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Chapter 14: Problem 29

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
According to the Arrhenius equation, the rate constant decreases with increasing activation energy because an increase in activation energy increases the magnitude of \( e^{-\frac{E_a}{RT}} \) reducing the rate constant. On the other hand, the rate constant increases with increasing temperature because an increase in temperature decreases the magnitude of \( e^{-\frac{E_a}{RT}} \), thereby increasing the rate constant.

Step by step solution

01

Understanding the Arrhenius Equation

Begin by understanding the Arrhenius equation. It is an equation that shows the temperature dependence of reaction rates. The Arrhenius equation is given by \( k = A e^{-\frac{E_a}{RT}} \) where k is the rate constant, A is the Arrhenius factor or frequency factor, E_a is the activation energy, R is the universal gas constant and T is the temperature.
02

Effect of Activation Energy on Rate Constant

On increasing the activation energy (E_a), the factor exponentiating \( e \) becomes larger. Remember, the exponential function \( e^-x \) decreases as \( x \) increases. Hence, the larger the value of \( E_a \), the smaller the value of \( e^{-\frac{E_a}{RT}} \), leading to a smaller rate constant. Therefore, the rate constant decreases with an increase in activation energy.
03

Effect of Temperature on Rate Constant

When the temperature (T) increases, \( \frac{E_a}{RT} \) becomes smaller because its denominator \( RT \) is increasing. As a result, the value of \( e^{-\frac{E_a}{RT}} \) increases, which in turn increases the value of the rate constant (k). Therefore, the rate constant increases with increasing temperature.

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Most popular questions from this chapter

Write the Arrhenius equation and define all terms.

What is meant by the order of a reaction?

Consider this elementary step: $$ X+2 Y \longrightarrow X Y_{2} $$ (a) Write a rate law for this reaction. (b) If the initial rate of formation of \(\mathrm{XY}_{2}\) is \(3.8 \times 10^{-3} \mathrm{M} / \mathrm{s}\) and the initial concentrations of \(X\) and \(Y\) are \(0.26 M\) and \(0.88 M\), what is the rate constant of the reaction?

Explain why termolecular reactions are rare.

The following gas-phase reaction was studied at \(290^{\circ} \mathrm{C}\) by observing the change in pressure as a function of time in a constant-volume vessel: $$ \mathrm{ClCO}_{2} \mathrm{CCl}_{3}(g) \longrightarrow 2 \mathrm{COCl}_{2}(g) $$ Determine the order of the reaction and the rate constant based on the following data: $$ \begin{array}{rc} \text { Time }(\mathrm{s}) & \mathrm{P}(\mathrm{mmHg}) \\ \hline 0 & 15.76 \\ 181 & 18.88 \\ 513 & 22.79 \\ 1164 & 27.08 \end{array} $$ where \(P\) is the total pressure.
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