To calculate the molality of a solution containing 20.7 g of potassium carbonate (K₂CO₃) dissolved in 500 mL of solution, we will follow these steps:
### Step 1: Calculate the number of moles of potassium carbonate (K₂CO₃).
The formula to calculate the number of moles (n) is:
\[
n = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}}
\]
Given:
- Mass of K₂CO₃ = 20.7 g
- Molar mass of K₂CO₃ = 138 g/mol (calculated from the atomic masses: K = 39.1, C = 12.0, O = 16.0)
Calculating the number of moles:
\[
n = \frac{20.7 \, \text{g}}{138 \, \text{g/mol}} \approx 0.150 \, \text{mol}
\]
### Step 2: Calculate the mass of the solution.
Given the density of the solution is 1 g/mL, we can calculate the mass of the solution using the formula:
\[
\text{mass of solution} = \text{density} \times \text{volume}
\]
Given:
- Volume of solution = 500 mL
- Density = 1 g/mL
Calculating the mass of the solution:
\[
\text{mass of solution} = 1 \, \text{g/mL} \times 500 \, \text{mL} = 500 \, \text{g}
\]
### Step 3: Calculate the mass of the solvent (water).
The mass of the solvent can be calculated by subtracting the mass of the solute from the mass of the solution:
\[
\text{mass of solvent} = \text{mass of solution} - \text{mass of solute}
\]
Given:
- Mass of solute (K₂CO₃) = 20.7 g
Calculating the mass of the solvent:
\[
\text{mass of solvent} = 500 \, \text{g} - 20.7 \, \text{g} = 479.3 \, \text{g}
\]
### Step 4: Convert the mass of the solvent to kilograms.
Since molality is expressed in moles of solute per kilogram of solvent, we need to convert grams to kilograms:
\[
\text{mass of solvent in kg} = \frac{479.3 \, \text{g}}{1000} = 0.4793 \, \text{kg}
\]
### Step 5: Calculate the molality of the solution.
The formula for molality (m) is:
\[
m = \frac{\text{moles of solute}}{\text{mass of solvent (kg)}}
\]
Substituting the values we calculated:
\[
m = \frac{0.150 \, \text{mol}}{0.4793 \, \text{kg}} \approx 0.313 \, \text{mol/kg}
\]
### Final Answer:
The molality of the solution is approximately **0.313 mol/kg**.
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