Question

Consider an octahedral complex \(\mathrm{MA}_{3} \mathrm{B}_{3} .\) How many geo- metric isomers are expected for this compound? Will any of the isomers be optically active? If so, which ones?

Short answer

There are two possible geometric isomers for the given \(\mathrm{MA}_{3}\mathrm{B}_{3}\) octahedral complex. Geometric isomer 1 is not optically active, while geometric isomer 2 is optically active, as it has an enantiomeric/non-superimposable mirror image.

Step-by-step solution

Determine the possible arrangements of ligands around the central atom

First, let's draw an octahedron with positions for ligands numbered 1 to 6. We have three A ligands and three B ligands to place around the central metal atom. We will start with A ligands and consider symmetry to avoid duplicating isomers. One possible arrangement is placing three A ligands in a plane, with the B ligands filling the other three positions. For example, we can place A ligands in positions 1, 3, and 4, and B ligands in positions 2, 5, and 6. We'll call this geometric isomer 1. The second possible arrangement is placing the A ligands at every other position, such as 1, 3, and 5. With B ligands in the other three positions, at 2, 4, and 6. We'll call this geometric isomer 2. There are no other unique arrangements possible for the \(\mathrm{MA}_{3}\mathrm{B}_{3}\) complex, so there are two possible geometric isomers.

Determine if any of the isomers are optically active

To investigate optical activity, we'll look for enantiomers (non-superimposable mirror images) for each geometric isomer. Geometric isomer 1 (three A ligands in a plane and three B ligands in a plane) has a mirror plane of symmetry containing M and dividing the A and B ligands. This means that any mirror image of the complex would still be superimposable on the original complex. Thus, geometric isomer 1 is not optically active. Geometric isomer 2 (three pairs of A and B ligands in alternate positions) does not have any symmetry planes. If we draw a mirror image of this isomer, we get an enantiomeric/non-superimposable complex. Thus, geometric isomer 2 has an enantiomer and is optically active. #Summary# There are two possible geometric isomers for the given \(\mathrm{MA}_{3}\mathrm{B}_{3}\) octahedral complex. Geometric isomer 1 is not optically active, while geometric isomer 2 is optically active, as it has an enantiomeric/non-superimposable mirror image.

Key concepts

Octahedral Complex

In the context of coordination chemistry, an octahedral complex is a compound where a central metal atom is surrounded by six ligands in a geometric arrangement that resembles an octahedron. Each ligand is connected to the metal by a coordinate bond, and they are spaced around the metal atom at 90-degree angles to one another along the Cartesian axes.

Understanding the structure of an octahedral complex is crucial, as the spatial arrangement of ligands can affect the complex's chemical properties, including reactivity and optical activity. In our exercise example, the octahedral complex is described by the formula \(\mathrm{MA}_{3} \mathrm{B}_{3}\), indicating that the central metal atom, M, is surrounded by three A ligands and three B ligands. The question is how these ligands can be arranged and what properties, such as geometric or optical isomerism, may result from different arrangements.

Geometric Isomers

Geometric isomers, also known as cis-trans isomers, are compounds with the same formula but different spatial arrangements of atoms. This type of isomerism is particularly pertinent to coordination complexes like the octahedral complex, where the arrangement of different ligands around the central metal can lead to isomers with distinctive properties.

For the \(\mathrm{MA}_{3} \mathrm{B}_{3}\) complex, we consider two different arrangements of the ligands A and B. If the three A ligands are adjacent to one another, forming a plane, and the B ligands occupy the opposite positions, we have one geometric isomer. Alternatively, if A and B alternate around the central atom, we have another distinct geometric isomer. Geometric isomers can result in vastly different chemical and physical behaviors, including their responses to polarized light, which is pivotal in the study of optical activity.

Enantiomers

Enantiomers are a type of stereoisomers that are mirror images of each other but cannot be superimposed. This non-superimposability makes enantiomers chiral, meaning they have handedness, much like left and right hands. Enantiomers often arise in coordination chemistry, where the spatial arrangement of ligands around the central metal can generate mirror-image forms.

In the \(\mathrm{MA}_{3} \mathrm{B}_{3}\) example, the second geometric isomer, with alternating A and B ligands, does not possess a plane of symmetry and therefore can exist as two enantiomers, which are optically active. Optical activity is detected when such compounds rotate the plane of polarized light, a characteristic exploited for applications ranging from pharmaceuticals to the food industry. In contrast, a compound like the first isomer, with a plane of symmetry, will not exhibit optical activity as its mirror image is superimposable on itself.