Here, the concentrations of acid and base are cA and cB, respectively, and the volumes of the two solutions are VAand VB, respectively. Note that the volumes can be in any units; for convenience we often take them in milliliters even though the concentrations are in moles/liter. Of course, you must use the same volume units on both sides--otherwise you are in trouble!
The equivalence point is a little more complicated with polyprotic acids. For instance, phosphoric acid titrated with NaOH would have three equivalence points! Any of these could be of use and, quite often, with polyprotic acids, some equivalence points are easier to see and/or measure than others. This aspect of things won't be discussed further here. For now, let us just concentrate on the very useful special case of a monoprotic acid, HA, being titrated with a simple strong base, NaOH. Such a simple titration is now shown in the following figure.
A little nomenclature is also useful here. We shall call this a "WASB" titration. What this means is, simply, "weak acid titrated by a strong base." This notation saves a lot of time. Another example would be "SASB" for "a strong base titrated by a strong acid." If using a pH meter in a titration, it is good technique to have the equivalence point occur at about 25 mL (i.e., half the volume of the buret). In the beaker below, you need enough solution to cover your electrode. What all this translates into is that the titrant (the solution in the beaker) be about one-tenth as concentrated as the solution in the buret (the titrating agent). Also, having the base more concentrated than the acid be at least a factor of ten gives better looking titration curves. Usually you want VA >> VB for really good looking curves. We won't delve into details here, but the best looking curves are obtained by plotting pH vs. VB/(VA + VB). Having the base much more concentrated than the acid thus has the best chance for "pretty" titration curves! For typical titrations, probably the ideal conditions would be 250 mL of 0.01M acid titrated by 0.1M base. (The endpoint is then at the mid-volume of the buret.)
We do make a few comments about the general aspects of the above curve. At the very start, there is a sharp rise in the pH which then slows when we get to the "buffer region." In the case shown above, this region is in the range 20% - 80% of the equivalence point (which is the range I previously indicated was the best buffer range on the average). The "dip" at the start of the titration curve is typical for WASB titration curves. The weaker the acid, larger the "dip." It should be noted here that, for SASB titrations, the "dip" is absent. This makes sense when you realize that the dip decreases as the acid strength increases. We show what we mean here with the following picture of a SASB titration (HCl titrated by NaOH).
As you can see, "all is in order." There is no dip and the titration proceeds as a very simple "S" curve. This is almost always the case for a SASB titration. (And, I really do not know any exceptions to the rule!)
Next, we look at a diprotic acid being titrated. Sometimes these are beautiful titration curves with clearly delineated equivalence points for both the first and second anions. More often than not, however, the curves are somewhat ambiguous. For instance, consider the following titration curve for oxalic acid:
In this case, the two equilbrium constants for acid dissociation are too close together to make as pretty a curve as desired. However, note that the second buffer region and equivalence point are clearly delineated. This is actually pretty good for the type of thing we need.
To conclude, we look at a WBSA titration. In this instance, it is ammonia being titrated with aqueous HCl. The curve is essentially an "upside-down version" of a WASB titration. In fact, this looks a lot like an upside-down version of the acetic acid/sodium hydroxide titration shown earlier. This is not really surprising since, in a rather strange coincidence from nature, the Ka value for acetic acid and the Kbvalue for ammonia are equal (to at least two significant figures); the value is 1.8 x 10-5 at 25oC. The curve we are about to show would look very much like the WASB curve for acetic acid if we replaced pH by pOH. But enought of that! Let's just look at the curve!
As you can see, the "dip" at the
start, the buffer region, and the general shape of what happens at the
equivalence point and thereafter is pretty much the same. The only
thing is that the curve is "upside-down" vis à vis what
we saw earlier. However, a buffer region is still definable and things,
otherwise, are quite normal!