Chapter 14, Problem 55GQ

When heated, tetrafluoroethylene dimerizes to form octafluorocyclobutane.

C₂F₄(g) → ½ C₄F₈(g)

To determine the rate of this reaction at 488 K, the data in the table were collected. Analysis was done graphically, as shown below:

1. (a) What is the rate law for this reaction?
2. (b) What is the value of the rate constant?
3. (c) What is the concentration of C₂F₄ after 600 s?
4. (d) How long will it take until the reaction is 90% complete?

Interpretation Introduction

Interpretation:

The rate law for the given reaction has to be determined.

Concept introduction:

In order to establish the plausibility of a mechanism, one must compare the rate law of the rate determining step to the experimentally determined rate law.

Rate law: It is generally the rate equation that consists of the reaction rate with the concentration or the pressures of the reactants and constant parameters.

Rate determining step: In a chemical reaction the rate determining step is the slowest step in which the rate of the reaction depends on the rate of that slowest step.

Activation energy: It is defined as the minimum energy required by the reacting species in order to undergo chemical reaction.

Reaction coordinate: It is the diagrammatic representation of a chemical reaction which depicts how the reactants gets transformed into product where the transition state and the intermediates present in the reaction are also depicted.

Intermediate species: It is the species formed during the middle of the chemical reaction between the reactant and the desired product.

Arrhenius equation:

• Arrhenius equation is a formula that represents the temperature dependence of reaction rates
• The Arrhenius equation has to be represented as follows

  $\begin{array}{l}k=A{e}^{-E_a/RT}\\ \ln k=\ln A{e}^{-E_a/RT}\\ \ln k=\left(-\frac{{\text{E}}_{\text{a}}}{\text{R}}\right)\left(\frac{1}{\text{T}}\right)+\ln \text{A}\end{array}$

• $E_a$ represents the activation energy and it’s unit is $\text{kJ/mol}$
• R represents the universal gas constant and it has the value of 8.314 $\text{J/K}\text{.mol}$
• T represents the absolute temperature
• A represents the frequency factor or collision frequency
• e represents the base of natural logarithm
•  Arrhenius equation equation was proposed by Svante Arrhenius in 1889.

Rate constant: The rate constant for a chemical reaction is the proportionality term in the chemical reaction rate law which gives the relationship between the rate and the concentration of the reactant present in the chemical reaction.

Rate order: The order of each reactant in a reaction is represented by the exponential term of the respective reactant present in the rate law and the overall order of the reaction is the sum of all the exponents of all reactants present in the chemical reaction. The order of the reaction is directly proportional to the concentration of the reactants.

Answer to Problem 55GQ

The rate law for the given reaction is as follows,

$Rate=k{\left[C_2{F}_4\right]}^2$

Explanation of Solution

The rate law is actually the expression that contains reaction rate with respect to the concentration of the reactants and constant parameters.

The rate law for the given reaction is determined as follows,

$\begin{array}{l}{\text{C}}_{\text{2}}{\text{F}}_{\text{4}\left(\text{g}\right)}\to \frac{\text{1}}{\text{2}}{\text{C}}_{\text{4}}{\text{F}}_{\text{8}\left(\text{g}\right)}\\ {\text{2C}}_{\text{2}}{\text{F}}_{\text{4}\left(\text{g}\right)}\to {\text{C}}_{\text{4}}{\text{F}}_{\text{8}\left(\text{g}\right)}\\ \text{Rate = k}{\left[{\text{C}}_{\text{2}}{\text{F}}_{\text{4}}\right]}^{\text{2}}\\ \text{since the reactant contains 2 as its coefficient}\text{.}\end{array}$

Interpretation Introduction

Interpretation:

The value of the rate constant has to be determined.

Concept introduction:

In order to establish the plausibility of a mechanism, one must compare the rate law of the rate determining step to the experimentally determined rate law.

Rate law: It is generally the rate equation that consists of the reaction rate with the concentration or the pressures of the reactants and constant parameters.

Rate determining step: In a chemical reaction the rate determining step is the slowest step in which the rate of the reaction depends on the rate of that slowest step.

Activation energy: It is defined as the minimum energy required by the reacting species in order to undergo chemical reaction.

Reaction coordinate: It is the diagrammatic representation of a chemical reaction which depicts how the reactants get transformed into product where the transition state and the intermediates present in the reaction are also depicted.

Intermediate species: It is the species formed during the middle of the chemical reaction between the reactant and the desired product.

