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Chapter 7: Problem 206

For a first order reaction \(\mathrm{A} \rightarrow \mathrm{P}\), the temperature (T) dependent rate constant (k) was found to follow the equation \(\log \mathrm{k}=-(2000) \mathrm{i} / \mathrm{T}+6.0 .\) The pre-exponential factor A and the activation energy Ea, respectively, are a. \(1.0 \times 10^{6} \mathrm{~s}^{-1}\) and \(9.2 \mathrm{~kJ} \mathrm{~mol}^{-1}\) b. \(6.0 \mathrm{~s}^{-1}\) and \(16.6 \mathrm{~kJ} \mathrm{~mol}^{-1}\) c. \(1.0 \times 10^{6} \mathrm{~s}^{-1}\) and \(16.6 \mathrm{~kJ} \mathrm{~mol}^{-1}\) d. \(1.0 \times 10^{6} \mathrm{~s}^{-1}\) and \(38.3 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Short Answer

Expert verified
The pre-exponential factor is \(1.0 \times 10^{6} \mathrm{~s}^{-1}\) and the activation energy is \(38.3 \mathrm{~kJ} \mathrm{~mol}^{-1}\), matching option (d).

Step by step solution

01

Understand the Arrhenius Equation

The Arrhenius equation describes the temperature dependence of the rate constant, given by \[ k = A e^{-\frac{E_a}{RT}} \] where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin. A logarithmic form of this equation is: \[ \log k = -\frac{E_a}{2.303RT} + \log A \].
02

Compare the Given Equation to Arrhenius Form

The given equation is \( \log k = -(2000) \frac{i}{T} + 6.0 \). Comparing this to the logarithmic form of the Arrhenius equation: \( \log k = -\frac{E_a}{2.303RT} + \log A \), we identify that \( \log A = 6.0 \) and \(-\frac{E_a}{2.303R} = -\frac{2000}{T}\).
03

Extract the Pre-exponential Factor \(A\)

Since \( \log A = 6.0 \), it follows that \( A = 10^6 \text{ s}^{-1} \).
04

Calculate the Activation Energy \(E_a\)

From the equation \(-\frac{E_a}{2.303R} = -2000\), solve for \(E_a\) by substituting the gas constant, \(R = 8.314 \text{ J mol}^{-1} \text{K}^{-1}\):\[ E_a = 2000 \times 2.303 \times 8.314 = 38288.244 \text{ J mol}^{-1} = 38.3 \text{ kJ mol}^{-1} \].
05

Compare Calculated Values to Options

From our calculations, \( A = 1.0 \times 10^{6} \text{ s}^{-1} \) and \( E_a = 38.3 \text{ kJ mol}^{-1} \), which matches option (d).

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Key Concepts

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

Pre-exponential Factor
In the context of the Arrhenius equation, the pre-exponential factor, often denoted as \( A \), is a crucial component. It provides the frequency of collisions with the correct orientation that lead to a reaction. This factor is specific to each reaction and is used to estimate the rate constant \( k \) over different temperatures.
The Arrhenius equation summarizes this relationship as: \[ k = A e^{-\frac{E_a}{RT}}\]
  • \( A \) is typically expressed in units of \( s^{-1} \) for first order reactions.
  • The value of \( A \) gives insight into the likelihood of a successful collision leading to a reaction.
  • In our exercise, \( \log A = 6.0 \) which, when converted, becomes \( A = 10^6 \; \text{s}^{-1} \).
Understanding \( A \) helps predict how often molecules are expected to successfully interact at a molecular level, ignoring the energy barrier鈥攖he focus of the next concept: activation energy.
Activation Energy
Activation energy, denoted as \( E_a \), represents the minimum energy required for a chemical reaction to occur. Think of it as the energy hurdle that reactant molecules must overcome to transform into products.
According to the Arrhenius equation's logarithmic form: \[ \log k = -\frac{E_a}{2.303RT} + \log A\]
  • \( E_a \) is typically measured in \( \text{kJ/mol} \).
  • This value influences the sensitivity of reaction rates to temperature changes.
  • Using this equation, our calculations yielded \( E_a = 38.3 \; \text{kJ mol}^{-1} \).
Knowing the activation energy allows chemists to understand how energy barriers affect the speed of chemical reactions and how these can be altered under different conditions to either speed up or slow down the reaction.
First Order Reaction
Reactions are classified based on the order, which indicates how the rate is affected by the concentration of reactants. In a first order reaction, the rate depends linearly on the concentration of one reactant. This means that any change in this reactant's concentration directly and proportionally changes the reaction rate. For a first order reaction like \( \mathrm{A} \rightarrow \mathrm{P} \):
  • The rate law is given by \( \, \text{Rate} = k[A] \), where \( [A] \) is the concentration of reactant A.
  • The unit of the rate constant \( k \) is \( \text{s}^{-1} \).
  • The rate of reaction is directly proportional to the amount of A present at a given time.
Understanding first order kinetics is valuable in predicting how reactions behave over time, especially in reactions where one reactant is involved, such as radioactive decay or some enzymatic reactions.

