Having trouble understanding pH = pKa + log([A-]/[HA])

  • #1

Main Question or Discussion Point

So I have a problem with the equation: pH = pKa + log([A-]/[HA])

The textbook states that when [H+] = Ka, there will be 50% A- and 50% HA.

However, this won't work when the coefficients of the reactants to the products are not the same. The equation only works for reactions where it is a 1:1 mole ratio of reactants to products.

So, my question is, why is it generalized and said that pH = pKa at half-equivalence?

What if we have H3PO4?
Wouldn't the overall equation be: H3PO4 + H2O <-> 3 [H3O+] +[PO4 3-]
and then that equation won't work?
 

Answers and Replies

  • #2
Borek
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It is about a single step of dissociation, and Ka used in the Henderson-Hasselbalch equation is a stepwise contant, not the overall one.
 
  • #3
It is about a single step of dissociation, and Ka used in the Henderson-Hasselbalch equation is a stepwise contant, not the overall one.
That makes a lot of sense. Then for all acids (and bases) it will be a 1:1 <-> 1:1 mole ratio, right? Otherwise this falls apart?
 
  • #4
epenguin
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Well yes, just look at the definition of an acid dissociation constant Ka. The ratio of concentration of acid and its conjugate base is 1 when [H+] = Ka. Or pH = pKa, same thing.

This then is true for each of the three equilibria you can write for the successive dissociations.

What is not true is that this gives 50% of total phosphate to each of the two forms - that is true only when there are only two forms. (Though actually when, as with phosphate and often, when the pK's are well separated then it is still a pretty good approximation, e.g. when making phosphate buffers you only need to consider one dissociation.)
 
  • #5
Borek
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That makes a lot of sense. Then for all acids (and bases) it will be a 1:1 <-> 1:1 mole ratio, right? Otherwise this falls apart?
Depends on what you mean by "all".

It works for each conjugate acid/base pair separately. Each conjugate pair has its own Ka value, and for pH=pKa concentrations of these particular conjugate acid and base will be identical.
 
  • #6
Well yes, just look at the definition of an acid dissociation constant Ka. The ratio of concentration of acid and its conjugate base is 1 when [H+] = Ka. Or pH = pKa, same thing.

This then is true for each of the three equilibria you can write for the successive dissociations.

What is not true is that this gives 50% of total phosphate to each of the two forms - that is true only when there are only two forms. (Though actually when, as with phosphate and often, when the pK's are well separated then it is still a pretty good approximation, e.g. when making phosphate buffers you only need to consider one dissociation.)
I see.
So in reality, when you have 3 successive deprotonations of H3PO4 in the same solution, such as:
H3PO4 <-> [H2PO4-] + [H+]
[H2PO4-] <-> [HPO4 2-] + [H+]
[HPO4 2-] <-> [PO4 3-] + [H+]

At the last dissociation, when pH = pKa, we actually have some of the H3PO4 and [H2PO4]- still present right? So the it is not exactly 50% to 50%, but close enough due to the difference in the pKa values (which makes sense, but I want to make sure I have this right).
 
  • #7
Borek
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Ah, I think I finally got what your problem is. Textbooks is a bit wrong. What Henderson-Hasselbalch equation says is that at pH=pKa the ratio of concentrations of the acid and conjugate base equals 1. For monoprotic acid that's equivalent to saying they are 50:50. For multiprotic acid that's no longer the case - while the ratio of concentrations in the conjugate pair is still 1:1, concentrations of other forms are not zero, so the 50:50 doesn't have to hold (although - depending on the acid - it can be quite close to that).

For example, for 0.1 M phosphoric acid partially neutralized so that pH=pKa2=7.2

H3PO44.46×10-7 M0.0%
H2PO4-0.05 M50.0%
HPO42-0.05 M50.0%
PO43-3.53×10-7 M0.0%

but for citric acid with much closer pKa values (again, 0.1 M and pH=pKa2=4.76)

C6H8O71.13×10-3 M1.1%
C6H7O7-0.0488 M48.8%
C6H6O72-0.0488 M48.8%
C6H5O73-1.13×10-3 M1.1%
 
  • #8
epenguin
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In the case of polyprotic acids confusion is usually avoided by naming the successive dissociation constants with number: Ka1, Ka2, Ka3 and pKa1, pKa2, pKa3 &c.

In the case of 0.1 M phosphate when pH = pKa2 then [H3PO4] = 4.3×10-7 and [PO43-] = 3.6 × 10-7, while [H2PO4-] is 0.0499992 but exactly equal to [HPO42-]. :oldsmile:
 
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