To find the pressure in the vessel containing a mixture of gases, we can use the ideal gas law equation:
\[ PV = nRT \]
Where:
- \( P \) = pressure (in N/m²)
- \( V \) = volume (in m³)
- \( n \) = total number of moles of gas
- \( R \) = universal gas constant (8.31 J/(mol·K))
- \( T \) = absolute temperature (in Kelvin)
### Step 1: Convert the volume from liters to cubic meters
The volume of the vessel is given as 3 liters. To convert this to cubic meters:
\[ V = 3 \, \text{L} = 3 \times 10^{-3} \, \text{m}^3 \]
### Step 2: Calculate the number of moles of each gas
We will calculate the number of moles for each gas using the formula:
\[ n = \frac{\text{mass}}{\text{molar mass}} \]
1. **For Oxygen (O₂)**:
- Mass = 6 g
- Molar mass = 32 g/mol
\[
n_{O_2} = \frac{6 \, \text{g}}{32 \, \text{g/mol}} = \frac{6}{32} = 0.1875 \, \text{mol}
\]
2. **For Nitrogen (N₂)**:
- Mass = 8 g
- Molar mass = 28 g/mol
\[
n_{N_2} = \frac{8 \, \text{g}}{28 \, \text{g/mol}} = \frac{8}{28} = 0.2857 \, \text{mol}
\]
3. **For Carbon Dioxide (CO₂)**:
- Mass = 5 g
- Molar mass = 44 g/mol
\[
n_{CO_2} = \frac{5 \, \text{g}}{44 \, \text{g/mol}} = \frac{5}{44} = 0.1136 \, \text{mol}
\]
### Step 3: Calculate the total number of moles
Now, we can find the total number of moles \( n \):
\[
n = n_{O_2} + n_{N_2} + n_{CO_2}
\]
\[
n = 0.1875 + 0.2857 + 0.1136 = 0.5868 \, \text{mol}
\]
### Step 4: Convert the temperature to Kelvin
The temperature is given as 27°C. To convert this to Kelvin:
\[
T = 27 + 273 = 300 \, \text{K}
\]
### Step 5: Substitute values into the ideal gas law equation
Now we can substitute the values into the ideal gas law to find the pressure \( P \):
\[
P = \frac{nRT}{V}
\]
Substituting the known values:
\[
P = \frac{(0.5868 \, \text{mol}) \times (8.31 \, \text{J/(mol \cdot K)}) \times (300 \, \text{K})}{3 \times 10^{-3} \, \text{m}^3}
\]
Calculating the numerator:
\[
P = \frac{(0.5868) \times (8.31) \times (300)}{3 \times 10^{-3}}
\]
\[
= \frac{1458.036}{3 \times 10^{-3}} = 486012 \, \text{N/m}^2
\]
### Step 6: Approximate the pressure
The calculated pressure is approximately:
\[
P \approx 4.86 \times 10^5 \, \text{N/m}^2
\]
### Final Answer
Thus, the pressure in the vessel is approximately \( 5 \times 10^5 \, \text{N/m}^2 \).
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