The number of optically active isomer for $\left[\mathrm{Pt}{\left({\mathrm{NH}}_3\right)}_2\right(\mathrm{F}\left)\right(\mathrm{Cl}\left)\right(\mathrm{Br}\left)\right(\mathrm{I}\left)\right]° \mathrm{is} \mathrm{\_\_\_\_\_\_\_\_\_\_\_}.$

To determine the number of optically active isomers for the complex [Pt(NH3)₂(F)(Cl)(Br)(I)]°, we need to consider the factors that influence optical activity in coordination complexes, such as:

1. Geometry of the complex

The platinum ion (Pt²⁺) is typically octahedral in coordination with six ligands. However, in this case, there are only 5 ligands—2 ammonia molecules (NH₃) and 4 halide ligands (F, Cl, Br, I).

2. Ligand arrangement

The 4 different halide ligands (F, Cl, Br, I) can be arranged in different positions around the platinum center. Since the ligands are different, the spatial arrangement matters, and symmetry will affect the number of optically active isomers. If the complex has no plane of symmetry or center of symmetry, it will be optically active.

3. Optical activity

A complex will exhibit optical activity if it has chirality, meaning it does not possess any symmetry elements like a mirror plane or inversion center. In this case, the presence of different halides creates asymmetry, leading to possible chirality and optically active isomers.

Step-by-Step Approach:

1. Possible arrangements of ligands

The complex [Pt(NH3)₂(F)(Cl)(Br)(I)]° has 4 different monodentate ligands (F, Cl, Br, I) and 2 NH₃ ligands. The 4 halide ligands can be arranged in ℓ_arrangements = 4! / 2! = 12 distinct ways (since two NH₃ ligands are identical).

2. Chirality check

With 4 different halides, the spatial arrangement of these ligands is such that the complex may not have any symmetry elements like a mirror plane or inversion center, implying chirality. Each distinct arrangement leads to two possible optical isomers: one right-handed (R) and one left-handed (S).

Thus, for each of the 12 distinct arrangements, there are two optical isomers, leading to:

Total optically active isomers = 12 × 2 = 24

However, the complex is octahedral with some symmetry restrictions, and not all 24 isomers are optically active. The actual number of optically active isomers is typically 12.