Which one of the following ions exhibit highest magnetic moment? (a) \(\mathrm{Cu}^{2+}\) (b) \(\mathrm{Ti}^{3+}\) (c) \(\mathrm{Ni}^{2+}\) (d) \(\mathrm{Mn}^{2+}\)

Understanding the concept of unpaired electrons is crucial because they play a pivotal role in determining an atom's magnetic properties.

Electrons are arranged in various energy levels known as orbitals, and each electron has a property called 'spin'. An orbital can hold a maximum of two electrons, which must have opposite spins, according to the Pauli exclusion principle. An unpaired electron is an electron in an orbital by itself, not accompanied by another with opposite spin. These unpaired electrons are paramount when it comes to magnetism because they create a magnetic field around the atom.

When considering ions as in the exercise, we observe that ions with more unpaired electrons will generally exhibit a higher magnetic moment. As demonstrated through the step-by-step solution, determining the number of these unpaired electrons is essential and is usually the first step in solving problems related to magnetic moments of ions.

The electron configuration of an atom or ion is the distribution of electrons in the energy levels and orbitals. It is essential since it informs us not only about the energy of an electron but also its relation to the rest of the electrons in an atom.

For ions, the electron configuration can be deduced by adding or removing electrons from the neutral atom's configuration, based on its charge, and paying close attention to the orbital energy hierarchy and order of filling. The configuration can reveal which orbitals contain unpaired electrons, hence a central step in solving for magnetic moments of ions.

For instance, Mn in its neutral state has a configuration of [Ar]3d^54s^2. When it forms Mn^2+, it loses two electrons from the 4s orbital, thus becoming [Ar]3d^5 with all five electrons in the 3d orbitals being unpaired.

Hund's Rule gives us insight into how electrons populate subshells. According to Hund's Rule, for any subshell, electrons will fill empty orbitals with parallel spins before they pair up. This minimizes repulsion between electrons, as electrons in the same subshell but in different orbitals are not as closely influenced by each other's charge.

By applying Hund's Rule to the ions in the exercise, it helps us accurately determine the number of unpaired electrons. For Ni^2+, with an electron configuration of 3d^8, Hund's Rule dictates first to fill all five 3d orbitals with one electron each. Only after each orbital has one electron do the remaining electrons start to pair up in the available orbitals. This results in Ni^2+ having two unpaired electrons.

The application of these principles not only contributes to the correct prediction of the magnetic properties of ions but also gives us a broader understanding of the arrangement of electrons in atoms and ions.