Question

Thioglycolic acid, \(\mathrm{HSCH}_{2} \mathrm{CO}_{2} \mathrm{H}\), a substance used in depilatory agents (hair removers) has \(\mathrm{p} K_{\mathrm{a}}=3.42 .\) What is the percent dissociation of thioglycolic acid in a buffer solution at \(\mathrm{pH}=3.0 ?\)

Short answer

The percent dissociation of thioglycolic acid in the buffer is approximately 27.7%.

Step-by-step solution

Understanding the Concept

Thioglycolic acid is a weak acid that partially dissociates in solution. The given values are the acid's \( \mathrm{p}K_{\mathrm{a}} = 3.42 \) and the \( \mathrm{pH} \) of the buffer solution, which is 3.0. We want to find the percent dissociation of the acid in this buffer.

Using the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation defines the relationship \( \mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log \left( \frac{[A^-]}{[HA]} \right) \), where \([A^-]\) is the concentration of the dissociated acid and \([HA]\) is the concentration of the undissociated acid.

Rearrange the Equation

First, rearrange the Henderson-Hasselbalch equation to solve for the ratio \( \frac{[A^-]}{[HA]} \): \[ \frac{[A^-]}{[HA]} = 10^{(\mathrm{pH} - \mathrm{p}K_{\mathrm{a}})} \] Substitute the given values: \( \frac{[A^-]}{[HA]} = 10^{(3.0 - 3.42)} \).

Calculate the Ratio

Calculate the ratio \( \frac{[A^-]}{[HA]} \): \[ \frac{[A^-]}{[HA]} = 10^{-0.42} \approx 0.383 \].

Determine Percent Dissociation

Define the percent dissociation as \( \frac{[A^-]}{[HA] + [A^-]} \times 100% \). Using \( \frac{[A^-]}{[HA]} = 0.383 \), we find the percent dissociation by calculating: \[ \text{Percent dissociation} = \frac{0.383}{1 + 0.383} \times 100% \approx 27.7\% \].

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is an essential concept in acid-base chemistry, especially when it comes to understanding buffers and how they resist changes in pH. It provides a way to estimate the pH of a solution containing a weak acid and its conjugate base. The equation takes the form:
\[pH = pK_a + \log\left( \frac{[A^-]}{[HA]} \right)\]Where:

• \(pH\) is the measure of acidity or basicity of the solution.
• \(pK_a\) is the negative logarithm of the acid dissociation constant, a characteristic value that reflects the strength of the weak acid.
• \([A^-]\) is the concentration of the conjugate base (dissociated form).
• \([HA]\) is the concentration of the undissociated acid.

The equation helps us understand how a buffer solution maintains its pH within certain limits despite the addition of acids or bases. By rearranging the equation, we can calculate the ratio of conjugate base to undissociated acid, giving insights into the system's behavior. This concept is especially useful when dealing with biological systems or industrial processes that require a stable pH.

Weak Acid Dissociation

Weak acid dissociation is a fundamental principle of acid-base chemistry that explains how certain acids do not completely dissociate in solution. Unlike strong acids, which dissociate nearly 100%, weak acids only partially dissociate into their ions. This is described through their acid dissociation constant, \(K_a\), which is a measurement of the extent of dissociation.
For a generic weak acid \(HA\) that dissociates into \(H^+\) and \(A^-\), the equilibrium can be represented as:
\[HA \rightleftharpoons H^+ + A^-\]The equilibrium expression for this dissociation is given by:
\[K_a = \frac{[H^+][A^-]}{[HA]}\]Where:

• \([H^+]\) is the concentration of hydrogen ions contributing to the solution's acidity.
• \([A^-]\) is the concentration of the conjugate base, the product of dissociation.
• \([HA]\) is the concentration of the undissociated acid remaining in the solution.

Weak acids like thioglycolic acid, used in hair removal products, have an associated \(pK_a\), which is derived from the \(K_a\) value. Understanding this helps us estimate how much of the acid dissociates under various conditions, vital for calculating parameters like the percent dissociation.

Buffer Solutions

Buffer solutions are a significant concept in maintaining the stable pH of a system. They are particularly important in biological systems and chemical applications where pH control is crucial. A buffer solution consists of a mixture of a weak acid and its conjugate base (or a weak base and its conjugate acid). This system can resist changes in pH when small amounts of acid or base are added.
The mechanism of a buffer is based on the equilibrium between the weak acid \(HA\) and its conjugate base \(A^-\). When additional H\(^+\) ions are introduced, the conjugate base \(A^-\) reacts with them to form more undissociated \(HA\), minimizing pH changes. Conversely, when OH\(^-\) ions are added, \(HA\) dissociates to form \(H^+\) and \(A^-\), neutralizing the added base.
Key characteristics of buffer solutions include:

• Resisting drastic pH changes with the addition of small amounts of acids or bases.
• The buffer capacity, which determines how much acid or base the buffer can neutralize before a significant pH change occurs.

In the example of thioglycolic acid, a buffer allows us to determine the percent dissociation by maintaining a known pH value. This makes practical applications possible, such as in pharmaceutical formulations and cosmetic products, ensuring desired acidity levels are met and maintained.