Inner/ Outer sphere

Difference between inner and outer sphere

Firstly, electron transfer reactions between metal complexes can be divided into two categories: outer sphere and inner sphere. In the former, the two complexes come together and an electron is transferred without any bond being formed between the complexes. In the latter, the two complexes form a definite intermediate where at least one ligand is shared by both metal ions.



In inner sphere reactions, of course, a ligand must be substituted on one metal ion by the ligand on the other metal ion that forms the bridge. Therefore if electron transfer is observed to occur faster than either complex undergoes ligand exchange reactions, this is good evidence that the mechanism for electron transfer is outer sphere. If neither complex has any ligands capable of bridging (that is, with extra lone pairs), the mechanism must be outer sphere.



The free energy of activation DG for (I) is 33 kJ mol–1, and the second order rate constant is 3 L mol–1 s–1 at 25 ?C. It is worth considering why there is a barrier, even though quite small, to a reaction in which there is no net chemical change, and for which therefore DG must be zero.

Basically, what is happening is that an electron from the Fe(II) t2g orbital is transferred to the Fe(III) t2g orbital. For this to happen without energy input, the energies of the two orbitals must be the same (Franck–Condon principle). But clearly, they will not be; [Fe(H2O)6]2+ is different from [Fe*(H2O)6]3+. Its metal–ligand bonds will be longer because the Fe2+–O electrostatic interaction will be less than that of Fe3+–O. If electron transfer were able to take place without energy input, we would end up with [Fe(H2O)6]3+ with bond lengths typical of [Fe(H2O)6]2+ and [Fe*(H2O)6]2+ with bond lengths typical of [Fe*(H2O)6]3+. Both could then relax with the release of energy, violating the first law of thermodynamics.



What has to happen to allow electron transfer is that the Fe(II) complex has to be in an excited vibrational state. At some time, its Fe–O bonds will be shortened somewhat, compared with the ground state. This will raise the energy of the t2g orbitals as there will then be greater repulsion of electrons in these orbitals by the water lone pairs. The Fe(III) complex has also to be in an excited vibrational state so that at some times, its bonds will be lengthened compared with the ground state, and the energy of the t2g orbitals will be lower than in the ground state. This is the origin of some of the activation energy for electron transfer.

Contributions to energy of activation

There are other contributions to the energy of activation. For example, both the ions in our Fe(II)/Fe(III) example are positively charged, so this would present a barrier to them coming together close enough to allow electron transfer to occur. There is also energy required to adjust the ‘second coordination sphere’ of solvent (i.e. the solvent which is strongly hydrogen-bonded to the coordinated waters) around the participating ions, from the ground state arrangement to the transition state arrangement. This contribution to the free energy of activation is called the ‘solvent reorganisation term’, and can be quite large.


DG (total) = DG(bond length changes) + DG(solvent reorg.) + DG(Coulombic)


The bigger the difference in metal–ligand bond lengths between oxidant and reductant, the slower will be the electron transfer reaction, because the free energy of activation will be higher. In the Fe(II)/Fe(III) case mentioned above, the barrier is quite small because both Fe(II) and Fe(III) give high spin complexes with water. Fe(II) is therefore t2g4 eg2 and Fe(III) is t2g3 eg2. The difference in bond lengths is comparatively small and is mostly due to the difference in oxidation state.


An example where there is a much bigger difference concerns reaction (III) below.


          [Co(NH3)6]3+ + [Co*(NH3)6]2+ ® [Co(NH3)6]2+ + [Co*(NH3)6]3+ (III)

Co–N   1.936(15) A        2.114(9) A



The second order rate constant for this reaction is about 10–6 dm3 mol–1 s–1 (i.e. very slow) and DGis over 100 kJ mol–1. This is because the Co(III) complex is low spin, and therefore t2g6 eg0. However, the Co(II) complex is high spin, t2g5 eg2. The presence of two extra electrons in the eg orbitals, repelling the ligands, means that the Co–N bonds in the Co(II) complex are much longer than in the Co(III) complex, so the two complexes must both be highly vibrationally activated before electron transfer can occur.

