Question

The pH at on-half the equivalence point in an acid-base titration was found to be \[~5.67.\] What is the value of Ka for this unknown acid?

Answer 1

Hint: We know that the equivalence point is a point in titration at which the amount of titrant added can completely neutralize the analyte solution. At this equivalence point in an acid-base titration, the number of moles of base becomes equal to the number of moles of acid and then the solution is only left with salt and water.

Complete answer:
As we know that the equivalence point in a titration is a point at which the amount of titrant added is enough to completely neutralize the analyte solution. At this point, the number of moles of titrant standard solution is equal to the moles of the solution of unknown concentration. Equivalent point and endpoint are two confusing terms, you should not get confused with these terms. The theoretical completion of the reaction is given by equivalence point. Equivalence is the volume of added titrant at which the number of moles of titrant is equal to the number of moles of the analyte. The endpoint is actually the measured physical change in the solution. There is a slight difference between the equivalence point and the endpoint in the titration is referred to as indicator error and is indeterminate.
\[pH=p{{K}_{a}}+\log \left( \dfrac{Conjugate\text{ }Base}{\text{ }Weak\text{ }Acid} \right)\Rightarrow \left[ HA \right]=\left[ {{A}^{-}} \right]\] which implies that the $\log \left( \dfrac{HA}{{{A}^{-}}} \right)=\log \left( 1 \right)=0$ thus, the half-equivalence point, the pH of the solution is equal to the $p{{K}_{a}}$ of the weak acid. Also, that at half equivalence point: $pH=p{{K}_{a}}.$
Also, pH value in chemistry is a method of measurement of the acidic or the basic nature of a solution or substance. A substance or solution is acidic if it has high concentration of hydrogen ions and is basic if it has high concentration of hydroxide ions or the low concentration of hydrogen ions. pH has an important role in reflecting the chemical properties of the solutions. pH has various activities such as control of the microbial or the biological functions, properties of certain chemicals and their behaviour.
The $p{{K}_{a}}$ is given by the acid dissociation constant of the weak acid, \[{{K}_{a}}\Rightarrow p{{K}_{a}}=-\log \left( {{K}_{a}} \right)\] and we know that the pH here is $11.34$ which implies that the ${{K}_{a}}={{10}^{-p{{K}_{a}}}}$
The half equivalence point, the solution will contain equal numbers of moles of the weak acid and of its conjugate base, which implies that you're now dealing with a buffer solution. The pH of a weak acid-conjugate base buffer can be calculated using the Henderson-Hasselbalch equation. So, at the half equivalence point, we will have \[{{K}_{a}}={{10}^{-pH}}={{10}^{-5.67}}=2.1\times {{10}^{-6}}\]

Note:
Remember that the method of quantitative analysis for determining the concentration of an acid or base by exactly neutralizing it with a standard solution of base or acid having known concentration is known as an acid-base titration. The theory behind acid-base titrations, let us consider a general acid-base reaction that is proceeding with a proton acceptor.