Question

(a) Consider two reactions, \(\mathrm{A}\) and \(\mathrm{B}\). If the rate constant for reaction B increases by a larger factor than that of reaction A when the temperature is increased from \(T_{1}\) to \(T_{2},\) what can you conclude about the relative values of the activation energies of the two reactions? (b) If a bimolecular reaction occurs every time an \(\mathrm{A}\) and a \(\mathrm{B}\) molecule collide, what can you say about the orientation factor and activation energy of the reaction?

Short answer

(a) Reaction B has a higher activation energy than reaction A. (b) The orientation factor is close to 1, and the activation energy is very low.

Step-by-step solution

Understanding the Effect of Temperature on Rate Constant

The relationship between the rate constant \( k \) and temperature \( T \) is given by the Arrhenius equation: \( k = A e^{-E_a/(RT)} \), where \( A \) is the frequency factor, \( E_a \) is the activation energy, and \( R \) is the universal gas constant. An increase in rate constant with temperature indicates the role of \( E_a \).

Comparing Rate Constants for Reactions A and B

For reaction B, the rate constant increases by a larger factor than that of reaction A when temperature increases from \( T_1 \) to \( T_2 \). This suggests that reaction B has a higher activation energy \( E_a \) than reaction A. Higher activation energies lead to larger changes in rate constants with temperature.

Rate of Bimolecular Reaction on Collision

If a bimolecular reaction occurs every time an \( \mathrm{A} \) and \( \mathrm{B} \) molecule collide, it means that all collisions are effective. This implies an orientation factor (also known as the steric factor) close to 1, indicating that the molecules do not require a specific orientation to react.

Considering Activation Energy for Bimolecular Reaction

For reactions that occur upon every collision, the activation energy \( E_a \) is very low or even negligible. This is because no additional energy barrier needs to be overcome beyond the initial collision energy.

Key concepts

Arrhenius Equation

The Arrhenius equation is key to understanding how temperature affects the rate of a chemical reaction. This equation is expressed as \( k = A e^{-E_a/(RT)} \), where:

• \( k \) is the rate constant, which tells us how fast a reaction progresses.
• \( A \) is the frequency factor that represents the likelihood of molecules colliding with the correct orientation.
• \( E_a \) symbolizes the activation energy, which is the minimum energy required for a reaction to occur.
• \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

The activation energy \( E_a \) can be thought of as a barrier that the reacting molecules need to overcome. As the temperature \( T \) increases, the term \(-E_a/(RT)\) becomes less negative, leading to an increase in the exponent \(-E_a/(RT)\), and thus the overall rate constant \( k \) becomes larger. This means the reaction happens quicker. The Arrhenius equation vividly shows how even a slight increase in temperature can lead to a significant increase in the reaction rate, especially when \( E_a \) is high.

Rate Constant

The rate constant \( k \) is a vital component of the rate equation, which quantifies the speed of a chemical reaction. For a given reaction, the rate constant is influenced by several factors:

• Temperature: An increase in temperature typically raises the rate constant, making reactions occur faster.
• Activation Energy \( E_a \): Higher activation energies reflect greater sensitivity of the rate constant to temperature changes.
• Frequency Factor \( A \): This embodies the frequency of collisions with the correct orientation for reaction.

In the exercise context, if Reaction B has a rate constant that increases more with temperature compared to Reaction A, this suggests that the activation energy \( E_a \) for Reaction B is higher. A higher \( E_a \) means that Reaction B's rate constant is more sensitive to temperature changes than that of Reaction A, resulting in a steeper increase in \( k \) with a rise in temperature. This can indicate a more significant barrier that needs to be overcome for Reaction B.

Bimolecular Reaction

A bimolecular reaction involves two reactant molecules colliding and reacting. In such reactions, the rate of reaction depends on the concentration of both reactants. The exercise mentions a scenario where a bimolecular reaction occurs every time an \( \mathrm{A} \) and \( \mathrm{B} \) molecule collide. This scenario provides insights into:

• Optimal Collisions: Every collision leads to a reaction, meaning collisions are effective. This is an ideal scenario where efficiency is maximal.
• Activation Energy: The activation energy for such a reaction is very low. Essentially, the molecular collision provides sufficient energy for the reaction to proceed, minimizing the need for an additional energy barrier.

Thus, such efficient bimolecular reactions suggest minimal constraints or preconditions for the reaction to progress, which simplifies the kinetics and theoretical models used to describe these reactions.

Orientation Factor

The orientation factor, also known as the steric factor, influences how often reacting molecules will successfully collide to form products. It is a dimensionless number ranging from 0 to 1:

• A factor close to 1 means that almost all collisions between reactant molecules result in a reaction. Such scenarios imply that molecules do not need any specific orientation to react.
• A lower factor suggests that only certain orientations will lead to effective reactions, meaning that many collisions are unsuccessful.

In cases where reactions happen with every collision (as with the bimolecular reaction described in the exercise), the orientation factor is nearly 1. This simplifies the dynamic as there are no additional orientation constraints needed for the reaction, indicating easy alignment or flexibility in the reacting molecules. This aligns with a low or negligible activation energy, as few barriers exist to hinder the reaction once molecules collide.