Distribution Diagram for speciation of Cr(VI)

[geostudent]

Hi, this is some Geochemistry homework. We were not really given much information on how to approach this problem. I treated it as a diprotic acid H2CrO4, and made an alpha plot using the equations given using the constants K only for the fully protonated and deprotonated species. I'm not sure how to approach this problem with four constants and plot the distributions of four species at different concentrations.

Any suggestions?





Make a distribution diagram for the various Cr(VI) species as a function of pH at CrT=1, 10-2 and 10-4M. Neglect activity corrections.
Equilibria: CrO42-+H+=HCrO4-; logK1=6.5
HCrO4-+H+=H2CrO4; logK2=-0.8
2 HCrO4-=Cr2O72-+H2O logK1a=1.52
Cr2O72-+ H+=H Cr2O7- logK2a=0.07

[Cr]T=[CrO42-]+[HCrO4-]+2[Cr2O72-]+2[HCr2O7-]

Homework Equations

[PLAIN]http://www.chem.uoa.gr/applets/appletacid/Images/text_d4.gif

[PLAIN]http://www.chem.uoa.gr/applets/appletacid/Images/text_d5.gif

The Attempt at a Solution

See the attached jpeg for my alpha plot.

 

[Borek]

Using just formulas you listed you can prepare a plot for dissociation, but not for dimerization. To be honest - I have no idea how to solve this particular problem in a simplified way. You can try to read about general approach to chemical equilibrium calculation, but it leads to sets of nonlinear equations that are difficult to solve manually - the only viable approach is numerical.

 

[geostudent]

Thanks Borek. I think I'll work on the numerical approach, maybe Newton-Raphson. This class, I think, was a mistake. It's a combined grad/undergrad class, and I've spent so much time reading Analytical Chem and Thermodynamics (classes I have NOT taken) that I've neglected my physics class and will have withdraw.

 

[geostudent]

Setting up equations for equilibria

The Attempt at a Solution

I have tried setting up mass balances and doing various substitutions to obtain an α₀ that I could solve for iteratively, but so far I've gotten nowhere. I've found alpha plots for the speciation of Cr(VI) in several scientific papers, but the authors always cite software for generating the diagrams and never show any quantitative procedures. Any pointers? If this were just the dissociation of a diprotic acid, I could use

α₀ = [H+]2/ [H+]2 + Ka1[H+]2+ Ka1Ka2



but that clearly won't work here.

 

[epenguin]

I see you asked almost the same question over a week ago.

It is a stage more difficult than the usual pH etc. questions asked here because you also have the dimerisation of the chromate.

It was not obvious what you mean by this

α₀ = [H+]2/ [H+]2 + Ka1[H+]2+ Ka1Ka2

It's confusing since
you do not say what a₀ means
you seem to have omitted brackets
I get the impression that by quoting from somewhere you have changed the meaning of the K's with respect to your quoted equations.

[H2CrO4]/[Total monomeric chromate] = [H⁺]²/ ([H⁺]² + K₁[H+] + K₁K₂) would make sense I think.

So, what to do?

Firstly you have written out most of the equations of the system (you could do so more clearly and please check the constants.). However there is still another one and that is the electroneutrality equation.

Secondly you could follow the recommendation of Polya, whom I often quote: if you cannot see how to solve a problem, try solving a simpler related one. This is not even artificial in your case - e.g. above say pH 3.5 you still have a problem, but you can see you can simplify - some of the concentrations become negligible. Maybe at some Cr_T you have simplifications too. At least you can do something even presentable and maybe then see better how to do the rest.

If I have time I will try and look again at this in next days but cannot promise, you should try to make a start along the suggested lines.

 

[geostudent]

Sorry to confuse, I should have been clearer.

α0 is the alpha value ( relative concentration) of the free acid. I had been attempting to use a generic formula in a text for the composition of polyprotic acids as a function of pH given by:

For a diprotic acid say:

α₀= [H⁺]² / C_T

α₁ = K₁[H⁺] / C_T

α₂ = K₁K_a2 / C_T





where C_T = [H⁺]² + K₁[H⁺] + K₁K₂

[H2CrO4]/[Total monomeric chromate] = [H+]2/ ([H+]2 + K1[H+] + K1K2)

With the original assignment, this is exactly what I did. The plot, of course, didn't account all of the species though, especially at the very low pH.

