[magician4]

ok, if we want for some oddball reason to define solubility in terms of sulphate ion, (...)

... which would, of course,  answer a slightly different question (as you're aware of, I take it), and I wouldn't denote it as "solubility of..", but as "final equilibrium concentration of..." instead (as the system won't be "saturated" with respect to sulfate / hydrogensulfate ions "stand alone" : you always could add more sulfuric acid, for example, and increase those values -  with traces of additional precipitate of lead sulfate as a consequence , that is)

but o.k., let's look at that approach for a change: of course this can be done / calculated, too

but even so: you can't avoid the full terror of charge balance calculation no matter what of the interconnected values you wish to calculate for - if you went for high precision, that is: c₀(H₂SO₄) , [HSO₄^-], [SO₄^2-], [H⁺] ,[Pb²⁺]_PbSO4 and c₀(PbCl₂) will remain coupled forever, no matter what value or combination of values you're going for, resulting in the same amount of calculation demand everytime.

.. where [SO₄2-]acid,eq, [HSO4-]acid,eq are equilibrium conc of these two species calculated as the system was before the adding of PbSO4. will this hold?

this, if I may point this out, is exactly what I did when I introduced path (a) to you ...
... and it is an approximation (and a good one, no question), but approximation it is
and yes, with this approximation instantly every major complication vanishes completely, the problem easily falls apart and suddenly can be calculated in a New York minute, as shown earlier.

that's why I'm campaigning for exactly this very approximation so much...


regards

Ingo

[magician4]

what just came to my mind:

if we neglected the influence of K_W completely (i.e. made the resulting pH a consequence of the dissociation of sulphuric acid, hydrogen sulfate exclusively), we might build up an even better approximation than before.

Just take a look at the equations derived thereof:

[tex]K_{a2} = \frac {(x+y) \cdot (c_0(H_2SO_4) + x )}{c_0(H_2SO_4) - x}[/tex]

[tex]K_{sp} = (c_0(PbCl_2) + y) \cdot (x + y ) [/tex]

(x = [SO₄^2-]_H2SO4 ; y = [Pb²⁺]_PbSO4 = [SO₄^2-]_PbSO4 )

two equations with two variables: easy goin'...

... as this will result in an equation of "only" to the fourth power in y , which still can be resolved analytically

regards

Ingo

[OmniReader]

this, if I may point this out, is exactly what I did when I introduced path (a) to you ...
... and it is an approximation (and a good one, no question), but approximation it is
and yes, with this approximation instantly every major complication vanishes completely, the problem easily falls apart and suddenly can be calculated in a New York minute, as shown earlier.

that's why I'm campaigning for exactly this very approximation so much...

thanks. but the approach seems same as for S = [Pb²⁺]_tot - 10^-6, which is calculation made in your high-precision method, so how come the new  S = [SO₄^2-]_tot,eq + [HSO₄^-]_tot,eq - ([SO₄^2-]_acid,eq + [HSO₄^-]_acid,eq) is approximate? Both approaches are just, total equilibrium concentration of all forms containing this ion, after addition of the salt, minus equilibrium concentrations of all forms containing this ion before addition of salt. in your high precision method from (iv) to (D), you employ that S = [Pb²⁺]_tot - 10^-6, so how come this is high precision but second solubility equation is not?

[magician4]

if you just added PbSO₄ to the point of saturation (and no further), lead ions do have no other options but "to be": hence, they simply behave additive, no matter what the source.
sulfate ions on the other hand do have other options: add "fresh" sulfate to a solution already containing sulfate/hydrogensulfate at equilibrium, and the sulfates won't just add up afterwards:
instead, the whole system will shift, with some hydrogensulfate being reestablished (and the pH shifting, hence)

the latter is the problem if you went for high precision, and hence would want to include the influence of the solvent, i.e. water, i.e. the K_W

if, on the other hand, you didn't include K_W, and hence state that the pH is approximately governed by sulfuric acid / hydrogensulfate exclusively, you'll end up with those two equations shown in my last post (which, later on, seem to be reduceable to a cubic equation on further treatment, as I realised after my post)

now, "cubic" still is in the ballpark of equations chemists try to avoid in their calculations, and most of us hence would go for my original proposal (a), as this is square at maximum.
... and hence acceptable


regards

Ingo

[OmniReader]

I'm ok with cubic or more because I just want numerical solution at the end of the day. which is easy for any polynomial using mathematica or even wolfram

I think I get what youre saying about solubility now. we cannot write [SO42-]added = [SO42-]after + [HSO4-]after - [SO42-]before - [HSO42-] before because of shifts in ratio of sulphate to hydrogensulfate.

Thanks very much for help

[magician4]

I think I get what youre saying about solubility now. we cannot write [SO42-]added = [SO42-]after + [HSO4-]after - [SO42-]before - [HSO42-] before because of shifts in ratio of sulphate to hydrogensulfate.

exactly!

Thanks very much for help

You're welcome


regards
Ingo