Phenylacetic acid 1C6H5CH2COOH2 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 M solution of C6H5CH2COOH has a pH of 2.68. Calculate the Ka value for this acid.

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Identify the given information: the concentration of phenylacetic acid \([C_6H_5CH_2COOH]\) is 0.085 M, and the pH of the solution is 2.68.

Calculate the concentration of hydrogen ions \([H^+]\) using the pH value: \([H^+] = 10^{-\text{pH}}\).

Write the expression for the acid dissociation constant \(K_a\) for phenylacetic acid: \(K_a = \frac{[H^+][A^-]}{[HA]}\), where \([A^-]\) is the concentration of the conjugate base and \([HA]\) is the concentration of the undissociated acid.

Assume that the initial concentration of the acid \([HA]\) decreases by \([H^+]\) due to dissociation, and \([A^-] = [H^+]\) at equilibrium.

Substitute the values into the \(K_a\) expression: \(K_a = \frac{[H^+]^2}{[HA] - [H^+]}\), and solve for \(K_a\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Acid Dissociation Constant (Ka)

The acid dissociation constant, Ka, quantifies the strength of an acid in solution. It is defined as the equilibrium constant for the dissociation of an acid into its conjugate base and a proton. A higher Ka value indicates a stronger acid, as it dissociates more completely in solution. For weak acids, like phenylacetic acid, Ka can be calculated using the concentrations of the products and reactants at equilibrium.

pH and its Relationship to Concentration

pH is a measure of the hydrogen ion concentration in a solution, calculated as pH = -log[H+]. In the case of weak acids, the pH can provide insight into the degree of dissociation of the acid. For a 0.085 M solution of phenylacetic acid with a pH of 2.68, the concentration of hydrogen ions can be determined, which is essential for calculating the Ka value.

Equilibrium Expressions

Equilibrium expressions describe the relationship between the concentrations of reactants and products at equilibrium. For the dissociation of a weak acid, the expression is given by Ka = [H+][A-]/[HA], where [H+] is the concentration of hydrogen ions, [A-] is the concentration of the conjugate base, and [HA] is the concentration of the undissociated acid. Understanding how to manipulate this expression is crucial for calculating Ka from pH and initial concentrations.