[Rutherford]

In a charge balance equation:
[X⁺]+[Y⁺]=[A^-]+2[B^2-]+[C^-]
If 2[B^2-]≈[C^-] and both concentrations are much smaller than [A^-], can I make an approximation to exclude just one species from the charge balance equation, e.g. B? So the equation becomes:
[X⁺]+[Y⁺]=[A^-]+[C^-]
Is this valid (to exclude just one species with very small concentration, while leaving the other)? Or both species must be excluded?

[Borek]

Not enough information.

But in general - the only way to make sure is to solve the problem and see if the answer makes sense. That's a general rule when checking which approximations work and which don't.

[Big-Daddy]

I am sure it would be ok in most systems. It will certainly not lower the accuracy of your value for [A-] or any concentration that makes the smaller two seem negligible. Whether reasonably accurate calculation of those two would be possible after making this approximation probably depends on the system and values, but I suspect it would not be (much) less accurate than if you just removed one of them from the charge balance (and also might not be much more accurate).

In short I think it would be at least as ok as removing both - though they this is just mathematical intuition and I've never tried it - in most scenarios.

[Rutherford]

Okay, but what if [A^-] is much larger than 2[B^2-] which is much larger than [C^-], would it be okay to exclude just 2[B^2-] from the equation, even though [C^-] is much smaller than 2[B^2-]?

[Borek]

Okay, but what if [A^-] is much larger than 2[B^2-] which is much larger than [C^-], would it be okay to exclude just 2[B^2-] from the equation, even though [C^-] is much smaller than 2[B^2-]?

I still think there is not enough information. If [B^2-] is just a spectator, and its concentration doesn't depend on the equilibrium position situation is quite different than if [B^2-] takes part in equilibrium.

Assuming it IS part of the equilibrium, idea doesn't sound good to me. If [C^-] << 2[B^2-] leaving it in the equation is counterproductive (you want to simplify it by ignoring negligible), plus IMHO you should ignore the smallest first.

[Big-Daddy]

Okay, but what if [A^-] is much larger than 2[B^2-] which is much larger than [C^-], would it be okay to exclude just 2[B^2-] from the equation, even though [C^-] is much smaller than 2[B^2-]?

I think it may work but you should try some cases to check. It is interesting as a mathematical question but not as a chemical one - if you can remove the larger, you can remove both.

The way I think it might work out can be visualized by considering, say, a number like 10,203. 10,203 = 10,000 + 200 + 3 and 10,000>>200 and 200>>3 in my example. Now let's say we want to approximate this in its parts - if we say 10,203 ≈ 10,000 then that is removing both of the smaller ones. If we remove 3, as is a logical step, this is a "very good" approximation because 10,203 ≈ 10,200 is a very good approximation. If we instead leave 3 and remove 200, we get 10,203 ≈ 10,003 - still a working approximation, better in fact than removing both, but nowhere near as good as just removing the smaller.

If you do try it with some numbers and proper simultaneous equations I would be interested to hear what you find.