Which of the following ion will have maximum magnetic moment? (a) \(\mathrm{Fe}^{3+}(Z=26)\) (b) \(\mathrm{Cr}^{3+}(Z=24)\) (c) \(\mathrm{Ti}^{4+}(Z=22)\) (d) \(\mathrm{Co}^{3+}(Z=27)\)

Unpaired electrons are the cornerstone of magnetism in atoms and ions. They are the electrons in an atom or ion that are not paired with another electron in an orbital. Each electron has a magnetic field due to its spin. In a pair, the magnetic fields of two electrons with opposite spins cancel each other out. However, when electrons are unpaired, their magnetic fields do not cancel and thus contribute to the overall magnetic moment of the atom or ion.

Understanding unpaired electrons is vital when determining an ion's magnetic properties. For instance, in the exercise, we need to look at various ions and determine which has the most unpaired electrons to figure out which would have the maximum magnetic moment. The more unpaired electrons there are, the stronger the magnetic moment will be, leading to a more magnetically active ion.

The electronic configuration of an ion tells us how electrons are distributed among the various orbitals of an atom. When an atom gains or loses electrons, it becomes an ion, and its electronic configuration changes accordingly. In the context of our exercise, we need to understand the electronic configuration of the ions because it will help us count the number of unpaired electrons by considering removed or added electrons due to ionization.

Removing electrons to get the configuration of a positive ion (cation), as done in the exercise, typically starts from the outermost shell. This process is crucial in solving the problem at hand because only by knowing the correct electronic configuration can we determine the number of unpaired electrons and subsequently calculate the magnetic moment of each ion.

Hund's rule plays a pivotal role when it comes to defining the electronic configuration of ions. This rule states that electrons will occupy all the degenerate (equal energy) orbitals singularly before pairing up. The logic behind this is to minimize electron repulsion and maximize the total spin resulting in a more stable electronic arrangement.

In terms of our exercise, applying Hund's rule when determining the electronic configuration of the ions gives us the number of unpaired electrons. Remember, unpaired electrons lead to magnetism. If we failed to apply Hund's rule correctly, we might incorrectly pair up electrons, which would lead us to underestimate the magnetic moment of an ion.

The magnetic moment of an ion is quantitatively estimated using the magnetic moment formula, which is \(\mu = \sqrt{n(n+2)}\), where \(n\) is the number of unpaired electrons. This formula is derived from quantum mechanics and it quantifies the magnitude of the magnetic moment associated with the unpaired electrons within the ion.

Applying this formula to each ion after determining the number of unpaired electrons through their electronic configurations and Hund's rule allows us to calculate and compare their magnetic moments. In the given exercise, after obtaining the number of unpaired electrons for each ion, you'd use this formula to ascertain which ion has the maximum magnetic moment. The ion with the highest number of unpaired electrons will typically have the highest magnetic moment and will be the most magnetically active among the options provided.