What will be the pressure exerted by a mixture of `3.2 g` of methane and `4.4g` of carbon dixide contained in a `9 dm^(3)` flask at `27^(@)C` ? .

Options

A. 0 . 82 atm
B. `8 314 xx 10^(4)` atm
C. 1 atm
D. 1.8 atm

Explanation

To find the pressure exerted by a mixture of methane and carbon dioxide in a flask, we can use the ideal gas law, which is given by the equation:

\[ PV = nRT \]

Where:
- \( P \) = pressure (in atm)
- \( V \) = volume (in liters)
- \( n \) = number of moles of gas
- \( R \) = ideal gas constant \( (0.0821 \, \text{atm} \cdot \text{L} / \text{mol} \cdot \text{K}) \)
- \( T \) = temperature (in Kelvin)

### Step 1: Convert the temperature to Kelvin
The temperature is given as \( 27^\circ C \). To convert Celsius to Kelvin, we use the formula:

\[ T(K) = T(°C) + 273 \]

So,

\[ T = 27 + 273 = 300 \, K \]

### Step 2: Convert the volume to liters
The volume is given as \( 9 \, dm^3 \). Since \( 1 \, dm^3 = 1 \, L \), we have:

\[ V = 9 \, L \]

### Step 3: Calculate the number of moles of methane (CH₄)
The molar mass of methane (CH₄) is calculated as follows:

- Carbon (C) = 12 g/mol
- Hydrogen (H) = 1 g/mol × 4 = 4 g/mol

Thus, the molar mass of CH₄ is:

\[ 12 + 4 = 16 \, g/mol \]

Now, using the formula for moles:

\[ n = \frac{\text{mass}}{\text{molar mass}} \]

For methane:

\[ n_{CH₄} = \frac{3.2 \, g}{16 \, g/mol} = 0.2 \, mol \]

### Step 4: Calculate the number of moles of carbon dioxide (CO₂)
The molar mass of carbon dioxide (CO₂) is calculated as follows:

- Carbon (C) = 12 g/mol
- Oxygen (O) = 16 g/mol × 2 = 32 g/mol

Thus, the molar mass of CO₂ is:

\[ 12 + 32 = 44 \, g/mol \]

Now, using the formula for moles:

\[ n = \frac{\text{mass}}{\text{molar mass}} \]

For carbon dioxide:

\[ n_{CO₂} = \frac{4.4 \, g}{44 \, g/mol} = 0.1 \, mol \]

### Step 5: Calculate the total number of moles
The total number of moles in the mixture is the sum of the moles of methane and carbon dioxide:

\[ n_{total} = n_{CH₄} + n_{CO₂} = 0.2 \, mol + 0.1 \, mol = 0.3 \, mol \]

### Step 6: Use the ideal gas law to find the pressure
Now, we can substitute the values into the ideal gas law equation \( PV = nRT \):

\[ P = \frac{nRT}{V} \]

Substituting the known values:

- \( n = 0.3 \, mol \)
- \( R = 0.0821 \, \text{atm} \cdot \text{L} / \text{mol} \cdot \text{K} \)
- \( T = 300 \, K \)
- \( V = 9 \, L \)

\[ P = \frac{(0.3 \, mol)(0.0821 \, \text{atm} \cdot \text{L} / \text{mol} \cdot \text{K})(300 \, K)}{9 \, L} \]

Calculating this gives:

\[ P = \frac{(0.3)(0.0821)(300)}{9} = \frac{7.389}{9} \approx 0.821 \, atm \]

### Final Answer
The pressure exerted by the gaseous mixture is approximately \( 0.821 \, atm \).

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