Counting Isomers of MA₂B₂C₂

Introduction

Octahedral MA₂B₂C₂ is a classic test of systematic isomer counting. Three different ligand types each occur twice, so there are more possibilities than ordinary cis/trans. The clean answer is five geometrical arrangements, one of which is chiral and exists as two enantiomers. If every distinct mirror image is counted separately, the total is six stereoisomers.

Core explanation

Use the three opposite axes of an octahedron as the counting framework. First, each identical pair A–A, B–B and C–C can be trans. Put A on one opposite axis, B on a second and C on the third. The axes are equivalent under rotation, so this is one all-trans geometrical class, not six permutations of x, y and z.

Next, exactly one identical pair can be trans. If A is trans, the two B ligands and two C ligands occupy the remaining four positions, each as a cis pair. Up to rotation, this produces one class with trans A and cis B/C. The corresponding class with trans B and the class with trans C are chemically distinct because A, B and C are different ligand types. That adds three classes. There cannot be exactly two identical trans pairs: if A fills one opposite axis and B another, the two remaining sites are opposite, forcing C trans as well.

Finally, no identical pair need be trans. Each A, B and C pair is cis: the all-cis class. It can be drawn in two mirror-related spatial arrangements. Unlike the all-trans and one-trans classes, the mirror image of all-cis cannot be superimposed by rotating the octahedron. These two forms are enantiomers, often distinguished by handedness rather than by a simple cis/trans word.

The tally is therefore 1 all-trans +3 one-trans +1 all-cis =5 geometric classes. Four of those classes are achiral and each contributes one stereoisomer. The all-cis class contributes two enantiomers, giving 1+3+2=6 individual stereoisomers. A worksheet that asks for “five isomers” may be counting geometrical types; a question asking for every stereoisomer, including optical partners, expects six. State the convention explicitly rather than arguing from an unlabeled number.

The ligand identities must be genuinely different. If B and C are actually the same ligand, the formula reduces to MA₂B₄ and only cis/trans remains. Conversely, if one of A, B or C has its own stereogenic centre or is bidentate, extra stereochemistry or adjacency constraints can arise. The five/six result assumes six simple monodentate ligands in an ideal octahedral arrangement.

For a concrete composition, [Co(NH₃)₂Cl₂(H₂O)₂]⁺ has A=NH₃, B=Cl⁻ and C=H₂O. Cobalt is +3 because two chlorides contribute −2 and the ion has charge +1. Its oxidation state and d count remain unchanged among the stereoisomers. Magnetic properties are therefore not a substitute for the ligand-position count.

Step-by-step reasoning

List the three identical pairs. Count the one all-trans case. Choose which single pair, A, B or C, is trans to get three one-trans cases. Add the all-cis case. Rule out a two-trans case because the third pair would necessarily be trans. Finally test all-cis for chirality and count its two enantiomers in the total.

Visual explanation

Draw three intersecting opposite axes. In the all-trans picture, assign one colour pair to each axis. In a one-trans picture, keep only one colour on an entire axis and distribute the other two colours around the equator as adjacent pairs. Draw the all-cis arrangement and its mirror image as two handed colour sequences around the metal.

Real-world analogy

Three pairs of coloured socks are placed at opposite ends of three rods. All pairs can be matched across rods, or one chosen colour can be opposite while the others sit adjacent. A twisted all-adjacent placement can have left- and right-handed versions that a simple rotation cannot exchange.

Real-world example

An octahedral cobalt(III) complex with two ammines, two chlorides and two waters has the MA₂B₂C₂ ligand pattern. A chemist designing a separation must consider five geometric families and, for the all-cis family, the possibility of resolving two optical partners.

Why?

Why is a “two-trans, one-cis” arrangement impossible? Two complete opposite axes use four of the six octahedral sites. The remaining two sites are the third opposite axis, so the final identical pair must also be trans.

Common misconception

“Five geometrical isomers means five total stereoisomers.” The all-cis geometrical class has two nonsuperimposable mirror forms, so the count of individual stereoisomers is six under the stated assumptions.

Worked example

Count stereoisomers of ideal octahedral [Co(NH₃)₂Cl₂(H₂O)₂]⁺. Label A=NH₃, B=Cl and C=H₂O. There is one all-trans arrangement, three one-trans arrangements distinguished by which pair is opposite, and one all-cis geometric class. The all-cis class is chiral and contributes a pair, so there are five geometrical classes and six individual stereoisomers.

Quick check

1. How many one-trans geometrical classes exist for MA₂B₂C₂? Answer: Three, depending on whether A, B or C is the trans identical pair. 2. Which class is chiral in the simple octahedral case? Answer: The all-cis class, which has two enantiomers.

Exam focus

Report both numbers—five geometric classes and six total stereoisomers—when the wording is ambiguous. Show the one-plus-three-plus-one grouping and explain why two-trans cannot occur.

Advanced insight

The all-cis pair illustrates why geometry labels alone can conceal stereochemical multiplicity. A classification by trans-pair signatures identifies geometric classes; orientation-preserving rotations, rather than reflections, determine whether a class splits into enantiomers.

Summary

MA₂B₂C₂ has one all-trans, three one-trans and one all-cis geometric class. The all-cis arrangement is chiral, so five geometric classes correspond to six individual stereoisomers.

Practice questions

1. Why does choosing trans A versus trans B make distinct MA₂B₂C₂ geometrical isomers? Answer: A and B are chemically different ligands. Rotation cannot change which identity forms the opposite pair, so the two arrangements are distinct. 2. What count should be reported if a question explicitly includes optical isomers? Answer: Six individual stereoisomers: four achiral classes counted once plus the two all-cis enantiomers. 3. What happens to the MA₂B₂C₂ count if B and C turn out to be the same ligand? Answer: The formula becomes MA₂B₄, which has only cis and trans octahedral geometrical arrangements under the simple monodentate assumptions.