Draw all the geometrical isomers of \(\left[\mathrm{Cr}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{2} \mathrm{BrCl}\right]^{+} .\) Which of these isomers also have an optical isomer? Draw the various isomers.

The given complex ion is \(\left[\mathrm{Cr}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{2}\mathrm{BrCl}\right]^{+}\), which consists of one Cr(III) central metal ion bonded to one ethylenediamine (en) ligand, two ammonia (NH_3) ligands, and two different halides (Br and Cl). Since we are asked to find the geometrical isomers of the complex, we will be focusing on cis/trans or fac/mer isomerism, depending on the arrangement of ligands around the central metal ion.

Since there are six ligand sites, the complex has a coordination number of six and is octahedral in shape. The Cr(III) metal ion is at the center, surrounded by the ligands.

In this octahedral complex, we will examine the possible arrangements of the ligands, keeping in mind the following types of isomerism: 1. Cis/Trans isomerism: Occurs when two ligands of the same type are either on the same plane (cis) or are opposite to each other (trans) on the central metal ion. 2. Fac/Mer isomerism: Refers to the arrangement of three ligands of the same type in either a facial (fac) manner, where they occupy one triangular face of the octahedron, or in a meridional (mer) manner, where they occupy a plane that contains the metal ion. For the given complex ion, the geometrical isomers we need to consider are: 1. Cis or trans to the two NH_3 ligands. 2. Fac or mer for the arrangement of the en ligand and the two halides (Br and Cl). Considering the geometrical isomers, we can draw the following: a) Trans isomer: NH_3 ligands are trans to each other, and the en ligand and the two halides (Br and Cl) are in a mer arrangement. b) Cis isomer: NH_3 ligands are cis to each other, and the en ligand and the two halides (Br and Cl) are in a fac arrangement.

To decide if the geometrical isomers have optical isomers, we need to check if the complex molecule has any plane of symmetry or center of symmetry. If a geometrical isomer has no plane or center of symmetry, then it will have an optical isomer (the molecule and its mirror image cannot be superimposed). a) Trans isomer: This isomer has a plane of symmetry, and thus, it does not have any optical isomers. b) Cis isomer: This isomer does not have a plane of symmetry or a center of symmetry, and thus, it has optical isomers (enantiomers). In conclusion, the cis isomer of the complex ion \(\left[\mathrm{Cr}(\mathrm{en})\left(\mathrm{NH}_{3}\right)_{2}\mathrm{BrCl}\right]^{+}\) has an optical isomer, while the trans isomer does not.