In a mixing of acetic acid and sodium acetate the ratio of concentration of the salts to the acid is increased ten times . Then the pH of the solution

Options

A. Increases by one unit
B. Decreases by one unit
C. Decrease ten fold
D. Increases ten fold

Answer

To solve the problem, we need to determine the pH of a solution formed by mixing acetic acid and sodium acetate when the ratio of the concentration of the salt (sodium acetate) to the acid (acetic acid) is increased ten times.

### Step 1: Understand the Components
- **Acetic Acid (CH₃COOH)**: This is a weak acid.
- **Sodium Acetate (CH₃COONa)**: This is the salt of acetic acid and a strong base (NaOH), and it acts as the conjugate base of acetic acid.

### Step 2: Recognize the Buffer System
When acetic acid and sodium acetate are mixed, they form a buffer solution. A buffer solution resists changes in pH when small amounts of acid or base are added.

### Step 3: Use the Henderson-Hasselbalch Equation
The pH of a buffer solution can be calculated using the Henderson-Hasselbalch equation:
\[ \text{pH} = \text{pKa} + \log\left(\frac{[\text{Salt}]}{[\text{Acid}]}\right) \]
Where:
- \([\text{Salt}]\) is the concentration of sodium acetate.
- \([\text{Acid}]\) is the concentration of acetic acid.
- \(\text{pKa}\) is the negative logarithm of the acid dissociation constant (Ka) of acetic acid.

### Step 4: Set Up the Initial Ratio
Let’s denote the initial concentrations as:
- \([\text{Salt}] = [\text{Salt}]_0\)
- \([\text{Acid}] = [\text{Acid}]_0\)

The initial pH can be expressed as:
\[ \text{pH}_0 = \text{pKa} + \log\left(\frac{[\text{Salt}]_0}{[\text{Acid}]_0}\right) \]

### Step 5: Increase the Ratio by 10 Times
According to the problem, the ratio of the concentration of salt to acid is increased ten times:
\[ \frac{[\text{Salt}]}{[\text{Acid}]} = 10 \cdot \frac{[\text{Salt}]_0}{[\text{Acid}]_0} \]

Now, substituting this into the Henderson-Hasselbalch equation gives:
\[ \text{pH}_1 = \text{pKa} + \log\left(10 \cdot \frac{[\text{Salt}]_0}{[\text{Acid}]_0}\right) \]

### Step 6: Simplify the Equation
Using the properties of logarithms:
\[ \text{pH}_1 = \text{pKa} + \log(10) + \log\left(\frac{[\text{Salt}]_0}{[\text{Acid}]_0}\right) \]
Since \(\log(10) = 1\), we can rewrite this as:
\[ \text{pH}_1 = \text{pKa} + 1 + \log\left(\frac{[\text{Salt}]_0}{[\text{Acid}]_0}\right) \]

### Step 7: Compare the Two pH Values
Now, comparing the initial pH and the new pH:
\[ \text{pH}_1 = \text{pH}_0 + 1 \]
This means that the pH of the solution increases by 1 unit when the ratio of the salt to acid is increased ten times.

### Conclusion
Thus, the final pH of the solution is:
\[ \text{pH} = \text{pKa} + 1 + \log\left(\frac{[\text{Salt}]_0}{[\text{Acid}]_0}\right) \]