Chapter 16: Problem 61
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
Expert verified
The \( K_a \) of the weak acid is \( 4.8 \times 10^{-9} \).
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
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.
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.
