Ksp with significant anion hydrolysis

[Borek]

I never suggested using Ksp in the form given to calculate solubility directly.

Argument about sig figs is completely off. We calculate concentration of PO₄^3- left not by subtraction, but by multiplication (he did it himself!), so knowing total concentration of all forms of phosphates with 3 sig figs we can calculate concentrations of all forms of phosphates with the same accuracy (assuming we know all three Ka values with accuracy high enough). What he says is "we know concentration is 1.0 M, and as only 1.0% dissociated, we can't calculate concentration of the dissociated part, because we have not enough sig figs". Really? What about 0.010 M?

To some extent this is just semantics - looks like he doesn't differentiate between "dissolution equilibrium" and "solubility product". I do. Fact that you were confused seems to suggest his approach can be misguiding.

Edit: I just checked the IUPAC definition (as published in the orange book). It says

The product of the ion activities raised to appropriate powers of an ionic solute in its saturated solution expressed with due reference to the dissociation equilibria involved and the ions present.

so (unless my English fails me) it looks like it allows using the "solubility product" name for other formulas than just simple ions concentrations product. Didn't know. But if so, it requires listing the reaction itself, otherwise it is ambiguous.

 

[Qube]

I never suggested using Ksp in the form given to calculate solubility directly.

No, we're going from solubility to Ksp, not the other way around. Solubility values are taken from Wikipedia.

Argument about sig figs is completely off. We calculate concentration of PO₄^3- left not by subtraction, but by multiplication (he did it himself!), so knowing total concentration of all forms of phosphates with 3 sig figs we can calculate concentrations of all forms of phosphates with the same accuracy (assuming we know all three Ka values with accuracy high enough). What he says is "we know concentration is 1.0 M, and as only 1.0% dissociated, we can't calculate concentration of the dissociated part, because we have not enough sig figs". Really? What about 0.010 M?

Huh? Well, the Ka values we were working with only had 2 sig figs. So right from the beginning that limited us in how many sig figs we can extract for phosphate ion hydrolysis. Most (make that all) Ka values I've seen only have two sig figs - i.e. #.# * 10^#. Plus the amount of phosphate ion we calculated to have been converted through hydrolysis is 100% to 2 or 3 sig figs.

I'm not sure if I follow the second part of your quote above.

To some extent this is just semantics - looks like he doesn't differentiate between "dissolution equilibrium" and "solubility product". I do. Fact that you were confused seems to suggest his approach can be misguiding.

Edit: I just checked the IUPAC definition (as published in the orange book). It says
so (unless my English fails me) it looks like it allows using the "solubility product" name for other formulas than just simple ions concentrations product. Didn't know. But if so, it requires listing the reaction itself, otherwise it is ambiguous.

Okay, looks like he's in the clear then, because he definitely listed the reactions before doing any calculations.

 

[Borek]

No, we're going from solubility to Ksp, not the other way around. Solubility values are taken from Wikipedia.

OK. It doesn't change anything. I never suggested to calculate Ksp from the solubility directly either. Ksp is not calculated using total (formal) concentrations, but equilibrium concentrations.

Huh? Well, the Ka values we were working with only had 2 sig figs. So right from the beginning that limited us in how many sig figs we can extract for phosphate ion hydrolysis. Most (make that all) Ka values I've seen only have two sig figs - i.e. #.# * 10^#.

pKa1 = 2.148, pKa2 = 7.199, pKa3 = 12.35. 4 sig figs for each. I have a reference for that. But it doesn't matter at all.

3 sig figs for the molar solubility would be a problem if the only way to calculate concentration of PO₄^3- left in the solution was "total minus eaten by hydrolysis", and the amount of PO₄^3- left was below 0.1%. Then yes, we don't have enough accuracy for the calculations. However, it is not the only way. Actually it is the worst way I can think of, there are much better methods of calculating PO₄^3- concentration.

You can start with assumption that hydrolysis was complete, use it to calculate pH and concentration of HPO₄^2-, plug these numbers together with pKa3 into the dissociation constant definition

K_{a3} = \frac{[H^+][PO_4^{3-}]}{[HPO_4^{2-}]}

and solve for PO₄^3-. You can then check if the assumption was correct - and as long as [PO₄^3-] << [HPO₄^2-] it holds well enough. That means concentration that you calculated is perfectly valid, and its accuracy is limited by either accuracy of molar solubility or accuracy of pKa3 - 3 sig figs vs 4 sig figs, so we can safely assume we got 3 sig figs from these data.

Note that if the assumption doesn't hold we are in the area where 3 sig figs of the molar solubility are not a problem for the "total minus eaten by hydrolysis" approach.

 

[Qube]

OK. It doesn't change anything. I never suggested to calculate Ksp from the solubility directly either. Ksp is not calculated using total (formal) concentrations, but equilibrium concentrations.

The values taken from Wikipedia are necessarily not the equilibrium concentrations? But why does my prof ask us to go from the Wiki values to Ksp? I'm guessing he's wrong in his approach?

Also what's the distinction between formal concentration and molar concentration?

[Edit] Okay I seem to have arrived at the fact that formal concentration does not take into account the actual chemical species which are solvated but rather only gives you the concentration of the moles of the original chemical formula in solution.

So if indeed Wiki gave us a formal (F) value rather than a molal (M) value ... how does that change anything? From the F value in grams of silver phosphate per liter we can go to moles of silver phosphate per liter. And since silver phosphate is stoichiometric, we can go to moles of silver ion in solution.

 

[Borek]

The values taken from Wikipedia are necessarily not the equilibrium concentrations?

No, they are not. I explained it at least once, your prof explained it as well, you are still surprised?

But why does my prof ask us to go from the Wiki values to Ksp? I'm guessing he's wrong in his approach?

You can calculate Ksp from the molar solubility, just not directly. Knowing just molar solubility you can easily calculate formal concentrations, but not equilibrium concentrations - for these you need additional information about dissociation constants. Once you find equilibrium concentrations, you can calculate Ksp value.

"Naive" approach - treating formal concentrations as equilibrium concentrations - is a sure way of getting wrong Ksp value.

We are going in circles, I feel like I am wasting my time repeating the same again and again.

 

[Qube]

No, they are not. I explained it at least once, your prof explained it as well, you are still surprised?



You can calculate Ksp from the molar solubility, just not directly. Knowing just molar solubility you can easily calculate formal concentrations, but not equilibrium concentrations - for these you need additional information about dissociation constants. Once you find equilibrium concentrations, you can calculate Ksp value.

"Naive" approach - treating formal concentrations as equilibrium concentrations - is a sure way of getting wrong Ksp value.

We are going in circles, I feel like I am wasting my time repeating the same again and again.

I see what you're saying. We have to take in account Kb (of the phosphate anion) in addition to the formal concentration. Is that your point?

 

[Borek]

We have to take into account ALL possible reactions that can change equilibrium concentrations of the ions produced in the dissolution.

 

[Qube]

Thanks for clarifying everything. I think I understand now and if anything we agree that the original Ksp value for Silver Phosphate is likely unreliable. And the other things we've learned:

1) The IUPAC definition of Ksp as being not the product of "constituent" ions but the products of dissolution equilibria with due reference to said dissolution equilibria.

2) Analytical equilibrium: always consider all relevant equilibria.