Step 1: Understanding the Concept:
Geometrical isomerism, or stereoisomerism, occurs in coordination complexes when ligands have different relative spatial arrangements around the central metal atom.
In square planar complexes, the central metal is at the center of a square, and four ligands are at the corners.
Geometrical isomerism is possible because the positions are not all equivalent relative to one another.
Two ligands can be either adjacent to each other (90 degrees apart, called the 'cis' isomer) or opposite to each other (180 degrees apart, called the 'trans' isomer).
For a square planar complex of the type \( [MA_4] \), all four ligands are the same, so no isomers exist.
For the type \( [MA_3B] \), replacing one ligand doesn't create different arrangements that cannot be superimposed by rotation.
Step 2: Key Formula or Approach:
The standard conditions for geometrical isomerism in square planar complexes (Coordination Number = 4) are:
1. \( [MA_2B_2] \) type: This yields two isomers (cis and trans).
2. \( [MA_2BC] \) type: This yields two isomers.
3. \( [MABCD] \) type: This yields three isomers.
Complexes of type \( [MA_4] \) or \( [MA_3B] \) do not show geometrical isomerism.
Given the formula \( [Pt(Cl)_x(Br)_y]^{2-} \), the total coordination number must be 4 for square planar geometry, meaning \( x + y = 4 \).
Step 3: Detailed Explanation:
Let's evaluate the options provided:
- (A) \( x = 1, y = 3 \): This gives \( [Pt(Cl)(Br)_3]^{2-} \). This is an \( [MAB_3] \) type complex. In this case, no matter which position the Cl ligand occupies, its relationship with the three Br ligands is identical. No cis/trans distinction can be made.
- (B) \( x = 3, y = 1 \): This gives \( [Pt(Cl)_3(Br)]^{2-} \). This is an \( [MA_3B] \) type complex. Similar to the previous case, rotating the molecule allows any configuration to match any other. No geometrical isomers are possible.
- (C) \( x = 2, y = 2 \): This gives \( [Pt(Cl)_2(Br)_2]^{2-} \). This is an \( [MA_2B_2] \) type complex.
In this arrangement, two Cl ligands can be adjacent (at 90 degrees), forming the cis-isomer.
Alternatively, the two Cl ligands can be across the metal from each other (at 180 degrees), forming the trans-isomer.
Since two distinct spatial arrangements exist that are not interconvertible by simple rotation, this complex shows geometrical isomerism.
- (D) \( x = 1, y = 1 \): This combination implies only two ligands, which does not satisfy the requirement for a coordination number of 4 in square planar complexes.
Step 4: Final Answer:
Geometrical isomerism in square planar coordination compounds occurs when there are at least two different types of ligands present in a specific ratio, most commonly 2:2.
Thus, for \( x=2 \) and \( y=2 \), the complex \( [PtCl_2Br_2]^{2-} \) will exhibit cis and trans isomers.
The correct answer is (C).