Question

Write the equilibrium-constant expression for the equilibrium $$ \mathrm{C}(s)+\mathrm{CO}_{2}(g) \rightleftharpoons 2 \mathrm{CO}(g) $$ The table that follows shows the relative mole percentages of \(\mathrm{CO}_{2}(g)\) and \(\mathrm{CO}(g)\) at a total pressure of 1 atm for several temperatures. Calculate the value of \(K_{p}\) at each temperature. Is the reaction exothermic or endothermic?

Short answer

The equilibrium constant expression for the given reaction is: \[ K_p = \frac{[\mathrm{CO}(g)]^2}{[\mathrm{CO_{2}(g)]} \] To calculate K_p at different temperatures, first calculate the partial pressures of CO2 and CO using their relative mole percentages and a total pressure of 1 atm. Then, substitute the partial pressures into the equilibrium constant expression and calculate K_p for each temperature. Observe how K_p changes with temperature: If K_p increases as temperature increases, the reaction is endothermic; if K_p decreases as temperature increases, the reaction is exothermic.

Step-by-step solution

Write the equilibrium constant expression for the reaction

The reaction is a heterogeneous reaction involving both solids and gases, so the equilibrium constant expression will only include the concentrations of the gases involved. The equilibrium constant expression for the reaction, K_p, can be written as: \[ K_p = \frac{({[\mathrm{CO}(g)]})^2}{[\mathrm{CO_{2}}(g)]} \]

Calculate the partial pressures of the gas species

Given the relative mole percentages ofCO2(g) and CO(g) and a total pressure of 1 atm, we can calculate their partial pressures: Partial pressure of CO2 (g) = Relative mole percentage of CO2 × Total pressure Partial pressure of CO (g) = Relative mole percentage of CO × Total pressure Now, plug in the relative mole percentages for different temperatures to calculate the partial pressures.

Calculate the value of K_p at each temperature

Once the partial pressures for CO2 and CO are calculated, we can determine K_p at each temperature by substituting the partial pressures into the equilibrium constant expression: K_p = (Partial pressure of CO)^2 / (Partial pressure of CO2) Calculate K_p for each temperature using the partial pressures calculated in Step 2. After calculating the value of K_p at different temperatures, observe how K_p changes as the temperature increases.

Determine whether the reaction is exothermic or endothermic

If K_p increases with an increase in temperature, the reaction is endothermic, as a higher temperature favors the formation of products. If K_p decreases with an increase in temperature, the reaction is exothermic, since a higher temperature favors the formation of reactants. Based on the change in K_p values with respect to temperature, we can deduce if the reaction is exothermic or endothermic.

Key concepts

Chemical Equilibrium

When a chemical reaction occurs, it can reach a state where the rate of the forward reaction equals the rate of the reverse reaction. This state is known as a chemical equilibrium. At equilibrium, the concentrations of reactants and products remain constant over time, but this does not mean they are equal; it simply means they are no longer changing. Equilibrium can be reached from either direction of the reaction.
In the provided exercise, the reaction between solid carbon (C) and carbon dioxide (CO₂) gas to form carbon monoxide (CO) gas is an example. Here, the equilibrium constant expression represents the ratio of the products to reactants at equilibrium, considering only the gases since solids and liquids do not have concentrations in the same way that gases do. In our case, the expression for the equilibrium constant, K_p, only includes the partial pressures of the gaseous reactants and products.

Partial Pressures

The partial pressure of a gas in a mixture is the pressure that the gas would exert if it occupied the entire volume of the mixture alone. For gas-phase reactions, the partial pressure can be used as an alternative to concentration when calculating the equilibrium constant, leading to the term K_p.
In the step-by-step solution for the reaction provided, calculating K_p requires determining the partial pressures of CO₂ and CO gases. The partial pressure of each component is obtained by multiplying its relative mole percentage by the total pressure of the gas mixture. It's important to always check whether the pressure units used align with the conditions specified in the equilibrium constant expression.

Le Chatelier's Principle

Le Chatelier's principle is a cornerstone concept in chemistry that predicts how a system at equilibrium will respond to changes in concentration, temperature, or pressure. According to this principle, if an external change is imposed on a system at equilibrium, the system adjusts itself to counteract the change and re-establish equilibrium.
For example, increasing the pressure on a gaseous equilibrium may shift the position of equilibrium toward the side with fewer moles of gas. Conversely, reducing the pressure favors the direction with more moles of gas. Similarly, changes in temperature can also shift equilibrium: an increase in temperature favors the endothermic direction, while a decrease favors the exothermic direction. Applying this principle helps predict the behavior of the system in the face of external changes, which is crucial for understanding and controlling chemical reactions.

Endothermic and Exothermic Reactions

Chemical reactions can either release energy to their surroundings, known as exothermic reactions, or absorb energy from their surroundings, termed endothermic reactions. This energy is typically in the form of heat.
An exothermic reaction will release heat, causing the surroundings to feel warmer. On the other hand, an endothermic reaction will cause the surroundings to feel cooler as heat is absorbed. In the context of the exercise, whether the reaction is exothermic or endothermic can be inferred by observing how the equilibrium constant, K_p, changes with temperature. If K_p increases with temperature, the system absorbs heat, suggesting an endothermic reaction. Conversely, a decrease in K_p with increasing temperature indicates an exothermic reaction, which releases energy as heat.