When do the approximations for weak acids break down?

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and written by Robert J. Hamers
UW - Madison Dept. of Chemistry
Interactive Chemistry
Acid-Base Chemistry
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Contents




Effects of dilution: Top of page

Understanding the effects of dilution amounts to determining two things:
  1. Whether water autoprotolysis or dissociation of the acid represents the largest contribution to H+
  2. If acid dissociation dominates, is the extent of dissociation small or large

These two properties lead to three possible types of behavior for "weak" acids in aqueous solution:
  1. Classical weak acid behavior
  2. Classical strong acid behavior
  3. Pure water limit



Quick Approximation Top of page

Let's consider a monoprotic acid. The equations describing the system are:

The relevant question that needs to be asked here is, which process (dissociation of HA or autoprotolysis of water) produces more H+?

To find out, let's consider the extreme cases. If we completely neglect water autoprotolysis, then equation (1) is the only source of H+. Under these conditions, we have:







This can be solved using the normal quadratic equation to give:



If the fractional dissociation is small so that x<<CHA, then this reduces to
or






Complete solution using the systematic method: Top of page

As above, the primary equations are:



We have both mass balance and charge balance equations.

Mass balance:



Charge balance:



From the CB equation and the definition of Ka, we can write:



Rearranging gives:



Substituting this into the charge balance equation gives:






Approximations: Top of page

Depending on the values of Ka and CHA, there are three different approximations that can be made:

Approximations A and B assume CHA is large, so the first term can always be made to dominate the second. We then have:



This can be solved exactly as a quadratic equation. However, in many cases further simplification can be made, depending on whether Ka<<[H+ ]or Ka>>[H+].




Approximation A: Top of page

Ka<<[H+]. In this case the first term in the denominator is much smaller than the second, and the equation reduces to:

, or

. This is the most common equation given for the dissociation of a "weak" acid.




Approximation B: Top of page

Ka>>[H+]. In this case, we have:

, or



Note that the result is the same as we typically have for a "strong" acid. We need to understand under what conditions approximations A and B will be valid.




Why do "weak" acids sometimes behave like "strong" acids, and "strong" acids sometimes behave like "weak" acids? Top of page

If we make the assumption that Ka>>[H+], we find that a "weak" acid actually behaves like a "strong" acid. Why ? Looking back at the equations we can see that the approximation Ka>>[H+] will be valid when Ka is reasonably large but H+ is small. This is most likely to occur when we have a "weak" acid present at low concentration. Note that the "standard" weak acid equation assumes that the fractional dissociation of the weak acid HA is small. That is, in the equation



We assumed that the amount of the weak acid that dissociates (represented by "x") is much smaller than the amount of HA that we initially put in solution (represented by CHA), so that . As the solution is diluted, Le Chatelier's principle drives the equilibrium to the right, and the fractional dissociation increases. In the limit of low concentration, Le Chatelier's principle says that we get complete dissociation. Although complete dissociation is usually associated with "strong" acids, the above equations show that at low concentration, "weak" acids behave like "strong" acids. Also, at high concentrations acids that we typically think of as being "strong" acids (sulfuric, phosphoric, etc.) undergo only partial dissociation, and therefore exhibit behavior typically attributed to "weak acids".

The fundamental distinction between strong acid and weak acid behavior is not simply the magnitude of Ka; rather, the distinction is whether dissociation of the acid is nearly complete (strong acid behavior) or very small (weak acid behavior).

To see whether an acid behaves like a "strong" acid or a "weak" acid, we can compare the predictions resulting from the two approximations. If it behaves like a "weak" acid, we have . If it behaves like a "strong" acid, we have

There will be a "crossover" between weak acid and strong acid behavior when these two approximations predict the same answer. This crossover occurs when



Or, equivalently, when



Then we get weak acid behavior when

and strong acid behavior when




Approximation C: Top of page

The last approximation we need to deal with is the assumption that water autoprotolysis can be neglected as the predominant source of H+. To see when this breaks down, we go back to the full equation describing the chemical system:



Thus far we have assumed that the first term is much larger than the second. At sufficiently low concentration, the second term must be dominant. In the limit of low concentration when CHA=0, we must recover the pure water limit.

How low does the concentration of weak acid need to be before we approach the "pure water" limit? It depends on the value of Ka. Looking at the full equation, we note that the pure water limit will be reached when the amount of H+ produced by dissociation of HA is just equal to that produced by water autoprotolysis. That occurs, of course, when the two terms on the right hand side of this equation are equal . We can then write:

. If , then this amounts to:

, or



If [H+] is greater than this value, then acid dissociation dominates; if [H+] is smaller than this value, water autoprotolysis dominates and we are approaching the infinite-dilution (or "pure water") limit.

We can equate the two terms (representing dissociation and water autoprotolysis) and substitute in this value of [H+]crossover to get:

or,





So, for we have the pure-water limit and




Summary: Top of page

Approximation A
High concentrations :
where water autoprotolysis is negligible and the fractional dissociation of HA is small, we obtain the common "weak" acid equation
This is the appropriate equation when: or, equivalently, when:
Approximation B
Lower concentrations :
where water autoprotolysis is still negligible but the fractional dissociation becomes large, the "weak" acid behaves like a "strong" acid.
This is the appropriate equation when:
both conditions are satisfied simultaneously.
For values of Ka < 10-7, these two conditions cannot be satisfied simultaneously. 		(or equivalently, when: )
AND
Approximation C
Very low concentrations :
where water autoprotolysis is the dominant source of H+, we have:
This is the appropriate equation when:




Exercises: Top of page
  1. Make a plot of pH vs. log(CHA) for acetic acid.
    1. What is the slope of the line at high concentration? Explain, based on the equilibrium equations, why this slope is obtained.
    2. What is the slope of the line at low concentration? Explain.
    3. At what concentration does acetic acid undergo a transition between these two limiting behaviors? Explain, based on the equilibrium expressions, and explain what is happening chemically that distinguishes these regions.
    4. Does acetic acid ever behave like a "strong" acid? Explain why or why not.

  2. Make a plot of pH vs. log(CHA) for hydrofluoric acid(HF). Compare your result with that obtained in (1). Is there an intermediate behavior? Explain.
    1. Under what conditions does HF behave like a "weak" acid? Explain
    2. Under what conditions does HF behave like a "strong" acid? Explain
    3. Under what conditions does a solution of HF behave like pure water?



Exercises with the Virtual Titrator Top of page

First look at approximations
  1. Open the Virtual Titrator
  2. Choose a relatively strong monoprotic weak acid (1e-5 < ka < 1)
    • Choose the <Acid> menu from the menu bar of the main window
    • Next, choose the submenu <Monoprotic>
    • Finally, choose the acid of your choice from the list
    • These are some good examples:
      • Chloroacetic Acid
      • Chlorous Acid
      • Iodic Acid
      • Pyruvic Acid
  3. Switch to a graph of pH vs. log(Acid Concentration)
    • Choose the <Graphs> menu from the menu bar of the main window
    • Then, choose the graph named: pH vs. log Acid Concentration
  4. Figure out the regions of this graph that correspond to each of the approximations

Very weak acids -- that can never use approximation B
  1. Open the Virtual Titrator
  2. Choose a very weak monoprotic weak acid (ka < 1e-7)
  3. These are some good examples:
    • Ammonia
    • Hydrogen Peroxide
    • Phenol
    • Tham
  4. Switch to a graph of pH vs. log(Acid Concentration)
  5. Notice that the graph has no middle area that would correspond to approximation B