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Chapter 8: Problem 192

The rate constant of a reaction is given by In \(\mathrm{k}\left(\mathrm{sec}^{-1}\right)\) \(=14.34-\left(1.25 \times 10^{4}\right) / \mathrm{T}\) What will be the energy of activation? a. \(24.83 \mathrm{kcal} \mathrm{mol}^{-1}\) b. \(49.66 \mathrm{kcal} \mathrm{mol}^{-1}\) c. \(12.42 \mathrm{kcal} / \mathrm{mol}\) d. none

Short Answer

Expert verified
The activation energy is 24.83 kcal/mol, which corresponds to option a.

Step by step solution

01

Identify the Arrhenius Equation

The Arrhenius equation is given by: \[ k = A e^{-E_a/(RT)} \] 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

Compare Given Equation to Arrhenius Form

The given equation is \( \ln k = 14.34 - \frac{1.25 \times 10^4}{T} \). This matches the linear form of the Arrhenius equation: \( \ln k = \ln A - \frac{E_a}{R} \frac{1}{T} \). By comparison, we can see that \( 14.34 \) represents \( \ln A \) and \( 1.25 \times 10^4 \) represents \( \frac{E_a}{R} \).
03

Calculate Activation Energy

To find the activation energy \( E_a \), we use the relation \( E_a = \left(1.25 \times 10^4\right) \times R \). The gas constant \( R \) is \( 1.987 \) \( \mathrm{cal} / \mathrm{mol} \cdot \mathrm{K} \). Thus, \[ E_a = 1.25 \times 10^4 \times 1.987 = 24837.5 \text{ cal/mol} = 24.83 \text{ kcal/mol} \].
04

Match with Given Choices

The calculated activation energy \( 24.83 \text{ kcal/mol} \) corresponds to option a. \( 24.83 \text{ kcal/mol} \).

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Key Concepts

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

Activation Energy
Activation energy is a crucial concept in the study of chemical reactions. It represents the minimum energy required for reactants to transform into products. You can think of it as a hill that reactants need to overcome to start the reaction.

A higher activation energy means that fewer molecules possess the necessary energy to react at a given temperature, rendering the reaction slower. Conversely, a lower activation energy implies that more molecules can react, speeding up the process. In the Arrhenius equation, the activation energy is denoted as \(E_a\). Understanding how to extract \(E_a\) from the Arrhenius equation involves recognizing its relationship with temperature and the rate constant. In our exercise, \(1.25 \times 10^4\) in the equation represents \(\frac{E_a}{R}\). Given that \(R\), the gas constant, is known, you can determine \(E_a\) by multiplying these values.
Rate Constant
The rate constant \(k\) is an essential part of the Arrhenius equation. It indicates how fast a reaction proceeds. A crucial component of reaction kinetics, it varies with temperature and the presence of a catalyst. In the given equation, the natural logarithm of the rate constant \(\ln k\) is expressed linear with respect to \(1/T\), consistent with the Arrhenius formulation. This highlights the dependency of the rate constant on temperature.

By understanding \(k\), you can predict how changing conditions like temperature will impact reaction speed. For instance, increasing the temperature raises \(k\), making reactions faster. This increase is due to more molecules having enough energy to surpass the activation energy threshold.
Gas Constant
The gas constant, denoted \(R\), is used in many equations relating to gases and thermodynamics. In chemistry, it provides a connection between energy scales and other physical quantities like temperature.Here, \(R\) is given as \(1.987\) \(\mathrm{cal/mol \cdot K}\). It links activation energy and temperature in the Arrhenius equation:

\[ k = A e^{-E_a/(RT)} \]Understanding \(R\) helps in translating temperature into the energy needed for a reaction. In practice, it serves as a key part in calculating activation energy, as seen in the exercise where multiplying \(R\) by \(1.25 \times 10^4\) yields the activation energy. Don't confuse the gas constant with the ideal gas law's \(R\). Even though they use the same symbol, each context determines its exact value.

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Most popular questions from this chapter

Which of the following statement about the Arrhenius equation is/are correct? a. On raising temperature, rate constant of the reaction of greater activation energy increases less rapidly than that of the reaction of smaller activation energy. b. The term \(\mathrm{e}^{-E a / \mathrm{RT}}\) represents the fraction of the molecules having energy in excess of threshold value. c. The pre-exponential factor becomes equal to the rate constant of the reaction at extremely high temperature. d. When the activation energy of the reaction is zero, the rate becomes independent of temperature

The activation energy for a simple chemical reaction \(\mathrm{X} \rightarrow \mathrm{Y}\) is Ea for forward direction. The value of Ea for backword direction may be a. \(-\mathrm{Ea}\) b. \(2 \mathrm{Ea}\) \(\mathbf{c}_{*}>\) or \(<\mathrm{Ea}\) d. Zero

The aquation of tris-(1,10-phenanthroline) iron (II) in acid solution takes place according to the equation: $$ \begin{aligned} &\mathrm{Fe}(\mathrm{phen})_{3}^{2}+3 \mathrm{H}_{3} \mathrm{O}^{+}+3 \mathrm{H}_{2} \mathrm{O} \rightarrow \\ &\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}^{2+}+3 \text { (phen) } \mathrm{H}^{+} \end{aligned} $$ If the activation energy is \(126 \mathrm{~kJ} / \mathrm{mol}\) and frequency factor is \(8.62 \times 10^{17} \mathrm{~s}^{-1}\), at what temperature is the rate constant equal to \(3.63 \times 10^{-3} \mathrm{~s}^{-1}\) for the first order reaction? a. \(0^{\circ} \mathrm{C}\) b. \(50^{\circ} \mathrm{C}\) c. \(45^{\circ} \mathrm{C}\) d. \(90^{\circ} \mathrm{C}\)

For a first order reaction, which of the following are not correct? a. \(t_{3 / 8}=2 t_{3 / 4}\) b. \(t_{3 / 4}=2 t_{1 / 2}\) c. \(t_{15 / 6}=4 t_{1 / 2}\) d. \(t_{15 / 16}=3 t_{3 / 4}\)

In the following question two statements Assertion (A) and Reason (R) are given Mark. a. If \(\mathrm{A}\) and \(\mathrm{R}\) both are correct and \(\mathrm{R}\) is the correct explanation of \(\mathrm{A}\); b. If \(A\) and \(R\) both are correct but \(R\) is not the correct explanation of \(\mathrm{A}\); c. \(\mathrm{A}\) is true but \(\mathrm{R}\) is false; d. \(\mathrm{A}\) is false but \(\mathrm{R}\) is true, e. \(\mathrm{A}\) and \(\mathrm{R}\) both are false. (A): The rate constant increases exponentially with the increase in temperature. ( \(\mathbf{R}\) ): With the rise in temperature, the average kinetic energy of the molecules increases.
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