p⁶'.

Electrons fill the orbitals from lower to higher energy states. The order is defined by the Aufbau principle, which suggests that electrons occupy the lowest energy orbitals available. As a helpful mnemonic, many students use the 'diagonal rule' to remember the order of filling: 1s, 2s, 2p, 3s, 3p, 4s, 3d, and so on.

Knowing this order is crucial when dealing with chemical reactions or the formation of ions, as it allows you to predict which electrons will be involved based on their energy levels and the overall stability of the atom or ion after electron removal or addition.

When an atom gains or loses electrons, it becomes an ion. The charge of an ion is adjusted by the differential number of electrons compared to its neutral state. Cations, or positive ions, are formed when atoms lose electrons, resulting in a net positive charge. Anions, on the other hand, result from the gain of electrons and have a net negative charge.

For instance, to transition from a neutral atom to a cation with a greater positive charge, we remove the electron(s) from the outermost orbital as per Hund's rule, which indicates the filling order of electrons within the same subshell. The reverse is true for anions: we add electrons to the last energy level following the 'octet rule', which suggests that atoms are generally most stable when they have eight electrons in their valence shell.

The periodic table is a fundamental tool in chemistry, providing a wealth of information at a glance about the elements—including electron configurations. Elements are arranged in order of increasing atomic number and are grouped based on similar properties. The table itself is divided into blocks (s, p, d, f), corresponding to the sublevel that is being filled with electrons for elements in that block.

Following periodic trends, such as electron affinities, ionization energies, and atomic radius, is helpful for predicting behavior during ion formation. For instance, elements in the same group (vertical columns) typically form ions with similar charges, which can be seen in their similar valence electron configurations. This systematic setup aids in quickly determining the base electron configuration of an element, which is the starting point for figuring out from which orbital an electron will be removed.

Quantum numbers are vital in pinpointing the location and energy of electrons within an atom. These include the principal quantum number (\( n \)), the angular momentum quantum number (\( l \)), the magnetic quantum number (\( m_l \)), and the spin quantum number (\( m_s \)).

\( n \) determines the energy level or shell of an electron, and as this number increases, so does the energy of the electron. \( l \) indicates the subshell or shape of the orbital, with \( l = 0 \) for 's', \( l = 1 \) for 'p', \( l = 2 \) for 'd', and \( l = 3 \) for 'f'. The magnetic quantum number defines the orbital's orientation in space, and finally, the spin quantum number designates the direction of the electron's spin within the orbital. By understanding these quantum numbers, we explain why electrons are removed from the highest-energy orbital (\( n \) and \( l \) values) during ion formation, as these electrons are less firmly held by the nucleus and therefore easier to remove.