Problem 69 - Chemistry The Science in Context

Question

The value of \(K_{\mathrm{p}}\) 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 exothermic, as it releases heat.

Step 1: Introduce the van't Hoff Equation

The van't Hoff equation expresses the temperature dependence of the equilibrium constant, which allows us to relate the change in \(K_\mathrm{p}\) to a change in temperature for a given reaction. The equation is written as: $$\frac{d \ln K_\mathrm{p}}{dT}=\frac{\Delta H_{\mathrm{rxn}}}{R T^{2}}$$ where \(d \ln K_\mathrm{p}\) is the change in the natural logarithm of the equilibrium constant, \(dT\) is the change in temperature, \(\Delta H_{\mathrm{rxn}}\) is the change in enthalpy of the reaction, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Step 2: Analyze the Relationship between \(K_\mathrm{p}\) and Temperature

We are given that the value of \(K_\mathrm{p}\) increases as the temperature decreases. This means that \(d \ln K_\mathrm{p}\) is positive and \(dT\) is negative. If we look at the expression in the van't Hoff equation: $$\frac{d \ln K_\mathrm{p}}{dT} = \frac{\Delta H_{\mathrm{rxn}}}{R T^{2}}$$ We can deduce that for this reaction, \(\Delta H_{\mathrm{rxn}}\) must also be negative, since a positive \(d \ln K_\mathrm{p}\) divided by a negative \(dT\) will result in a negative expression on the right-hand side of the equation.

Step 3: Determine if the Reaction is Exothermic or Endothermic

The sign of \(\Delta H_{\mathrm{rxn}}\) determines if the reaction is exothermic or endothermic. A negative \(\Delta H_{\mathrm{rxn}}\) signifies an exothermic reaction, meaning heat is released during the reaction. Conversely, a positive \(\Delta H_{\mathrm{rxn}}\) signifies an endothermic reaction, meaning heat is absorbed during the reaction. Since we found that for this water-gas shift reaction, \(\Delta H_{\mathrm{rxn}}\) is negative (based on the relationship between \(K_\mathrm{p}\) and temperature), we can conclude that the reaction is exothermic, as heat is released.

Key Concepts

van't Hoff equation

The van't Hoff equation provides a powerful insight into how equilibrium constants change with temperature. It allows us to determine whether a reaction absorbs or releases heat—crucial for understanding thermodynamic behavior. To dive deeper, let's start with the equation itself: \(\frac{d \ln K_{\mathrm{p}}}{dT}=\frac{\Delta H_{\mathrm{rxn}}}{R T^{2}}\). Here, \(K_{\mathrm{p}}\) is the equilibrium constant and \(T\) is the temperature in Kelvin.

Several key components make this equation valuable. For instance:

• \(\Delta H_{\mathrm{rxn}}\) represents the change in enthalpy or heat content during a reaction.
• \(R\) is the ideal gas constant.
• The expression is derived from the logarithmic change in the equilibrium constant (\(K_{\mathrm{p}}\)) relative to temperature.

If changes in temperature lead to an increase in \(K_{\mathrm{p}}\), the enthalpy change \(\Delta H_{\mathrm{rxn}}\) must have a specific sign, indicating whether the reaction is exothermic or endothermic. Understanding this equation helps predict how reactions behave under different thermal conditions.

exothermic reactions

Exothermic reactions are a fascinating type of chemical reactions where heat is released. This release of energy usually manifests as an increase in temperature of the surroundings. For the water-gas shift reaction, we determined that it is exothermic. This conclusion was drawn because the equilibrium constant \(K_{\mathrm{p}}\) increased as the temperature decreased; this aligns with the behavior of exothermic reactions based on the van't Hoff equation.

In exothermic reactions:

• The enthalpy change \(\Delta H_{\mathrm{rxn}}\) is negative.
• The products generally have lower energy than the reactants.
• Heat is a byproduct, potentially increasing the efficiency of energy-dependent processes.

These characteristics make exothermic reactions crucial in many industrial processes and natural phenomena, such as combustion, where understanding their thermodynamics can optimize energy use.

water-gas shift reaction

The water-gas shift reaction is a cherished process in both industrial chemistry and environmental science. It's depicted as: \(\mathrm{CO}(g)+\mathrm{H}_{2} \mathrm{O}(g) \rightleftharpoons \mathrm{H}_{2}(g)+\mathrm{CO}_{2}(g)\). This reaction is exothermic, meaning it releases heat, as determined using the van't Hoff equation.

Key points about the water-gas shift reaction include:

• It's a crucial reaction for hydrogen production, with widespread implications in energy sectors, particularly in fuel cells.
• The exothermic nature of the reaction means it's favored at lower temperatures, making it vital for creating hydrogen efficiently.
• The ability of this reaction to convert carbon monoxide and water into hydrogen and carbon dioxide facilitates the reduction of toxic emissions, playing a vital role in clean energy initiatives.

Understanding the water-gas shift reaction's thermodynamics and equilibrium behavior is essential for harnessing its full potential in advancing sustainable technology.