Mostly, I don't want to use a plug and chug formula from a textbook. I haven't taken above Gen. Chem II and I'm trying to acquaint myself with handling problems like this. The relationships and mathematics sometimes seem very cryptic to me.


Thanks so much for the feedback. I will setup an electroneutrality eq. and look into the dimerization and see what I can figure out.

 

[epenguin]

Mostly, I don't want to use a plug and chug formula from a textbook. I haven't taken above Gen. Chem II and I'm trying to acquaint myself with handling problems like this. The relationships and mathematics sometimes seem very cryptic to me.Thanks so much for the feedback. I will setup an electroneutrality eq. and look into the dimerization and see what I can figure out.

That is the right (not to mention only!) approach. Do not worry about having done only gen. Chem II. This sort of problem involves only a very limited set of ideas, not wide knowledge, and on the other hand general chemists and everyone who is not dealing with this sort of problem every day finds them difficult and confusing. In fact more questions on this come in the chemistry/biology sections here than on anything else. There has to be a bit of building up habit and intuition - a lot of it is 'at this pH that or that species will be negligibly present' or 'at pH = this pK, that and that are equal, most of the change happens over about 1 pH unit either way'.

The distinctive thing in your problem is that it involves both ionisations and dimerisation. If you look at it in the pH > 3 or so region I think you reduce to a worthy problem of principle that incorporates this essential feature of coupled dimerisation and ionisation equilibria. You ought to be able to see (qualitatively without math) that this coupling means that the total concentration should affect the pH range over which an ionisation happens, and in what direction. Yes, the math will probably involve a quadratic equation at least.

Then the total problem later is that plus a complication of more ionisation equilibria.

Sorry to confuse, I should have been clearer.

α0 is the alpha value ( relative concentration) of the free acid. I had been attempting to use a generic formula in a text for the composition of polyprotic acids as a function of pH given by:

For a diprotic acid say:

α₀= [H⁺]² / C_T

α₁ = K₁[H⁺] / C_T

α₂ = K₁K_a2 / C_T

where C_T = [H⁺]² + K₁[H⁺] + K₁K₂

Thanks so much for the feedback. I will set up an electroneutrality eq. and look into the dimerization and see what I can figure out.

OK you have set up some of the equations. (Though I don't much like your notation - I personally would prefer one that is more explicit at the same time avoiding writing out full chemical formulae all the time. To see what I was doing I would call the species H₂Cr, HCr^-, Cr^2-, Cr₂^2-, HCr₂^- and forget the α's.)

I advise writing out all the equilibrium equations explicitly, with the pK's beside them.

As well as ionisation equilibria there is the dimerisation equilibrium equation which you haven't yet written.It worried me how your dimerisation was quoted with a pK_a notation same as an acid-base, seems inconsistent - check and possibly reproduce onto here the original info you are working from.

On second thoughts I am not sure the electroneutrality eq is all that useful in this case, but it might help eliminate 1 variable.

Edit: On third thoughts it probably does nothing - it would bring in a thing we have managed to not mention and hopefully we can leave out of consideration (not that its amounts are negligible) - the counter-ion, e.g. Na⁺ that we would have to add to chromic acid to get the pH range. If the question concerned the amounts of NaOH we had to add to get those pH's then we'd have to consider it.

 

[geostudent]

The Na⁺ involved another problem I presented a while back. I think it was bubbling CO₂ through an NaCl solution, then adding Na(OH).

The logK's I've given are the same as those given by the professor.

You are totally right about being explicit. It's easy to lose track of what I'm doing using alphas.

I will try to come back to this problem. Problem set due now involves isotopic fractionations in the mantle.

 

[epenguin]

OK I found this problem of interest and I have done it for the simplified system, the figs would be very accurate above pH 3.5. It is retarded because of confusion and mistakes in the numerical calculation/plotting plus also I was being thrown out of the site every time I tried to view the post.

