You have an equimolar mixture of the gases \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2},\) along with some \(\mathrm{He}\), in a container fitted with a piston. The density of this mixture at STP is 1.924 \(\mathrm{g} / \mathrm{L}\) . Assume ideal behavior and constant temperature and pressure. a. What is the mole fraction of He in the original mixture? b. The \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\) react to completion to form \(\mathrm{SO}_{3} .\) What is the density of the gas mixture after the reaction is complete?

Recall that density is mass per unit volume. For an ideal gas mixture, we can use the formula: \[Density = \frac{M * P}{R * T}\] Where \(Density\) is given by the problem (1.924 g/L), \(M\) is the molar mass of the gas mixture, \(P\) is the pressure (STP) of 1 atm, \(R\) is the ideal gas constant (0.0821 L * atm/(mol * K)), and \(T\) is the temperature (STP) of 273 K. Plug in the relevant values: \[1.924 \frac{g}{L} = \frac{M * 1 \ atm}{0.0821 \frac{L \cdot atm}{mol \cdot K} * 273 \ K}\] Now, solve for the molar mass of the original gas mixture. #Step 2: Write the equimolar reaction between SO₂ and O₂#

The reaction between SO₂ and O₂ is given by the balanced chemical equation: \[SO_{2} + \frac{1}{2} O_{2} \rightarrow SO_{3}\] Because the mixture is equimolar, for every mole of SO₂, there is half a mole of O₂. #Step 3: Calculate the mole fraction of He#

To find the mole fraction of He, we first need the molar masses of all three gases in the mixture: - SO₂: 32 (S) + 2 * 16 (O) = 64 g/mol - O₂: 2 * 16 (O) = 32 g/mol - He: 4 g/mol Define the mole fractions of SO₂, O₂, and He using 1 mol in total: \[x_{SO_{2}} = \frac{64}{M}\] \[x_{O_{2}} = \frac{32}{M}\] \[x_{He} = \frac{4}{M}\] We are looking for the molar mass of the original gas mixture. From step 1, we can calculate \(M\): \[M = \frac{1.924 \times (0.0821) * 273}{1} \approx 43.3 \ g/mol\] Now calculate the mole fraction of He: \[x_{He} = \frac{4}{43.3} \approx 0.0924\] This means there are approximately 0.0924 moles of He in every mole of the mixture. #Step 4: Find the density of the gas mixture after the reaction is complete#

After the reaction between SO₂ and O₂ is complete, the remaining mixture in the container is made of SO₃ and He. Calculate the molar mass of SO₃: - SO₃: 32 (S) + 3 * 16 (O) = 80 g/mol Now, calculate the molar mass of the new mixture considering the mole fraction of He (0.0924): \[Mnew = (1 - 0.0924) * 80 + 0.0924 * 4 = 75.04 \ g/mol\]

Now we can use this molar mass to find the density of the new mix containing only He and SO₃ at STP, using the density formula: \[Density = \frac{Mnew * P}{R * T} = \frac{75.04 \times 1}{0.0821 * 273} = 3.14 \frac{g}{L}\] The density of the gas mixture after the reaction between SO₂ and O₂ is complete and SO₃ is formed is 3.14 g/L.