Question

4 moles each of \(\mathrm{SO}_{2}\) and \(\mathrm{O}_{2}\) gases are allowed to react to form \(\mathrm{SO}_{3}\) in a closed vessel. At equilibrium \(25 \%\) of \(\mathrm{O}_{2}\) is used up. The total number of moles of all the gases at equilibrium is: (a) \(6.5\) (b) \(7.0\) (c) \(8.0\) (d) \(2.0\)

Short answer

The total number of moles at equilibrium is 7.

Step-by-step solution

Understand the Reaction

The chemical reaction involved is \[ 2\text{SO}_2 (g) + \text{O}_2 (g) \rightleftharpoons 2\text{SO}_3 (g) \].In this reaction, two moles of sulfur dioxide gas react with one mole of oxygen gas to produce two moles of sulfur trioxide gas.

Calculate Initial Moles

Initially, there are 4 moles of \( \text{SO}_2 \) and 4 moles of \( \text{O}_2 \).

Determine Moles of Oxygen Used

At equilibrium, 25% of the initial \( \text{O}_2 \) is used. This is calculated as:\[ \text{moles of } \text{O}_2 \text{ used} = 0.25 \times 4 = 1 \text{ mole} \].

Use Stoichiometry to Find Moles of SO3 Formed

Since the reaction consumes \( 1 \text{ mole} \) of \( \text{O}_2 \), 2 moles of \( \text{SO}_2 \) are used and produce 2 moles of \( \text{SO}_3 \) according to the stoichiometry of the reaction: \[ 2\text{SO}_2 (g) + \text{O}_2 (g) \rightarrow 2\text{SO}_3 (g) \].Therefore, the moles of \( \text{SO}_3 \) formed is 2.

Calculate Moles at Equilibrium

- Moles of \( \text{SO}_2 \) left = initial moles - moles reacted = 4 - 2 = 2 moles.- Moles of \( \text{O}_2 \) left = initial moles - moles reacted = 4 - 1 = 3 moles.- Moles of \( \text{SO}_3 \) = 2 moles (formed).

Compute Total Moles at Equilibrium

The total moles of gases at equilibrium is the sum of moles of \( \text{SO}_2 \), \( \text{O}_2 \), and \( \text{SO}_3 \):\[ 2 + 3 + 2 = 7 \text{ moles} \].

Key concepts

Stoichiometry

Stoichiometry is a crucial concept in chemistry that helps understand the quantitative relationships within a chemical reaction. It involves using balanced chemical equations to calculate the amounts of reactants and products. In the reaction given by the exercise, we have \( 2\text{SO}_2 (g) + \text{O}_2 (g) \rightarrow 2\text{SO}_3 (g) \). This equation tells us that:

• Two moles of sulfur dioxide \( \text{SO}_2 \) react with one mole of oxygen \( \text{O}_2 \).
• Two moles of sulfur trioxide \( \text{SO}_3 \) are formed as products.

Given this reaction, if \( 1 \text{ mole} \) of \( \text{O}_2 \) is consumed, it directly implies that \( 2 \text{ moles} \) of \( \text{SO}_2 \) are also consumed, producing \( 2 \text{ moles} \) of \( \text{SO}_3 \). This relationship is the essence of stoichiometry, illustrating how moles of different substances relate to each other in a reaction.

Reaction Mechanisms

Understanding reaction mechanisms is important in predicting how a chemical reaction proceeds, often depicted by a series of steps.These steps provide insight into the transformation of reactants to products. The given reaction \( 2\text{SO}_2 (g) + \text{O}_2 (g) \rightarrow 2\text{SO}_3 (g) \) can be considered a single-step process based on stoichiometry, but the underlying mechanism might involve multiple sub-steps.

• In the context of this exercise, knowing that the reaction involves a crucial step where \( \text{O}_2 \) reacts with \( \text{SO}_2 \) is key.
• Each step in a mechanism involves breaking and forming specific bonds, resulting in intermediates that eventually lead to the products.

The detailed understanding of such mechanisms helps in controlling reaction rates and conditions to optimize product formation.

Equilibrium Calculations

Chemical equilibrium refers to the state in which the concentrations of reactants and products remain constant over time, although the reaction continues to occur in both directions. In this exercise:

• The reaction initially starts with 4 moles of both \( \text{SO}_2 \) and \( \text{O}_2 \).
• Equilibrium is reached when 25% of \( \text{O}_2 \) is used, meaning \( 1 \text{ mole} \) is consumed.
• The equilibrium position was calculated by determining the moles of substances left and formed based on stoichiometry.

To find the equilibrium state, we calculated:- Moles of \( \text{SO}_2 \) left: \( 4 - 2 = 2 \text{ moles} \)- Moles of \( \text{O}_2 \) left: \( 4 - 1 = 3 \text{ moles} \)- Moles of \( \text{SO}_3 \) produced: \( 2 \text{ moles} \)The total number of moles at equilibrium amounts to 7, reflecting how both reactants and products stabilize in a closed system.