This is simplified to the system ½H₂Cr₂O₇^2-$\leftrightarrow$ HCrO₄^-$\leftrightarrow$ CrO₄^2- + H⁺.

red is 1M dark blue is 10^-2M azure is 10^-4M

However I could not upload my computation directly and had to scan a printout – for some reason a red curve (uppermost on left) looks blue!

I suggest you will find it easiest to go by stages. First just formulate and solve for [X], or better for [X]/[X_T] in a simple monomer dimer equilibrium ½X₂ $\leftrightarrow$ X. This does involve solving a simple quadratic. On reflection and remembering previous comments this is nothing special – for any general bimolecular process like that, or A + B $\leftrightarrow$ C even A + B $\leftrightarrow$ C + D to calculate the equilibrium position if all you are given is the eq. constant and total amounts e.g. of starting material you always have to solve a quadratic I think. (It’s true pH and buffer problems that are a major part of this forum are bimolecular H⁺ + A^- $\leftrightarrow$ HA and you manage mostly without quadratics – that is because you are given extra information about the equilibrium point in those problems – either [H⁺] or [A^-] by the concentration of counter-ion and electroneutrality or added alkali etc.)

Then when this is clear you can add an extra equilibrium of a deprotonation of the X and you have the system above. You will see you can easily do the whole system and these ionisations are kind of pendants to the polymerisation. Reversing my previous opinion something like your a’s can be helpful after all.

Now I understand you meant the pK’s of bichromate are 1.52 and 0.07; they do not come into my simplified model where it is always 2^-. You do not give a constant for dimerisation K_D. I have used the value 224 M^-1, you use whatever you are given, but I find a difficulty.

What does molarity of a dimer or polymer mean? - I do not actually know whether there is a convention about this. I prefer to use the ‘atomic molarity’ I,e, for instance the molarity of chromium atoms in the two forms. That allows me always to use a simple conservation equation in all cases, like

[A₂] + [A] = A_T

without thinking about multiples all the time. But then I have to convert a given molecular K_D multiplying it by two. I took the constant from here http://en.wikipedia.org/wiki/Predominance_diagram because at least the constant is defined there which it wasn’t in two other sources I found. Also there is a motivation for preferring to express things as dissociation constants, in this case about 4.4X10^-2 M as this gives the idea of the concentrations over which things are in balance and the greatest changes happen.

I will add some other comments later but have to go now, also the site is playing up and I can't see what I'm writing etc.

 

[epenguin]

OK I will nearly complete this with a few points I think worth noting.

1. Students should generally be able to reason, even beforehand, that qualitatively the results are those to be expected.

2. One should have an eye for thermodynamic reciprocity. E.g. the fact that the pH profile is different at different pH’s means that varying the concentration, e.g diluting a 1M solution around pH 7 with water would change the pH, in contrast to ordinary buffer systems where (ideally, without ionic strength effects) it would not vary.

3. At low molarity the system simplifies to only the HCrO₄^-$\leftrightarrow$ CrO₄^2- + H⁺ system. The curve is then of the form of a simple titration curve. The black curve (which I did not previously mention) in the fig of the last post is such a curve for pK 6.5 chromate assuming no dimerisation and has that form - and is quite close to the curve at high dilution (10^-4M). Though it is not very obvious the curves at higher concentration do not have the same shape as a titration curve and are not parallel to it but steeper. This can be better seen by checking, e.g. with a ruler, the horizontal distance between the curves which decreases as you go upwards. This is an example of co-operativity, the deprotonation gets easier as more is deprotonated.

4. The calculation can be easily completed for the low pH’s with their extra protonations. The low pKs are noted to be ‘not well known’ with widely varying estimates. I never had to deal with things at pH’s as low as 1 or 0 and as far as I remember anything happening there was considered with scepticism. However to able to do that sort of calculation and not be afraid of it is the point of this exercise. For that matter there seem to be varying estimates even of the mid-range pK given as 6.5, perhaps a matter of varying conditions. This should stimulate a reflection – how would you actually determine the various constants experimentally?

5. Perhaps there are more complicate calculations of the same general kind when you consider cation complexes. Much more complicated and algebraic solutions become no longer possible or useful I guess.

6. I do not know what the geophysical applications of this sort of thing are, you might ask. It sounds an interesting and useful course.