Arrhenius equation:

• Arrhenius equation is a formula that represents the temperature dependence of reaction rates
• The Arrhenius equation has to be represented as follows

  $\begin{array}{l}k=A{e}^{-E_a/RT}\\ \ln k=\ln A{e}^{-E_a/RT}\\ \ln k=\left(-\frac{{\text{E}}_{\text{a}}}{\text{R}}\right)\left(\frac{1}{\text{T}}\right)+\ln \text{A}\end{array}$

• $E_a$ represents the activation energy and it’s unit is $\text{kJ/mol}$
• R represents the universal gas constant and it has the value of 8.314 $\text{J/K}\text{.mol}$
• T represents the absolute temperature
• A represents the frequency factor or collision frequency
• e represents the base of natural logarithm
•  Arrhenius equation equation was proposed by Svante Arrhenius in 1889.

Rate constant: The rate constant for a chemical reaction is the proportionality term in the chemical reaction rate law which gives the relationship between the rate and the concentration of the reactant present in the chemical reaction.

Rate order: The order of each reactant in a reaction is represented by the exponential term of the respective reactant present in the rate law and the overall order of the reaction is the sum of all the exponents of all reactants present in the chemical reaction. The order of the reaction is directly proportional to the concentration of the reactants.

Answer to Problem 55GQ

The value for rate constant for the given reaction is $\text{0}\text{.045M/s}$

Explanation of Solution

From the rate law it is clear that the given reaction is a second order reaction and hence its slope value is equal to the rate constant value therefore from the given graph that is the slope of the graph will give the value for rate constant.

$slope=\frac{25-33.3}{335-520}=\frac{-8.3}{-185}=0.045M/s$

Interpretation Introduction

Interpretation:

The concentration of $C_2{F}_4$ after $600s$ has to be calculated.

Concept introduction:

In order to establish the plausibility of a mechanism, one must compare the rate law of the rate determining step to the experimentally determined rate law.

Rate law: It is generally the rate equation that consists of the reaction rate with the concentration or the pressures of the reactants and constant parameters.

Rate determining step: In a chemical reaction the rate determining step is the slowest step in which the rate of the reaction depends on the rate of that slowest step.

Activation energy: It is defined as the minimum energy required by the reacting species in order to undergo chemical reaction.

Reaction coordinate: It is the diagrammatic representation of a chemical reaction which depicts how the reactants gets transformed into product where the transition state and the intermediates present in the reaction are also depicted.

Intermediate species: It is the species formed during the middle of the chemical reaction between the reactant and the desired product.

Arrhenius equation:

• Arrhenius equation is a formula that represents the temperature dependence of reaction rates
• The Arrhenius equation has to be represented as follows

  $\begin{array}{l}k=A{e}^{-E_a/RT}\\ \ln k=\ln A{e}^{-E_a/RT}\\ \ln k=\left(-\frac{{\text{E}}_{\text{a}}}{\text{R}}\right)\left(\frac{1}{\text{T}}\right)+\ln \text{A}\end{array}$

• $E_a$ represents the activation energy and it’s unit is $\text{kJ/mol}$
• R represents the universal gas constant and it has the value of 8.314 $\text{J/K}\text{.mol}$
• T represents the absolute temperature
• A represents the frequency factor or collision frequency
• e represents the base of natural logarithm
•  Arrhenius equation equation was proposed by Svante Arrhenius in 1889.

Rate constant: The rate constant for a chemical reaction is the proportionality term in the chemical reaction rate law which gives the relationship between the rate and the concentration of the reactant present in the chemical reaction.

Rate order: The order of each reactant in a reaction is represented by the exponential term of the respective reactant present in the rate law and the overall order of the reaction is the sum of all the exponents of all reactants present in the chemical reaction. The order of the reaction is directly proportional to the concentration of the reactants.

Answer to Problem 55GQ

The concentration of $C_2{F}_4$ after 600s is $0.03M$ .

Explanation of Solution

The concentration for given reactant after 600s is determined as follows,

The given reaction is a second order reaction hence the following formula is used.