Marcus theory

The ideas shown above were expressed quantitatively by R.A. Marcus (Nobel Prize for Chemistry 1992). He showed that for a given electron transfer reaction, such as

          [Fe(CN)6]4–  + [Mo(CN)8]3–  ®      [Fe(CN)6]3–  + [Mo(CN)8]4– (II)

                             (k12)2 = fk1k2K12

                             – the Marcus cross–relation

where k12 is the rate constant for reaction (II), K12 is the equilibrium constant for (II), and k1 and k2 are the rates of the corresponding two electron self exchange reactions,

[Fe(CN)6]4–  + [Fe*(CN)6]3–  ®     [Fe(CN)6]3–  + [Fe*(CN)6]4–  (IV), and

[Mo(CN)8]4–   + [Mo*(CN)8]3–    ®   [Mo(CN)8]3–    +   [Mo*(CN)8]4–  (V)


[The factor f is not a constant, but a complex parameter that takes into account the encounter complex formation process, but for our purposes it can be taken as being almost 1.] The following Table gives an idea of the success of the simple version of Marcus theory dealt with here, assuming f = 1:





(Reducing agent on left, oxidant on right)

log K12

k12 obs

k12 calc. (Marcus)

[Ru(NH3)6]2+ + [Ru(NH3)5(py)]3+   ® products


1.4 ? 106

4 ? 106

[Ru(NH3)5(py)]2+ + [Ru(NH3)4(bipy)]3+   ® products


1.1 ? 108

4 ? 107

V2+(aq) + [Co(en)3]3+   ® products


5.8 ? 10–4

7 ? 10–4

[Fe(CN)6]4–  + [Mo(CN)8]3–    ® products


3.0 ? 104

4 ? 104


So: (1)   There is quite good agreement between the observed rate constants k12 and the calculated ones from the theory and (2) there is good correlation between the magnitude of K12 and k12.


Following on from point (2), in fact there should be a linear free energy relationship between rates of electron transfer, and driving force for the reactions. This follows from the fact that ln K is proportional to DG and ln k is proportional to DG, so:


k122 = fk1k2K12             (Marcus relation)

Assume f = 1

Express as ln:    

2 ln k = ln k1 + ln k2 + ln K12

Implies that[image] 2 DG12 = DG1 + DG2 + DG12


This has been tested for the oxidation of a series of complexes [Fe(R–phen)3]2+ by Ce(IV) ions to [Fe(R–phen)3]3+ (R–phen = substituted 1,10–phenanthroline) (left).



The more electron–withdrawing the R group, the worse the phen ligand is at stabilizing low spin Fe(III) and the better it is at stabilizing low spin Fe(II), so the less driving force there is for [Fe(R–phen)3]2+ oxidation, and the slower is the rate.

Inner Sphere Reactions

In this class of electron transfer reaction, a ligand–bridged intermediate is formed. Proof that such intermediates were involved was obtained by Carol Creutz, Henry Taube and their co–workers. Taube won the Nobel prize in the early 1980’s for this. For example, consider the reaction (VI), done in acid conditions.


[Co(NH3)5Cl]2+ + [Cr(H2O)6]2+ ® [Cr(H2O)5Cl]2+ + [Co(H2O)6]2+ + 5 NH4+ (VI)  


The first thing that is worth noting about this reaction is that, if it proceeded by an outer-sphere reaction mechanism, we would expect the rate to be slow, because the two electron self-exchange processes involved, k1 and k2 in the Marcus equation, i.e. Co(II)/Co(III), and Cr(II)/Cr(III), will themselves be very slow (the Co(II)/Co(III) case we have already discussed, but there is also a large bond length difference expected for Cr(II)/Cr(III) because Cr(II) is high-spin d4 whereas Cr(III) is d3).


It is noted that (1) the observed rate constant k12 for this reaction is much faster than that predicted by the Marcus equation, and (2) unlike outer-sphere mechanisms, a ligand seems to have ‘changed places’ in the course of the reaction.


