Chapter 6, Problem 6.53P

The conversion of ${\left({\text{CH}}_{\text{3}}\right)}_{\text{3}}\text{Cl}$ to ${\left({\text{CH}}_{\text{3}}\right)}_{\text{2}}\text{C}={\text{CH}}_{\text{2}}$ can occur by either a one-step or a two-step mechanism, as shown in Equations [1] and [2].

a. What rate equation would be observed for the mechanism in Equation [1]?

b. What rate equation would be observed for the mechanism in Equation [2]?

c. What is the order of each rate equation (i.e., first, second, and so forth)?

d. How can these rate equations be used to show which mechanism is the right one for this reaction?

e. Assume Equation [1] represents an endothermic reaction and draw an energy diagram for the reaction. Label the axes, reactants, products, $E_{\text{a}}$, and $\Delta H°$. Draw the structure for the transition state.

f. Assume Equation [2] represents an endothermic reaction and that the product of the rate-determining step is higher in energy than the reactants or products. Draw an energy diagram for this two-step reaction. Label the axes, reactants and products for each step, and the $E_{\text{a}}$, and $\Delta H°$ for each step. Label $\Delta H{°}_{\text{overall}}$. Draw the structure for the transition state.

Interpretation Introduction

(a)

Interpretation: The rate equation for the mechanism in Equation [1] is to be stated.

Concept introduction: The rate equation is given as,

$\text{Rate}=k{\left[\text{A}\right]}^x{\left[\text{B}\right]}^y$

The order of reaction depends on the exponents $x$ and $y$. It is calculated as the sum of exponents. Rate equation gives the information regarding the mechanism of the reaction.

The rate equation for the reactions in which rate of a reaction depends on one reactant only is given as,

$\text{Rate}=k{\left[\text{A}\right]}^{\text{n}}$

Answer to Problem 6.53P

The rate law equation for the mechanism in Equation [1] is given as,

$\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]\left[{\text{OH}}^{-}\right]$

Explanation of Solution

The mechanism of Equation [1] is shown below.

Figure 1

The above reaction occurs in one step. The hydroxide ion attacks on proton of the starting material. As a result, new double bond formed between two carbon atoms. Iodide ion is expelled out as a leaving group. Therefore, the rate of the reaction relies on the concentration of the starting material and as well as nucleophile. The rate equation for this mechanism is given as,

$\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]\left[{\text{OH}}^{-}\right]$

Conclusion

(a) The rate equation for the mechanism in Equation [1] is given as, $\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]\left[{\text{OH}}^{-}\right]$

Interpretation Introduction

(b)

Interpretation: The rate equation for the mechanism in Equation [2] is to be stated.

Concept introduction: The rate equation is given as,

$\text{rate}=k{\left[\text{A}\right]}^x{\left[\text{B}\right]}^y$

The order of reaction depends on the exponents $x$ and $y$. It is calculated as the sum of exponents. Rate equation gives the information regarding the mechanism of the reaction.

Answer to Problem 6.53P

The rate equation for the mechanism in Equation [2] is given as,

$\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]$

Explanation of Solution

The mechanism of Equation [2] is shown below.

Figure 2

The above reaction occurs in twosteps. In the first step, $\text{C}-\text{I}$ bond is broken and the formation of carbocation takes place. In second step, hydroxide ion abstracts hydrogen. The first step is slow step. Hence, it determines the rate of a reaction.

Therefore, the rate of the reaction relies on the concentration of the starting material. The rate equation for this mechanism is given as,

$\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]$

Conclusion

The rate equation for the mechanism in Equation [2] is given as,

$\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]$

Interpretation Introduction

(c)

Interpretation: The order of each rate equation is to be identified.

Concept introduction: The rate equation is given as,

$\text{rate}=k{\left[\text{A}\right]}^x{\left[\text{B}\right]}^y$

The order of reaction depends on the exponents $x$ and $y$. It is calculated as the sum of exponents. Rate equation gives the information regarding the mechanism of the reaction.

Answer to Problem 6.53P

Theorder of first rate equation is second and the order of second rate equation is first.

Explanation of Solution

The rate law equation for the mechanism in Equation [1] is given as,

$\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]\left[{\text{OH}}^{-}\right]$

The rate equation implies that rate of a reaction relies on the concentration of the starting material and nucleophile. Therefore, it is a second ordered rate equation.

The rate equation for the mechanism in Equation [2] is given as,

$\text{Rate}=k\left[{\left({\text{CH}}_{\text{3}}\right)}_3\text{C}-\text{I}\right]$

The rate equation implies that rate of a reaction relies on the concentration of the starting material only. Therefore, it is a first ordered rate equation.

Conclusion

The order of first rate equation is second and the order of second rate equation is first.

