Question

The spin-only magnetic moments of ${\left[ {{\text{Fe}}{{\left( {{\text{N}}{{\text{H}}_3}} \right)}_6}} \right]^{3 + }}$ and ${\left[ {{\text{Fe}}{{\text{F}}_6}} \right]^{3 - }}$ in BM are respectively:
A. $1.73$ and $1.73$
B. $5.92$ and $1.73$
C. $1.73$ and $5.92$
D. $5.92$ and $5.92$

Answer 1

Valence bond theory could not explain about the magnetic moments and electronic spectra of most complexes. So a more radical approach was put forward which had only room for electrostatic forces. Thus crystal field theory was introduced.

Complete step by step solution:
The electron pairs interact with the d orbitals on the central metal. The central metal cation is surrounded by ligands which contain one or more lone pairs of electrons. The ionic ligands like ${{\text{F}}^ - }{\text{,C}}{{\text{l}}^ - }$ are regarded as the point negative charges and neutral molecules like ${{\text{H}}_2}{\text{O, N}}{{\text{H}}_3}$ as point dipoles. There is no orbital overlap. The bonding between metal and ligand is purely electrostatic.
With these information, we can calculate the magnetic moment of ${\left[ {{\text{Fe}}{{\left( {{\text{N}}{{\text{H}}_3}} \right)}_6}} \right]^{3 + }}$ and ${\left[ {{\text{Fe}}{{\text{F}}_6}} \right]^{3 - }}$
In ${\left[ {{\text{Fe}}{{\left( {{\text{N}}{{\text{H}}_3}} \right)}_6}} \right]^{3 + }}$compound, since ${\text{N}}{{\text{H}}_3}$ is a neutral molecule, it has no charge. Thus the oxidation state of iron is $+ 3$.
Electronic configuration of ${\text{Fe - }}\left[ {{\text{Ar}}} \right]3{{\text{d}}^6}4{{\text{s}}^2}$
Electronic configuration of ${\text{F}}{{\text{e}}^{3 + }}{\text{ - }}\left[ {{\text{Ar}}} \right]3{{\text{d}}^5}$

$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$

${\text{N}}{{\text{H}}_3}$ is a strong ligand and ${{\text{F}}^ - }$ is a weaker ligand than ${\text{N}}{{\text{H}}_3}$. Weak field ligands have low energy thereby forming a high spin complex. While strong field ligands have high energy thereby forming low spin complexes. In strong fields, the electrons are jumped to ${{\text{e}}_{\text{g}}}$ levels after pairing in ${{\text{t}}_{{\text{2g}}}}$ level. But in weak ligands, the electrons are paired only after filling one electron each in ${{\text{t}}_{{\text{2g}}}}$ and ${{\text{e}}_{\text{g}}}$ levels.
Thus in ${\left[ {{\text{Fe}}{{\left( {{\text{N}}{{\text{H}}_3}} \right)}_6}} \right]^{3 + }}$ molecule, the filling of electrons will be as given below:

$\uparrow \downarrow$| $\uparrow \downarrow$| $\uparrow$| |
---|---|---|---|---

$3{\text{d}}$
There is only one unpaired electron. Thus ${\text{n = 1}}$.
Magnetic moment is dependent on the unpaired electrons from the given formula below:
Magnetic moment,$\mu = \sqrt {{\text{n}}\left( {{\text{n}} + 2} \right)}$ , where ${\text{n}}$ is the number of unpaired electrons.
i.e. $\mu = \sqrt {1\left( {1 + 2} \right)} = \sqrt {1 \times 3} = \sqrt 3 = 1.73{\text{BM}}$
Similarly in ${\left[ {{\text{Fe}}{{\text{F}}_6}} \right]^{3 - }}$, the electrons are filled as given below:

$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$	$\uparrow$

$3{\text{d}}$
There are five unpaired electrons, i.e. ${\text{n = 5}}$
Magnetic moment, $\mu = \sqrt {5\left( {5 + 2} \right)} = \sqrt {5 \times 7} = \sqrt {35} = 5.92{\text{BM}}$
So the magnetic moment of ${\left[ {{\text{Fe}}{{\left( {{\text{N}}{{\text{H}}_3}} \right)}_6}} \right]^{3 + }}$ and ${\left[ {{\text{Fe}}{{\text{F}}_6}} \right]^{3 - }}$ are $5.92$ and $1.73$

Hence, the correct option is B.

Note: Magnetic moment is one of the applications of crystal field theory. The color of complexes can also be determined using crystal field theory. In ${{\text{d}}^1}{\text{,}}{{\text{d}}^2}{\text{,}}{{\text{d}}^3}{\text{,}}{{\text{d}}^8}{\text{,}}{{\text{d}}^9}$ complexes have same spin state and all are paramagnetic. The low spin state ${{\text{d}}^6}{\text{,}}{{\text{d}}^{10}}$ complexes are diamagnetic. In ${{\text{d}}^4}{\text{,}}{{\text{d}}^5}{\text{,}}{{\text{d}}^6}{\text{,}}{{\text{d}}^7}$ complexes, the number of unpaired electrons are different in high spin and low spin octahedral complexes.