Co(III) complexes are extremely inert to ligand substitution because they are low spin d6. Cr(II) complexes are very labile (high spin d4). On the right hand side, Cr(III) complexes are also inert, but Co(II) complexes are labile (high spin d7). The Cr(III) complex could be isolated from the reaction because it is inert, and proof obtained that it has a Cl ligand, although the Cr(II) reactant did not. If the reaction is performed in the presence of free radiolabelled Cl ions, no radiolabelled Cl ends up coordinated to the Cr(III), proving that the Cl ligand must have originated from the Co(III) complex. The Co(II)–ammine complex generated decomposes in the presence of acid, giving the hexaaquo ion as shown. The separate steps in the process are:


Substitution of one water ligand on the labile Cr(II) ion by a second lone pair on the Cl ligand from the Co(III) complex:


[Cr(H2O)6]2+ +[Co(NH3)5Cl]2+  ® [(H2O)5CrII–(m–Cl)–CoIII(H2O)5]4+ + H2O


Electron transfer from Cr(II) to Co(III) over the Cl bridge:

[(H2O)5CrII–Cl–CoIII(NH3)5]4+ ® [(H2O)5CrIII–Cl–CoII(NH3)5]4+ 


Hydrolysis of the now labile Co(II)–Cl bond:

[(H2O)5CrIII–Cl–CoII(NH3)5]4+ ® [(H2O)5CrIII–Cl]2+ + [CoII(NH3)5(H2O)]2+   


Finally, the Co(II)–ammine complex, now labile to ligand substitution, hydrolyses rapidly under the acidic conditions to give [Co(H2O)6]2+ and 5 NH4+


It is interesting to note that electron transfer reactions can sometimes be used in synthesis. One brief example will now be given.


The thiocyanate ion NCS is ambidentate. Usually, 3d metal ions prefer to bond to the nitrogen, M–NCS giving an N–thiocyanato complex. For example,


          [Cr(H2O)6]3+ + NCS ® [Cr(NCS)(H2O)5]2+ (very slow!)


However, by using a redox reaction exactly like the one above, the Cr–SCN (S–thiocyanato) complex can be made, because the thiocyanate ion can bridge by bonding to the Cr(II) via the sulfur:


[Cr(H2O)6]2+ + [SCN–Co(NH3)5]2+ ® [NCS–Cr(H2O)5] + [Co(H2O)6]2+ + 5 NH4+



Good rules-of-thumb for deciding whether electron transfer goes by IS or OS

1.           If the rate of electron transfer is faster than the rate of ligand exchange for both of the reactants – must be outer sphere.

2.           If the rate of electron transfer is much faster than Marcus theory would predict (based on measuring K12, k1 and k2), there is a good chance that the reaction is inner sphere.

3.           However, for inner sphere to work, there must be at least one ligand, in the reactants, capable of bridging, i.e. with an additional lone pair (e.g. a halide ion, hydroxide, thiocyanate, 4,4’-bipyridine, etc.); water is a very poor bridging ligand and will not suffice.


4.           Just because a reaction goes via the IS mechanism does not mean that the bridging ligand necessarily ends up transferring to the second metal – which metal it remains coordinated to, once the intermediate has decomposed, depends upon which metal centre is more labile – it is more likely to end up on the less labile metal centre.

Mixed valence complexes

There has always been great interest in bi– or polynuclear metal complexes in which two metals with different oxidation states are connected by a bridging ligand. These could be regarded as ‘frozen’ inner sphere electron transfer intermediates. An insoluble, polymeric, example is Prussian Blue, FeIII4{[FeII(CN)6]3·nH2O. This can be regarded as a cubic lattice of CN ions, arranged so that the low spin d6 Fe(II) ions are surrounded by the C ends of 6 CN ligands, and the high spin d5 Fe(III) ions are surrounded by, on average, four CN nitrogens and two water molecules. The blue colour arises because although this is the stable form, it is possible for the material to absorb photons in the red region of the spectrum, to promote electrons from the Fe(II) across the CN bridging ligands to the Fe(III). Absorption of this red light leads to the intense blue colour seen for this substance.


More recently, soluble, monomeric complexes have been constructed, deliberately, to study the rate of electron transfer between the metal ions. The structure of some typical examples is shown above.



Notice that (1) the further the electron has to travel, the slower the rate constant and (2) an unsaturated bridge allows more rapid electron transfer. For instance, pyrazole is about the same length as Ph2PCH2PPh2, but the rate constant for the unsaturated pyrazole is 30 times greater than for the saturated diphosphine.