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Chapter 18: Problem 39

List the information needed to calculate the \(K_{\mathrm{a}}\) of a weak acid.

Short Answer

Expert verified
To calculate the \(K_{\mathrm{a}}\) of a weak acid, gather information on the initial concentration of the weak acid (\(C_{HA}\)), calculate the degree of dissociation (伪), and find the equilibrium concentrations of the weak acid, hydrogen ions, and the conjugate base. Use these concentrations in the formula \(K_{\mathrm{a}} = \frac{[H^+]_{equilibrium} [A^-]_{equilibrium}}{[HA]_{equilibrium}}\) to determine the \(K_{\mathrm{a}}\) value.

Step by step solution

01

1. Identifying the weak acid and its dissociation reaction

Write the dissociation reaction of the weak acid. Let's denote the weak acid as HA, where H represents the hydrogen ion to be released, and A is the conjugate base. The reaction can be written as: \[HA \rightleftharpoons H^+ + A^-\]
02

2. Noting the initial concentration of the weak acid

Before dissociation starts, determine the initial concentration of the weak acid, HA (\(C_{HA}\)), which is usually given in the problem statement. Remember that the initial concentrations of the products (\(H^+\) and \(A^-\)) are typically 0.
03

3. Calculating the degree of dissociation

Obtain the degree of dissociation (伪) of the weak acid, which represents the proportion of dissociated molecules. This value can be calculated using information on pH or otherwise given in the problem statement.
04

4. Calculating the change in concentrations using the degree of dissociation

Use the degree of dissociation (伪) to find the change in concentrations of the weak acid (HA), the hydrogen ions (\(H^+\)), and the conjugate base (\(A^-\)). For Example, Change in concentration of HA = -伪 * \(C_{HA}\) Change in concentration of \(H^+\) = 伪 * \(C_{HA}\) Change in concentration of \(A^-\) = 伪 * \(C_{HA}\)
05

5. Setting up the equilibrium concentrations

Determine the equilibrium concentrations of the weak acid, hydrogen ions, and the conjugate base using the initial concentration and the change in concentration found in step 4. For Example, \([HA]_{equilibrium}\) = \(C_{HA}\) - 伪 * \(C_{HA}\) \([H^+]_{equilibrium}\) = 伪 * \(C_{HA}\) \([A^-]_{equilibrium}\) = 伪 * \(C_{HA}\)
06

6. Calculating the \(K_{\mathrm{a}}\) constant

Use the equilibrium concentrations of the weak acid, hydrogen ions, and the conjugate base to calculate \(K_{\mathrm{a}}\) using the formula: \[K_{\mathrm{a}} = \frac{[H^+]_{equilibrium} [A^-]_{equilibrium}}{[HA]_{equilibrium}}\] Plug in the equilibrium concentrations from step 5 into the formula and evaluate the expression to find the \(K_{\mathrm{a}}\) value of the weak acid. These are the steps required to calculate the \(K_{\mathrm{a}}\) of a weak acid using the given information.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dissociation Reaction
Understanding the dissociation reaction of a weak acid is the first step in calculating its acid dissociation constant, known as \(K_{\mathrm{a}}\). A weak acid, such as HA, only partially ionizes in a solution. This means that when HA dissolves in water, it splits into hydrogen ions \((H^+)\) and its conjugate base \((A^-)\).
The reaction is represented as:
\[HA \rightleftharpoons H^+ + A^-\]
This equation shows a state of dynamic equilibrium, where the acid does not release all its hydrogen ions at once, making it 'weak.' The double arrows indicate that the reaction is reversible, meaning products can convert back into reactants.
Degree of Dissociation
The degree of dissociation, denoted by \(\alpha\), is a measure of how much of the weak acid has dissociated in the solution. It is crucial for determining the concentrations of all species at equilibrium.
Mathematically, \(\alpha\) is calculated by the formula:
\[ \alpha = \frac{\text{Concentration of dissociated acid}}{\text{Initial concentration of acid}} \]
A higher \(\alpha\) value indicates more dissociation. Since weak acids partially dissociate, \(\alpha\) usually takes on a value less than 1.\(\alpha\) can often be derived from the pH or given directly in a problem statement.
Equilibrium Concentrations
After identifying the degree of dissociation, the next step involves calculating equilibrium concentrations. Initially, only the weak acid is present, while the concentrations of hydrogen ions and conjugate base are zero.
To find the equilibrium concentrations, use:
  • Change in concentration of HA is \(-\alpha \times C_{HA}\)
  • Change in concentration of \(H^+\) is \(\alpha \times C_{HA}\)
  • Change in concentration of \(A^-\) is \(\alpha \times C_{HA}\)
Substitute these into the equilibrium expressions:
\[ [HA]_{equilibrium} = C_{HA} - \alpha \times C_{HA} \]
\[ [H^+]_{equilibrium} = \alpha \times C_{HA} \]
\[ [A^-]_{equilibrium} = \alpha \times C_{HA} \]These formulas help us understand the balance at equilibrium.
Weak Acid Calculation
The goal of these calculations is to determine the \(K_{\mathrm{a}}\), or acid dissociation constant, of the weak acid. This constant is a quantitative measure of the strength of an acid in solution. The formula for \(K_{\mathrm{a}}\) is:
\[K_{\mathrm{a}} = \frac{[H^+]_{equilibrium} [A^-]_{equilibrium}}{[HA]_{equilibrium}}\]
Plug in the equilibrium concentrations calculated in the previous section to find \(K_{\mathrm{a}}\).
Having a higher \(K_{\mathrm{a}}\) value means the acid dissociates more in the solution, indicating a stronger weak acid. Understanding \(K_{\mathrm{a}}\) helps predict the behavior of acids in various environments and is essential in chemistry.

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