Mathematically prove that the pH at the halfway point of a titration of a weak acid with a strong base (where the volume of added base is half of that needed to reach the equivalence point) is equal to pKa for the acid.

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insert step 1> Start by understanding that at the halfway point of a titration of a weak acid with a strong base, half of the weak acid has been converted to its conjugate base. This means the concentrations of the weak acid (HA) and its conjugate base (A-) are equal.

insert step 2> Use the Henderson-Hasselbalch equation, which is given by: \( \text{pH} = \text{pKa} + \log \left( \frac{[A^-]}{[HA]} \right) \).

insert step 3> At the halfway point, since \([A^-] = [HA]\), the ratio \( \frac{[A^-]}{[HA]} \) becomes 1.

insert step 4> Substitute this ratio into the Henderson-Hasselbalch equation: \( \text{pH} = \text{pKa} + \log(1) \).

insert step 5> Since \( \log(1) = 0 \), the equation simplifies to \( \text{pH} = \text{pKa} \). This shows that at the halfway point, the pH is equal to the pKa of the weak acid.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Titration

Titration is a quantitative analytical technique used to determine the concentration of a solute in a solution. In an acid-base titration, a solution of known concentration (the titrant) is added to a solution of unknown concentration until the reaction reaches its equivalence point. The halfway point occurs when half of the acid has been neutralized, which is crucial for understanding the relationship between pH and pKa.

pH and pKa

pH is a measure of the acidity or basicity of a solution, defined as the negative logarithm of the hydrogen ion concentration. pKa is the negative logarithm of the acid dissociation constant (Ka) and indicates the strength of an acid; lower pKa values correspond to stronger acids. At the halfway point of a titration of a weak acid with a strong base, the pH equals the pKa because the concentrations of the weak acid and its conjugate base are equal.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation relates the pH of a solution to the pKa of an acid and the ratio of the concentrations of its conjugate base and acid. It is expressed as pH = pKa + log([A-]/[HA]). At the halfway point of a titration, the concentrations of the weak acid and its conjugate base are equal, making the log term zero, thus simplifying the equation to pH = pKa, which mathematically proves the relationship.

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Henderson-Hasselbalch Equation

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Textbook Question

A sample of 0.2140 g of an unknown monoprotic acid was dissolved in 25.0 mL of water and titrated with 0.0950 M NaOH. The acid required 30.0 mL of base to reach the equivalence point. (a) What is the molar mass of the acid?

Textbook Question

A sample of 0.1687 g of an unknown monoprotic acid was dissolved in 25.0 mL of water and titrated with 0.1150 M NaOH. The acid required 15.5 mL of base to reach the equivalence point. (a) What is the molar mass of the acid?

Textbook Question

Suppose you want to do a physiological experiment that calls for a pH 6.50 buffer. You find that the organism with which you are working is not sensitive to the weak acid H2A 1Ka1 = 2 * 10-2; Ka2 = 5.0 * 10-72 or its sodium salts. You have available a 1.0 M solution of this acid and a 1.0 M solution of NaOH. How much of the NaOH solution should be added to 1.0 L of the acid to give a buffer at pH 6.50? (Ignore any volume change.)