Problem 115 - Chemistry The Science in Context

Question

On the basis of the frequency factors and activation energy values of the following two reactions, determine which one will have the larger rate constant at room temperature \((298 \mathrm{K})\) \(\mathrm{O}_{3}(g)+\mathrm{O}(g) \rightarrow \mathrm{O}_{2}(g)+\mathrm{O}_{2}(g)\) \(A=8.0 \times 10^{-18} \mathrm{cm}^{3} /(\text { molecules } \cdot \mathrm{s}) \quad E_{\mathrm{a}}=17.1 \mathrm{kJ} / \mathrm{mol}\) \(\mathrm{O}_{3}(g)+\mathrm{Cl}(g) \rightarrow \mathrm{ClO}(g)+\mathrm{O}_{2}(g)\) \(A=2.9 \times 10^{-11} \mathrm{cm}^{3} /(\text { molecules } \cdot \mathrm{s}) \quad E_{\mathrm{a}}=2.16 \mathrm{kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

Answer: Reaction 2 will have the larger rate constant at room temperature (298 K).

Step 1: Write down the Arrhenius equation

The Arrhenius equation is given by: $$k = A \times e^{-\frac{E_a}{RT}}$$ where k is the rate constant, A is the frequency factor, \(E_a\) is the activation energy, R is the gas constant, and T is the temperature.

Step 2: Convert activation energies to the appropriate units

The activation energies are given in kJ/mol. We need to convert them to J/mol to match the gas constant, which has units of J/(mol·K). For Reaction 1: \(E_{a1} = 17100 \;\text{J}/\text{mol}\) For Reaction 2: \(E_{a2} = 2160 \;\text{J}/\text{mol}\)

Step 3: Calculate the rate constants for both reactions using the Arrhenius equation

For Reaction 1: \( k_1 = A_1 \times e^{-\frac{E_{a1}}{RT}}\) \( k_1 = (8.0 \times 10^{-18}) \times e^{-\frac{17100}{(8.314 \times 298)}}\) \( k_1 = 3.745 \times 10^{-20}\) For Reaction 2: \( k_2 = A_2 \times e^{-\frac{E_{a2}}{RT}}\) \( k_2 = (2.9 \times 10^{-11}) \times e^{-\frac{2160}{(8.314 \times 298)}}\) \( k_2 = 1.482 \times 10^{-10}\)

Step 4: Compare the rate constants

Now, we compare the calculated rate constants for both reactions: \(k_1 = 3.745 \times 10^{-20}\) \(k_2 = 1.482 \times 10^{-10}\) Since the rate constant for Reaction 2 (\(k_2\)) is larger than that for Reaction 1 (\(k_1\)), Reaction 2 will have the larger rate constant at room temperature (298 K).

Key Concepts

Rate Constant

The rate constant, often denoted by the symbol \(k\), is a crucial parameter in chemical kinetics that determines the speed at which a reaction occurs. In the Arrhenius equation, it expresses the unique rate of a particular reaction under specified conditions. The equation itself is represented as:\[k = A \times e^{-\frac{E_a}{RT}}\]where:

• \(k\) is the rate constant.
• \(A\) is the frequency factor.
• \(E_a\) is the activation energy.
• \(R\) is the gas constant (8.314 J/mol·K).
• \(T\) is the absolute temperature in Kelvin.

The rate constant is greatly influenced by temperature and the presence of catalysts. A higher rate constant means the reaction will proceed faster under the given conditions. Comparing rate constants allows chemists to predict which of many potential reactions will be more efficient at a given temperature.

Activation Energy

Activation energy, represented as \(E_a\), is the minimum energy that reacting molecules must have for a reaction to occur. This energy barrier determines how quickly a reaction can proceed. In the context of the Arrhenius equation, it's a significant part of the exponential term:\[k = A \times e^{-\frac{E_a}{RT}}\]The value of \(E_a\) affects the rate constant dramatically.

• Lower \(E_a\) values facilitate a quicker reaction because the necessary threshold energy is low, allowing more molecules to undergo the reaction at any given time.
• Higher \(E_a\) values slow down the reaction because only a few molecules manage to reach the necessary energy level.

Factors such as catalysts can lower the activation energy, enabling the reaction to proceed at a faster rate without a change in temperature. Smaller \(E_a\) leads to a larger rate constant, reflecting a faster chemical reaction.

Frequency Factor

The frequency factor \(A\), also known as the pre-exponential factor, reflects how often molecules collide with the proper orientation for a reaction to occur. It is a component of the Arrhenius equation:\[k = A \times e^{-\frac{E_a}{RT}}\]This factor is associated with the probability that collisions result in a successful reaction.

• A larger \(A\) indicates a higher likelihood that collisions result in reactions, given the correct activation energy is achieved.
• It does not depend on temperature, unlike the exponential term.

In comparing chemical reactions, a higher frequency factor supports a higher rate constant, assuming \(E_a\) remains similar or lower. It signifies the inherent reactivity of the reactants independent of other influences like temperature or energy barriers.