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Electrochemistry calculations (redox potentials and cells)

Dear Reader,

I was recently teaching some electrochemistry to some students, now before we get going it is important to note that the conventions on writing electrode potentials changed years ago. So if thou gets one’s ye olde text book out or ye olde almanack of chemie be ready for some possible problems.

For example in the 1950s paper on plutonium redox chemistry by Sherman W. Rabideau and  Joe F. Lemon, Journal of the American Chemical Society, 1951, 73, 2895-2899 the redox couple for Pu(IV)/Pu(III) is listed as being -0.953 volts vs the standard hydrogen electrode. Now in modern text books it is listed as about + 0.95 volts. This modern use is an example of the “European” convention while the example I gave as an example of the “American” convention. I would say that both are equally valid but if you want to do a redox calculation be careful that you do not mix unwittingly mix data from both conventions up.

I have not seen the american convention being used much in modern text books, but please be aware that it does exist.

OK Health warning over lets get on with some chemistry. Now as a brain teaser (or brain expanded) I asked my students to consider the question of will a solution of iron(II) tend to reduce a solution of plutonium(IV) to plutonium(III) thus forming iron(III) in the process.

Now the redox couple of iron(II) / iron (III) is +0.77 volts (European convention). So we can calculate the cell voltage for our cell under “standard” conditions (1 mole per litre of everything, at 25 ºC and 1 bar).

We can combine the following two half equations

Fe2+ → Fe3+ + e

e + Pu4+ → Pu3+

To give us

Fe2++ Pu4+ → Fe3+ + Pu3+

The emf of the cell will be under standard conditions equal to 0.18 volts, but which thing will be reduced and what will be oxidized. The plutonium couple is higher (more positive) than the iron one in our European type text. So the plutonium will tend to oxidize the iron to form plutonium(III).

We get the emf of the cell from the difference between the two redox couples. All redox couples are expressed relative to hydrogen gas / hydrogen ions in 1 M acid. This choice of standard is simply a convention. Like the Greenwich meridian we need some arbitrary point to call zero. Hydrogen is a good choice as it features in so many reactions.

From the value of the emf of our cell we can get the ΔG of the reaction, to do this we use the following equation.

ΔG = -nFE

n is the number of electrons which are transferred in the cell reaction (1) and F is Faraday’s constant (charge on a mole of electrons) which is equal to 96485 C. Using this we can get a value for the Gibbs free energy of the reaction. Keep in mind that a cell with a positive emf is a cell which is able to do work, thus ΔG for the reaction in the cell must be negative.

This works out as -17367.3 joules per mole.

We can now get the equilibrium constant for this reaction, when a cell has a emf of zero then it has reached equilibrium. Now we need to use a different equation.

ΔG = ΔGº + RT Ln Q

In our case

Q = aPu3+ aFe3+ / aPu4+ aFe2+

Now in the ideal world (nice place I wish I lived there) the activity coefficient is equal to one, the closest we get to an ideal world is a dilute solution. So for nice and dilute solutions we can write

Q = [Pu3+] [Fe3+] / [Pu4+] [Fe2+]

If ΔG = 0 (zero) then

ΔGº = -RT ln ([Pu3+] [Fe3+] / [Pu4+] [Fe2+])

Rearrange to

-ΔGº / RT = ln ([Pu3+] [Fe3+] / [Pu4+] [Fe2+])

exp (-ΔGº / RT) = ([Pu3+] [Fe3+] / [Pu4+] [Fe2+])

Now do the maths

exp (7) = ([Pu3+] [Fe3+] / [Pu4+] [Fe2+]) = k

Now k = 1107

we can rewrite the equations to give us

ΔGº = -RT ln k

Now it should be clear to my readers that a solution of iron(II) will reduce tetravalent plutonium into trivalent plutonium. You might be interested to read that the classic method of adjusting the oxidation state of plutonium from +4 to +3 in the PUREX process is to use ferrous sulfamate.

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Molecular boxes

OK a bit more about prussian blue and what are known as coordination polymers. The idea of a coordination polymer is that it is a repeating network where bridging ligands link each metal centre to the next, thus making a polymer.

To understand them we need to build up to it, it is not a good idea to try to run before you walk. Lets start with a metal complex which has lone pairs poking out into space ready to bind to something else. Err Oh Err how about [CpFe(PR3)(CN)2]this is a 18 VE complex. For those of you who do not know what the 18 VE rule is then I suggest that you read Tony Hill‘s book which will explain the 18 VE rule. A copy of organotransition metal chemistry by Anthony Hill can be obtained from the RSC.

Here is a picture of the anion, you can see it has the two nitrile groups which each have a lone pair on the nitrogen atom.

The cyclopentadienyl iron triphenylphosphine dicyanide anion

Now here is a complex where the Cu(PCy3) group is used, this group is a weak lewis acid so it has the ability to bind to lewis bases such as the lone pairs of the cyanide groups of the iron fragment.

You can now see the square-ish arrangement of two copper atoms (green) and two irons (yellow), what you are looking at may be a bit of an “atomic fog” but do not worry yourselves too much.

The diiron dicopper square complex

As I feel kind here is a picture of the inner core of the complex, I have removed the carbon groups attached to the phosphorus atoms.

Copper iron square complex without some of the groups to give you a better look

Now if we use three cyanides to link a metal to three other metals we can make a cube, well it is a bit of a distorted cube. I am sure that the more able minded persons reading this can understand how this is step towards a infinite solid with lots of cubes in it. Here is a picture of it. Before you ask the cage is anionic, and the atom at the centre of the cage is a potassium. This complex was published by T.B.Rauchfuss in the Journal of the American Chemical Society, 2007, 129, 1931.

The cube of eight cobalt atoms bridged by 12 cyanide ligands

The cube is capped with cyclopentadienyl and tetramethylcyclobutadiene ligands. For those of you who can not see through the atomic soup (or atomic fog) here is the cube without the capping ligands.

The cobalt cage without the capping groups

* If as a university teacher you do not get on with Tony Hill’s book then you can always replace this with another book.

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