Problem 134 - Chemistry: An Atoms-Focused Approach

Question

People with diabetes can have elevated levels of two acids in their blood: \(\beta\) -hydroxybutyric acid \(\left(\mathrm{p} K_{\mathrm{a}}=4.72\right)\) and acetoacetic acid \(\left(\mathrm{p} K_{\mathrm{a}}=3.58\right)\) (EQUATION CAN'T COPY) The presence of these acids lowers the \(\mathrm{pH}\) of blood and can serve to diagnose diabetes. a. Which acid yields a solution with the lower \(\mathrm{pH}\) if \(0.100 M\) solutions are prepared? b. Which acid will dissociate to a greater degree? c. What is the pH of a solution that contains \(15.8 \mathrm{mM}\) of acetoacetic acid and \(10.8 \mathrm{m} M\) sodium acetoacetate? d. What is the \(\mathrm{pH}\) of a solution of \(90 \mathrm{mg} / \mathrm{L}\) \beta-hydroxybutyric acid and 90 mg/L \(\beta\) -hydroxybutyrate anion after \(100 \mu \mathrm{L}\) of \(0.100 M\) HCl has been added?

Step-by-Step Solution

Answer

Answer: Acetoacetic acid has a lower pH in 0.100 M solutions and dissociates to a greater degree than β-hydroxybutyric acid.

Step 1: a. Comparing the pH of 0.100 M solutions

To find which acid yields a solution with a lower pH when 0.100 M solutions are prepared, we need to compare the pH for both solutions. Remember that the lower the pH, the more acidic the solution is. For a weak acid, the pH can be estimated using the following equation: \(pH = pK_a + log\Big(\frac{[A^-]}{[HA]}\Big)\) However, since the problem is only asking which one is more acidic, we can just compare the \(pK_a\) values. The acid with the lower \(pK_a\) will yield a solution with a lower pH. For \(\beta\)-hydroxybutyric acid, \(pK_a = 4.72\). For acetoacetic acid, \(pK_a = 3.58\). Thus, acetoacetic acid yields a solution with a lower pH (more acidic) than a \(\beta\)-hydroxybutyric acid solution.

Step 2: b. Determining which acid dissociates to a greater degree

The degree of dissociation of an acid is related to its acidity constant, \(K_a\). The higher the \(K_a\) value, the more it will dissociate into its ions, making the solution more acidic. We can find the \(K_a\) values for both acids using the \(pK_a\) values: \(K_a = 10^{-pK_a}\) For \(\beta\)-hydroxybutyric acid: \(K_{a_1} = 10^{-4.72}\) For acetoacetic acid: \(K_{a_2} = 10^{-3.58}\) Comparing the two values, we find that \(K_{a_2}\) is greater than \(K_{a_1}\). This means that acetoacetic acid dissociates to a greater degree than \(\beta\)-hydroxybutyric acid.

Step 3: c. Calculating the pH of a solution containing acetoacetic acid and sodium acetoacetate

To find the pH of the solution, we use the Henderson-Hasselbalch equation: \(pH = pK_a + log\Big(\frac{[A^-]}{[HA]}\Big)\) For a solution containing acetoacetic acid (\(pK_a=3.58\)) and sodium acetoacetate: The concentration of acetoacetic acid is \(15.8\) mM, which is the same as [HA]. The concentration of sodium acetoacetate is \(10.8\) mM, which is the same as [A\(^-\)]. Using these values in the equation: \(pH = 3.58 + log\Big(\frac{10.8}{15.8}\Big)\) Solve for pH to get the final value.

