A 5.0 L flask contains 19.5 g of `SO_(3)` and 1.0 g of He gas at `20^(@)C`. Calculate the partial pressures exerted by `SO_(3)` and He and the total pressure of the gaseous mixture.

Explanation

To solve the problem, we need to calculate the partial pressures of \( SO_3 \) and He, and then find the total pressure of the gas mixture. We will 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 (\( 0.0821 \, \text{L} \cdot \text{atm} / \text{K} \cdot \text{mol} \))
- \( T \) = temperature (in Kelvin)

### Step 1: Convert the temperature to Kelvin
Given that the temperature is \( 20^\circ C \):

\[ T = 20 + 273 = 293 \, K \]

### Step 2: Calculate the number of moles of \( SO_3 \)
The molar mass of \( SO_3 \) can be calculated as follows:
- Sulfur (S) = 32 g/mol
- Oxygen (O) = 16 g/mol, and there are 3 oxygen atoms in \( SO_3 \)

\[ \text{Molar mass of } SO_3 = 32 + (3 \times 16) = 32 + 48 = 80 \, g/mol \]

Now, using the mass of \( SO_3 \) given (19.5 g), we can calculate the number of moles (\( n \)):

\[ n_{SO_3} = \frac{\text{mass}}{\text{molar mass}} = \frac{19.5 \, g}{80 \, g/mol} = 0.24375 \, mol \approx 0.244 \, mol \]

### Step 3: Calculate the number of moles of He
The molar mass of He is 4 g/mol. Using the mass of He given (1.0 g):

\[ n_{He} = \frac{1.0 \, g}{4 \, g/mol} = 0.25 \, mol \]

### Step 4: Calculate the partial pressure of \( SO_3 \)
Using the ideal gas law, we can find the partial pressure of \( SO_3 \):

\[ P_{SO_3} = \frac{n_{SO_3}RT}{V} \]

Substituting the values:

\[ P_{SO_3} = \frac{0.244 \, mol \times 0.0821 \, \frac{L \cdot atm}{K \cdot mol} \times 293 \, K}{5.0 \, L} \]

Calculating this:

\[ P_{SO_3} = \frac{0.244 \times 0.0821 \times 293}{5.0} \approx 1.19 \, atm \]

### Step 5: Calculate the partial pressure of He
Now, we can calculate the partial pressure of He:

\[ P_{He} = \frac{n_{He}RT}{V} \]

Substituting the values:

\[ P_{He} = \frac{0.25 \, mol \times 0.0821 \, \frac{L \cdot atm}{K \cdot mol} \times 293 \, K}{5.0 \, L} \]

Calculating this:

\[ P_{He} = \frac{0.25 \times 0.0821 \times 293}{5.0} \approx 1.21 \, atm \]

### Step 6: Calculate the total pressure of the gas mixture
The total pressure \( P_{total} \) is the sum of the partial pressures:

\[ P_{total} = P_{SO_3} + P_{He} \]

Substituting the values we calculated:

\[ P_{total} = 1.19 \, atm + 1.21 \, atm = 2.40 \, atm \]

### Final Answers:
- Partial pressure of \( SO_3 \): \( 1.19 \, atm \)
- Partial pressure of He: \( 1.21 \, atm \)
- Total pressure of the gas mixture: \( 2.40 \, atm \)