Problem 70 - Chemistry: The Central Science in SI Units, Global Edition

Question

True or false: When the temperature of an exothermic reaction increases, the rate constant of the forward reaction decreases, which leads to a decrease in the equilibrium constant, \(K_{c}\)

Step-by-Step Solution

Answer

The given statement is partially true and partially false. When the temperature of an exothermic reaction increases, the rate constant of the forward reaction actually increases, not decreases, as given by the Arrhenius equation. However, it is true that an increase in temperature results in a decrease in the equilibrium constant, \(K_{c}\), for an exothermic reaction, as described by the van't Hoff equation.

Step 1: Understanding Exothermic Reactions

An exothermic reaction is a chemical reaction that releases energy in the form of heat. According to Le Chatelier's principle, increasing the temperature of an exothermic reaction will shift the equilibrium position to favor the reverse (endothermic) reaction. This is because the system will try to counteract the increased temperature by absorbing the extra heat.

Step 2: Effect on Rate Constants

The rate constant, denoted by \(k\), is affected by temperature changes according to the Arrhenius equation: \[k = A \exp\left(\frac{-E_{a}}{RT}\right)\] where \(A\) is the pre-exponential factor, \(E_{a}\) is the activation energy, \(R\) is the gas constant, and \(T\) is the temperature in Kelvin. As the temperature increases, the exponential term in the equation will generally decrease, which will result in an increase in the rate constant. Hence, the statement "When the temperature of an exothermic reaction increases, the rate constant of the forward reaction decreases" is false.

Step 3: Effect on the Equilibrium Constant

The equilibrium constant, \(K_{c}\), also depends on temperature, as given by the van't Hoff equation: \[\frac{d\ln K_{c}}{dT} = \frac{\Delta H_{rxn}}{RT^2}\] where \(\Delta H_{rxn}\) is the enthalpy change of the reaction. Since the reaction is exothermic, \(\Delta H_{rxn}\) is negative. Therefore, for an exothermic reaction, increasing the temperature will result in a decrease in the equilibrium constant. Thus, the statement "which leads to a decrease in the equilibrium constant, \(K_{c}\)" is true.

Step 4: Conclusion

The initial statement "When the temperature of an exothermic reaction increases, the rate constant of the forward reaction decreases, which leads to a decrease in the equilibrium constant, \(K_{c}\)" is partially true and partially false. The rate constant of the forward reaction actually increases with increasing temperature. However, it is true that increasing the temperature of an exothermic reaction results in a decrease in the equilibrium constant, \(K_{c}\).

Key Concepts

Le Chatelier's Principle

Le Chatelier's Principle is a fundamental concept in chemistry that helps predict how changes in conditions can affect chemical equilibrium. It states that when a system at equilibrium is disturbed by a change in concentration, temperature, or pressure, it will shift to counteract this change and restore a new equilibrium.

For exothermic reactions, which release heat, increasing the temperature adds extra energy to the system. According to Le Chatelier’s Principle, the equilibrium will shift to absorb this added heat.

This means the system will favor the reverse reaction — moving from products back to reactants. In simpler words:

• If you increase the temperature, the system will try to cool down by favoring the endothermic direction (the opposite of exothermic).
• If you decrease the temperature, the system will generate heat by favoring the exothermic reaction.

This principle helps us understand why changing temperature can affect the yield of products in a chemical reaction.

Arrhenius Equation

The Arrhenius Equation is a critical formula that describes how the rate constant of a reaction, represented by \(k\), changes with temperature. It is expressed as:
\[k = A \exp\left(\frac{-E_{a}}{RT}\right)\]
where:

• \(A\) is the pre-exponential factor, or frequency factor, which is a constant for each chemical reaction.
• \(E_{a}\) is the activation energy required for the reaction.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.

The Arrhenius Equation shows us that as the temperature increases, the value of \(k\) generally increases as well.
This is because a higher temperature provides the necessary energy to overcome the activation energy barrier faster and more frequently.

For exothermic reactions, this means that both the forward and reverse reactions can become faster, but the backward reaction may become more favored energetically depending on the temperature shift, as indicated by Le Chatelier's Principle.

Equilibrium Constant

The equilibrium constant, \(K_{c}\), is a value that expresses the ratio of concentrations of products to reactants at equilibrium for a given reaction.

In other words, it provides insights into the position of equilibrium. The equilibrium constant is sensitive to temperature changes, especially in exothermic reactions, and its dependency can be quantified using the van't Hoff equation:
\[\frac{d\ln K_{c}}{dT} = \frac{\Delta H_{rxn}}{RT^2}\]
Where:

• \(\Delta H_{rxn}\) represents the enthalpy change of the reaction.
• \(R\) is the gas constant.
• \(T\) is the temperature in Kelvin.

For an exothermic reaction, where \(\Delta H_{rxn}\) is negative, increasing the temperature leads to a decrease in \(K_{c}\).
This decrease occurs because the shift in equilibrium favors the reverse reaction, leading to fewer products and more reactants being present at the new equilibrium.

Understanding how \(K_{c}\) changes with temperature helps chemists control reactions more effectively.