Lesson 2 · 30 min

Geometry and isomerism

Cisplatin treats cancer while transplatin does not; [Co(en)₃]³⁺ comes in left- and right-handed forms. Turn the 3D models and count how many isomers a complex really has.

In this lesson

Match coordination numbers 2, 4 and 6 with linear, tetrahedral, square-planar and octahedral geometries
Recognise cis/trans and fac/mer isomers, and explain how Werner proved the octahedron by counting isomers
Decide whether a complex is chiral and recognise Δ/Λ enantiomers
Count the stereoisomers of common complex types

In 1965 Rosenberg set out to study how electric fields affect bacterial division, and found that E. coli stopped dividing and grew into long filaments. The culprit was not the field but platinum–ammine–chloride complexes formed from the platinum electrodes. Further tests showed that only the “cis” isomers, with two chlorides side by side, were active; the “trans” isomers, with them opposite, barely worked. In 1969 cis- $\ce{[PtCl2(NH3)2]}$ proved active against tumours in mice, and cisplatin became one of the most important anticancer drugs. One formula, two spatial arrangements, and utterly different behaviour.

Predict first

How many different stereoisomers does $\ce{[CoCl2(NH3)4]+}$ have?

Pick an answer and lock it in before you look. A wrong guess teaches more than no guess.

1 2 3 4

Coordination number sets the shape

Coordination number 2: linear, e.g. $\ce{[Ag(NH3)2]+}$ , $\ce{[CuCl2]-}$ .
Coordination number 4: tetrahedral ( $\ce{[Zn(NH3)4]^2+}$ , $\ce{[CoCl4]^2-}$ ) or square planar. d⁸ ions — Ni(II) with strong-field ligands, Pd(II), Pt(II), Au(III) — are almost always square planar, e.g. $\ce{[Ni(CN)4]^2-}$ and $\ce{[PtCl4]^2-}$ .
Coordination number 6: octahedral, the commonest of all.

VSEPR does not transfer to transition-metal complexes: d electrons do not occupy directions the way lone pairs do, and the square plane is itself a result of the d electrons meeting the ligand field (lesson 3).

Werner’s argument: how many geometric isomers each candidate geometry predicts
Type	Planar hexagon	Trigonal prism	Octahedron	Werner found
MA₅B	1	1	1	1
MA₄B₂	3	3	2	2
MA₃B₃	3	3	2	2

Source: Geometric isomers only (enantiomers not counted separately), after Housecroft & Sharpe, Inorganic Chemistry, ch. 19; the octahedral column is reproduced by this site’s enumeration kernel.

Werner only ever found two forms of $\ce{[CoCl2(NH3)4]+}$ and two of $\ce{[CoCl3(NH3)3]}$ , exactly as an octahedron predicts. For $\ce{MA3B3}$ the two isomers are called fac (facial: the three A on one face of the octahedron) and mer (meridional: the three A along a “meridian”, two of them trans).

The rigorous way to count isomers is to list every arrangement and count as one all those that a rotation superimposes. Mathematically this is Burnside’s lemma.

N = \frac{1}{|G|}\sum_{g \in G} \left|X^{g}\right|

Burnside’s lemma: the number of distinct arrangements is the average number of arrangements left unchanged by each rotation. The octahedron’s rotation group has 24 operations.

$G$: rotation group of the polyhedron (24 for an octahedron, 12 for a tetrahedron, 8 for a square)
$X^{g}$: arrangements left unchanged by the operation g

Complexes in the mirror

A molecule that cannot be superimposed on its mirror image, however it is turned, is chiral, and the two forms are enantiomers. The three chelate rings of $\ce{[Co(en)3]^3+}$ are like the blades of a propeller, left- or right-handed, labelled Λ and Δ (by the IUPAC rule, a right-handed helix is Δ). The cis form of $\ce{[CoCl2(en)2]+}$ is chiral too, while the trans form has a mirror plane and is not — three stereoisomers in all. Enantiomers have nearly identical physical properties and differ only towards polarised light and in chiral surroundings.

Enumerated by symmetry, one by one · 3D

Isomers of complexes

Complex Show the mirror image

$\ce{[Co(en)3]^3+}$ 2 stereoisomers: 1 geometric isomers, 1 of them as pairs of enantiomers

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Drag to rotate · scroll or pinch to zoom · arrow keys rotate too

Chiral: not superimposable on its mirror image

With the mirror image on, rotate the models and try to superimpose the two propellers. Then switch to [CoCl₂(en)₂]⁺: the trans form matches its mirror image, the cis form does not. Finally compare square-planar cisplatin with tetrahedral [ZnCl₂(NH₃)₂]: a tetrahedral MA₂B₂ has only one form.

Worked example

How many stereoisomers does [CoCl₂(en)₂]⁺ have?

Each ethylenediamine spans one edge of the octahedron (necessarily cis), and the two Cl take the remaining two vertices. Count all the stereoisomers.

Concept check

Which of these complexes exists as a pair of enantiomers?

trans-

\ce{[CoCl2(en)2]+}

cis-

\ce{[CoCl2(en)2]+}

cis-

\ce{[PtCl2(NH3)2]}

\ce{[Co(NH3)6]^3+}

Try the calculation

How many stereoisomers (counting enantiomers separately) does the octahedral complex $\ce{[CoCl2(H2O)2(NH3)2]+}$ (type $\ce{MA2B2C2}$ ) have?

Key takeaways

Coordination number 2 is linear, 4 tetrahedral or square planar (d⁸), 6 octahedral.
MA₄B₂ and MA₃B₃ each have 2 geometric isomers (cis/trans, fac/mer); Werner used this to prove the octahedron.
Complexes not superimposable on their mirror images are chiral: [Co(en)₃]³⁺ exists as Δ and Λ, and cis-[CoCl₂(en)₂]⁺ is chiral too.
To count isomers, group all arrangements by rotation (Burnside’s lemma), then check each class against its mirror image.

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Data and references

B. Rosenberg, L. Van Camp, T. Krigas, “Inhibition of cell division in Escherichia coli by electrolysis products from a platinum electrode”, Nature 205, 698–699 (1965).
A. von Zelewsky, Stereochemistry of Coordination Compounds, Wiley (1996).
C. E. Housecroft, A. G. Sharpe, Inorganic Chemistry, 5th ed., Pearson (2018), ch. 19 (d-block chemistry: coordination geometries and isomerism).
B. Rosenberg, L. Van Camp, J. E. Trosko, V. H. Mansour, “Platinum compounds: a new class of potent antitumour agents”, Nature 222, 385–386 (1969).
IUPAC, Nomenclature of Inorganic Chemistry (2005), IR-9.3.4 (configuration index; skew-line convention for Δ/Λ).