Question

The \(K_{\mathrm{a}}\) for dichloroacetic acid is \(3.32 \times 10^{-2} .\) Approximately what percentage of the acid is dissociated in a \(0.10 \mathrm{M}\) aqueous solution?

Short answer

Approximately 57.6% of the acid is dissociated.

Step-by-step solution

Express the Acid Dissociation

Let the initial concentration of dichloroacetic acid be denoted as \([HA]_0 = 0.10\,M\). The acid dissociates according to the equation: \(HA \rightleftharpoons H^+ + A^-\). We set up the equilibrium expression: \(K_{\mathrm{a}} = \frac{[H^+][A^-]}{[HA]}\).

Set Up the ICE Table

In an ICE (Initial, Change, Equilibrium) table, we have:- Initial: \([HA]_0 = 0.10\,M, [H^+]_0 = 0, [A^-]_0 = 0\).- Change: \([-x]\) for \([HA]\), \(+x\) for \([H^+]\), and \(+x\) for \([A^-]\).- Equilibrium: \([HA] = 0.10 - x\), \([H^+] = x\), \([A^-] = x\).

Substitute into the Equilibrium Expression

Substitute the equilibrium concentrations into the expression: \[K_{\mathrm{a}} = \frac{x \times x}{0.10 - x} = 3.32 \times 10^{-2}\].This simplifies to: \(x^2 = (3.32 \times 10^{-2})(0.10 - x)\).

Assume Small x Approximation

Since \(K_a\) is relatively small, we assume \(x\) to be small compared to the initial concentration, i.e., \(0.10 - x \approx 0.10\). Thus, the equation simplifies to \(x^2 = 3.32 \times 10^{-3}\).

Solve for x

Solve for \(x\) by taking the square root: \[x = \sqrt{3.32 \times 10^{-3}} = 0.0576\].This is the concentration of \([H^+]\) and \([A^-]\) at equilibrium.

Calculate Percentage Dissociation

Percentage dissociation is calculated as the ratio of dissociated acid concentration to initial concentration, multiplied by 100:\[\text{Percentage dissociation} = \frac{x}{[HA]_0} \times 100 = \frac{0.0576}{0.10} \times 100 = 57.6\%\].

Key concepts

Dichloroacetic Acid

Dichloroacetic acid is a type of carboxylic acid, and the term "dichloro" refers to the presence of two chlorine atoms in its structure. This acid is known for its dissociative properties in water, where it partially breaks down into ions, influencing its acidity. The acid dissociation constant, represented as \( K_a \), measures the strength of the acid in the solution. A higher \( K_a \) indicates a stronger acid, which dissociates more readily. For dichloroacetic acid, this value is \( 3.32 \times 10^{-2} \), suggesting moderate strength among weak acids.
Dichloroacetic acid reacts with water by donating a hydrogen ion \( H^+ \) to the water, forming a hydronium ion \( H_3O^+ \), and resulting in chloride ions \( Cl^- \).

• The chemical equation for the dissociation is \( HA \rightleftharpoons H^+ + A^- \).
• The initial concentration of the acid solution plays a crucial role in understanding how much of it actually dissociates.

Understanding these properties help predict the behavior of dichloroacetic acid in various chemical contexts, especially in buffering and titration scenarios.

ICE Table

The ICE table is a vital tool in chemistry, especially in equilibrium calculations. ICE stands for Initial, Change, and Equilibrium, and it’s used to systematically track the concentrations of reactants and products in a reaction over time. This structured approach simplifies the calculations needed to predict the outcome of reactions.
For dichloroacetic acid in this exercise, the initial concentrations are set before any dissociation occurs. At the initial stage, the concentration of dichloroacetic acid \([HA]_0\) is \(0.10 \, M\), while the concentrations of \(H^+\) and \(A^-\) are \(0\).

• As the dissociation proceeds, a change occurs: -\(x\) for \([HA]\), and +\(x\) for both \([H^+]\) and \([A^-]\).
• At equilibrium, the concentrations adjust to \([HA] = 0.10 - x\), \([H^+] = x\), and \([A^-] = x\).

This table simplifies solving the equilibrium expression and determining the unknowns, which in this case is \(x\), representing the concentration of the dissociated ions at equilibrium.

Equilibrium Expression

Once we set up an ICE table, the next step involves formulating the equilibrium expression, which is derived from the law of mass action for the dissociation reaction of dichloroacetic acid. This is represented by the equation \( K_a = \frac{[H^+][A^-]}{[HA]} \). It defines the relationship between the concentrations of products and reactants at equilibrium.
Substituting the equilibrium concentrations from the ICE table into this expression allows us to solve for \(x\). Using the given \( K_a \) value of \(3.32 \times 10^{-2}\), we set up the equation:

• \[ K_{ ext{a}} = \frac{x^2}{0.10 - x} = 3.32 \times 10^{-2} \]
• Assuming \(0.10 - x \approx 0.10\), simplifies it to \[ x^2 = 3.32 \times 10^{-3} \], from which \(x\) can be solved.
• Solving gives \(x = 0.0576\), indicating the concentration of \(H^+\) ions at equilibrium.

Understanding these relationships is crucial to predicting how much of the acid is dissociated and how it affects the overall acidity of the solution.