Question

Calculate the \(K_{\mathrm{a}}\) of a weak acid if a \(0.19-M\) aqueous solution of the acid has a \(\mathrm{pH}\) of 4.52 at \(25^{\circ} \mathrm{C}\).

Short answer

The \( K_a \) of the weak acid is \( 4.8 \times 10^{-9} \).

Step-by-step solution

Understand the Problem

The problem gives us the concentration of a weak acid solution and its pH, and asks us to calculate the acid dissociation constant, \( K_{\mathrm{a}} \). To find \( K_{\mathrm{a}} \), we need to connect the pH of the solution to the concentration of hydrogen ions, \( [\mathrm{H}^+] \), and use this to calculate \( K_{\mathrm{a}} \).

Convert pH to \( [\mathrm{H}^+] \) Concentration

Use the formula \( [\mathrm{H}^+] = 10^{-\mathrm{pH}} \) to convert the given pH of 4.52 to the hydrogen ion concentration:\[ [\mathrm{H}^+] = 10^{-4.52} = 3.02 \times 10^{-5} \; \mathrm{M} \]

Set Up the Expression for \( K_a \)

For the dissociation of a weak acid \( HA \), the equilibrium expression is:\[ HA \rightleftharpoons H^+ + A^- \]\( K_{\mathrm{a}} = \frac{[H^+][A^-]}{[HA]} \)Assume \([A^-] = [H^+] = 3.02 \times 10^{-5} \; \mathrm{M} \) since they are produced in a 1:1 ratio.

Calculate Equilibrium Concentrations

The initial concentration of \( HA \) is 0.19 M. At equilibrium, the concentration of \( HA \) becomes:\[ [HA] = 0.19 - 3.02 \times 10^{-5} \approx 0.19 \; \mathrm{M} \](since \( 3.02 \times 10^{-5} \) is so small compared to 0.19, we can approximate it as 0.19 M).

Calculate \( K_a \)

Substitute the equilibrium concentrations into the \( K_{\mathrm{a}} \) expression:\[ K_{\mathrm{a}} = \frac{(3.02 \times 10^{-5})^2}{0.19} \]Calculate:\[ K_{\mathrm{a}} = \frac{9.12 \times 10^{-10}}{0.19} = 4.8 \times 10^{-9} \]

Verify Calculation and Approximation

Reassess the approximation made by checking if \( 3.02 \times 10^{-5} \) is indeed negligible compared to 0.19. If it is, our calculations are valid, confirming \( K_a = 4.8 \times 10^{-9} \).

Key concepts

Weak Acid

Weak acids are substances that only partially dissociate in water. They exist in a state of equilibrium between the undissociated acid and the ions formed. Unlike strong acids, which completely dissociate in solution, weak acids do not release all of their hydrogen ions into the solution.
This incomplete dissociation is crucial when calculating their properties, such as the acid dissociation constant, or \(K_{\text{a}}\). The value of \(K_{\text{a}}\) helps quantify the strength of the acid by indicating the extent to which an acid ionizes in solution.
The smaller the \(K_{\text{a}}\), the weaker the acid, as it suggests fewer hydrogen ions are produced given the same concentration of acid. Understanding this helps in analyzing acidity and predicting the behavior of the acid in different chemical reactions.

pH Calculation

The measurement of acidity, pH, is an important concept when discussing acids and bases. pH is calculated as the negative logarithm of the hydrogen ion concentration: \[ \text{pH} = -\log [H^+] \]
In this exercise, the given pH was 4.52. Converting this pH value back to the hydrogen ion concentration involves using the inverse of the logarithm function.
Specifically, the hydrogen ion concentration \([H^+]\) is found using: \[ [H^+] = 10^{-\text{pH}} \]
Thus, with a pH of 4.52, the \([H^+]\) is calculated to be \(3.02 \times 10^{-5}\,\text{M}\). This conversion is fundamental to understanding the relationship between pH and the concentration of hydrogen ions in a solution.

Equilibrium Concentration

In the context of weak acids, equilibrium concentration refers to the concentrations of reactants and products when the reaction has reached a state where the rates of the forward and reverse reactions are equal. For a weak acid dissociation as given in:\[ HA \rightleftharpoons H^+ + A^- \]
the equilibrium expression incorporates these concentrations:\[ K_{\text{a}} = \frac{[H^+][A^-]}{[HA]} \]We often make the assumption that the initial concentration of the acid is only slightly decreased by the acid ionizing to reach equilibrium, especially if the amount ionized is very small compared to the initial concentration.
This assumption simplifies calculations and allows us to approximate that the initial and equilibrium concentrations of the acid are almost the same, aiding in calculating \(K_{\text{a}}\). Understanding equilibrium concentration helps in practically determining the behavior of weak acids in solution.

Hydrogen Ion Concentration

The concentration of hydrogen ions in a solution, represented as \([H^+]\), is a key determinant of the solution's acidity. In weak acids, the concentration of \([H^+]\) is important for calculating the acid dissociation constant \(K_{\text{a}}\).
It directly impacts the pH, which provides insight into the solution's acidity level. The relationship between \([H^+]\) and pH is inverse and logarithmic, as given by:\[ \text{pH} = -\log [H^+] \]
Even more, \([H^+]\) gives information about the extent of dissociation of the weak acid. By equating \([H^+]\) to \([A^-]\) in simple weak acid cases (assuming these concentrations are equal at equilibrium), effective methods to assess the overall dissociation process are provided.
Knowing the hydrogen ion concentration allows chemists to gauge how an acid will behave under different circumstances and to predict the pH levels accordingly.