Question

I have a question related to finding the theoretical buffer capacity of a diprotic system, specifically about the formula for the concentration of the strong titrant. In this post:

What is the formula for theoretical buffer capacity for a diprotic buffer system?

The top answer, given by grsousajunior, gives the equation for a buffer capacity, calculated by taking the derivative of the concentration of strong base as a function of pH. All of the math makes sense to me, except for the given equation for the concentration of strong base, CB
. Here’s how my math for that equation worked out (most of this is reiterated in grsousajunior’s answer):

The first ionization is represented by the equation:

H2A+H2O⇌HA−+H3O+
, and therefore has a Ka1
of Ka1=[H3O+][HA−][H2A]

The second ionization is represented by the equation:

HA−+H2O⇌A2−+H3O+
, and therefore has a Ka2
of Ka2=[H3O+][A2−][HA−]

Shifting around the equation for Ka1
to find [HA−]=Ka1[H2A][H3O+]
and putting that into the equation [A2−]=Ka2[HA−][H3O+]
, [A2−]=Ka1Ka2[H2A][H3O+]2
, at least by my math.

As far as I can tell, grsousajunior after that used the equation for the charge balance:

[H3O+]+[B+]=[OH−]+[HA−]+2[A2−]

Which I think is correct, solved it for [B+]
, and got [B+]=CB=[OH−]−[H3O+]+[HA−]+2[A2−]
. Again, this is just speculation on my part. This brings me to my source of confusion. In the answer, grsousajunior gives the equation CB=Kw[H3O+]−[H3O+]+Ka1[H2A]([H3O+]+2Ka2)[H3O+]2+Ka1[H3O+]+Ka1Ka2
. To me, it looks like [OH−]
from the charge balance equation became Kw[H3O+]
, which makes perfect sense.

What I don’t get is how [HA−]+2[A2−]
became Ka1[H2A]([H3O+]+2Ka2)[H3O+]2+Ka1[H3O+]+Ka1Ka2
.

When I add the equations that I have for those two concentrations, doubling the latter, I instead get Ka1[H2A]([H3O+]+2Ka2)[H3O+]2
. Everything is the same, except for the denominator. To me, it looks like the denominator somehow ended up as a quadratic, and I have know idea why. My best guess is that the concentrations of hydronium are separate between the two ionizations, but I don’t know how to wrangle out the math for that.

Does anyone know if or how the math I did is wrong? Thank you!

P.S. - Everything after this step makes sense to me, but the equation for buffer capacity that I would end up with using my equation for the concentration of strong base would look a bit different.

Answer 1

The question revolves around the derivation of the concentration of strong titrant (CB) in a theoretical buffer capacity calculation for a diprotic system, with confusion arising from the algebraic manipulation of the charge balance equation.

Step-by-step explanation:

The question concerns the calculation of the theoretical buffer capacity for a diprotic buffer system, with specific interest in understanding the correct mathematical derivation of the concentration of a strong titrant, denoted as CB, when considering both ionizations of the diprotic acid.

For a diprotic system, the ionization reactions and the associated equilibrium constants (Ka1 and Ka2) are critical to determining buffer capacity, which can be calculated using the Henderson-Hasselbalch equation. The result is a set of complex expressions that take into account the ionization states of the diprotic acid (H2A, HA−, and A2−) and the various species present in solution, such as H3O+ and OH−. It's crucial to properly account for these species in the charge balance equation.

Regarding the specifics of the mathematical derivation raised by the student, there is an apparent ambiguity in the algebraic manipulation of the charge balance equation and the final expression for CB. It's essential to properly justify each step of this derivation, which might involve recognizing the interdependency of hydronium ion concentrations between the first and second ionizations. The student's confusion seems to stem from this point in the calculation.