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Chapter 14: Problem 36

Use the Arrhenius equation to show why the rate constant of a reaction (a) decreases with increasing activation energy and (b) increases with increasing temperature.

Short Answer

Expert verified
Rate constant decreases with higher activation energy and increases with higher temperature.

Step by step solution

01

Write down the Arrhenius Equation

The Arrhenius equation is given by the formula: \[ k = A e^{-\frac{E_a}{RT}} \] where: \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.
02

Analyze Effect of Activation Energy

According to the Arrhenius Equation, the term \( e^{-\frac{E_a}{RT}} \) signifies the exponential decrease of the rate constant \( k \) with an increase in \( E_a \). This is because, as \( E_a \) increases, the exponent \(-\frac{E_a}{RT}\) becomes more negative, resulting in a smaller \( k \).
03

Analyze Effect of Temperature

In the Arrhenius equation, increasing temperature \( T \) affects the term \(-\frac{E_a}{RT}\) by decreasing its magnitude (since \( T \) is in the denominator). This results in a less negative exponent, which increases the rate constant \( k \).
04

Conclusion on Effects

(a) The rate constant decreases with increasing activation energy because it enhances the negative effect of the exponent, reducing \( k \). (b) Conversely, the rate constant increases with increasing temperature as it reduces the magnitude of the negative exponent, increasing \( k \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rate Constant
The rate constant, denoted as \( k \), is a central concept in chemical kinetics. It defines the speed at which a reaction proceeds. Different reactions will have different rate constants, which are influenced by conditions such as temperature and activation energy. The Arrhenius equation expresses this relationship mathematically as \[ k = A e^{-\frac{E_a}{RT}} \] where \( A \) and \( E_a \) are specific to each reaction, and \( R \), the gas constant, is a universal value.
  • A high rate constant signifies a faster reaction.
  • A low rate constant indicates a slower reaction.
Understanding how the rate constant changes helps in managing and designing chemical processes to achieve desired speeds and outcomes.
Activation Energy
Activation energy, represented by \( E_a \), is the minimum energy required for a reaction to occur. It is the barrier that reactants need to overcome to transform into products. Within the Arrhenius equation, this energy plays a pivotal role as the exponent \( e^{-\frac{E_a}{RT}} \).
  • Higher activation energy implies a greater energy barrier, leading to a smaller rate constant \( k \).
  • Lower activation energy results in fewer barriers, enabling a higher \( k \).
Therefore, reducing activation energy through catalysts or other means can enhance the rate of a chemical reaction by allowing more molecules to surpass this barrier.
Temperature Effect
Temperature has a direct influence on the rate at which chemical reactions occur. In the Arrhenius equation, temperature \( T \) is part of the exponential expression in \( e^{-\frac{E_a}{RT}} \). This implies that as the temperature increases, the value of \(-\frac{E_a}{RT}\) becomes less negative.
  • This change reduces the impact of the negative exponent, leading to an increase in the rate constant \( k \).
  • Thus, at higher temperatures, reactions typically proceed faster.
This temperature dependency is why heating substances often speeds up reactions, making them more efficient and ushering reactions towards completion.
Chemical Kinetics
Chemical kinetics is the study of reaction speeds and the factors affecting these rates. Understanding kinetics involves analyzing how different conditions alter the speed of reactions, which is crucial for controlling industrial processes, laboratory experiments, and even biological systems.
  • The Arrhenius equation serves as a fundamental expression in kinetics, showing how \( k \) changes with \( E_a \) and \( T \).
  • By controlling variables like temperature and activation energy, chemists can influence reaction rates.
Knowledge of chemical kinetics allows scientists to develop new materials, optimize chemical syntheses, and increase the efficiency and safety of processes involving chemical reactions.

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Most popular questions from this chapter

The rate law for the reaction: $$ 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \longrightarrow 2 \mathrm{NOCl}(g) $$ is given by rate \(=k[\mathrm{NO}]\left[\mathrm{Cl}_{2}\right]\). (a) What is the order of the reaction? (b) A mechanism involving the following steps has been proposed for the reaction: $$ \begin{aligned} \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) & \longrightarrow \mathrm{NOCl}_{2}(g) \\ \mathrm{NOCl}_{2}(g)+\mathrm{NO}(g) & \longrightarrow 2 \mathrm{NOCl}(g) \end{aligned} $$ If this mechanism is correct, what does it imply about the relative rates of these two steps?

A factory that specializes in the refinement of transition metals such as titanium was on fire. The firefighters were advised not to douse the fire with water. Why?

How does a catalyst increase the rate of a reaction?

(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?

The rate constants of some reactions double with every \(10^{\circ}\) rise in temperature. Assume that a reaction takes place at \(295 \mathrm{~K}\) and \(305 \mathrm{~K}\). What must the activation energy be for the rate constant to double as described?
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