Question

Equilibrium is established in the reaction \(2 \mathrm{SO}_{2}(\mathrm{g})+\) \(\mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{3}(\mathrm{g}) \quad\) at \(\quad \mathrm{a} \quad\) temperature \(\quad\) where \(K_{\mathrm{c}}=100 .\) If the number of moles of \(\mathrm{SO}_{3}(\mathrm{g})\) in the equilibrium mixture is the same as the number of moles of \(\mathrm{SO}_{2}(\mathrm{g}),\) (a) the number of moles of \(\mathrm{O}_{2}(\mathrm{g})\) is also equal to the number of moles of \(\mathrm{SO}_{2}(\mathrm{g}) ;\) (b) the number of moles of \(\mathrm{O}_{2}(\mathrm{g})\) is half the number of moles of \(\mathrm{SO}_{2} ;\) (c) \(\left[\mathrm{O}_{2}\right]\) may have any of several values; (d) \(\left[\mathrm{O}_{2}\right]=0.010 \mathrm{M}\)

Short answer

The answers that match with \(K_c = 100\) are (c) \([O_2]\) may have any of several values and (d) \([O_2] = 0.010 M\).

Step-by-step solution

Exploring the known facts

We know that SO3 is produced from the reaction of SO2 and O2, with the reactant and product concentrations expressed through the equilibrium constant \(K_c = 100\). Also, we know the moles of SO2 and SO3 are equal.

Understanding the equilibrium constant

The equilibrium constant, Kc, is defined as the ratio of the concentrations of products to the reactants. For the reaction \(2SO_2 + O_2 \rightleftharpoons 2SO_3\), Kc would be represented as: \(K_c = [SO_3]^2 / ([SO_2]^2*[O_2])\). Because we know that the number of moles of \(SO_3\) is equal to the number of moles of \(SO_2\), we can simplify this formula to: \(K_c = 1 / [O_2]\)

Evaluating the options

With our simplified equation, we can check each of the suggested answers: (a) If the number of moles of \(O_2\) is equal to the number of moles of \(SO_2\), then \(K_c = 1\). This does not match the known \(K_c = 100\). (b) If the number of moles of \(O_2\) is half the number of moles of \(SO_2\), then \(K_c = 4\). This does not match the known \(K_c = 100\). (c) If \([O_2]\) has any value, this implies the possibility that \(K_c = 100\), which matches the known \(K_c = 100\). (d) If \([O_2] = 0.010 M\), then \(K_c = 1 / 0.010 = 100\), which matches the known \(K_c = 100\)

Key concepts

Equilibrium Constant

The equilibrium constant, denoted as \( K_c \), is a critical component in understanding chemical equilibrium. It provides a quantitative measure of the concentrations of reactants and products in a reaction at equilibrium. For a balanced chemical reaction, the expression for \( K_c \) depends on the stoichiometry of the reaction.
For the given reaction \(2 \text{SO}_2 (g) + \text{O}_2 (g) \rightleftharpoons 2 \text{SO}_3 (g)\), the equilibrium constant is expressed as:\[K_c = \frac{[\text{SO}_3]^2}{[\text{SO}_2]^2[\text{O}_2]}\]This equation means \( K_c \) is calculated by taking the concentration of products raised to the power of their respective coefficients, divided by the concentration of reactants raised to the power of their coefficients. In this example, if the concentrations of \( \text{SO}_3 \) and \( \text{SO}_2 \) are equal, the expression simplifies, helping us evaluate the options given in the problem.
Understanding \( K_c \) is essential, as it informs us whether the reactants or products are favored in the equilibrium and the extent of the reaction at a specific temperature.

Reaction Stoichiometry

Reaction stoichiometry refers to the relationship between the quantities of reactants and products in a chemical reaction. It is determined by the balanced chemical equation, which indicates the proportions required for the reaction to occur.
In the reaction \(2 \text{SO}_2 (g) + \text{O}_2 (g) \rightleftharpoons 2 \text{SO}_3 (g)\), the stoichiometry implies that two moles of \( \text{SO}_2 \) and one mole of \( \text{O}_2 \) are needed to produce two moles of \( \text{SO}_3 \). This ratio is crucial when calculating how changes in amounts or concentrations of reactants or products affect the reaction.
The problem states that the moles of \( \text{SO}_3 \) and \( \text{SO}_2 \) are equal, indicating equal concentrations at equilibrium. Using stoichiometry, we analyze how these equal concentrations impact the concentration of \( \text{O}_2 \). When working through such problems, always ensure the balanced equation guides the calculations and understanding of reactant-product relationships.

Le Chatelier's Principle

Le Chatelier's Principle is a concept used to predict the effect of a change in conditions on chemical equilibria. Simply put, it suggests that if an external change is applied to a system at equilibrium, the system adjusts to counteract that change and re-establish equilibrium.
For example, consider our reaction \(2 \text{SO}_2 (g) + \text{O}_2 (g) \rightleftharpoons 2 \text{SO}_3 (g)\). If the concentration of \( \text{O}_2 \) changes, the system will shift the equilibrium position to balance itself. Increasing \( \text{O}_2 \) would favor the forward reaction, producing more \( \text{SO}_3 \), while decreasing \( \text{O}_2 \) would favor the reverse reaction, forming more \( \text{SO}_2 \) and \( \text{O}_2 \).
This principle helps chemists control reactions to optimize desired product formation. By manipulating conditions such as pressure, temperature, and concentration, chemists can push reactions in favorable directions to achieve higher yields. Understanding Le Chatelier's Principle is key to mastering dynamic systems and equilibrium adjustments.