Problem 37 - Chemistry and Chemical Reactivity

Question

Which of the following correctly relates the two equilibrium constants for the two reactions shown? \(\mathrm{NOCl}(\mathrm{g}) \rightleftharpoons \mathrm{NO}(\mathrm{g})+1 / 2 \mathrm{Cl}_{2}(\mathrm{g}) \quad K_{1}\) \(2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) \rightleftarrows 2 \mathrm{NOCl}(\mathrm{g}) \quad K_{2}\) (a) \(K_{2}=-K_{1}^{2}\) (c) \(K_{2}=1 / K_{1}^{2}\) (b) \(K_{2}=1 /\left(K_{1}\right)^{1 / 2} \quad\) (d) \(K_{2}=2 K_{1}\)

Step-by-Step Solution

Answer

Option (c) \( K_2 = \frac{1}{K_1^2} \) is correct.

Step 1: Understand the Reaction Directions

Two reactions are given: 1. The first reaction is the decomposition of NOCl into NO and Cl2, represented as \( \mathrm{NOCl}(\mathrm{g}) \rightleftharpoons \mathrm{NO}(\mathrm{g})+\frac{1}{2}\mathrm{Cl}_{2}(\mathrm{g}) \). The equilibrium constant for this reaction is \( K_1 \).2. The second reaction is the formation of NOCl from NO and Cl2, represented as \( 2 \mathrm{NO}(\mathrm{g}) + \mathrm{Cl}_{2}(\mathrm{g}) \rightleftarrows 2 \mathrm{NOCl}(\mathrm{g}) \). The equilibrium constant for this reaction is \( K_2 \).

Step 2: Express Equilibrium Constants Mathematically

For each reaction, the equilibrium constant expression is derived based on the reaction's stoichiometry.1. For the first reaction: \[ K_1 = \frac{[\mathrm{NO}][\mathrm{Cl}_2]^{1/2}}{[\mathrm{NOCl}]} \]2. For the second reaction, reversing and doubling the first reaction: \[ K_2 = \frac{[\mathrm{NOCl}]^2}{[\mathrm{NO}]^2[\mathrm{Cl}_2]} \]

Step 3: Relate the Equilibrium Constants

The second reaction is effectively the reverse of the first reaction, with all coefficients multiplied by 2. The relationship between the equilibrium constants of a reversed reaction is \( K' = \frac{1}{K} \), and doubling the coefficients squares this relationship: \( K_2 = \left(\frac{1}{K_1}\right)^2 = \frac{1}{K_1^2} \).

Step 4: Match with Given Options

The relationship derived from the calculation is \( K_2 = \frac{1}{K_1^2} \). Comparing this with the options given:- (a) \( K_2 = -K_1^2 \) is incorrect.- (b) \( K_2 = \frac{1}{(K_1)^{1/2}} \) is incorrect.- (c) \( K_2 = \frac{1}{K_1^2} \) matches our derived expression.- (d) \( K_2 = 2K_1 \) is incorrect.

Key Concepts

Chemical equilibrium

Let's start by understanding what chemical equilibrium really means. It's reached in a chemical reaction when the rates of the forward and reverse reactions are equal. This doesn't mean the concentrations of reactants and products are equal, just that they're formed at the same rate.

At equilibrium, the concentrations of products and reactants remain constant over time. This balance is a dynamic process, where molecules continue to react, yet their amounts stay unchanged. We've all seen examples of this when a bottle of soda is capped and after shaking, the fizz seems to settle. At this point, the rate of carbon dioxide escaping is equal to the rate of dissolution back into the soda.

In the context of our given reactions, one involves the breakdown of NOCl, while the other involves the formation of NOCl. Equilibrium allows these reactions to reach a state where the production and consumption of each species balance out.

Stoichiometry

Stoichiometry involves using balanced chemical equations to calculate the relative quantities of reactants and products involved in a reaction. It's all about the relationship between the amounts of substances involved in chemical reactions, usually in the form of balanced equations.

In a stoichiometric equation, coefficients are used to indicate the ratio of moles of each substance involved. For the first reaction, we have NOCl decomposing to NO and Cl extsubscript{2}. Important to notice is the coefficient of \(\frac{1}{2}\) for Cl extsubscript{2}, which indicates that 0.5 moles of Cl extsubscript{2} are produced for each mole of NOCl decomposed.

In the second reaction, coefficients are doubled, meaning 2 moles of NO react with 1 mole of Cl extsubscript{2} to produce 2 moles of NOCl. Doubling the coefficients in the stoichiometry affects the calculation of equilibrium constants, highlighting its significant role in determining how much product forms from reactants.

Reversed reactions

Reversed reactions are just what they sound like — the original reaction going in the opposite direction. This is a common occurrence in equilibrium situations, as reactions can proceed in both forward and reverse directions.

When reversing a chemical reaction, the reactants become products and vice versa. The equilibrium constant for the reverse reaction is the reciprocal of the equilibrium constant for the forward reaction. This is because at equilibrium, the reciprocal relationship accounts for the changed direction of formation and consumption.

In our problem, the first equation describes the decomposition of NOCl, while the second is the formation of NOCl. When interpreting these reversed reactions, remember how NOCl forms from NO and Cl extsubscript{2}, and apply the reciprocal aspect when relating their equilibrium constants.

Equilibrium expressions

Equilibrium expressions are mathematical representations that help calculate the equilibrium constant for a chemical reaction. They are based on the concentrations of reactants and products at equilibrium.

For a general reaction like \ aA + bB \rightleftharpoons cC + dD \ , the equilibrium constant expression is \[ K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \].

For our first reaction, the equilibrium expression is given by \[ K_1 = \frac{[\text{NO}][\text{Cl}_2]^{1/2}}{[\text{NOCl}]} \]. This describes the relationship between the reactants and products when NOCl decomposes. When the reaction is reversed and doubled, the expression shifts to \[ K_2 = \frac{[\text{NOCl}]^2}{[\text{NO}]^2[\text{Cl}_2]} \], illustrating how changes in coefficients impact equilibrium calculations.

Understanding equilibrium expressions helps us predict how a change in concentrations will affect the reaction system, leading us to conclusions about reaction dynamics and equilibria.