Question

Given the equilibrium concentrations for each gas at \(850^{\circ} \mathrm{C}\), calculate the value of \(K_{\text {eq }}\) for the manufacture of sulfur trioxide. $$ \begin{array}{ccc} 2 \mathrm{SO}_{2}(g)+\mathrm{O}_{2}(g) & \rightleftarrows & 2 \mathrm{SO}_{3}(g) \\ 1.75 \mathrm{M} & 1.50 \mathrm{M} & 2.25 \mathrm{M} \end{array} $$

Short answer

The equilibrium constant \(K_{eq}\) is approximately 1.1025.

Step-by-step solution

Identify the Reaction and Its Components

Examine the chemical equation: \(2 \text{SO}_2 (g) + \text{O}_2 (g) \rightleftharpoons 2 \text{SO}_3 (g)\). Note the stoichiometry: 2 moles of \(\text{SO}_2\), 1 mole of \(\text{O}_2\), and 2 moles of \(\text{SO}_3\).

Write the Expression for the Equilibrium Constant

The equilibrium constant \(K_{eq}\) is expressed as: \[K_{eq} = \frac{[\text{SO}_3]^2}{[\text{SO}_2]^2 [\text{O}_2]}\] where \([\text{SO}_2]\), \([\text{O}_2]\), and \([\text{SO}_3]\) are the molar concentrations of \(\text{SO}_2\), \(\text{O}_2\), and \(\text{SO}_3\) at equilibrium.

Substitute Equilibrium Concentrations into the Expression

Insert the given equilibrium concentrations into the equilibrium expression: \[K_{eq} = \frac{(2.25)^2}{(1.75)^2 \times (1.50)}\] which corresponds to the concentrations of \(\text{SO}_3=2.25\text{ M}\), \(\text{SO}_2=1.75\text{ M}\), and \(\text{O}_2=1.50\text{ M}\).

Simplify and Calculate \(K_{eq}\)

Carry out the calculations: \[K_{eq} = \frac{5.0625}{3.0625 \times 1.50}\] This becomes \[K_{eq} = \frac{5.0625}{4.59375}\] which simplifies to approximately \(1.1025\).

Key concepts

Chemical Equilibrium

Chemical equilibrium is a fascinating concept in chemistry and occurs when a chemical reaction reaches a state where the concentration of reactants and products remains unchanged over time. In other words, the rate of the forward reaction equals the rate of the reverse reaction. This doesn't mean that the reactants and products are in equal concentrations; rather, their concentrations remain constant over time.

Understanding chemical equilibrium is essential because it allows chemists to predict how a system reacts to different conditions, such as changes in temperature, pressure, or concentration. An equilibrium position is part of this and can shift when the conditions change, following Le Chatelier's Principle. When you're exploring reactions like the formation of sulfur trioxide (\(\text{SO}_3\)), you'll witness this dynamic balance.

It's important to carry experiments in controlled environments to maintain equilibrium, especially in industrial settings where the production of specific compounds like \(\text{SO}_3\) is needed consistently. By understanding chemical equilibrium, scientists can optimize reactions for better yields and more efficient processes.

Reaction Stoichiometry

Reaction stoichiometry is the quantitative relationship between the amounts of reactants and products in a chemical reaction. When dealing with equilibrium reactions like the conversion of \(\text{SO}_2\) and \(\text{O}_2\) to \(\text{SO}_3\), understanding stoichiometry ensures precise calculations. This is based on the balanced chemical equation:

• 2 moles of \(\text{SO}_2\)
• 1 mole of \(\text{O}_2\)
• 2 moles of \(\text{SO}_3\)

The coefficients of the balanced equation indicate the molar ratio of the reactants and products.

In practical terms, stoichiometry allows chemists to determine the amount of one substance needed to react with a given amount of another substance. It also helps calculate the amounts of products formed in a reaction. This is crucial for both laboratory experiments and industrial chemical processes, where precise measurements and proportions are key to success.

By understanding these relationships, chemists maximize efficiency and reduce waste, in both small-scale and large-scale chemical production.

Equilibrium Calculations

Equilibrium calculations are a mathematical evaluation of the concentrations in a reaction at equilibrium. For this, chemists use an equilibrium constant, \(K_{eq}\), which provides a numerical value representing the ratio of concentrations of products to reactants at equilibrium for a given reaction.

For example, in calculating \(K_{eq}\) for the formation of sulfur trioxide, you start with the expression:
\[K_{eq} = \frac{[\text{SO}_3]^2}{[\text{SO}_2]^2 [\text{O}_2]}\] This formula results from the stoichiometry of the balanced chemical equation.

Inserting known equilibrium concentrations, like \([\text{SO}_3] = 2.25\) M, \([\text{SO}_2] = 1.75\) M, and \([\text{O}_2] = 1.50\) M, allows for accurate calculation. You perform arithmetic operations to solve for \(K_{eq}\).

These calculations help predict how a system responds to changes and determine how far the reaction proceeds, which is imperative when optimizing conditions for desired outcomes in chemical production processes. Understanding equilibrium and correctly performing these calculations is an integral skill in advancing chemical research and industry applications.