Counting Isomers of MA₄B₂ and MA₃B₃

Introduction

Octahedral MA₄B₂ and MA₃B₃ each have two geometrical arrangements, but the names describe different features. For MA₄B₂, the question is whether the two B ligands are adjacent or opposite: cis or trans. For MA₃B₃, three A ligands can occupy one triangular face or stretch around a meridian: fac or mer. Drawing all six positions systematically shows why there are exactly two of each.

Core explanation

An octahedron has six vertices at ±x, ±y and ±z, forming three pairs of opposite positions. In MA₄B₂, fix one B at +z. Put the other at −z and the B pair is trans. Put it at any of ±x or ±y and the pair is cis. The four cis placements are related by rotations around the first B axis and therefore represent one arrangement. These two possibilities exhaust the available positions because any pair of distinct octahedral vertices is either adjacent or opposite.

The cis and trans forms can have different properties. Their dipoles, ligand substitution pathways and interactions with other molecules may differ. Yet both retain the same ligand connectivity and formal metal oxidation state. In the elementary point-field approximation, an identical d count does not tell which geometric arrangement is present. Spectroscopy or structural analysis may distinguish them.

In MA₃B₃, focus on the three A ligands. If every A–A pair is cis, the A vertices form a triangular face of the octahedron. The three B vertices form the opposite face. This is the facial , or fac, arrangement. If one A–A pair is trans, the three A sites trace a meridian through the metal-containing octahedral framework; this is meridional , or mer. All other drawings with the same pattern can be rotated into one of these two classes.

There cannot be two different trans A–A pairs among only three A ligands, because each vertex has only one opposite vertex. Thus the classification “zero trans A pairs” or “one trans A pair” is exhaustive. The fac form has no A–A trans pair; the mer form has exactly one. This trans-pair test is faster and safer than relying on whether a perspective drawing looks like a face.

For these simple arrangements with monodentate A and B ligands, neither geometric class adds a distinct optical partner; mirror views can be superimposed by symmetry. Thus two geometrical isomers also mean two individual stereoisomers for each formula class. This statement assumes an ideal octahedral coordination sphere and simple identical ligands, not additional chirality within a ligand itself.

Examples include cis- and trans-[Co(NH₃)₄Cl₂]⁺, and fac- and mer-[Co(NH₃)₃Cl₃]. For the first, Co is +3 because two chloride ligands contribute −2 and the ion has charge +1. For the second, three chloride ligands contribute −3 and the complex is neutral, again giving Co³⁺. Isomer labels do not alter that charge calculation.

Step-by-step reasoning

For MA₄B₂, fix one B and classify the second as cis or trans. For MA₃B₃, fix the A set and ask whether any pair of A ligands lies opposite; zero means fac, one means mer. Remove rotational duplicates and test mirror images before reporting total stereoisomers.

Visual explanation

Draw six octahedral vertices as three opposite pairs. Colour two vertices B for MA₄B₂, first adjacent and then opposite. For MA₃B₃, colour three A vertices: in fac choose one from each opposite pair forming a face; in mer choose one complete opposite pair plus a third A on a perpendicular axis.

Real-world analogy

Two coloured lights on opposite corners of a room are trans, while neighbouring corners are cis. For three lights, they can cluster on one triangular face or span a line through opposite points. Turning the room model does not create a new arrangement.

Real-world example

Cis- and trans-[Co(NH₃)₄Cl₂]⁺ are historically useful examples of octahedral geometrical isomerism. Their existence helps show that the six positions are not simply a flat hexagon; a three-dimensional octahedral arrangement naturally gives the observed pair of forms.

Why?

Why is fac/mer not the preferred label for MA₄B₂? It describes the arrangement of three identical ligands relative to an octahedral face or meridian. With only two B ligands, their adjacency or opposition is captured directly by cis/trans.

Common misconception

“MA₃B₃ has three isomers because there are three pairs of opposite positions.” The three axes are equivalent under rotation. What matters is whether the A set contains a trans pair, giving only fac or mer.

Worked example

Count isomers of neutral [Co(NH₃)₃Cl₃]. NH₃ is A and Cl is B. If the three chlorides occupy one face, none is trans to another chloride: fac. If two chlorides are opposite and the third is perpendicular, the arrangement is mer. Rotations account for every other drawing, and neither class has a distinct mirror partner, so there are two geometrical and two total stereoisomers.

Quick check

1. How many A–A trans pairs occur in fac-MA₃B₃? Answer: None; all three A ligands are mutually cis. 2. What distinguishes cis- from trans-MA₄B₂? Answer: The two B ligands are adjacent in cis and opposite in trans.

Exam focus

Use trans-pair counting rather than a drawing’s orientation. State that simple monodentate MA₄B₂ and MA₃B₃ each yield two geometrical forms and no extra enantiomeric pair.

Advanced insight

Octahedral symmetry makes all six unlabeled vertices equivalent. The two isomer classes arise only after ligand labels break that symmetry; cis/trans and fac/mer are invariant relationships under rotation.

Summary

Octahedral MA₄B₂ has cis and trans forms. MA₃B₃ has fac, with no identical-ligand trans pair, and mer, with one. Each simple class yields two total stereoisomers across its pair of geometries.

Practice questions

1. Classify an MA₃B₃ arrangement with one A–A trans pair. Answer: It is meridional, or mer. A fac arrangement has no A–A trans pair. 2. How many geometric isomers does octahedral MA₄B₂ have, and why? Answer: Two: its B ligands can be adjacent or opposite. All drawings within each category are related by octahedral rotations. 3. What is cobalt’s oxidation state in [Co(NH₃)₄Cl₂]⁺, and does cis/trans change it? Answer: x+2(−1)=+1 gives Co³⁺. Cis/trans changes ligand positions, not formula or oxidation state.