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Most popular questions from this chapter

In the formation of sulphur trioxide by the contact process,\(2 \mathrm{SO}_{2}+\mathrm{O}_{2}=2 \mathrm{SO}_{3}\), the rate of reaction can be measured as \(-\mathrm{d}\left(\mathrm{SO}_{2}\right)=6.0 \times 10^{-4} \mathrm{~mol} \mathrm{~L}^{-1} \frac{\mathrm{s}^{-1}}{\mathrm{dt}}\). Here the incorrect statements are a. The rate of reaction expressed in terms of \(\mathrm{O}_{2}\) will be \(4.0 \times 10^{-4}\) mole \(\mathrm{L}^{-1} \mathrm{~s}^{-1}\) b. The rate of reaction expressed in terms of \(\mathrm{O}_{2}\) will be \(6.0 \times 10^{-6}\) mole \(\mathrm{L}^{-1} \mathrm{~s}^{-1}\) c. The rate of reaction expressed in terms of \(\mathrm{SO}_{3}\) will be \(6.0 \times 10^{-4} \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~s}^{-1}\). d. The rate of reaction expressed in terms of \(\mathrm{O}_{2}\) will be \(3.0 \times 10^{-4}\) mole \(\mathrm{L}^{-1} \mathrm{~s}^{-1}\)

The first order isomerization reaction: Cyclopropane \(\rightarrow\) Propene, has a rate constant of \(1.10 \times 10^{-4} \mathrm{~s}^{-1}\) at \(470^{\circ} \mathrm{C}\) and \(5.70 \times 10^{-4} \mathrm{~s}^{-1}\) at \(500^{\circ} \mathrm{C}\). What is the activation energy (Ea) for the reaction? a. \(340 \mathrm{~kJ} / \mathrm{mol}\) b. \(260 \mathrm{~kJ} / \mathrm{mol}\) c. \(160 \mathrm{~kJ} / \mathrm{mol}\) d. \(620 \mathrm{~kJ} / \mathrm{mol}\)

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 \(\mathrm{e}^{-\mathrm{Ea} \mathrm{RT}}\) 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 indenendent of temnerature

Which of the following statements are correct? (1) Order of a reaction can be known from experimental results and not from the stoichiometry of reaction. (2) Molecularity a reaction refers to (i) each of the elementary steps in (an overall mechanism of) a complex reaction or (ii) a single step reaction (3) Overall molecularity of a reaction may be determined in a manner similar to overall order of reaction(4) Overall order of a reaction \(\mathrm{A}^{\mathrm{m}}+\mathrm{B}^{\mathrm{n}} \rightarrow \mathrm{AB}_{\mathrm{x}}\) is \(\mathrm{m}+\mathrm{n} .\) Select the correct answer using the following codes: a. 2 and 3 b. 1,3 and 4 c. 2,3 and 4 d. 1,2 and 3

A complex reaction, \(2 \mathrm{~A}+\mathrm{B} \rightarrow \mathrm{C}\) takes place in two steps as follows: \(\mathrm{A}+\mathrm{B} \stackrel{\mathrm{k}_{1}}{\longrightarrow} 2 \mathrm{C}, \mathrm{A}+2 \mathrm{~B} \stackrel{\mathrm{k}_{2}}{\longrightarrow} \mathrm{C}\) If \(\mathrm{K}_{1}<<\mathrm{K}_{2}\), order of reaction is a. Zero order b. One c. Two d. Three
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