To solve the problem step by step, we will follow the outlined approach based on the information provided.
### Step 1: Understand the Given Data
- Number of moles of gas (n) = 0.1 moles
- Initial volume of gas (V1) = 1 dm³
- Initial pressure of gas (P1) = unknown
- Final pressure of gas (P2) = 1 atm (after the piston is released)
- R (ideal gas constant) = 0.08 atm L mol⁻¹ K⁻¹
- Molality of ethylene glycol solution = 0.5 molal
- Freezing point depression constant (K_f) = 2.0 K molal⁻¹
- Initial temperature (T_initial) = 0°C = 273 K (freezing point of pure water)
### Step 2: Calculate the Freezing Point of the Solution
Using the formula for freezing point depression:
\[
\Delta T_f = K_f \cdot m
\]
Where:
- \(m\) = molality of the solution = 0.5 mol/kg
Calculating \(\Delta T_f\):
\[
\Delta T_f = 2.0 \, \text{K molal}^{-1} \times 0.5 \, \text{mol/kg} = 1 \, \text{K}
\]
Now, calculate the freezing point of the solution:
\[
T_f = T_{initial} - \Delta T_f = 273 \, \text{K} - 1 \, \text{K} = 272 \, \text{K}
\]
### Step 3: Calculate the Pressure of the Gas Using the Ideal Gas Law
Using the ideal gas equation:
\[
PV = nRT
\]
Rearranging for pressure (P):
\[
P = \frac{nRT}{V}
\]
Substituting the known values:
\[
P = \frac{(0.1 \, \text{mol}) \times (0.08 \, \text{atm L mol}^{-1} K^{-1}) \times (272 \, \text{K})}{1 \, \text{dm}^3}
\]
Calculating:
\[
P = \frac{0.1 \times 0.08 \times 272}{1} = 2.176 \, \text{atm}
\]
### Step 4: Apply Boyle's Law to Find Final Volume
According to Boyle's Law:
\[
P_1 V_1 = P_2 V_2
\]
Where:
- \(P_1 = 2.176 \, \text{atm}\)
- \(V_1 = 1 \, \text{dm}^3\)
- \(P_2 = 1 \, \text{atm}\)
- \(V_2 = ?\)
Rearranging for \(V_2\):
\[
V_2 = \frac{P_1 V_1}{P_2}
\]
Substituting the known values:
\[
V_2 = \frac{(2.176 \, \text{atm}) \times (1 \, \text{dm}^3)}{1 \, \text{atm}} = 2.176 \, \text{dm}^3
\]
### Step 5: Round the Final Answer
Rounding \(V_2\) to two decimal places gives:
\[
V_2 \approx 2.18 \, \text{dm}^3
\]
### Final Answer
The final volume of the gas after the piston is released is approximately **2.18 dm³**.
---
