Question

What is the percentage dissociation of \(0.1 \mathrm{M}\) solution of acetic acid? \(\left(K_{a}=10^{-5}\right)\) (a) \(10 \%\) (b) \(100 \%\) (c) \(1 \%\) (d) \(0.01 \%\)

Short answer

The percentage dissociation of a 0.1 M solution of acetic acid is 1%.

Step-by-step solution

Write down the expression for the dissociation constant

The dissociation constant for acetic acid \text{(CH}_3\text{COOH}) can be expressed as: \[ K_a = \frac{[\text{CH}_3\text{COO}^-][\text{H}^+]}{[\text{CH}_3\text{COOH}]} \]

Set up the expression with initial concentrations and changes

Let x be the concentration of H+ and CH3COO- ions produced from the dissociation of acetic acid. The initial concentration of acetic acid is 0.1 M, and it decreases by x. Then we can write the new concentrations at equilibrium as: [CH3COOH] = 0.1 - x M[CH3COO-] = x M[H+] = x M

Substitute the equilibrium concentrations into the Ka expression

Substitute the above values into the dissociation constant expression:\[ K_a = \frac{x^2}{0.1 - x} \]

Solve for x assuming \((0.1 - x) \approx 0.1\)

Since K_a is small, we can assume that x will be much smaller than 0.1 M and therefore, \((0.1 - x) \approx 0.1\). Now, the dissociation constant equation simplifies to \[ K_a = \frac{x^2}{0.1} \]Solving for x gives: \[ x = \sqrt{K_a \times 0.1} \]\[ x = \sqrt{10^{-5} \times 0.1} \]\[ x = \sqrt{10^{-6}} \]\[ x = 10^{-3}\text{ M} \]

Calculate the percentage dissociation

Percentage dissociation is calculated as the ratio of dissociated acetic acid to the initial concentration multiplied by 100: \[ \text{Percentage dissociation} = \left(\frac{x}{0.1}\right) \times 100\% \]\[ \text{Percentage dissociation} = \left(\frac{10^{-3}}{0.1}\right) \times 100\% = 1\% \]

Key concepts

Chemical Equilibrium

Understanding chemical equilibrium is crucial when analyzing the behavior of substances in a closed system. It is the state in which the concentrations of reactants and products remain constant over time because the rates of the forward and reverse reactions are equal. This does not imply that the reactants and products are in equal concentrations, but rather that their ratios don't change over time.

In the case of acetic acid in aqueous solution, chemical equilibrium refers to the point where the rate at which acetic acid molecules dissociate into acetate ions and hydrogen ions equals the rate at which these ions recombine to form acetic acid. Achieving equilibrium is an intrinsic property of chemical reactions and is vital for the calculation of the percentage dissociation in acetic acid solutions.

Acid Dissociation Constant

The acid dissociation constant, abbreviated as Ka, is a quantifiable measure of the strength of an acid in solution. It is used to describe the extent of the dissociation of an acid into ions. A larger Ka value indicates a stronger acid that dissociates more completely in solution. Conversely, a smaller Ka value suggests a weaker acid, which doesn't dissociate as much.

For acetic acid, a relatively weak acid, the Ka value is low (\(10^{-5}\)), indicating that not all of the acetic acid molecules will dissociate into acetate and hydrogen ions. This information is essential to calculate the percentage dissociation and understand the equilibrium behavior of the acid in the solution.

Ka Expression

The Ka expression for acetic acid is derived from the equilibrium condition of its dissociation reaction in water. Specifically, the expression is \[ K_a = \frac{[\text{CH}_3\text{COO}^-][\text{H}^+]}{[\text{CH}_3\text{COOH}]} \].

The square brackets indicate the concentrations of the substances at equilibrium. The numerator contains the product of the concentrations of the acetate ion (CH3COO-) and the hydrogen ion (H+), while the denominator is the concentration of undissociated acetic acid (CH3COOH). This expression allows us to relate the concentration of dissociated ions to the strength of the acid, as indicated by its Ka value.

Equilibrium Concentration Calculation

To find the equilibrium concentrations necessary for calculating the percentage dissociation, an initial concentration is set for acetic acid, and the amount of dissociation, represented by variable x, is considered. After establishing the equilibrium concentrations of the ions and undissociated acid, we can use the Ka expression to calculate the value of x.

When we solve for x, taking into account the assumption that \(0.1 - x \approx 0.1\) due to the small Ka value, we can determine the amount of acetic acid that has dissociated. After calculating the dissociated concentration, we translate this value into a percentage dissociation which gives an indication of how much acetic acid has been converted into ions at the chemical equilibrium in the solution.