Question

Equal moles of sulfur dioxide gas and oxygen gas are mixed in a flexible reaction vessel and then sparked to initiate the formation of gaseous sulfur trioxide. Assuming that the reaction goes to completion, what is the ratio of the final volume of the gas mixture to the initial volume of the gas mixture if both volumes are measured at the same temperature and pressure?

Short answer

The ratio of the final volume of the gas mixture to the initial volume of the gas mixture is \(\frac{5}{4}\), or 1.25 when both volumes are measured at the same temperature and pressure.

Step-by-step solution

(1) Write down the balanced chemical equation for the reaction

The balanced chemical equation for the reaction between sulfur dioxide and oxygen to form sulfur trioxide is: 2SO2 (g) + O2 (g) -> 2SO3 (g)

(2) Determine the number of moles of reactants and products in the initial state

We are given that the reaction mixture initially contains equal moles of sulfur dioxide and oxygen. Let us assume that the number of moles of each reactant is 'n'. n moles of SO2 and n moles of O2 are present at the beginning.

(3) Determine the number of moles of reactants and products in the final state

The reaction goes to completion, which means all available reactants are consumed entirely, and only products are left. According to the balanced chemical equation, 2 moles of SO2 react with 1 mole of O2 to form 2 moles of SO3. Therefore, after the reaction is complete, SO2: 0 moles O2: n - n/2 = n/2 moles (This is because half of the oxygen moles react with the sulfur dioxide) SO3: 2n moles (Since 2 moles of SO3 are formed from 2 moles of SO2 reacting with 1 mole of O2)

(4) Determine the initial and final volumes of the gas mixture using Avogadro's law

According to Avogadro's law, the volume of a gas is directly proportional to the number of moles present (at constant temperature and pressure). Let V_initial be the initial volume of the gas mixture and V_final be the final volume. We can write these expressions for the initial and final volumes: V_initial = (n + n) * V (Initial volume contains equal moles of SO2 and O2) V_final = (n/2 + 2n) * V (Final volume contains n/2 moles of O2 and 2n moles of SO3)

(5) Calculate the ratio of the final volume to the initial volume

To find the ratio of the final volume to the initial volume, we need to divide V_final by V_initial: Ratio = V_final / V_initial = [(n/2 + 2n) * V] / [(n + n) * V] We can simplify the ratio expression as follows: Ratio = (n/2 + 2n) / (2n) = (n/2 + 2n)/(2n) = (n + 4n) / (4n) = 5n / 4n = 5/4

(6) Final answer

The ratio of the final volume of the gas mixture to the initial volume of the gas mixture is 5/4, or 1.25 when both volumes are measured at the same temperature and pressure.