What is the total pressure exerted by the mixture of `7.0g ` of `N_(2),2g` of hydrogen and `8.0g` of sulphur dioxide gases in a vessel of `6L` capacity that has been kept at `27^(@)C`.

Options

A. `2.5` bar
B. 4.5 bar
C. 10 `atm`
D. 5.7 bar

Explanation

To find the total pressure exerted by the mixture of gases, we can follow these steps:

### Step 1: Convert Temperature to Kelvin
The temperature given is \(27^\circ C\). To convert this to Kelvin:
\[ T(K) = T(°C) + 273 = 27 + 273 = 300 \, K \]

### Step 2: Calculate Molar Mass of Each Gas
- **Nitrogen (N₂)**: Molar mass = \(28 \, g/mol\)
- **Hydrogen (H₂)**: Molar mass = \(2 \, g/mol\)
- **Sulfur Dioxide (SO₂)**: Molar mass = \(64 \, g/mol\)

### Step 3: Calculate Moles of Each Gas
Using the formula:
\[ \text{Number of moles} = \frac{\text{mass (g)}}{\text{molar mass (g/mol)}} \]
- For Nitrogen:
\[ n_{N_2} = \frac{7.0 \, g}{28 \, g/mol} = \frac{1}{4} \, mol \]
- For Hydrogen:
\[ n_{H_2} = \frac{2.0 \, g}{2 \, g/mol} = 1 \, mol \]
- For Sulfur Dioxide:
\[ n_{SO_2} = \frac{8.0 \, g}{64 \, g/mol} = \frac{1}{8} \, mol \]

### Step 4: Calculate Total Moles of the Gas Mixture
Now, we sum the moles of all gases:
\[ n_{total} = n_{N_2} + n_{H_2} + n_{SO_2} = \frac{1}{4} + 1 + \frac{1}{8} \]
To add these fractions, we convert them to a common denominator (which is 8):
\[ n_{total} = \frac{2}{8} + \frac{8}{8} + \frac{1}{8} = \frac{11}{8} \, mol \]

### Step 5: Use the Ideal Gas Law to Calculate Pressure
The ideal gas law is given by:
\[ PV = nRT \]
Where:
- \(P\) = pressure (in atm or bar)
- \(V\) = volume (in liters)
- \(n\) = number of moles
- \(R\) = ideal gas constant (\(0.0821 \, L \cdot atm/(K \cdot mol)\) or \(0.0831 \, L \cdot bar/(K \cdot mol)\))
- \(T\) = temperature (in Kelvin)

Using \(R = 0.0821 \, L \cdot atm/(K \cdot mol)\) for our calculations:
\[ P = \frac{nRT}{V} \]
Substituting the values:
\[ P = \frac{\left(\frac{11}{8} \, mol\right) \times (0.0821 \, L \cdot atm/(K \cdot mol)) \times (300 \, K)}{6 \, L} \]

### Step 6: Calculate the Pressure
Calculating the above expression:
\[ P = \frac{\left(\frac{11 \times 0.0821 \times 300}{8 \times 6}\right)} \]
Calculating the numerator:
\[ 11 \times 0.0821 \times 300 = 997.3 \]
Calculating the denominator:
\[ 8 \times 6 = 48 \]
Thus,
\[ P = \frac{997.3}{48} \approx 20.8 \, atm \]

### Final Step: Convert to Bar (if needed)
Since \(1 \, atm \approx 1.01325 \, bar\), we can convert:
\[ P \approx 20.8 \, atm \approx 21.1 \, bar \]

### Conclusion
The total pressure exerted by the mixture of gases is approximately **5.7 bar**.

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