Question

Thioglycolic acid, \(\mathrm{HSCH}_{2} \mathrm{CO}_{2} \mathrm{H}\), a substance used in depilatory agents (hair removers) has \(\mathrm{pK}_{\mathrm{a}}=3.42 .\) What is the percent dissociation of thioglycolic acid in a buffer solution at \(\mathrm{pH}=3.00 ?\)

Short answer

The percent dissociation of thioglycolic acid at pH 3.00 is approximately 27.54%.

Step-by-step solution

Understand the Problem Requirements

We're asked to find the percent dissociation of thioglycolic acid in a buffer solution where the pH is 3.00, given that the acid has a pK_a of 3.42. The percent dissociation is related to how much the acid ionizes in solution.

Use the Henderson-Hasselbalch Equation

The percent dissociation can be estimated using the Henderson-Hasselbalch equation, which relates pH, pK_a, and the ratio of the concentration of the conjugate base to the acid. The equation is: \[ \text{pH} = \text{pK}_a + \log \left( \frac{[A^-]}{[HA]} \right), \] where \([A^-]\) is the concentration of the ionized form and \([HA]\) is the concentration of the non-ionized form.

Rearrange for Concentration Ratio

Rearrange the Henderson-Hasselbalch equation to solve for the concentration ratio \([A^-]/[HA]\): \[ \frac{[A^-]}{[HA]} = 10^{\text{pH} - \text{pK}_a} = 10^{3.00 - 3.42} = 10^{-0.42}. \]

Calculate the Concentration Ratio

Calculate the value of the ratio: \( 10^{-0.42} \approx 0.38. \) This implies that for every 1 molecule of HA, there are 0.38 molecules of A^-.

Calculate Percent Dissociation

Percent dissociation (%) is calculated using: \[ \text{Percent Dissociation} = \left( \frac{[A^-]}{[HA] + [A^-]} \right) \times 100. \] Using the ratio obtained, \[ \frac{[A^-]}{[HA] + [A^-]} = \frac{0.38}{1 + 0.38}. \]

Solve for Percent Dissociation

Substitute the ratio into the percent dissociation formula: \[ \text{Percent Dissociation} = \left( \frac{0.38}{1.38} \right) \times 100 \approx 27.54\%. \]

Key concepts

Thioglycolic Acid

Thioglycolic acid, also known as mercaptoacetic acid, is an organic compound with the formula \(\text{HSCH}_2\text{CO}_2\text{H}\). It is notable for its use in various cosmetic products, prominently as an active component in depilatory agents and hair treatment solutions. This substance facilitates hair removal by breaking down the disulfide bonds in keratin, the primary protein in hair. This chemical reaction weakens the hair structure, making it easy to remove hair with physical means.

It is important for students working with thioglycolic acid to understand its acidic properties. The compound has a \(\text{pK}_a\) value of 3.42, which is a measure of its tendency to donate a proton (\(\text{H}^+\)). A lower \(\text{pK}_a\) indicates a stronger acid that dissociates more readily.

Overall, thioglycolic acid serves as an excellent case study for applying concepts of acid dissociation and understanding buffer systems.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental tool in the study of buffer solutions. It relates the pH of a solution to the \(\text{pK}_a\) of the acid and the concentrations of the acid (\([HA]\)) and its conjugate base (\([A^-]\)). The equation is given by:
\[\text{pH} = \text{pK}_a + \log \left( \frac{[A^-]}{[HA]} \right)\]
The usefulness of this equation cannot be overstated. It allows you to calculate either the pH of a solution or the ratio of the concentrations of acid and conjugate base, provided you know two of the three parameters.

In the context of thioglycolic acid, the equation provides insight into how much the acid ionizes in a buffer solution with a given pH. This is crucial for determining the acid's behavior in different chemical environments.

Buffer Solutions

Buffer solutions play a critical role in maintaining the pH of solutions despite the addition of small amounts of acids or bases. They do so by using a combination of weak acid and its conjugate base. Understanding buffer solutions is essential in various scientific fields, especially in chemistry and biology, where pH stability is often a necessity.

When dealing with thioglycolic acid in a buffer solution, you are essentially examining how the weak acid component and the conjugate base work together to resist changes in pH upon the introduction of additional acidic or basic elements. This property of buffer solutions is crucial for processes like enzymatic reactions in biochemical environments.

The ability of buffer solutions to maintain a relatively stable pH is why they are utilized in laboratory settings, such as in biochemical experiments and cellular cultures, to ensure that reactions occur under optimal and consistent conditions.

pK_a and pH Relationship

The relationship between \(\text{pK}_a\) and pH is central to understanding how acids and bases behave in solutions. The \(\text{pK}_a\) is a constant that reflects the intrinsic acidity of a compound, indicating how easily an acid can donate its proton. Lower \(\text{pK}_a\) values correspond to stronger acids.pH, on the other hand, is a measure of the hydrogen ion concentration in a solution, essentially the acidity of the solution itself.

When analyzing thioglycolic acid in a buffer solution, the \(\text{pK}_a\) helps determine how much the acid dissociates at a given pH. If the pH is less than \(\text{pK}_a\), the acid primarily exists in its non-dissociated form. Conversely, if the pH exceeds the \(\text{pK}_a\), the acid mainly dissociates into its conjugate base.

Thus, understanding these concepts is vital for predicting and controlling the behavior of weak acids and bases in various chemical environments, which is instrumental in fields ranging from pharmaceuticals to environmental science.