Problem 21 - Chemistry and Chemical Reactivity

Question

Consider the following equilibria involving \(\mathrm{SO}_{2}(\mathrm{g})\) and their corresponding equilibrium constants. \(\mathrm{SO}_{2}(\mathrm{g})+1 / 2 \mathrm{O}_{2}(\mathrm{g}) \rightleftarrows \mathrm{SO}_{3}(\mathrm{g}) \quad K_{1}\) \(2 \mathrm{sO}_{3}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \quad K_{2}\) 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

The correct expression relating \(K_1\) to \(K_2\) is (e) \(K_{2} = \frac{1}{K_{1}^{2}}\).

Step 1: Identify the Reaction Relations

We are given two reactions and their corresponding equilibrium constants:1. \( \mathrm{SO}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \rightleftharpoons \mathrm{SO}_{3}(\mathrm{g}) \) with constant \( K_{1} \).2. \( 2 \mathrm{SO}_{3}(\mathrm{g}) \rightleftharpoons 2 \mathrm{SO}_{2}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{g}) \) with constant \( K_{2} \).

Step 2: Expressing Equilibrium Constants

For reaction 1, the equilibrium constant expression is:\[ K_{1} = \frac{\left[\mathrm{SO}_{3}\right]}{\left[\mathrm{SO}_{2}\right]\left[\mathrm{O}_{2}\right]^{1/2}} \]For reaction 2, the equilibrium constant expression is:\[ K_{2} = \frac{\left[\mathrm{SO}_{2}\right]^{2}\left[\mathrm{O}_{2}\right]}{\left[\mathrm{SO}_{3}\right]^{2}} \]

Step 3: Identifying the Relationship

Notice reaction 2 is the reverse and an expanded form (multiplied by 2) of reaction 1. Therefore, if reaction 1 is reversed and squared, it will give reaction 2.From reaction 1 to 2, equilibrium constant gets squared reciprocally: \[ K_{2} = \frac{1}{K_{1}^{2}} \] (because reversing an equation requires taking the reciprocal of the equilibrium constant and squaring due to multiplication by 2).

Step 4: Choosing the Correct Option

From our identified relationship \( K_{2} = \frac{1}{K_{1}^{2}} \), the correct expression is option (e). \[ K_{2} = \frac{1}{K_{1}^{2}} \]

Key Concepts

Chemical Equilibrium

In the world of chemistry, chemical equilibrium is a fundamental concept that describes the state of a reversible reaction where the rate of the forward reaction equals the rate of the reverse reaction. This means that the concentrations of products and reactants remain constant over time.

It is crucial to understand that equilibrium does not mean the quantities of reactants and products are equal. Rather, it indicates a balance in their formation and decomposition rates.

When representing chemical equilibrium, we use the double-headed arrow symbol (\(\rightleftharpoons\)) in chemical equations. This depicts that the reactions can proceed both forwards and backwards.

Understanding equilibrium is essential in predicting how changes in conditions like temperature, pressure, or concentration can affect the chemical system.

• Increased temperature may shift the equilibrium position.
• Changing the concentration of either reactants or products can also lead the system to adjust to a new equilibrium.
• Pressure changes, primarily in gaseous reactions, can affect the direction of equilibrium.

Hence, the concept of equilibrium is crucial for controlling chemical processes and optimizing conditions in industry and laboratories.

Reaction Reversibility

Reversible reactions are reactions that can occur in both directions: from reactants to products and from products back to reactants. Unlike irreversible reactions, which proceed to completion, reversible reactions reach a state of equilibrium where the forward and reverse reactions happen at the same rate.

Reversibility is an important aspect in understanding chemical processes because it offers insights into the dynamic nature of chemical reactions. It allows chemists to manipulate and control reactions by shifting the equilibrium position, influencing which products or reactants are favored at certain conditions.

For example, in the equilibrium involving \(\mathrm{SO}_{2}\) and \(\mathrm{SO}_{3}\), their reactions can proceed both ways as demonstrated in the given equilibria. Such control over reaction pathways is essential for industrial chemists aiming to maximize yield in processes like the production of sulfuric acid.

When working with reversible reactions, consider:\

• How pressure changes affect gaseous equilibria.
• The role of catalysts in speeding up the attainment of equilibrium without favoring either reaction direction.
• Removing or adding products/reactants to drive the reaction one way or another.

In summary, understanding how reactions can be reversed and controlled is key for efficient and successful chemical experimentation and production.

Equilibrium Expressions

Equilibrium expressions are mathematical representations that quantitatively describe the ratio of product concentrations to reactant concentrations at equilibrium for a chemical reaction. These are commonly expressed using the equilibrium constant \(K\).

The equilibrium constant expression is derived from the balanced chemical equation and has the general form: \[K = \frac{[\text{products}]}{[\text{reactants}]}\]For each component, concentrations are raised to the power of their coefficients in the balanced equation.

In our example, the equilibrium expression for reaction 1 (\(\mathrm{SO}_{2} + \frac{1}{2} \mathrm{O}_{2} \rightleftharpoons \mathrm{SO}_{3}\)) is:\[K_{1} = \frac{[\mathrm{SO}_3]}{[\mathrm{SO}_2][\mathrm{O}_2]^{1/2}}\]This formula allows chemists to calculate \(K\) under various conditions and predict the system's behavior when disturbed.

For reaction 2 (\(2 \mathrm{SO}_3 \rightleftharpoons 2 \mathrm{SO}_2 + \mathrm{O}_2\)), the equilibrium constant \(K_2\) is:\[K_{2} = \frac{[\mathrm{SO}_2]^2[\mathrm{O}_2]}{[\mathrm{SO}_3]^2}\]By understanding these expressions, we can determine how \(K\) values relate when reactions are reversed or modified. For instance, in this exercise, \(K_2 = \frac{1}{K_1^2}\), illustrating the impact of reversing and scaling the initial reaction. These expressions are pivotal in predicting reaction directions and concentrations at equilibrium, making them essential tools in chemistry.