The correct order of magnetic moment (spin only values in BM) among the following is (a) \(\left[\mathrm{MnCl}_{4}\right]^{2-}>\left[\mathrm{CoCl}_{4}\right]^{2-}>\mathrm{Fe}\left(\mathrm{CN}_{6}\right)^{4-}\) (b) \(\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4}>\left[\mathrm{MnCl}_{4}\right]^{2-}>\left[\mathrm{CoCl}_{4}\right]^{2-}\) (c) \(\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4}>\left[\mathrm{CoCl}_{4}\right]^{2-}>\left[\mathrm{MnCl}_{4}\right]^{2-}\) (d) \(\left[\mathrm{MnCl}_{4}\right]^{2-}>\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}>\left[\mathrm{CoCl}_{4}\right]^{2-}\) (Atomic number of \(\mathrm{Mn}=25, \mathrm{Fe}=26, \mathrm{Co}=27, \mathrm{Ni}=28\) )

Understanding the spin-only magnetic moment is critical in predicting the magnetic behavior of coordination complexes. The formula \(\mu = \sqrt{n(n+2)}\) is essential for calculating the magnetic moment, where \(\mu\) represents the magnetic moment in Bohr magnetons (BM), and \(n\) is the number of unpaired electrons in the complex.

Each unpaired electron has an intrinsic spin that contributes to the total magnetic moment of the complex. Since paired electrons have opposite spins that cancel out each other's magnetic moment, they do not contribute to the magnetic moment. Hence, the significance of unpaired electrons becomes evident when using this formula to solve problems like the exercise presented.

By substituting the appropriate value of \(n\), one can compute the magnetic moment, which will then allow comparison between different complexes. This concept is instrumental in explaining the magnetic properties observed in coordination chemistry.

Crystal Field Theory (CFT) is a model that describes the interaction between transition metal ions and their surrounding ligands. The theory provides insights into how the ligands affect the energy levels (d-orbitals) of the metal ion.

According to CFT, the d-orbitals split into different energy levels upon the approach of the ligands due to repulsion. In an octahedral field, for instance, the d-orbitals split into two groups: a lower-energy set of three orbitals (t2g) and a higher-energy set of two orbitals (eg). The energy gap between these two sets is termed as the crystal field splitting energy (\(\Delta_o\)).

Ligands that cause a large \(\Delta_o\) lead to low-spin configurations as electrons pair up in the lower-energy orbitals to minimize the energy, while ligands that cause a small \(\Delta_o\) lead to high-spin configurations because electrons occupy the higher-energy orbitals before pairing.

Unpaired electrons play a pivotal role in determining the magnetic properties of coordination complexes. An unpaired electron is an electron that occupies an orbital alone, i.e., it does not have another electron with opposite spin within the same orbital.

Transition metal ions often have electrons in their d-orbitals which can remain unpaired depending on the nature of the metal ion and the type of ligands present in the complex. The step by step solution involves identifying these unpaired electrons to predict magnetic moments. The more unpaired electrons a complex has, the greater its magnetic moment. In the exercise, it's crucial to first determine the electron configuration of each metal ion and then take into account the effect of ligands to establish the correct number of unpaired electrons.

High-spin and low-spin complexes refer to the possible configurations that can result from the arrangement of electrons in the d-orbitals of a transition metal ion under the influence of ligands, as explained by Crystal Field Theory.

A high-spin complex occurs when the ligands generate a small crystal field splitting energy (\(\Delta_o\)), resulting in the occupation of the higher-energy orbitals by the electrons before they start to pair up. This usually happens with weak-field ligands like halides (Cl^-, for example). On the other hand, a low-spin complex forms when strong-field ligands, such as cyanide (CN^-), produce a large \(\Delta_o\) and cause electrons to pair up in the lower-energy orbitals first, leading to fewer unpaired electrons and, consequently, a smaller magnetic moment.

In the exercise, understanding the nature of the ligands and their impact on the metal's electron configuration is key to solving for the correct order of magnetic moments, as strong-field ligands induce fewer unpaired electrons and weak-field ligands result in more unpaired electrons.