[Big-Daddy]

Let's say I have put a known concentration c₀ of H2SO4 in water solution, and, calling the first dissociation strong, I want to find unknown K_a2. I measure [H⁺], and room temperature to get K_w.

I started from c₀=[HSO₄^-]+[SO₄^2-] (mass balance), the charge balance and equilibrium constant expressions, found the fractions of acid in each of the two forms, and placed these into the charge balance. The cubic I ended up with (in [H⁺]³) finds accurate roots of [H⁺] with a given K_a2, but if I instead rearrange it to solve for K_a2 and plug the [H⁺] in it fails. Why? And how do I find K_a2 from [H⁺] at a certain c₀, with known K_w and K_a1=∞.

(For the record the example I tried was c₀=1*10^-7 M, [H+]=2.414*10^-7 M is one of the solutions of my cubic and should be correct, using Ka2=1.1*10^-2, but putting this value back in gives me Ka2=8.745*10^-3).

[Borek]

Let's say I have put a known concentration c₀ of H2SO4 in water solution, and, calling the first dissociation strong, I want to find unknown K_a2. I measure [H⁺], and room temperature to get K_w.

I started from c₀=[HSO₄^-]+[SO₄^2-] (mass balance), the charge balance and equilibrium constant expressions, found the fractions of acid in each of the two forms, and placed these into the charge balance. The cubic I ended up with (in [H⁺]³) finds accurate roots of [H⁺] with a given K_a2, but if I instead rearrange it to solve for K_a2 and plug the [H⁺] in it fails. Why? And how do I find K_a2 from [H⁺] at a certain c₀, with known K_w and K_a1=∞.

(For the record the example I tried was c₀=1*10^-7 M, [H+]=2.414*10^-7 M is one of the solutions of my cubic and should be correct, using Ka2=1.1*10^-2, but putting this value back in gives me Ka2=8.745*10^-3).

Try to read you post and judge if anyone - not seeing your work and not knowing all formulas you have derived - will be able to help.

BTW, I got 42. I am just not entirely sure what the question was.

[Big-Daddy]

Ok I will rewrite here rather than modifying the original post, since that is more by the forum spirit:

At a certain, known initial/analytical concentration of diprotic acid H2A, I have measured [H+]. I know that the first dissociation of H2A is always strong (goes to completion) and the second is weak.

Can I use this data-point (analytical concentration, c₀(H2A), of the acid, and the [H⁺] it produces) to correctly calculate K_a2, given that K_a1 is (known to be) infinity?

My original method for doing so - tested against H2A=H2SO4, c₀(H2SO4) = 1 · 10^-7 M and [H+] = 2.41 · 10^-7 M - gives a K_a2 of 9.6456 · 10^-4.

I know this to be wrong because K_a2 = 1.1*10^-2 for H2SO4 and this reading of [H+] (that [H+] = 2.41 · 10^-7 M when c₀(H2A) = 1 · 10^-7 M, if K_a2 = 1.1*10^-2) was actually given as a calculated value in a problem I found.

My full method can be found below. Since the steps are the same for all such problems I have omitted rearrangements etc.

------------------------

From K_a2 = [A^2-][H+] / [HA-] and c₀(H2A) = [HA^-] + [A^2-], we substitute to find:

[A^2-] = c₀(H2A) * K_a2 / ([H⁺] + K_a2)

and

[HA^-] = c₀(H2A) * [H⁺] / ([H⁺] + K_a2)

Now we substitute from above into the charge balance [H+] = [HA^-] + 2[A^2-] + [OH-], along with [OH^-]=K_w/[H⁺]. This leaves us with a cubic equation in [H⁺]. I rearranged this for K_a2 to get:

[tex]K_{a2} = \frac{H^3 - H^2c_0 - K_wH}{K_w+2c_0H-H^2}[/tex]

where H stands for [H⁺] and the rest should be obvious. I plugged in the values c₀(H2SO4) = 1 · 10^-7 M, [H+] = 2.41 · 10^-7 M, K_w = 10^-14 and got roughly 9.65 · 10^-4 as the K_a2.

Why is it so different from the correct value? What procedure should I follow, to find K_a2, knowing [H+] for a given c₀(H2A) as well as K_w, given that K_a1 = ∞.

[Borek]

[tex]K_{a2} = \frac{H^3 - H^2c_0 - K_wH}{K_w+2c_0H-H^2}[/tex]

Looks like this equation is very sensitive to rounding errors.

Edit: especially when you are using such a low concentration. Try c around 0.01 M.

[Big-Daddy]

Ok thanks a lot. Surprising result for a physical situation, errors in the sixth significant figure changed the result by an order of magnitude (!) But yes, the equation does work, with the most accurate proton concentration I can get my hands on.

Thanks for the help.