Problem 116 - 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 \((298 \mathrm{K})\) \(\mathrm{O}_{3}(g)+\mathrm{Cl}(g) \rightarrow \mathrm{ClO}(g)+\mathrm{O}_{2}(g)\) \(A=2.9 \times 10^{-11} \mathrm{cm}^{3} /(\text { molecules } \cdot \mathrm{s}) \quad E_{2}=2.16 \mathrm{kJ} / \mathrm{mol}\) \(\mathrm{O}_{3}(g)+\mathrm{NO}(g) \rightarrow \mathrm{NO}_{2}(g)+\mathrm{O}_{2}(g)\) \(A=2.0 \times 10^{-12} \mathrm{cm}^{3} /(\text { molecules } \cdot \mathrm{s}) \quad E_{2}=11.6 \mathrm{kJ} / \mathrm{mol}\)

Step-by-Step Solution

Answer

The reaction O3(g) + Cl(g) -> ClO(g) + O2(g) has the larger rate constant at room temperature (298 K) with a value of 6.86 x 10^-12 cm³/(molecules⋅s).

Step 1: Write down the Arrhenius equation

The Arrhenius equation expresses the dependency of the reaction rate constant \(k\) on temperature \(T\) and activation energy \(E_a\) like this: $$k=Ae^{-\frac{E_a}{RT}}$$ where: - \(A\) is the pre-exponential factor (or frequency factor) with units \(\mathrm{cm}^{3} /(\text { molecules } \cdot \mathrm{s})\), - \(E_a\) is the activation energy in \(\mathrm{kJ/mol}\), - \(R=0.008314\ \mathrm{kJ /(mol\cdot K)}\) is the universal gas constant, and - \(T\) is the temperature in Kelvin.

Step 2: Calculate the rate constants for each reaction using the Arrhenius equation

We are given the values of \(A\) and \(E_a\) for both reactions, and we know that the temperature is \(T=298\ \mathrm{K}\). Now, we will plug in these values into the Arrhenius equation and calculate the rate constants \(k_1\) and \(k_2\) for both reactions. For the first reaction, O3(g) + Cl(g) -> ClO(g) + O2(g): $$k_1=A_1e^{-\frac{E_{a1}}{RT}}=2.9 \times 10^{-11}\ e^{-\frac{2.16}{(0.008314)(298)}}$$ $$k_1=2.9 \times 10^{-11}e^{-0.87108}=6.86 \times 10^{-12}\ \mathrm{cm}^{3}/(\text { molecules } \cdot \mathrm{s})$$ For the second reaction, O3(g) + NO(g) -> NO2(g) + O2(g): $$k_2=A_2e^{-\frac{E_{a2}}{RT}}=2.0 \times 10^{-12} e^{-\frac{11.6}{(0.008314)(298)}}$$ $$k_2=2.0 \times 10^{-12}e^{-4.6923}=1.15 \times 10^{-14} \mathrm{cm}^{3}/(\text { molecules } \cdot \mathrm{s})$$

Step 3: Compare the rate constants

Now, we have the rate constant values for both reactions at the given temperature: $$k_1=6.86 \times 10^{-12}\ \mathrm{cm}^{3}/(\text { molecules } \cdot \mathrm{s})$$ $$k_2=1.15 \times 10^{-14}\ \mathrm{cm}^{3}/(\text { molecules } \cdot \mathrm{s})$$ Comparing these values, we can clearly see that the rate constant for the first reaction \((k_1)\) is larger than the rate constant for the second reaction \((k_2)\) at room temperature \((298\ \mathrm{K})\). Thus, the reaction \(\mathrm{O}_{3}(g)+\mathrm{Cl}(g) \rightarrow\mathrm{ClO}(g)+\mathrm{O}_{2}(g)\) will have the larger rate constant at room temperature.

Key Concepts

Reaction Rate Constant

The reaction rate constant, commonly denoted as \( k \), is a crucial parameter in the study of chemical kinetics. It quantifies the speed of a chemical reaction and is influenced by various factors such as temperature, pressure, and the nature of reactants. More specifically, the Arrhenius equation provides a mathematical relationship describing how \( k \) varies with temperature, which is particularly important for predicting reaction behavior under different conditions.

For any chemical reaction, the value of \( k \) affects how fast the reactants are converted into products. When \( k \) is high, the reaction is typically fast, while a low \( k \) indicates a slower reaction. This is why, in comparing two reactions under identical conditions, the one with the higher rate constant will generally proceed more rapidly. As such, understanding and calculating the reaction rate constant using the Arrhenius equation gives chemists the ability to anticipate reaction speed and design processes more effectively.

Activation Energy

Activation energy, represented as \( E_a \) in chemistry, is a fundamental concept that refers to the minimum amount of energy needed for reactants to transform into products during a chemical reaction. It can be thought of as an energy barrier that reactants must overcome for the reaction to proceed.

The concept of activation energy is critical because it helps explain why certain reactions occur spontaneously at high temperatures while remaining unfeasible at lower temperatures. The Arrhenius equation incorporates \( E_a \) to show the significant impact temperature has on reaction rates. When the values of \( E_a \) are lower, it indicates that less energy is required for the reaction to take place; hence, these reactions are often faster, especially at higher temperatures. Conversely, a high \( E_a \) suggests that a greater input of energy is necessary, which results in slower reactions unless sufficient thermal energy is provided.

In practice, knowing the activation energy enables chemists to control the rate of reactions through temperature management, which is vital for numerous industrial and laboratory processes.

Frequency Factor

The frequency factor, denoted as \( A \) in the Arrhenius equation, is also called the pre-exponential factor. It is associated with the frequency of collisions and the proper orientation of the reacting molecules. Essentially, it's a measure of how often particles collide in the right way to cause a reaction.

The frequency factor encompasses the likelihood of a reaction occurring when molecules collide, and it's influenced by physical aspects of the reactants like geometric and steric considerations. In practical terms, a higher value of \( A \) usually implies that there are more successful collisions per unit of time, leading to a potentially faster reaction. Thus, when comparing two reactions with similar activation energies, the one with the higher frequency factor will typically have a larger rate constant.

This concept can offer insights when optimizing reactions. By focusing on factors that increase \( A \), such as the concentration of reactants or adding substances that facilitate proper molecular alignment, chemists can enhance the efficiency and speed of reactions.