Problem 21 - Chemistry and Chemical Reactivity

Question

Consider the following equilibria involving \(\mathrm{SO}_{2}(\mathrm{g})\) and their corresponding equilibrium constants. \(\mathrm{SO}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightleftarrows \mathrm{SO}_{3}(\mathrm{g}) \quad K_{1}\) \(2 \mathrm{SO}_{3}(\mathrm{g}) \rightleftarrows 2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \quad K_{2}\) Which of the following expressions relates \(K_{1}\) to \(K_{2} ?\) (a) \(K_{2}=K_{1}^{2}\) (d) \(K_{2}=1 / K_{1}\) (b) \(K_{2}^{2}=K_{1}\) (e) \(K_{2}=1 / K_{1}^{2}\) (c) \(K_{2}=K_{1}\)

Step-by-Step Solution

Answer

Option (e) \( K_{2} = \frac{1}{K_{1}^{2}} \) relates \( K_{1} \) to \( K_{2} \).

Step 1: Write the Given Equilibria

We are given two equilibria: 1. \( \text{SO}_{2}(\text{g}) + \frac{1}{2} \text{O}_{2}(\text{g}) \rightleftarrows \text{SO}_{3}(\text{g}) \) with an equilibrium constant \( K_{1} \).2. \( 2\text{SO}_{3}(\text{g}) \rightleftarrows 2\text{SO}_{2}(\text{g})+\text{O}_{2}(\text{g}) \) with an equilibrium constant \( K_{2} \).

Step 2: Examine the Relation of Reactions

The first reaction can be expressed as occurs once to form \( \text{SO}_{3}(\text{g}) \) from \( \text{SO}_{2}(\text{g}) \) and \( \text{O}_{2}(\text{g}) \). The second reaction involves the reverse of this process but doubled, forming 2 \( \text{SO}_{3}(\text{g}) \).

Step 3: Determine the Relationship between Equilibrium Constants

From equilibrium theory, when the stoichiometry of an equation is multiplied, the equilibrium constant changes as the original constant raised to the power of the integer of the multiplier. So, if we reverse and double the reaction: two times the first reaction (reversed), \\[ K_{2} = \left( \frac{1}{K_{1}} \right)^{2} = \frac{1}{K_{1}^{2}}.\]The \( K \) value for the doubled reaction is \( K^{2} \), since each concentration term is squared due to doubling all coefficients.

Key Concepts

Equilibrium Constants

Equilibrium constants, often denoted by the letter "K," play an essential role in chemical reactions, particularly for reversible reactions. They provide a quantitative measure of the position of equilibrium in a chemical reaction. The equilibrium constant is calculated as the product of the concentrations of the products, each raised to the power of their coefficients in the balanced equation, divided by the product of the concentrations of the reactants, similarly raised to the power of their coefficients. It tells us how far a reaction proceeds at a given temperature before reaching equilibrium.

• If the value of the equilibrium constant is large (greater than 1), it indicates that the equilibrium favors the formation of products.
• A small equilibrium constant (less than 1) suggests the reactants are favored at equilibrium.

Understanding equilibrium constants is crucial because they allow chemists to predict how a reaction will behave under certain conditions. Knowing how these constants transform when reactions are reversed or coefficients changed helps in predicting the behavior of complex reactions.

Reversible Reactions

Reversible reactions occur when the reactants form products, which can then revert back into the initial reactants. This indicates a dynamic state where the reaction can proceed in both forward and backward directions. In the context of the original exercise, we have the formation of \[ \mathrm{SO}_{3}(\mathrm{g}) \rightleftarrows \mathrm{SO}_{2}(\mathrm{g}) + \frac{1}{2}\mathrm{O}_{2}(\mathrm{g}) \]The double arrow signifies the reversibility of the reaction.In practice, reversible reactions never proceed to complete consumption of reactants. Instead, they reach a point where the rate of the forward reaction equals the rate of the reverse reaction. At this stage, the concentrations of reactants and products remain constant, although the reactions continue to occur. This state is referred to as dynamic equilibrium. Reversible reactions are essential in nature and industry, contributing to various processes, including metabolic pathways and chemical manufacturing.

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the amounts of reactants and products in a chemical reaction. It ensures that a chemical equation is balanced, meaning that there are equal numbers of each type of atom on both sides of the equation. In equilibrium reactions, stoichiometry helps in determining the equilibrium constant expression.When a chemical equation is manipulated, such as doubling the stoichiometric coefficients, the equilibrium constant needs to be adjusted accordingly. For example, in the exercise, the second reaction \[ 2 \mathrm{SO}_{3}(\mathrm{g}) \rightleftarrows 2 \mathrm{SO}_{2}(\mathrm{g}) + \mathrm{O}_{2}(\mathrm{g}) \]is essentially the first reaction doubled and reversed. This doubling impacts the related equilibrium constant, requiring it to be squared as per the stoichiometric principles. Understanding stoichiometry is key for chemists to accurately measure and predict the outcomes of reactions, giving insights into how changes in concentration or pressure could impact the equilibrium position.

Equilibrium Theory

Equilibrium theory encompasses the principles and calculations that describe the state of chemical equilibrium, where a reaction and its reverse occur at the same rate. At this point, the concentrations of all reactants and products remain constant. This concept allows us to predict the concentrations of substances involved in a reaction at equilibrium and understand how different factors can shift the equilibrium state. Factors such as temperature, pressure, and concentration changes can affect the equilibrium position, described by Le Chatelier's Principle. For example, increasing the concentration of reactants or decreasing the concentration of products will generally shift the equilibrium to the right, favoring the formation of more products. Equilibrium theory also explains how equilibrium constants change when a reaction is reversed or when stoichiometric coefficients are changed. In the exercise example, reversing the reaction leads to reciprocal changes in the equilibrium constant, and doubling the coefficients requires squaring the equilibrium constant. This theoretical foundation is crucial for chemists to manage and optimize reactions, impacting fields ranging from environmental science to pharmaceutical development.