Interpretation Introduction

(d)

Interpretation: The use of rate equations to describe which mechanism is the right one for the given reaction is to be stated.

Concept introduction: The rate equation is given as,

$\text{Rate}=k{\left[\text{A}\right]}^x{\left[\text{B}\right]}^y$

The order of reaction depends on the exponents $x$ and $y$. It is calculated as the sum of exponents. Rate equation gives the information regarding the mechanism of the reaction.

The rate equation for the reactions in which rate of a reaction depends on one reactant only is given as,

$\text{Rate}=k{\left[\text{A}\right]}^{\text{n}}$

Answer to Problem 6.53P

The equation [2] is the right one for the given mechanism because it involves two steps to complete the reaction.

Explanation of Solution

The equation showing the right mechanism is,

Figure 3

In the above reaction, the starting material is tertiary alkyl halide. In the first step, $\text{C}-\text{I}$ bond is broken. It leads to the formation of more stable carbocation. In the second step, hydroxide ion abstracts hydrogen and new double bond between two carbon atoms is generated. Hence, the first step is rate-determining step.

Therefore, the equation [2] is the right one for the given mechanism.

Conclusion

The equation [2] is the right one for the given mechanism because it involves two steps to complete the reaction.

Interpretation Introduction

(e)

Interpretation: An energy diagram and the structure of the transition state are to be drawn by assuming Equation [1] as an endothermic reaction. The axes, reactants, products, $E_{\text{a}}$, and $\Delta H°$ in an energy diagram are to be labeled.

Concept introduction: The transition state is formed during the conversion of reactants into products in the chemical reaction. In an energy level diagram, it corresponds to the high potential energy along the y-axis. In this state, the dashed bond implies that bonds are partially broken and partially formed in the reaction.

Answer to Problem 6.53P

An energy diagram and the structure of the transition state by assuming Equation [1] as an endothermic reaction is drawn in Figure 4 and Figure 5 respectively The axes, reactants, products, $E_{\text{a}}$, and $\Delta H°$ in an energy diagram are labeled in energy diagram.

Explanation of Solution

An energy diagram in which Equation [1] is considered as endothermic reaction is drawn as,

Figure 4

In the energy diagram, reaction coordinate is represented by x-axis and energy is represented by y-axis. The reaction coordinate describes the progress of the reaction. The energy of starting material is low in the diagram as compared to the energy of the product, which indicates that this reaction is endothermic. The minimum energy that is required to form the product from the starting material is represented by $E_{\text{a}}$. The change in enthalpy in the reaction is labeled as $\Delta H°$.

The structure of transition state is,

Figure 5

The transition state is formed during the conversion of reactants into products in the chemical reaction. This state cannot be isolated. The dashed bond indicates that bonds are partially broken or formed during the reaction.

Conclusion

An energy diagram and the structure of the transition state by assuming Equation [1] as an endothermic reaction is drawn in Figure 4 and Figure 5 respectively The axes, reactants, products, $E_{\text{a}}$, and $\Delta H°$ in an energy diagram are labeled in energy diagram.

Interpretation Introduction

(f)

Interpretation: An energy diagram and the structure of the transition state are to be drawn by assuming Equation [2] as an endothermic reaction and the energy of product of the rate-determining step is higher than the reactants or products. The axes, reactants, products, $E_{\text{a}}$, and $\Delta H°$ in an energy diagram are to be labeled.

Concept introduction: The transition state is formed during the conversion of reactants into products in the chemical reaction. In an energy level diagram, it corresponds to the high potential energy along the y-axis. In this state, the dashed bond implies that bonds are partially broken and partially formed in the reaction.

Answer to Problem 6.53P

An energy diagram and the structure of the transition state for Equation [2] is drawn in Figure 6 and Figure 7 respectively. The axes, reactants, products, $E_{\text{a}}$, and $\Delta H°$ in an energy diagram are labeled.

Explanation of Solution

Figure 6

In the energy diagram, reaction coordinate is represented by x-axis and energy is represented by y-axis. The reaction coordinate describes the progress of the reaction. The energy of starting material is low in the diagram as compared to the energy of the product, which indicates that this reaction is endothermic. The minimum energy that is required to form the product from the starting material is represented by $E_{\text{a}}$. The change in enthalpy in the reaction is labeled as $\Delta H°$.

The structure of transition states are,

Figure 7

The transition state is formed during the conversion of reactants into products in the chemical reaction. This state cannot be isolated. The dashed bond indicates that bonds are partially broken or formed during the reaction.

Conclusion

An energy diagram and the structure of the transition state for Equation [2] is drawn in Figure 6 and Figure 7 respectively. The axes, reactants, products, $E_{\text{a}}$, and $\Delta H°$ in an energy diagram are labeled.