Step 4: d. Calculating the pH of a solution containing β-hydroxybutyric acid and β-hydroxybutyrate anion after adding HCl

First, we need to convert the mass concentration of \(\beta\)-hydroxybutyric acid and its anion to molarity. The molecular weight of \(\beta\)-hydroxybutyric acid is 104.11 g/mol, and the anion form has the same weight. Thus, the concentrations in moles per liter are: \([\beta\text{-hydroxybutyric acid}] = \frac{90\text{ mg}}{104.11\text{ g/mol} \times 1\text{ L}}\) \([\beta\text{-hydroxybutyrate anion}] = \frac{90\text{ mg}}{104.11\text{ g/mol} \times 1\text{ L}}\) Next, calculate the concentration of added HCl: \([HCl] = \frac{0.100\text{ M} \times 100 \times 10^{-6}\text{ L}}{1\text{ L}}\) Subtract HCl concentration from the anion concentration, as the HCl will react with the anions. Similarly, add the concentration of HCl to the acid concentration, as the reaction forms more \(\beta\)-hydroxybutyric acid: \([\beta\text{-hydroxybutyric acid}]_{\text{new}} = [\beta\text{-hydroxybutyric acid}]_{\text{old}} + [HCl]\) \([\beta\text{-hydroxybutyrate anion}]_{\text{new}} = [\beta\text{-hydroxybutyrate anion}]_{\text{old}} - [HCl]\) Finally, use the Henderson-Hasselbalch equation to find the pH: \(pH = pK_a + log\Big(\frac{[\beta\text{-hydroxybutyrate anion}]_{\text{new}}}{[\beta\text{-hydroxybutyric acid}]_{\text{new}}}\Big)\) \(pH = 4.72 + log\Big(\frac{[\beta\text{-hydroxybutyrate anion}]_{\text{new}}}{[\beta\text{-hydroxybutyric acid}]_{\text{new}}}\Big)\) Solve for pH to get the final value.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a mathematical formula used to estimate the pH level of buffer solutions, which contain weak acids and their conjugate bases. This equation is expressed as:\[ pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right) \]In this equation:

• pH represents the acidity or alkalinity of the solution.
• pK_a is the negative logarithm of the acid dissociation constant (K_a), specific to each acid.
• [A^-] denotes the concentration of the conjugate base (deprotonated form).
• [HA] signifies the concentration of the weak acid (protonated form).

This equation is especially useful in calculating the pH of solutions where a weak acid is mixed with its salt, typically seen in buffer solutions. Buffers help to maintain a stable pH when small amounts of acids or bases are added, making them crucial in biological and chemical systems. Utilizing this equation requires knowing the concentrations of both the weak acid and its conjugate base in the solution.

Weak Acids

Weak acids are central players in many chemical equilibria and biological processes. When dissolved in water, weak acids only partially dissociate into hydrogen ions and anions. This partial ionization is why they are termed 'weak.'Some key characteristics of weak acids include:

• They have higher pK_a values compared to strong acids, indicating lower acidity.
• Their dissociation in water is represented by equilibrium: \[ HA \rightleftharpoons H^+ + A^- \] This equilibrium shows that only a fraction of HA (the acid) dissociates into H⁺ (hydrogen ions) and A^- (conjugate base).

Understanding the properties of weak acids is important for interpreting their role in reactions and solutions. They are common in biological systems, such as in the form of lactic acid and acetic acid, and help regulate internal pH, contributing to the stability of enzymes and other biological molecules.

pKa and Ka Relationship

The pK_a and K_a relationship is crucial in understanding acid strength and behavior in solutions. The acid dissociation constant, K_a, quantifies the strength of an acid in solution by measuring its tendency to lose a proton:\[ K_a = \frac{[H^+][A^-]}{[HA]} \]Here, [H⁺] stands for the concentration of hydrogen ions, [A^-] for the concentration of the conjugate base, and [HA] for the concentration of the undissociated acid. The pK_a is the negative logarithm of K_a:\[ pK_a = -\log(K_a) \]Key points about their relationship include:

• The smaller the pK_a, the stronger the acid. This stems from a larger K_a, meaning a greater tendency to donate protons.
• K_a provides a direct measure of acidity, while pK_a offers a more manageable number for calculations and comparisons.
• Both values are used to predict the behavior of acids in reactions, including buffer systems and titrations.

Understanding the pK_a and K_a relationship allows for the determination of acid strength and the extent of their dissociation, making them invaluable tools in both theoretical and practical chemistry.