Question

Calculate \(K\) for the reaction $$ \mathrm{Fe}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \rightleftarrows \mathrm{FeO}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{g}) $$ given the following information: $$ \begin{aligned} \mathrm{H}_{2} \mathrm{O}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) & \rightleftarrows \mathrm{H}_{2}(\mathrm{g})+\mathrm{CO}_{2}(\mathrm{g}) & & K=1.6 \\ \mathrm{FeO}(\mathrm{s})+\mathrm{CO}(\mathrm{g}) & \rightleftarrows \mathrm{Fe}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{g}) & & K=0.67 \end{aligned} $$

Short answer

The equilibrium constant \(K\) is approximately 2.384.

Step-by-step solution

Identify the Target Reaction

The reaction for which we need to find the equilibrium constant \(K\) is \( \mathrm{Fe}(\,\mathrm{s})+\mathrm{H}_2 \mathrm{O}(\,\mathrm{g}) \rightleftarrows \mathrm{FeO}(\,\mathrm{s})+\mathrm{H}_2(\,\mathrm{g}) \).

Write the Given Reactions and Constants

We have two provided reactions with their equilibrium constants: 1. \( \mathrm{H}_2 \mathrm{O}(\,\mathrm{g})+\mathrm{CO}(\,\mathrm{g}) \rightleftarrows \mathrm{H}_2(\,\mathrm{g})+\mathrm{CO}_2(\,\mathrm{g}) \) with \( K = 1.6 \).2. \( \mathrm{FeO}(\,\mathrm{s})+\mathrm{CO}(\,\mathrm{g}) \rightleftarrows \mathrm{Fe}(\,\mathrm{s})+\mathrm{CO}_2(\,\mathrm{g}) \) with \( K = 0.67 \).

Reverse the Second Reaction

Reverse the second reaction to match the terms with the products and reactants in the target reaction:\( \mathrm{Fe}(\,\mathrm{s}) + \mathrm{CO}_2(\,\mathrm{g}) \rightleftarrows \mathrm{FeO}(\,\mathrm{s}) + \mathrm{CO}(\,\mathrm{g}) \).The equilibrium constant for the reversed reaction is the reciprocal of the original: \( K' = \frac{1}{0.67} \approx 1.49 \).

Combine the Adjusted Reactions

Add the first given reaction and the reversed second reaction:- \( \mathrm{H}_2 \mathrm{O}(\,\mathrm{g})+\mathrm{CO}(\,\mathrm{g}) \rightleftarrows \mathrm{H}_2(\,\mathrm{g})+\mathrm{CO}_2(\,\mathrm{g}) \)- \( \mathrm{Fe}(\,\mathrm{s}) + \mathrm{CO}_2(\,\mathrm{g}) \rightleftarrows \mathrm{FeO}(\,\mathrm{s}) + \mathrm{CO}(\,\mathrm{g}) \)This results in:\( \mathrm{Fe}(\,\mathrm{s}) + \mathrm{H}_2 \mathrm{O}(\,\mathrm{g}) \rightleftarrows \mathrm{FeO}(\,\mathrm{s}) + \mathrm{H}_2(\,\mathrm{g}) \).

Calculate the New Equilibrium Constant

When combining reactions, the new equilibrium constant \( K_{new} \) is the product of the equilibrium constants for the individual steps:\[ K_{new} = (1.6) \times (1.49) \approx 2.384 \].

Key concepts

Chemical Equilibrium

Chemical equilibrium refers to a state in a chemical reaction where the concentrations of reactants and products remain constant over time. This is because the forward and reverse reactions occur at the same rate. At this point, although the reactions continue to happen, there is no net change in the concentration of substances involved. This balance does not mean that the amounts are equal, but that their ratios remain stable.

• In a balanced chemical equation, arrows pointing both ways (<=>) indicate that equilibrium can be achieved.
• Equilibrium can be dynamic, with molecules constantly reacting but maintaining a constant macroscopic ratio.
• Environmental conditions such as temperature and pressure can shift a reaction between forward and reverse states, impacting equilibrium.

Understanding chemical equilibrium is crucial for calculating equilibrium constants, which provide insight into the favorability of a reaction under certain conditions.

Reaction Reversal

In chemical equilibrium, reactions can reverse direction, moving from products back to reactants. This flexibility allows systems to adjust and maintain balance under changing conditions. To reverse a reaction, one essentially swaps the positions of reactants and products.

• The equilibrium constant for a reversed reaction is the reciprocal of the original constant. If the constant for the forward reaction is \( K \), the reversed reaction will have \( K' = \frac{1}{K} \).
• Reversal is essential when using Hess's law to calculate equilibrium constants for complex reactions by breaking them into simpler steps.
• Being able to reverse a reaction allows greater control and predictability in chemical processes.

Equilibrium Constant Calculation

The equilibrium constant, \( K \), quantifies the ratio of products to reactants at equilibrium for a given reaction. It is calculated by using the concentrations of substances involved in the reaction.

• For the reaction \( aA + bB \rightleftarrows cC + dD \), the equilibrium constant is given by the expression: \( K = \frac{[C]^c[D]^d}{[A]^a[B]^b} \), where brackets denote concentration and letters represent coefficients.
• A \( K \) value greater than one indicates that the reaction favors products, while a value less than one indicates a tendency to favor reactants.
• Equilibrium constants allow prediction of the direction of a reaction and help in calculating concentrations of unknowns in a chemical equation.
• In reactions involving solids and pure liquids (like that in the original exercise), their concentrations are considered to be equal to one and are thus omitted from the \( K \) expression.

Le Chatelier's Principle

Le Chatelier's Principle states that if a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to counteract the change. This principle helps predict how changes such as concentration, temperature, and pressure will affect a reaction's equilibrium.

• Increasing the concentration of reactants typically shifts the equilibrium to the right, favoring product formation, while increasing products does the opposite.
• Temperature changes can drive the equilibrium direction depending on whether a reaction is endothermic (absorbs heat) or exothermic (releases heat).
• Pressure changes particularly affect reactions involving gases; increasing pressure usually favors the side with fewer gas molecules.
• Le Chatelier's Principle provides a framework for understanding chemical response to environmental shifts and assists in optimizing reactions in industrial processes for desired outcomes.