Determine the number of unpaired electrons expected in terms of crystal field theory:

a) [Fe(NO_2)_6]^3โˆ’

b) [Fe(H_2O)_6]^3+

To determine the number of unpaired electrons in a complex using crystal field theory, we need to consider the splitting of d-orbitals in the presence of ligands.

a) [Fe(NO_2)6]^3-

The complex has a charge of 3-. Since Fe is in the +3 oxidation state, it has 5 d-electrons (from Fe3+ = [Ar] 3d5). In an octahedral field, the d-orbitals split into two sets, with three orbitals (dxz, dyz, dz^2) at a higher energy level (termed "t2g") and two orbitals (dxy, dx^2-y^2) at a lower energy level (termed "eg").

Based on Hund's rule, we place electrons in separate orbitals of the same energy before pairing them up. In this case, we have 5 electrons to distribute.

Since each NO2 ligand is a strong-field ligand, it causes a large energy difference between t2g and eg sets. According to the spectrochemical series, NO2 is a weak field ligand. Therefore, it causes no pairing of electrons. So, all 5 electrons occupy the t2g orbitals, resulting in 5 unpaired electrons.

b) [Fe(H2O)6]^3+

Similarly, the complex has a charge of 3+. Fe in the +3 oxidation state has 5 d-electrons. In an octahedral field, the d-orbitals split into two sets, with three orbitals (t2g) at a higher energy level and two orbitals (eg) at a lower energy level.

In this case, H2O is a weak-field ligand, so it causes less splitting of the d-orbitals. The energy difference between the two sets (t2g and eg) is relatively small. Nevertheless, according to Hund's rule, we still distribute electrons one by one before pairing them up.

Since we have 5 electrons to distribute, we fill all three t2g orbitals with one electron each, and then pair up the remaining two electrons in the eg orbitals. As a result, there are 3 unpaired electrons in [Fe(H2O)6]^3+.