Banach–Alaoglu theorem: Difference between revisions

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{{merge|Equivalence point|discuss=Talk:Equivalence point|date=September 2011}}


[[Image:Oxalic acid titration grid.png|thumb|A typical titration curve of a diprotic acid, [[oxalic acid]], titrated with a strong base, [[sodium hydroxide]]. Each of the two equivalence points is visible]]


[[Titration]]s are often recorded on graphs called '''titration curves''', which generally contain the volume of the [[titrant]] as the [[independent variable]] and the [[pH]] of the solution as the [[dependent variable]] (because it changes depending on the composition of the two solutions).
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The [[equivalence point]] on the graph is where all of the starting solution (usually an [[acid]]) has been neutralized by the titrant (usually a [[Base (chemistry)|base]]). It can be calculated precisely by finding the [[second derivative]] of the '''titration curve''' and computing the [[point of inflection|points of inflection]] (where the graph changes [[concave function|concavity]]); however, in most cases, simple visual inspection of the curve will suffice (in the curve given to the right, both equivalence points are visible, after roughly 15 and 30 [[milliliter|mL]] of [[Sodium hydroxide|NaOH solution]] has been titrated into the [[oxalic acid]] solution. To calculate the [[acid dissociation constant]] (pK<sub>a</sub>), one must find the volume at the half-equivalence point, that is where half the amount of titrant has been added to form the next compound (here, sodium hydrogen oxalate, then [[disodium oxalate]]). Halfway between each equivalence point, at 7.5 mL and 22.5 mL, the pH observed was about 1.5 and 4, giving the pK<sub>a</sub>.
 
In [[monoprotic acid]]s, the point halfway between the beginning of the curve (before any titrant has been added) and the equivalence point is significant: at that point, the concentrations of the two species (the acid and conjugate base) are equal. Therefore, the [[Henderson-Hasselbalch equation]] can be solved in this manner:
:<math>pH = pK_a + \log \left( \frac{[\mbox{base}]}{[\mbox{acid}]} \right)</math>
:<math>pH = pK_a + \log(1)\,</math>
:<math>pH = pK_a \,</math>
 
Therefore, one can easily find the pK<sub>a</sub> of the monoprotic acid by finding the pH of the point halfway between the beginning of the curve and the equivalence point, and solving the simplified equation. In the case of the sample curve, the ''K<sub>a</sub>'' would be approximately 1.78×10<sup>−5</sup> from visual inspection (the actual ''K<sub>a2</sub>'' is 1.7×10<sup>−5</sup>)
 
For [[polyprotic]] acids, calculating the acid dissociation constants is only marginally more difficult: the first acid dissociation constant can be calculated the same way as it would be calculated in a monoprotic acid. The second acid dissociation constant, however, is the point halfway between the first equivalence point and the [[second equivalence point]] (and so on for acids that release more than two protons, such as [[phosphoric acid]]).
 
==References==
{{Unreferenced|date=August 2008}}
{{Reflist}}
 
[[Category:Titration]]

Latest revision as of 01:57, 18 September 2014


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