Consider an octahedral complex, \(\mathrm{MA}_{2} \mathrm{~B}_{4}\). How many geometric isomers are expected for this compound? Will any of the isomers be optically active? If so, which ones?

An octahedral complex has a central metal atom (M) surrounded by six ligands. In this case, we have two ligands of type A and four ligands of type B. To determine the geometric isomers, we need to consider all the possible arrangements for the A and B ligands around the central metal atom.

To find the geometric isomers, we will consider the following scenarios: 1. Both A ligands are next to each other (cis configuration). 2. Both A ligands are opposite to each other (trans configuration). Here are the possible geometric isomers: - Cis configuration: The two A ligands are adjacent, and the remaining four positions are occupied by B ligands. There is only one way to arrange the ligands in this configuration. - Trans configuration: The two A ligands are opposite each other, and the remaining four positions are occupied by B ligands. In this configuration, we have three possibilities (A ligands in axial position, A ligands in the equatorial position, A ligands in the axial and equatorial position). Hence, we have a total of \(1+3 = 4\) geometric isomers.

An optically active substance has the ability to rotate plane-polarized light. A molecule is optically active if it is not superimposable on its mirror image (i.e., it lacks a plane of symmetry). For the octahedral complex \(\mathrm{MA}_{2}\mathrm{B}_{4}\): - Cis configuration: In this configuration, there is a plane of symmetry passing through the central metal atom and the two A ligands, making it optically inactive. Trans configuration: 1. Both A ligands are in axial positions: This configuration also has a plane of symmetry passing through the axial ligands and perpendicular to the equatorial plane, making this isomer optically inactive. 2. Both A ligands are opposite in the equatorial plane: This configuration has a plane of symmetry passing through the M atom and between A ligands; thus, this isomer is optically inactive. 3. A ligands are in one axial position and one equatorial position such that they are opposite: There is no plane of symmetry in this case, making this isomer optically active. In conclusion, there are four geometric isomers, and only one of them (one trans isomer with A ligands in one axial and one equatorial position) is optically active.