Problem 115 - Chemistry: An Atoms-Focused Approach

Question

The valuc of \(K_{p}\) for the water-gas shift reaction $$ \mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g) $$ increases as the temperature decreases. Is the reaction exothermic or endothermic?

Step-by-Step Solution

Answer

Answer: The water-gas shift reaction is endothermic.

Step 1: Understand Le Chatelier's Principle

Le Chatelier's Principle states that if a system at equilibrium is subjected to a change in temperature, pressure, or concentration of reactants and products, the system will react in a manner to counteract that change and reestablish a new equilibrium position. In the context of changing temperatures, if a reaction is exothermic (releasing heat), an increase in temperature will shift the equilibrium towards the reactants. Conversely, if a reaction is endothermic (absorbing heat), an increase in temperature will shift the equilibrium towards the products.

Step 2: Recall the relationship between Kp and ∆G

The relationship between the equilibrium constant Kp and the change in Gibbs Free Energy (∆G) under given conditions is described by the equation: $$ \Delta G=-RT \ln {K_p} $$ where R is the gas constant, T is the temperature in Kelvin, and Kp is the equilibrium constant.

Step 3: Consider the reaction in the context of Le Chatelier's principle

Since the value of Kp increases when the temperature decreases, this means that the equilibrium shifts towards the products as the temperature decreases. This behavior is consistent with an endothermic reaction because, as mentioned earlier, when a reaction is endothermic, decreasing the temperature will shift the equilibrium towards the products.

Step 4: Conclusion

Based on Le Chatelier's principle and the relationship between Kp and ∆G, we can conclude that the given water-gas shift reaction is endothermic, as the equilibrium shifts towards the products when the temperature decreases.

Key Concepts

Understanding Endothermic Reactions

An endothermic reaction is a chemical process where heat is absorbed from the surroundings. This means that the reaction requires energy input to proceed. The absorption of heat makes the surroundings cooler. When analyzing such a reaction using Le Chatelier's Principle, an increase in temperature shifts the equilibrium position toward the products. This occurs because the system absorbs the extra heat to counteract the imposed change, favoring product formation.
In the specific case of the water-gas shift reaction, the increase in the equilibrium constant, \(K_p\), as temperature decreases indicates an endothermic reaction. This suggests that the reaction absorbs heat to drive the forward process, turning carbon monoxide and water vapor into hydrogen gas and carbon dioxide.
This behavior is consistently observed in endothermic reactions, where cooler conditions tend to shift equilibria towards greater product formation due to the absorption effect.

Role of the Equilibrium Constant (\(K_p\))

The equilibrium constant, \(K_p\), is a critical factor in understanding chemical equilibria concerning gases. It is a dimensionless value that reflects the ratio of the concentrations of products to reactants at equilibrium, each raised to the power of their coefficients in the balanced equation. For a general reaction, where gases are involved:

• \(K_p\) > 1 indicates that products are favored at equilibrium.
• \(K_p\) < 1 indicates that reactants are favored.

When temperature changes occur, they affect the value of \(K_p\) depending on whether the reaction is endothermic or exothermic. In endothermic reactions, lowering the temperature typically increases \(K_p\), as seen in the water-gas shift reaction. This is because the system compensates by shifting towards more product formation, which aligns with energy absorption.
Understanding how \(K_p\) shifts with temperature changes is crucial in predicting the behavior of gas-phase reactions and adjusting conditions favorably for desired products.

Gibbs Free Energy and Chemical Equilibria

Gibbs Free Energy, often represented as \(\Delta G\), is a thermodynamic property that indicates the amount of available energy to do work during a chemical reaction. Specifically, \(\Delta G\) helps determine whether a reaction will proceed spontaneously:

• If \(\Delta G < 0\), the reaction is spontaneous and favors product formation.
• If \(\Delta G > 0\), the reaction is non-spontaneous and favors the reactants.

The relationship between Gibbs Free Energy and the equilibrium constant is expressed by the equation \(\Delta G = -RT \ln {K_p}\). Here, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, and \(K_p\) is the equilibrium constant.

• As \(K_p\) increases, \(\Delta G\) becomes more negative, indicating a more favorable reaction towards products.

In the context of the water-gas shift reaction, a decrease in temperature leading to an increase in \(K_p\) signifies a move towards lower \(\Delta G\). Therefore, the reaction is driven forward, highlighting the endothermic nature as the system seeks to achieve a lower energy state with more products formed.