The hydrogen ion concentration of a 0.006 M benzoic acid solution is `(K_(a) = 6 xx 10^(-5))`

Options

A. `0.6 xx 10^(-4)`
B. `6 xx 10^(-4)`
C. `6 xx 10^(-5)`
D. `3.6 xx 10^(-4)`

Answer

To find the hydrogen ion concentration of a 0.006 M benzoic acid solution, we can follow these steps:

### Step 1: Write the dissociation equation of benzoic acid.
Benzoic acid (C₆H₅COOH) dissociates in water as follows:
\[ \text{C}_6\text{H}_5\text{COOH} \rightleftharpoons \text{C}_6\text{H}_5\text{COO}^- + \text{H}^+ \]

### Step 2: Write the expression for the acid dissociation constant (Kₐ).
The expression for the acid dissociation constant (Kₐ) is given by:
\[ K_a = \frac{[\text{C}_6\text{H}_5\text{COO}^-][\text{H}^+]}{[\text{C}_6\text{H}_5\text{COOH}]} \]

### Step 3: Set up the equilibrium concentrations.
Let the initial concentration of benzoic acid be \( C = 0.006 \, M \). At equilibrium, if \( x \) is the concentration of \( \text{H}^+ \) ions produced, we have:
- \([\text{C}_6\text{H}_5\text{COO}^-] = x\)
- \([\text{H}^+] = x\)
- \([\text{C}_6\text{H}_5\text{COOH}] = C - x = 0.006 - x\)

### Step 4: Substitute the equilibrium concentrations into the Kₐ expression.
Substituting the equilibrium concentrations into the Kₐ expression gives:
\[ K_a = \frac{x \cdot x}{0.006 - x} = \frac{x^2}{0.006 - x} \]

### Step 5: Assume \( x \) is small compared to \( C \).
Since benzoic acid is a weak acid, we can assume that \( x \) is small compared to 0.006 M. Therefore, we can approximate:
\[ 0.006 - x \approx 0.006 \]

So the equation simplifies to:
\[ K_a \approx \frac{x^2}{0.006} \]

### Step 6: Solve for \( x \).
Now substituting the value of \( K_a \):
\[ 6 \times 10^{-5} = \frac{x^2}{0.006} \]

Rearranging gives:
\[ x^2 = 6 \times 10^{-5} \times 0.006 \]
\[ x^2 = 3.6 \times 10^{-7} \]

Taking the square root:
\[ x = \sqrt{3.6 \times 10^{-7}} \]
\[ x \approx 1.897 \times 10^{-4} \]

### Step 7: Conclusion.
Thus, the hydrogen ion concentration \([\text{H}^+]\) in the solution is approximately:
\[ [\text{H}^+] \approx 1.9 \times 10^{-4} \, M \]