20 mL of 0.1 M acetic acid is mixed with 50 mL of potassium acetate. `K_(a)` of acetic acid = `1.8 xx 10^(-5)` at `27^(circ)C`. Calculate concentration of potassium acetate if pH of the mixture is 4.8.

Options

A. 0.1 M
B. 0.04M
C. 0.4 M
D. 0.02M

Answer

To solve the problem step by step, we need to calculate the concentration of potassium acetate in the mixture given the pH of the solution and the concentration of acetic acid.

### Step 1: Calculate pKa from Ka
We know that:
\[ pK_a = -\log(K_a) \]
Given \( K_a = 1.8 \times 10^{-5} \):
\[ pK_a = -\log(1.8 \times 10^{-5}) \]

**Calculation:**
Using a calculator:
\[ pK_a = -(\log(1.8) + \log(10^{-5})) \]
\[ pK_a = -(0.255 + (-5)) \]
\[ pK_a = 4.745 \]

### Step 2: Use the Henderson-Hasselbalch equation
The Henderson-Hasselbalch equation is:
\[ pH = pK_a + \log\left(\frac{[Salt]}{[Acid]}\right) \]
Given \( pH = 4.8 \) and \( pK_a = 4.745 \):
\[ 4.8 = 4.745 + \log\left(\frac{[Salt]}{[Acid]}\right) \]

### Step 3: Rearranging the equation to find the ratio of concentrations
Subtract \( pK_a \) from both sides:
\[ 4.8 - 4.745 = \log\left(\frac{[Salt]}{[Acid]}\right) \]
\[ 0.055 = \log\left(\frac{[Salt]}{[Acid]}\right) \]

### Step 4: Convert from log to antilog
To find the ratio of concentrations:
\[ \frac{[Salt]}{[Acid]} = 10^{0.055} \]

**Calculation:**
Using a calculator:
\[ \frac{[Salt]}{[Acid]} \approx 1.138 \]

### Step 5: Calculate the concentration of acetic acid
We need to find the concentration of acetic acid in the mixture. The volume of acetic acid is 20 mL with a concentration of 0.1 M:
\[ [Acid] = \frac{moles}{volume(L)} \]
Moles of acetic acid:
\[ \text{Moles of acetic acid} = 0.1 \, \text{mol/L} \times 0.020 \, \text{L} = 0.002 \, \text{mol} \]

Total volume of the mixture:
\[ \text{Total volume} = 20 \, \text{mL} + 50 \, \text{mL} = 70 \, \text{mL} = 0.070 \, \text{L} \]

Concentration of acetic acid in the mixture:
\[ [Acid] = \frac{0.002 \, \text{mol}}{0.070 \, \text{L}} \approx 0.02857 \, \text{M} \]

### Step 6: Use the ratio to find the concentration of potassium acetate
From the ratio:
\[ \frac{[Salt]}{[Acid]} = 1.138 \]
Let \( [Salt] = x \):
\[ x = 1.138 \times [Acid] \]
\[ x = 1.138 \times 0.02857 \approx 0.0324 \, \text{M} \]

### Step 7: Calculate the concentration of potassium acetate
The concentration of potassium acetate in the total volume (70 mL):
\[ [Salt] = \frac{0.0324 \, \text{mol}}{0.050 \, \text{L}} \]
\[ [Salt] = 0.0324 \times 0.050 \, \text{L} = 0.00162 \, \text{mol} \]

Finally, to find the molarity of potassium acetate:
\[ \text{Molarity} = \frac{0.00162 \, \text{mol}}{0.050 \, \text{L}} = 0.0324 \, \text{M} \]

### Final Answer
The concentration of potassium acetate is approximately **0.0324 M**.