Methanol (CH3OH) is produced commercially by the catalyzed reaction of carbon monoxide and hydrogen: CO(g) + 2 H2(g) ⇌ CH3OH(g). An equilibrium mixture in a 2.00-L vessel is found to contain 0.0406 mol CH3OH, 0.170 mol CO, and 0.302 mol H2 at 500 K. Calculate Kc at this temperature.

Verified step by step guidance

Step 1: Write the balanced chemical equation for the reaction: CO(g) + 2 H2(g) ⇌ CH3OH(g).

Step 2: Identify the concentrations of each species in the equilibrium mixture. Use the formula \([\text{Concentration}] = \frac{\text{moles}}{\text{volume}}\) to calculate the concentrations: \([\text{CH}_3\text{OH}] = \frac{0.0406 \text{ mol}}{2.00 \text{ L}}\), \([\text{CO}] = \frac{0.170 \text{ mol}}{2.00 \text{ L}}\), \([\text{H}_2] = \frac{0.302 \text{ mol}}{2.00 \text{ L}}\).

Step 3: Write the expression for the equilibrium constant \(K_c\) for the reaction: \(K_c = \frac{[\text{CH}_3\text{OH}]}{[\text{CO}][\text{H}_2]^2}\).

Step 4: Substitute the calculated concentrations into the \(K_c\) expression: \(K_c = \frac{[\text{CH}_3\text{OH}]}{[\text{CO}][\text{H}_2]^2}\) using the values from Step 2.

Step 5: Simplify the expression to find the value of \(K_c\).

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, resulting in constant concentrations of reactants and products. In this state, the system is dynamic, meaning that reactions continue to occur, but there is no net change in the concentrations. Understanding this concept is crucial for analyzing reactions like the one involving methanol production, as it sets the foundation for calculating equilibrium constants.

Equilibrium Constant (Kc)

The equilibrium constant, Kc, quantifies the ratio of the concentrations of products to reactants at equilibrium, each raised to the power of their coefficients in the balanced equation. For the reaction CO(g) + 2 H2(g) ⇌ CH3OH(g), Kc is calculated using the formula Kc = [CH3OH] / ([CO][H2]^2). This constant provides insight into the position of equilibrium and the extent of the reaction under specific conditions.

Concentration Units and Molarity

Concentration is a measure of the amount of a substance in a given volume, typically expressed in moles per liter (Molarity, M). In the context of the equilibrium expression, the concentrations of the reactants and products must be calculated in molarity to accurately determine Kc. Understanding how to convert moles to molarity using the volume of the reaction vessel is essential for solving equilibrium problems.

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Related Practice

Textbook Question

The following equilibria were attained at 823 K:

CoO(s) + H₂(g) → Co(s) + H₂O(g) K_c = 67

CoO(s) + CO(g) → Co(s) + CO₂(g) K_c = 490

Based on these equilibria, calculate the equilibrium constant for H₂(g) + CO₂(g) → CO(g) + H₂O(g) at 823 K.

Textbook Question

Consider the equilibrium N₂(𝑔) + O₂(𝑔) + Br₂(𝑔) ⇌ 2 NOBr(𝑔) Calculate the equilibrium constant 𝐾𝑝 for this reaction, given the following information at 298 K:

2 NO(𝑔) + Br₂(𝑔) ⇌ 2 NOBr(𝑔) 𝐾_𝑐= 2.02

NO(𝑔) ⇌ N₂(𝑔) + O₂(𝑔) 𝐾_𝑐= 2.1×10³⁰

Textbook Question

The equilibrium 2 NO(𝑔) + Cl₂(𝑔) ⇌ 2 NOCl(𝑔) is established at 500.0 K. An equilibrium mixture of the three gases has partial pressures of 0.095 atm, 0.171 atm, and 0.28 atm for NO, Cl₂, and NOCl, respectively. (a) Calculate 𝐾_𝑝 for this reaction at 500.0 K.

Textbook Question

The equilibrium 2 NO(𝑔) + Cl₂(𝑔) ⇌ 2 NOCl(𝑔) is established at 500.0 K. An equilibrium mixture of the three gases has partial pressures of 0.095 atm, 0.171 atm, and 0.28 atm for NO, Cl₂, and NOCl, respectively. (b) If the vessel has a volume of 5.00 L, calculate Kc at this temperature.