Question

Phenylacetic acid \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{COOH}\right)\) is one of the substances that accumulates in the blood of people with phenylketonuria, an inherited disorder that can cause mental retardation or even death. A \(0.085 \mathrm{M}\) solution of \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{COOH}\) has a pH of \(2.68 .\) Calculate the \(K_{a}\) value for this acid.

Short answer

The \(K_a\) value for phenylacetic acid is approximately \(5.25 \times 10^{-5}\).

Step-by-step solution

Convert pH to the concentration of hydrogen ions (H+)

The first step is to convert the given pH value into the concentration of hydrogen ions in the solution. The formula to calculate the concentration of hydrogen ions from the pH is: \[ [\mathrm{H}^+] = 10^{-\mathrm{pH}} \] Plugging in the given pH value, we get: \[ [\mathrm{H}^+] = 10^{-2.68} \]

Calculate the concentration of hydrogen ions (H+)

Now, we need to calculate the actual concentration of hydrogen ions: \[ [\mathrm{H}^+] = 10^{-2.68} \approx 2.089 \times 10^{-3} \, \mathrm{M} \] So the concentration of hydrogen ions in the solution is \(2.089 \times 10^{-3} \, \mathrm{M}\).

Write the dissociation equation of phenylacetic acid

To find the \(K_{a}\), we need to set up the dissociation equation of phenylacetic acid. The dissociation equation is: \[ \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{COOH} \rightleftarrows \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{COO}^- + \mathrm{H}^+ \]

Write the expression for \(K_{a}\)

Now that we have the dissociation equation, we need to write the expression for \(K_{a}\). The expression is: \[ K_a = \frac{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{COO}^-][\mathrm{H}^+]}{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{COOH}]} \]

Find the concentrations of the dissociated species

Since phenylacetic acid is a weak acid, we can assume that the amount dissociated is equal to the concentration of 𝐻⁺ ions, which is \(2.089 \times 10^{-3} \, \mathrm{M}\). Thus, the concentration of phenylacetic acid anion, 𝐶₆𝐻₅𝐶𝐻₂𝐶𝑂𝑂⁻, is also \(2.089 \times 10^{-3} \, \mathrm{M}\). The concentration of undissociated phenylacetic acid remaining in the solution is given by: \[ 0.085\,\mathrm{M} - 2.089 \times 10^{-3} \, \mathrm{M} \approx 0.0829 \, \mathrm{M} \]

Calculate \(K_{a}\)

Now, we will plug in the concentrations of the dissociated species into the \(K_{a}\) expression: \[ K_{a} = \frac{(2.089 \times 10^{-3})(2.089 \times 10^{-3})}{0.0829} \] And solve for \(K_{a}\): \[ K_{a} \approx 5.25 \times 10^{-5} \] The \(K_a\) value for phenylacetic acid is approximately \(5.25 \times 10^{-5}\).

Key concepts

pH Calculation

Understanding how to calculate pH is crucial for anyone studying chemistry, especially when dealing with acids and bases. The pH scale is a measure of the acidity or basicity of an aqueous solution, with values ranging from 0 to 14. A pH less than 7 indicates an acidic solution, while a pH greater than 7 denotes a basic solution. Neutral solutions have a pH of approximately 7.

To convert a given pH value to the concentration of hydrogen ions (\text{[H+]}) in the solution, the following formula is used: \[ [\text{H}^+] = 10^{-\text{pH}} \] For example, if a solution has a pH of 2.68, the concentration of hydrogen ions can be calculated as \[ [\text{H}^+] = 10^{-2.68} \] which yields a result of approximately \(2.089 \times 10^{-3} \mathrm{M}\). This conversion is a foundational step in solving many problems related to chemical equilibrium and acid-base reactions in aqueous solutions.

Weak Acid Dissociation

Weak acids do not fully dissociate in water, meaning that at equilibrium, a significant amount of the acid remains undissociated. The dissociation of a weak acid in aqueous solution can be represented by an equilibrium equation, for instance:\[\text{HA} \rightleftarrows \text{A}^- + \text{H}^+\] where HA represents the weak acid, A- the conjugate base, and H+ the hydrogen ion released upon dissociation.

For a weak acid, the extent of dissociation can be expressed by the acid dissociation constant, Ka, which is a quantitative measure of its strength. This value is essential as it allows us to predict the behavior of the acid in solution and understand how it will react in various chemical contexts. The formula for Ka is defined by the ratio of the concentration of the products to the concentration of the reactants at equilibrium, excluding water:\[K_a = \frac{[\text{A}^-][\text{H}^+]}{[\text{HA}]}\] For phenylacetic acid, given in the exercise, the Ka value gives insight into how much the acid dissociates at a given pH.

Phenylketonuria

Phenylketonuria (PKU) is an inherited metabolic disorder characterized by the body's inability to properly break down the amino acid phenylalanine, which is found in many protein-rich foods. This inability is due to a deficiency of the enzyme phenylalanine hydroxylase. As a result, phenylalanine and its byproducts, such as phenylacetic acid, accumulate in the blood and can cause a range of health problems, including intellectual disability if not managed appropriately.

The study of phenylacetic acid and its dissociation constant is important not just for chemistry students but also for understanding the biochemistry of conditions like PKU. By managing dietary phenylalanine intake and monitoring the buildup of such compounds in the bloodstream, it is possible to manage PKU effectively, preventing the adverse consequences of the disorder.

Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions in a closed system are equal, resulting in no net change in the concentration of reactants and products over time. It is an important concept in chemistry because it helps predict the extent of a reaction and the concentrations of substances in a mixture at equilibrium.

In the context of acids like phenylacetic acid from our exercise, once equilibrium is reached, you can apply the equilibrium concept to find the Ka value. This calculation helps understand the balance between the associated and dissociated forms of the acid in solution, further explaining the acid's behavior and properties in a biological or chemical system. Knowing how to calculate this value is essential to manipulate and predict the outcome of many chemical reactions in both laboratory and real-world scenarios.