Question

An acid base indicator which is a weak acid has a \(\mathrm{p} K_{\mathrm{a}}\) value \(=5.5\). At what concentration ratio of sodium acetate to acetic acid would the indicator show a colour half way between those of its acid and conjugate base forms? \(\mathrm{p} K_{\mathrm{a}}\) of acetic acid \(=4.75 .[\) Antilog \((0.75)=5.62\), Antilog \((0.79)=6.3\), Antilog \((0.69)=4.93]\) (a) \(4.93: 1\) (b) \(6.3: 1\) (c) \(5.62: 1\) (d) \(2.37: 1\)

Short answer

The indicator will show a colour halfway between its acid and conjugate base forms at a concentration ratio of sodium acetate to acetic acid of 5.62:1, which is option (c).

Step-by-step solution

Understand the Henderson-Hasselbalch Equation

For an acid-base indicator which is a weak acid, the colour change midpoint occurs when the concentrations of the acid and its conjugate base are equal. This concept utilizes the Henderson-Hasselbalch equation, which for a weak acid (HA) is given by: \( \mathrm{pH} = \mathrm{p}K_a + \log(\frac{[A^-]}{[HA]}) \). The pH at the midpoint is equal to the pKa of the indicator. Knowing this, we can set up the equation to solve for the concentration ratio at this midpoint.

Calculate the Concentration Ratio

We're given the pKa value of the indicator is 5.5. At the midpoint where the indicator shows a colour halfway between its acid form and conjugate base form, the pH is equal to the pKa value. This means, for the Henderson-Hasselbalch equation, \( \mathrm{pH} = \mathrm{p}K_a \). Therefore, the concentration ratio of conjugate base [A^-] to the acid [HA] is 1:1 because log(1) equals 0, simplifying the equation to \( \mathrm{pH} = \mathrm{p}K_a \). However, we need the ratio of sodium acetate (which is the salt of acetic acid and acts as the conjugate base in solution) to acetic acid. Since we have the \( \mathrm{p}K_{\mathrm{a}} \) of acetic acid, we can compare the pKa values to find this ratio.

Find the Concentration Ratio Using Given pKa Values

The \( \mathrm{p}K_a \) of acetic acid is given as 4.75, and the pKa of the indicator is 5.5. The difference between these two pKa values is \( 5.5 - 4.75 = 0.75 \). The ratio of concentrations [A^-] to [HA] can be found by taking the antilog of this difference: Antilog(0.75) = 5.62. This indicates a concentration ratio of sodium acetate to acetic acid of 5.62:1.

Confirm the Correct Option

Based on the calculation, the correct concentration ratio of sodium acetate to acetic acid should be 5.62:1 for the indicator to show a colour halfway between its acid and conjugate base forms. This corresponds to option (c).

Key concepts

Acid-Base Indicator

An acid-base indicator is a substance that changes color in response to a change in pH. These indicators are typically weak acids or bases that exhibit different colors in their protonated and deprotonated forms. The point at which the color change is observed is called the endpoint, which should ideally coincide with the equivalence point in a titration. The pH at which the indicator changes color depends on its specific chemical properties.

Indicators are carefully chosen for titrations based on the pH of the equivalence point. For instance, the indicator in the exercise is described to change its color halfway between the pH values representative of its acid and conjugate base forms. This characteristic is pivotal as it simplifies the determination of concentration ratios with the Henderson-Hasselbalch equation, as the ratio will be 1:1 at the midpoint of the color change.

pKa Value

The pKa value is a measure of the strength of an acid in solution. It is the negative base-10 logarithm of the acid dissociation constant (Ka) of a solution. The lower the pKa value, the stronger the acid because it tends to donate its proton (H+) more readily. When the pH of the solution equals the pKa, the concentrations of the acid and its conjugate base are equal.

In the context of the exercise, knowing the pKa value is crucial for understanding at what pH the indicator will show an intermediate color. It is at the pH equal to the pKa that we can determine the desired concentration ratio using the Henderson-Hasselbalch equation.

Concentration Ratio

The concentration ratio refers to the proportion of acid to its conjugate base (or vice versa) in a buffer solution. This ratio is important because it dictates the pH of the solution. According to the Henderson-Hasselbalch equation, when the ratio is 1:1, the pH of the solution is equal to the pKa of the acid. A buffer solution resists changes in pH when small amounts of acid or base are added, and this resistance is governed by the concentration ratio.

In the exercise, we calculated the concentration ratio of sodium acetate to acetic acid using the difference between their pKa values. It is important to remember that the calculation of this ratio depends on the relative strengths of the acid and its conjugate base. Understanding and calculating this ratio accurately allows students to predict and control the pH of buffer solutions.

Acetic Acid

Acetic acid, also known as ethanoic acid, is a weak organic acid that is well-known for its presence in vinegar. As a weak acid, it only partially dissociates in water to form hydrogen ions and acetate ions. The strength of acetic acid is represented by its pKa value, which is 4.75. This value helps us understand its behavior in aqueous solutions and how it will interact when mixed with other substances.

In the example provided, acetic acid is part of a buffer system when combined with its salt, sodium acetate. The pKa value allows us to understand the degree of ionization of acetic acid in solution and helps in determining the concentration ratio necessary for the acid-base indicator to achieve its midpoint color change. This information is crucial when designing buffer solutions in both educational laboratory settings and real-world applications.