Question

(a) Is the dissociation of fluorine molecules into atomic fluorine, \(\mathrm{F}_{2}(g) \rightleftharpoons 2 \mathrm{~F}(g)\) an exothermic or endothermic process? (b) If the temperature is raised by \(100 \mathrm{~K}\), does the equilibrium constant for this reaction increase or decrease? (c) If the temperature is raised by \(100 \mathrm{~K},\) does the forward rate constant \(k_{f}\) increase by a larger or smaller amount than the reverse rate constant \(k_{r} ?\)

Short answer

(a)The dissociation of fluorine molecules into atomic fluorine is an endothermic process. (b)If the temperature is raised by 100 K, the equilibrium constant for this reaction will increase. (c)If the temperature is raised by 100 K, the forward rate constant (kf) increases by a larger amount than the reverse rate constant (kr).

Step-by-step solution

Identify the nature of the reaction

Since breaking chemical bonds requires energy, dissociation of fluorine molecules into atomic fluorine requires an input of energy. Therefore, the given reaction is an endothermic process. Answer (a): The dissociation of fluorine molecules into atomic fluorine is an endothermic process.

Analyze the effect of a temperature change on the equilibrium constant

According to Le Châtelier's principle, if we increase the temperature of an endothermic reaction, the equilibrium will shift towards the endothermic side (the right side) in order to absorb the added energy. As the reaction shifts to the right, the concentration of products (atomic fluorine) will increase, leading to an increase in the equilibrium constant. Answer (b): If the temperature is raised by 100 K, the equilibrium constant for this reaction will increase.

Analyze the effect of a temperature change on the forward and reverse rate constants

According to the Arrhenius equation, the rate constants of both forward and reverse reactions will increase with an increase in temperature: \(k = Ae^{-\frac{Ea}{RT}}\) Where k is the rate constant, A is the pre-exponential factor, Ea is the activation energy, R is the gas constant, and T is the temperature in Kelvin. As the temperature increases, the exponent term becomes smaller, and k increases. However, the increase in the forward rate constant (kf) will be greater than that of the reverse rate constant (kr) since the forward reaction (an endothermic process) better absorbs the added thermal energy than the reverse reaction (an exothermic process). Answer (c): If the temperature is raised by 100 K, the forward rate constant (kf) increases by a larger amount than the reverse rate constant (kr).

Key concepts

Equilibrium Constant

The equilibrium constant (\( K \)) is a mathematical expression that quantifies the ratio of concentrations of products to reactants at equilibrium for a reversible reaction. \[K = \frac{[\text{products}]^{\text{coefficients}}}{[\text{reactants}]^{\text{coefficients}}}\]The value of \( K \) depends on temperature:

• A high \( K \) indicates more products are present compared to reactants, meaning equilibrium favors the formation of products.
• A low \( K \) signifies more reactants are present than products, implying the equilibrium leans towards the reactants.

In an endothermic reaction, like the dissociation of fluorine molecules into atomic fluorine, increasing the temperature shifts the equilibrium towards the products. This shift occurs because the system absorbs heat, favoring the endothermic side according to Le Châtelier's principle. This results in an increased value of the equilibrium constant.

Rate Constants

Rate constants characterize how fast a reaction proceeds. They are represented by \( k_f \) for the forward reaction and \( k_r \) for the reverse reaction. These constants are impacted by the following factors:

• Temperature: Higher temperatures typically increase both \( k_f \) and \( k_r \) due to molecules moving faster and overcoming activation energy barriers more easily.
• Activation Energy: According to the Arrhenius equation, reactions with a lower activation energy have larger rate constants because reactants convert to products more readily.

For the given endothermic reaction, when the temperature increases by 100 K, both \( k_f \) and \( k_r \) increase. However, \( k_f \) will increase to a greater extent than \( k_r \). This is because the endothermic forward reaction benefits more from the added heat compared to the exothermic reverse reaction.

Le Châtelier's Principle

Le Châtelier's principle describes how a system at equilibrium responds to external changes. It states that if an external condition, such as temperature, pressure, or concentration, is altered, the system will adjust to minimize that change and re-establish equilibrium.
In the context of an endothermic reaction:

• Temperature Increase: For the reaction \( \mathrm{F}_{2}(g) \rightleftharpoons 2 \mathrm{F}(g) \), raising the temperature causes the system to handle the additional heat by shifting towards the products, thus absorbing the added energy.
• Equilibrium Shift: As the equilibrium shifts to favor the formation of atomic fluorine, the equilibrium constant \( K \) increases, reflecting the enhanced concentration of products.

This principle helps understand why certain reactions might favor products or reactants under different conditions and is a powerful tool for predicting the outcome of changes in reaction conditions.