Problem 6 - Biochemistry: Concepts and Connections

Question

The weak acid HA is \(2 \%\) ionized (dissociated) in a \(0.20 \mathrm{M}\) solution. (a) What is \(K_{\mathrm{a}}\) for this acid? (b) What is the pH of this solution?

Step-by-Step Solution

Answer

(a) \( K_a \approx 8.16 \times 10^{-5} \); (b) pH \( \approx 2.40 \).

Step 1: Identify Initial Concentrations

The initial concentration of the weak acid HA is given as \( [HA]_0 = 0.20 \text{ M} \). Since it is \( 2\% \) ionized, the concentrations that dissociate are important for calculating \( K_a \).

Step 2: Determine Concentration of Ionized Acid

Calculate the concentration of HA that ionizes: \( [HA]_0 \cdot \frac{2}{100} = 0.20 \cdot 0.02 = 0.004 \text{ M} \). This means, \( [H^+] = [A^-] = 0.004 \text{ M} \) at equilibrium.

Step 3: Calculate Equilibrium Concentrations

The remaining concentration of HA after ionization is \( [HA] = [HA]_0 - [H^+] = 0.20 - 0.004 = 0.196 \text{ M} \).

Step 4: Write the Expression for Ka

The expression for the acid dissociation constant, \( K_a \), is \[ K_a = \frac{[H^+][A^-]}{[HA]} \]. Substitute the equilibrium concentrations into this formula.

Step 5: Calculate Ka

Substitute known values into the equation: \( K_a = \frac{(0.004)(0.004)}{0.196} = \frac{0.000016}{0.196} \approx 8.16 \times 10^{-5} \).

Step 6: Calculate pH of the Solution

pH is calculated using the formula \( pH = -\log [H^+] \). Here, \( [H^+] = 0.004 \text{ M} \).

Step 7: Compute pH Value

Calculate \( pH = -\log(0.004) \approx 2.40 \). The solution has a pH of approximately 2.40.

Key Concepts

Ionization Percentage

Ionization percentage is a key concept in understanding how much of an acid ionizes or dissociates in a solution. When we talk about a 2% ionization for a weak acid like HA, it means that 2% of the acid molecules dissociate into ions in the solution.
The formula to calculate the ionization percentage is:

• Ionization Percentage = \( \left( \frac{[ ext{Dissociated Acid}]}{[ ext{Initial Acid}]} \right) \times 100 \)

Given that the initial concentration of HA is 0.20 M and it's 2% ionized, we can calculate the concentration of the ionized acid:

• The concentration of ionized HA = \( 0.20 \times \frac{2}{100} = 0.004 \) M

This calculation shows that 0.004 M of the acid is ionized into hydrogen ions \( [H^+] \) and conjugate base ions \( [A^-] \) in the solution.
In understanding ionization, remember that it tells how extensively an acid ionizes, giving insight into the acidity and strength of the acid in a given solution.

Acid Dissociation Constant

The acid dissociation constant, symbolized as \( K_a \), is a crucial parameter in evaluating the strength of a weak acid in solution. It reflects the equilibrium position of the acid dissociation reaction.
For an acid HA that ionizes into \( H^+ \) and \( A^- \), the expression for \( K_a \) is:

• \( K_a = \frac{[H^+][A^-]}{[HA]} \)

In our example, after ionization, we have:

• \([H^+] = [A^-] = 0.004 \text{ M} \)
• \([HA] = 0.196 \text{ M} \)

Let's substitute these into the formula:

• \( K_a = \frac{(0.004)(0.004)}{0.196} \approx 8.16 \times 10^{-5} \)

This value is the acid dissociation constant for HA in this scenario.
Understanding \( K_a \) helps chemists and students alike compare acids' strengths. A larger \( K_a \) value indicates a stronger acid, as it means more of the acid ionizes in the solution.

pH Calculation

Calculating pH is fundamental in assessing the acidity of solutions. For a weak acid solution, such as our 0.20 M HA, knowing the hydrogen ion concentration is key to determining the pH.
The formula to find pH is:

• \( pH = -\log [H^+] \)

With our earlier calculation showing that the hydrogen ion concentration \( [H^+] \) is 0.004 M, we compute the pH as follows:

• \( pH = -\log(0.004) \approx 2.40 \)

This indicates that the solution is acidic, as a pH below 7 signifies an acidic environment.
Remember, the pH scale ranges from 0 to 14:

• Poorly acidic solutions have a lower pH
• Neutral solutions, like pure water, have a pH of around 7
• Alkaline solutions have a pH higher than 7

Thus, understanding how to determine the pH is an important aspect of characterizing chemical solutions.