To solve the problem, we will follow these steps:
### Step 1: Identify the total pressure and the partial pressures
We know that the total pressure in the vessel is 1 atm. The vapor pressure of the unknown compound is given as 0.68 atm. Therefore, we can find the partial pressure of helium (He) using Dalton's Law of Partial Pressures.
\[
P_{\text{He}} = P_{\text{total}} - P_{\text{unknown}} = 1 \, \text{atm} - 0.68 \, \text{atm} = 0.32 \, \text{atm}
\]
### Step 2: Use the Ideal Gas Law to find the volume of helium
The Ideal Gas Law is given by the equation:
\[
PV = nRT
\]
We can rearrange this to find the volume \( V \):
\[
V = \frac{nRT}{P}
\]
For helium, we have:
- \( n = 0.1 \, \text{moles} \)
- \( R = 0.082 \, \text{L atm/(mol K)} \)
- \( T = 0^\circ C = 273 \, \text{K} \)
- \( P = 0.32 \, \text{atm} \)
Substituting these values into the equation:
\[
V_{\text{He}} = \frac{0.1 \, \text{mol} \times 0.082 \, \text{L atm/(mol K)} \times 273 \, \text{K}}{0.32 \, \text{atm}}
\]
Calculating this gives:
\[
V_{\text{He}} = \frac{2.2436}{0.32} \approx 7.015625 \, \text{L} \approx 7 \, \text{L}
\]
### Step 3: Use the Ideal Gas Law to find the volume of the unknown compound
Now, we will calculate the volume of the unknown compound. The pressure of the unknown compound is 0.68 atm.
Using the same Ideal Gas Law:
\[
V_{\text{unknown}} = \frac{nRT}{P}
\]
For the unknown compound, we have:
- \( n = 1.0 \, \text{mole} \)
- \( R = 0.082 \, \text{L atm/(mol K)} \)
- \( T = 273 \, \text{K} \)
- \( P = 0.68 \, \text{atm} \)
Substituting these values into the equation:
\[
V_{\text{unknown}} = \frac{1.0 \, \text{mol} \times 0.082 \, \text{L atm/(mol K)} \times 273 \, \text{K}}{0.68 \, \text{atm}}
\]
Calculating this gives:
\[
V_{\text{unknown}} = \frac{22.414}{0.68} \approx 32.94 \, \text{L}
\]
### Step 4: Calculate the total volume
Now, we can find the total volume by adding the volumes of helium and the unknown compound:
\[
V_{\text{total}} = V_{\text{He}} + V_{\text{unknown}} \approx 7 \, \text{L} + 32.94 \, \text{L} \approx 39.94 \, \text{L}
\]
Thus, the total volume of the gases at \( 0^\circ C \) is approximately **40 L**.
### Final Answer
The total volume (in litre) of the gases at \( 0^\circ C \) is close to **40 L**.
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