Considering \(\mathrm{H}_{2} \mathrm{O}\) as a weak field ligand, the number of unpaired electrons in \(\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}\) will be (Atomic no of \(\mathrm{Mn}=25\) ) (a) 2 (b) 3 (c) 4 (d) 5

Ligand Field Theory (LFT) is an extension of crystal field theory that offers a more detailed description of how ligands interact with the central transition metal ion in a coordination compound. It takes into account the covalent, as well as the ionic aspects of coordination bond formation. In essence, LFT explains the splitting of degenerate (energetically equal) d orbitals under the influence of an electric field produced by surrounding ligands. This is crucial for understanding the magnetic properties, such as the number of unpaired electrons, and the color of the compound. A weak field ligand, like water in the given exercise, produces a smaller splitting of the d orbitals and often does not lead to electron pairing, resulting in a high-spin configuration.

Crystal Field Splitting occurs when the five degenerate d orbitals of a transition metal ion experience an uneven distribution of electrons due to the different electric field strengths exerted by adjacent ligands in a coordination compound. This results in a separation into two energy levels: a lower-energy set called the t2g orbitals and a higher-energy set called the eg orbitals. The extent of this splitting effect is determined by the strength of the ligands: strong field ligands cause significant splitting and may induce the electron pairing, while weak field ligands, such as H2O mentioned in the exercise, cause minor splitting with less likelihood of electron pairing.

Transition metals have valence electrons in their d orbitals which leads to their unique chemical and physical properties. The electronic configuration is written by adding electrons to orbitals following the sequence predicted by the energy levels of those orbitals, which is provided by the aufbau principle. For a transition metal cation like Mn2+ in the exercise, the electronic configuration typically involves filling the 3d orbitals before the 4s. Hence, Mn2+ with a 3d^5 configuration implies that all five d orbitals contain one electron each, in absence of a strong field ligand.

Hund's Rule dictates how electrons are distributed among orbitals of the same energy. It states that every orbital in a subshell must be occupied by a single electron before any orbital is doubly occupied, and all electrons in singly occupied orbitals must have the same spin direction. This rule is relevant to the exercise, as it justifies why each of the five d orbitals in the Mn2+ ion contains a single electron, ultimately leading to high magnetic moments due to the presence of five unpaired electrons.

The aufbau principle provides a guideline for the order in which electrons fill subatomic orbitals, suggesting they occupy the lowest-energy orbitals first. This principle in combination with Hund's Rule and Pauli Exclusion Principle helps explain the electronic configurations of atoms. In our exercise, aufbau principle indicates that in the Mn2+ ion the 3d orbitals are filled with five electrons prior to the removal of the two 4s electrons upon ionization, resulting in 3d^5 configuration with unpaired electrons, attributable to the systematic building of electron occupancy.