The degree of dissociation of acetic acid in a 0.1 M solution is `1.0xx10^(-2)`. The `pK_(a)` of acetic acid value.

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the pK_a of acetic acid given its degree of dissociation in a 0.1 M solution, we can follow these steps:

### Step 1: Write the dissociation equation
The dissociation of acetic acid (CH₃COOH) can be represented as:
\[ \text{CH}_3\text{COOH} \rightleftharpoons \text{CH}_3\text{COO}^- + \text{H}^+ \]

### Step 2: Define initial concentrations
Let the initial concentration of acetic acid be \( C = 0.1 \, \text{M} \). At the start (time \( t = 0 \)):
- Concentration of CH₃COOH = \( C \)
- Concentration of CH₃COO⁻ = 0
- Concentration of H⁺ = 0

### Step 3: Define changes at equilibrium
Let \( \alpha \) be the degree of dissociation. At equilibrium:
- Concentration of CH₃COOH = \( C(1 - \alpha) \)
- Concentration of CH₃COO⁻ = \( C\alpha \)
- Concentration of H⁺ = \( C\alpha \)

Given that \( \alpha = 1.0 \times 10^{-2} \), we can substitute this value.

### Step 4: Substitute values into the equilibrium expression for K_a
The expression for the acid dissociation constant \( K_a \) is:
\[ K_a = \frac{[\text{CH}_3\text{COO}^-][\text{H}^+]}{[\text{CH}_3\text{COOH}]} \]
Substituting the equilibrium concentrations:
\[ K_a = \frac{(C\alpha)(C\alpha)}{C(1 - \alpha)} \]

### Step 5: Simplify the expression
This simplifies to:
\[ K_a = \frac{C^2\alpha^2}{C(1 - \alpha)} = \frac{C\alpha^2}{1 - \alpha} \]

### Step 6: Assume \( \alpha \) is small
Since \( \alpha \) is small (1.0 × 10⁻²), we can approximate \( 1 - \alpha \approx 1 \):
\[ K_a \approx C\alpha^2 \]

### Step 7: Substitute known values
Substituting \( C = 0.1 \, \text{M} \) and \( \alpha = 1.0 \times 10^{-2} \):
\[ K_a \approx 0.1 \times (1.0 \times 10^{-2})^2 \]
\[ K_a \approx 0.1 \times 1.0 \times 10^{-4} \]
\[ K_a \approx 1.0 \times 10^{-5} \]

### Step 8: Calculate pK_a
The pK_a is calculated using the formula:
\[ pK_a = -\log(K_a) \]
Substituting the value of \( K_a \):
\[ pK_a = -\log(1.0 \times 10^{-5}) \]
\[ pK_a = 5 \]

### Conclusion
Thus, the pK_a of acetic acid is 5.

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