An aqueous solution of `CH_3COOH` has a pH=3 and acid dissociation constant of `CH_3COOH` is `10^(-5)` What will be the concentration of acid taken initially ?

Options

A. 0.1 M
B. 0.11 M
C. 0.09 M
D. 0.101 M

Answer

To find the initial concentration of acetic acid (CH₃COOH) in an aqueous solution with a given pH and acid dissociation constant (Ka), we can follow these steps:

### Step 1: Calculate the concentration of hydrogen ions [H⁺]
The pH of the solution is given as 3. We can calculate the concentration of hydrogen ions using the formula:
\[ \text{pH} = -\log[H^+] \]
Rearranging this gives:
\[ [H^+] = 10^{-\text{pH}} \]
Substituting the value of pH:
\[ [H^+] = 10^{-3} \, \text{M} \]

### Step 2: Set up the equilibrium expression for the dissociation of acetic acid
The dissociation of acetic acid in water can be represented as:
\[ \text{CH}_3\text{COOH} \rightleftharpoons \text{H}^+ + \text{CH}_3\text{COO}^- \]
Let the initial concentration of acetic acid be \( C \) M. At equilibrium, if \( x \) is the concentration of H⁺ ions produced, we have:
- [CH₃COOH] = \( C - x \)
- [H⁺] = \( x \)
- [CH₃COO⁻] = \( x \)

From Step 1, we found that \( x = [H^+] = 10^{-3} \) M.

### Step 3: Write the expression for the acid dissociation constant (Ka)
The expression for the acid dissociation constant is:
\[ K_a = \frac{[H^+][CH_3COO^-]}{[CH_3COOH]} \]
Substituting the equilibrium concentrations:
\[ K_a = \frac{x \cdot x}{C - x} = \frac{x^2}{C - x} \]
Given that \( K_a = 10^{-5} \) and \( x = 10^{-3} \):
\[ 10^{-5} = \frac{(10^{-3})^2}{C - 10^{-3}} \]

### Step 4: Solve for C
Substituting the values into the equation:
\[ 10^{-5} = \frac{10^{-6}}{C - 10^{-3}} \]
Cross-multiplying gives:
\[ 10^{-5}(C - 10^{-3}) = 10^{-6} \]
Expanding this:
\[ 10^{-5}C - 10^{-8} = 10^{-6} \]
Rearranging the equation:
\[ 10^{-5}C = 10^{-6} + 10^{-8} \]
\[ 10^{-5}C = 10^{-6} + 0.01 \times 10^{-6} \]
\[ 10^{-5}C = 1.1 \times 10^{-6} \]
Now, dividing both sides by \( 10^{-5} \):
\[ C = \frac{1.1 \times 10^{-6}}{10^{-5}} = 0.11 \, \text{M} \]

### Final Answer
The initial concentration of acetic acid (CH₃COOH) is \( 0.11 \, \text{M} \).