What is the percentage dissociation of 0.1 M solution of acetic acid ? `(K_(a)=10^(-5)) (A)10 % (B)100 % (C)1 % (D)0.01 %`

Options

A. `10%`
B. `100%`
C. `1%`
D. `0.01%`

Explanation

To find the percentage dissociation of a 0.1 M solution of acetic acid (CH₃COOH) given that its dissociation constant (Kₐ) is 10⁻⁵, we can follow these steps:

### Step-by-Step Solution:

1. **Write the dissociation equation:**
The dissociation of acetic acid in water can be represented as:
\[ \text{CH}_3\text{COOH} \rightleftharpoons \text{CH}_3\text{COO}^- + \text{H}^+ \]

2. **Set up the initial concentrations:**
Let the initial concentration of acetic acid (CH₃COOH) be \( C = 0.1 \, \text{M} \).
At the start (t = 0):
- [CH₃COOH] = 0.1 M
- [CH₃COO⁻] = 0 M
- [H⁺] = 0 M

3. **Define the change in concentration:**
If \( \alpha \) is the degree of dissociation (the fraction that dissociates), then at equilibrium:
- [CH₃COOH] = \( C - C\alpha = 0.1 - 0.1\alpha \)
- [CH₃COO⁻] = \( C\alpha = 0.1\alpha \)
- [H⁺] = \( C\alpha = 0.1\alpha \)

4. **Write the expression for the equilibrium constant (Kₐ):**
The expression for the dissociation constant is:
\[ K_a = \frac{[\text{CH}_3\text{COO}^-][\text{H}^+]}{[\text{CH}_3\text{COOH}]} \]
Substituting the equilibrium concentrations:
\[ K_a = \frac{(0.1\alpha)(0.1\alpha)}{0.1 - 0.1\alpha} \]

5. **Simplify the equation:**
This simplifies to:
\[ K_a = \frac{0.01\alpha^2}{0.1(1 - \alpha)} = \frac{0.1\alpha^2}{1 - \alpha} \]

6. **Assume \( \alpha \) is small:**
Since \( K_a \) is small, we can assume \( \alpha \) is very small compared to 1. Therefore, \( 1 - \alpha \approx 1 \):
\[ K_a \approx 0.1\alpha^2 \]

7. **Substitute the value of Kₐ:**
Given \( K_a = 10^{-5} \):
\[ 10^{-5} = 0.1\alpha^2 \]

8. **Solve for \( \alpha \):**
Rearranging gives:
\[ \alpha^2 = \frac{10^{-5}}{0.1} = 10^{-4} \]
Taking the square root:
\[ \alpha = 10^{-2} = 0.01 \]

9. **Calculate the percentage dissociation:**
The percentage dissociation is given by:
\[ \text{Percentage dissociation} = \alpha \times 100 = 0.01 \times 100 = 1\% \]

### Final Answer:
The percentage dissociation of the 0.1 M solution of acetic acid is **1%** (Option C).