To solve this problem, we need to analyze the chemical reaction between sulfur dioxide (SO2) and oxygen (O2) and how the pressure changes as the reaction proceeds. The reaction can be represented as follows:
Understanding the Reaction
The balanced chemical equation for the reaction between sulfur dioxide and oxygen to form sulfur trioxide (SO3) is:
- 2 SO2(g) + O2(g) → 2 SO3(g)
From this equation, we can see that 2 moles of SO2 react with 1 mole of O2 to produce 2 moles of SO3. This stoichiometry will be crucial in determining the changes in pressure as the reaction proceeds.
Initial Conditions
Initially, we have:
- 8 moles of SO2
- 4 moles of O2
The total number of moles before the reaction starts is:
Total moles = 8 moles SO2 + 4 moles O2 = 12 moles
The initial pressure of the mixture is given as 2.96 atm. We can use the ideal gas law, which states that pressure is directly proportional to the number of moles when temperature and volume are constant.
Calculating Moles Reacted
Next, we need to determine how many moles of SO2 react. Since 80% of the initial moles of SO2 react:
Moles of SO2 reacted = 80% of 8 moles = 0.80 × 8 = 6.4 moles
Now, according to the stoichiometry of the reaction, for every 2 moles of SO2 that react, 1 mole of O2 is consumed. Therefore, the moles of O2 that react can be calculated as follows:
Moles of O2 reacted = (6.4 moles SO2) × (1 mole O2 / 2 moles SO2) = 3.2 moles O2
Remaining Moles After Reaction
After the reaction, we can determine the remaining moles of SO2 and O2:
- Remaining SO2 = 8 moles - 6.4 moles = 1.6 moles
- Remaining O2 = 4 moles - 3.2 moles = 0.8 moles
Now, we also need to account for the moles of SO3 produced. According to the reaction, the moles of SO3 produced will be equal to the moles of SO2 that reacted:
Moles of SO3 produced = 6.4 moles
Total Moles After Reaction
The total number of moles in the vessel after the reaction is:
Total moles = Remaining SO2 + Remaining O2 + SO3 produced
Total moles = 1.6 moles + 0.8 moles + 6.4 moles = 8.8 moles
Calculating Final Pressure
Since the temperature and volume of the vessel remain constant, we can use the initial pressure and the initial total moles to find the final pressure:
Using the ideal gas law, we know that:
P1 / n1 = P2 / n2
Where:
- P1 = initial pressure = 2.96 atm
- n1 = initial total moles = 12 moles
- P2 = final pressure (what we want to find)
- n2 = final total moles = 8.8 moles
Rearranging the equation to solve for P2 gives us:
P2 = P1 × (n2 / n1)
Substituting the values:
P2 = 2.96 atm × (8.8 moles / 12 moles)
P2 = 2.96 atm × 0.7333 = 2.17 atm
Final Result
The pressure of the mixture after 80% of the initial moles of SO2 have reacted is approximately 2.17 atm.