$\begin{array}{l}\frac{\text{1}}{{\left[\text{A}\right]}_{\text{t}}}\text{ = }\frac{\text{1}}{{\left[\text{A}\right]}_{\text{0}}}\text{+ kt}\\ \text{t = 600}\text{s}\\ {\left[\text{A}\right]}_{\text{t}}\text{= Concentration of reactant after some time t}\text{.}\\ {\left[\text{A}\right]}_{\text{0}}=Initialc\text{oncentration of reactant = 0}\text{.100M}\end{array}$

$\begin{array}{l}\frac{\text{1}}{{\left[\text{A}\right]}_{\text{t}}}\text{ = }\frac{\text{1}}{\text{0}\text{.100M}}\text{+0}\text{.045M/s}\times \text{600}\text{s}\\ \frac{\text{1}}{{\left[\text{A}\right]}_{\text{t}}}=10+27\\ \frac{\text{1}}{{\left[\text{A}\right]}_{\text{t}}}=37\\ {\left[\text{A}\right]}_{\text{t}}=\frac{1}{37}=0.03M\end{array}$

Interpretation Introduction

Interpretation:

The time take to complete $90\%$ of the reaction has to be given.

Concept introduction:

In order to establish the plausibility of a mechanism, one must compare the rate law of the rate determining step to the experimentally determined rate law.

Rate law: It is generally the rate equation that consists of the reaction rate with the concentration or the pressures of the reactants and constant parameters.

Rate determining step: In a chemical reaction the rate determining step is the slowest step in which the rate of the reaction depends on the rate of that slowest step.

Activation energy: It is defined as the minimum energy required by the reacting species in order to undergo chemical reaction.

Reaction coordinate: It is the diagrammatic representation of a chemical reaction which depicts how the reactants gets transformed into product where the transition state and the intermediates present in the reaction are also depicted.

Intermediate species: It is the species formed during the middle of the chemical reaction between the reactant and the desired product.

Arrhenius equation:

• Arrhenius equation is a formula that represents the temperature dependence of reaction rates
• The Arrhenius equation has to be represented as follows

  $\begin{array}{l}k=A{e}^{-E_a/RT}\\ \ln k=\ln A{e}^{-E_a/RT}\\ \ln k=\left(-\frac{{\text{E}}_{\text{a}}}{\text{R}}\right)\left(\frac{1}{\text{T}}\right)+\ln \text{A}\end{array}$

• $E_a$ represents the activation energy and it’s unit is $\text{kJ/mol}$
• R represents the universal gas constant and it has the value of 8.314 $\text{J/K}\text{.mol}$
• T represents the absolute temperature
• A represents the frequency factor or collision frequency
• e represents the base of natural logarithm
•  Arrhenius equation equation was proposed by Svante Arrhenius in 1889.

Rate constant: The rate constant for a chemical reaction is the proportionality term in the chemical reaction rate law which gives the relationship between the rate and the concentration of the reactant present in the chemical reaction.

Rate order: The order of each reactant in a reaction is represented by the exponential term of the respective reactant present in the rate law and the overall order of the reaction is the sum of all the exponents of all reactants present in the chemical reaction. The order of the reaction is directly proportional to the concentration of the reactants.

Answer to Problem 55GQ

The time required to complete 90% of the given reaction is $\text{2000 s}$.

Explanation of Solution

The time required by the reaction to complete by 90% is determined as follows,

$\begin{array}{l}\frac{\text{1}}{{\left[\text{A}\right]}_{\text{t}}}\text{ = }\frac{\text{1}}{{\left[\text{A}\right]}_{\text{0}}}\text{+ kt}\\ \text{t = Time taken}\\ {\left[\text{A}\right]}_{\text{t}}\text{= Concentration of reactant after some time t}\text{.}\\ {\left[\text{A}\right]}_{\text{0}}=Initialc\text{oncentration of reactant = 0}\text{.100M}\end{array}$

$\begin{array}{l}\frac{\text{1}}{{\left[\text{A}\right]}_{\text{t}}}\text{ = }\frac{\text{1}}{{\left[\text{A}\right]}_{\text{0}}}\text{+ kt}\\ \text{t = }\left[\frac{\text{1}}{{\left[\text{A}\right]}_{\text{t}}}\text{-}\frac{\text{1}}{{\left[\text{A}\right]}_{\text{0}}}\right]\text{×}\frac{\text{1}}{\text{k}}\\ \text{=}\left[\frac{\text{1}}{\text{0}\text{.01}}\text{-}\frac{\text{1}}{\text{0}\text{.1}}\right]\text{×}\frac{\text{1}}{\text{0}\text{.045}}\text{=}\text{2000}\text{s}\\ \\ \text{where, Initial concentration,}{\left[\text{A}\right]}_{\text{0}}\text{ = 0}\text{.100M}\\ \text{Concentration after 90\%,}{\left[\text{A}\right]}_{\text{t}}\text{=100\% - 90\%}\text{=}\text{10\%}\\ \text{=}\text{10\% of initial concentration =0}\text{.01M }\end{array}$