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What kind of isomerism exists between $\left[ {Cr{{\left( {{H_2}O} \right)}_6}} \right]C{l_3}$ (violet) and $\left[ {Cr{{\left( {{H_2}O} \right)}_5}Cl} \right]C{l_2}.{H_2}O$ (greyish-green)?
(A) Linkage isomerism
(B) Solvate isomerism
(C) Ionisation isomerism
(D) Coordination isomerism

Answer
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Hint: In these given complexes the molecular formula or composition remains the same but the difference is in the position of water or solvate ligands inside and outside of the coordination spheres. This difference is what is causing the change of colours.

Complete step by step solution:
First of all, we will see what these isomerisms are.
-For (A) Linkage isomerism: It is a type of isomerism in which the same ligand is attached or bonded to the central metal ion via different atoms. The molecular formula of the linkage isomers remains the same and they differ only in the linkage of the ligand to the central metal ion.
For example: $\left[ {Co{{\left( {N{H_3}} \right)}_5}N{O_2}} \right]C{l_2}$ (yellow coloured) and $\left[ {Co{{\left( {N{H_3}} \right)}_5}ONO} \right]C{l_2}$ (red coloured)
-For (B) Solvate isomerism: In this type of isomerism the composition of the isomers remains same but they differ in the number of solvate (${H_2}O$) ligand molecules and the counter ions in the complex.
For example: $\left[ {Cr{{\left( {{H_2}O} \right)}_6}} \right]C{l_3}$ (violet coloured), $\left[ {Cr{{\left( {{H_2}O} \right)}_5}Cl} \right]C{l_2}.{H_2}O$ (greyish-green) and $\left[ {Cr{{\left( {{H_2}O} \right)}_4}C{l_2}} \right]Cl.{\left( {{H_2}O} \right)_2}$ (dark green).
-For (C) Ionisation isomerism: In this type of isomerism, there is an exchange of ions between the inside and outside of the coordination sphere. For such isomers, the molecular formula remains the same but the ions produced in the solution are different.
For example: $\left[ {Co{{\left( {N{H_3}} \right)}_5}S{O_4}} \right]Br$ and $\left[ {Co{{\left( {N{H_3}} \right)}_5}Br} \right]S{O_4}$
-For (D) Coordination isomerism: It is a type of isomerism in which the composition of the complex ion varies because the ligands attached to a specific metal ion change but the ratio of total ligand to metal ions remains the same.
For example: $\left[ {Cr{{\left( {N{H_3}} \right)}_6}} \right]\left[ {Co{{\left( {CN} \right)}_6}} \right]$ and $\left[ {Cr{{\left( {CN} \right)}_6}} \right]\left[ {Co{{\left( {N{H_3}} \right)}_6}} \right]$. Here Cr and Co interchange their positions.
-From the above discussion we can see that for the given two complexes:
$\left[ {Cr{{\left( {{H_2}O} \right)}_6}} \right]C{l_3}$ (violet): In this complex the number of solvate or water molecules (${H_2}O$) inside the coordination sphere is 6 and outside is 0.
$\left[ {Cr{{\left( {{H_2}O} \right)}_5}Cl} \right]C{l_2}.{H_2}O$ (greyish-green): In this complex the solvate or water molecules (${H_2}O$) inside the coordination sphere is 5 and outside is 1.
In these 2 complexes, the difference is in the number of solvate (${H_2}O$) ligand molecules and the counter ions in the complex. So, they will show solvate isomerism.

Hence, the correct option will be: (B) Solvate isomerism

Note: If two coordination complexes have the same molecular formula but different structures, they are known as isomers. Isomerism is basically of 2 types which are further subdivided:
(1) Stereoisomerism: (a) Geometrical isomerism
(b) Optical isomerism
(2) Structural isomerism: (a) Coordination isomerism
(b) Ionisation isomerism
(c) Hydrate isomerism
(d) Linkage isomerism