Animated Solution for Chemistry - Coordination Compounds: Among the complex ions, $[\text{Co}({\text{NH}}_2-{\text{CH}}_2-{\text{CH}}_2-{\text{NH}}_2{)}_2{\text{Cl}}_2{]}^{+}$, $[{\text{CrCl}}_2({\text{C}}_2{\text{O}}_4{)}_2{]}^{3-}$, $[\text{Fe}({\text{H}}_2\text{O}{)}_4(\text{OH}{)}_2{]}^{+}$, $[\text{Fe}({\text{NH}}_3{)}_2(\text{CN}{)}_4{]}^{-}$, $[\text{Co}({\text{NH}}_2-{\text{CH}}_2-{\text{CH}}_2-{\text{NH}}_2{)}_2({\text{NH}}_3)\text{Cl}{]}^{2+}$ and $[\text{Co}({\text{NH}}_3{)}_4({\text{H}}_2\text{O})\text{Cl}{]}^{2+}$, the number of complex ion(s) that show(s) cis-trans isomerism is -

Imagine a central metal atom acting as the core of a star, with six ligands orbiting it like planets. In an octahedral complex, these six ligands occupy the vertices of a regular octahedron. But what happens when some of these 'planets' are identical? This is where the fascinating world of geometrical isomerism begins.

When two identical ligands are positioned adjacent to each other, separated by an angle of $90^{\circ }$, we call this the cis form. Conversely, if they are placed exactly opposite to each other, separated by $180^{\circ }$, they form the trans isomer.

General Formulas

To quickly identify if an octahedral complex exhibits cis-trans isomerism, we rely on a set of general formulas. The most common types that show this behavior are:

1. $[{\text{Ma}}_4{\text{b}}_2]$ and $[{\text{Ma}}_4\text{bc}]$: Here, 'a', 'b', and 'c' are monodentate ligands. The isomerism depends on the relative positions of the two 'b' ligands (or the 'b' and 'c' ligands). 2. $[\text{M}(\text{AA}{)}_2{\text{b}}_2]$ and $[\text{M}(\text{AA}{)}_2\text{bc}]$: Here, 'AA' represents a symmetrical bidentate ligand (like ethylenediamine or oxalate).

Crucial Constraint: A standard bidentate ligand has a short "bite angle" and can only span adjacent cis positions ($90^{\circ }$). It cannot stretch across the metal atom to occupy trans positions ($180^{\circ }$).

Let's systematically break down each of the six complexes provided in the problem:

1. $[\text{Co}(\text{en}{)}_2{\text{Cl}}_2{]}^{+}$ Here, $\text{en}$ (ethylenediamine) is our bidentate ligand ($\text{AA}$), and $\text{Cl}$ is our monodentate ligand ($\text{b}$). This perfectly matches the $[\text{M}(\text{AA}{)}_2{\text{b}}_2]$ type. The two chloride ions can be placed either adjacent (cis) or opposite (trans) to each other. Thus, it shows cis-trans isomerism.

2. $[{\text{CrCl}}_2({\text{C}}_2{\text{O}}_4{)}_2{]}^{3-}$ Similarly, ${\text{C}}_2{\text{O}}_4^{2-}$ (oxalate) is a bidentate ligand ($\text{AA}$), and $\text{Cl}$ is monodentate ($\text{b}$). This is another classic $[\text{M}(\text{AA}{)}_2{\text{b}}_2]$ complex. It will definitely exhibit cis-trans isomerism.

3. $[\text{Fe}({\text{H}}_2\text{O}{)}_4(\text{OH}{)}_2{]}^{+}$ In this complex, all ligands are monodentate. We have four water molecules ($\text{a}$) and two hydroxide ions ($\text{b}$). This fits the $[{\text{Ma}}_4{\text{b}}_2]$ profile. The two hydroxide ions can easily adopt cis or trans configurations.

4. $[\text{Fe}({\text{NH}}_3{)}_2(\text{CN}{)}_4{]}^{-}$ Again, all ligands are monodentate. We have four cyanide ions ($\text{a}$) and two ammonia molecules ($\text{b}$). This is another $[{\text{Ma}}_4{\text{b}}_2]$ type complex, guaranteeing the presence of cis-trans isomers.

5. $[\text{Co}(\text{en}{)}_2({\text{NH}}_3)\text{Cl}{]}^{2+}$ This one is slightly more diverse. We have the bidentate $\text{en}$ ($\text{AA}$), alongside two different monodentate ligands: ${\text{NH}}_3$ ($\text{b}$) and $\text{Cl}$ ($\text{c}$). This matches the $[\text{M}(\text{AA}{)}_2\text{bc}]$ type. The ammonia and chloride ligands can be positioned cis or trans relative to each other.

6. $[\text{Co}({\text{NH}}_3{)}_4({\text{H}}_2\text{O})\text{Cl}{]}^{2+}$ Finally, we have four ammonia molecules ($\text{a}$), one water molecule ($\text{b}$), and one chloride ion ($\text{c}$). This is an $[{\text{Ma}}_4\text{bc}]$ type complex. The water and chloride ligands can be arranged in cis or trans positions.

After carefully analyzing the geometry and ligand types of each complex, we find that every single one of the six given complexes possesses the structural prerequisites to exhibit cis-trans isomerism.

Therefore, the total number of complex ions showing cis-trans isomerism is $6$.