Chapter 10, Problem 10.3P

t-Butyl alcohol (TBA) is an important octane enhancer that is used to replace lead additives in gasoline [Ind. Eng. Chem. Res., 27, 2224 (1988)]. TBA was produced by the liquid-phase hydration (W) of isobutene (I) over an Amberlyst-15 catalyst. The system is normally a multiphase mixture of hydrocarbon, water, and solid catalysts. However, the use of cosolvents or excess TBA can achieve reasonable miscibility.

The reaction mechanism is believed to be

$\text{I}+\text{S}\rightleftarrows \text{I}\cdot \text{S}$        (P10-3.1)

$\text{W}+\text{S}\rightleftarrows \text{W}\cdot \text{S}$    (P10-3.2)

$\text{W}\cdot \text{S}+\text{I}\cdot \text{S}\rightleftarrows \text{TBA}\cdot \text{S}+\text{S}$    (P10-3.3)

$\text{TBA}\cdot \text{S}\rightleftarrows \text{TBA}+\text{S}$    (P10-3.4)

Derive a rate law assuming:

1. (a) The surface reaction is rate-limiting.
2. (b) The adsorption of isobutene is limiting.
3. (c) The reaction follows Eley–Rideal kinetics

   $\text{I}\cdot \text{S}+\text{W}\to \text{TBA}\cdot \text{S}$    (P10-3.5)

   and the surface reaction is limiting.

4. (d) Isobutene (I) and water (W) are adsorbed on different sites.

   $\text{I}+{\text{S}}_1\rightleftarrows \text{I}\cdot {\text{S}}_1$        (P10-3.6)

   $\text{W}+{\text{S}}_2\to \text{W}\cdot {\text{S}}_2$    (P10-3.7)

   TBA is not on the surface, and the surface reaction is rate-limiting.

5. (e) What generalization can you make by comparing the rate laws derived in parts (a) through (d)?

Interpretation Introduction

Interpretation: The rate laws corresponding to the surface reactions as the rate-limiting step in the liquid phase hydration of isobutene has to be determined.

Concept Introduction:

Rate law:

The rate law of the chemical reaction states that the rate of reaction is the function of the concentration of the reactants and the products present in that specific reaction. The rate is actually predicted by the slowest step of the reaction.

If there is a chemical reaction which has reactants A and B that reacts to form products then their rate law is given as follows.

    $r=k{\left[A\right]}^a{\left[B\right]}^b$

Here, $\left[A\right]$ is the concentration of the reactant A, $\left[B\right]$ is the concentration of the reactant B and $k$ is the rate constant.

Explanation of Solution

Liquid phase hydration of Isobutene follows the given mechanism as shown below:

$\begin{array}{l}\text{I}\text{+}\text{S}\stackrel{}{\rightleftarrows }\text{I}\text{.}\text{S}\\ \text{W}\text{+}\text{S}\stackrel{}{\rightleftarrows }\text{W}\text{.}\text{S}\\ \text{W}\text{.}\text{S}\text{+}\text{I}\text{.}\text{S}\stackrel{}{\rightleftarrows }\text{TBA}\text{.}\text{S}\text{+}\text{S}\\ \text{TBA}\text{.}\text{S}\stackrel{}{\rightleftarrows }\text{TBA}\text{+}\text{S}\end{array}$

Here

$\text{I}$-Isobutene

$\text{W}$- Liquid-phase hydration of Isobutene $\text{(I)}$

$\text{S}$- Surface.

The each step in the mechanism is a surface reaction. The rate laws corresponding to each step can be given as follows:

$\text{I}\text{+}\text{S}\stackrel{}{\rightleftarrows }\text{I}\text{.}\text{S}$

${\text{r}}_{\text{ADI}}\text{=}{\text{k}}_{\text{AI}}\left[{\text{C}}_{\text{I}}{\text{C}}_{\text{V}}\text{-}\frac{{\text{C}}_{\text{I}\text{.S}}}{{\text{k}}_{\text{I}}}\right]$

${\text{r}}_{\text{ADI}}$= Rate of Adsorption with respect to isobutene.

${\text{k}}_{\text{AI}}$ = Rate constant with respect to adsorption of isobutene.

$\text{W}\text{+}\text{S}\stackrel{}{\rightleftarrows }\text{W}\text{.}\text{S}$

${\text{r}}_{\text{ADW}}\text{=}{\text{k}}_{\text{AW}}\left[{\text{C}}_{\text{W}}{\text{C}}_{\text{V}}-\frac{{\text{C}}_{\text{W}\text{.S}}}{{\text{k}}_{\text{W}}}\right]$

${\text{r}}_{\text{ADW}}$= Rate of Adsorption with respect to the liquid phase of isobutene.

${\text{k}}_{\text{AW}}$ = Rate constant with respect to adsorption of liquid phase of isobutene.

$\text{W}\text{.}\text{S}\text{+}\text{I}\text{.}\text{S}\stackrel{}{\rightleftarrows }\text{TBA}\text{.}\text{S}\text{+}\text{S}$

${\text{r}}_{\text{S}}\text{=}{\text{k}}_{\text{S}}\left[{\text{C}}_{\text{W}\text{.S}}{\text{C}}_{\text{I}\text{.S}}-\frac{{\text{C}}_{\text{V}}{\text{C}}_{\text{TBA}\text{.S}}}{{\text{k}}_{\text{S}}}\right]$

${\text{r}}_{\text{S}}$= Rate of surface reaction.

${\text{k}}_{\text{S}}$= Rate constant of the surface reaction.

$\text{TBA}\text{.}\text{S}\stackrel{}{\rightleftarrows }\text{TBA}\text{+}\text{S}$

${\text{r}}_{\text{D}}\text{=}{\text{k}}_{\text{D}}\left[{\text{C}}_{\text{TBA}\text{.S}}\text{-}\frac{{\text{C}}_{\text{TBA}}{\text{C}}_{\text{V}}}{{\text{k}}_{\text{D}}}\right]$

${\text{r}}_{\text{D}}$= Rate of Desorption.

${\text{k}}_{\text{D}}$= Rate constant of Desorption.

$\text{C}$= Concentration in all the rate laws

Interpretation Introduction

Interpretation: The rate laws corresponding to the adsorption of isobutene as the rate-limiting step in the liquid phase hydration of isobutene has to be determined.

Concept Introduction:

The rate law of the chemical reaction states that the rate of reaction is the function of the concentration of the reactants and the products present in that specific reaction. The rate is actually predicted by the slowest step of the reaction.

If there is a chemical reaction which has reactants A and B that reacts to form products then their rate law is given as follows.

    $r=k{\left[A\right]}^a{\left[B\right]}^b$

Here, $\left[A\right]$ is the concentration of the reactant A, $\left[B\right]$ is the concentration of the reactant B and $k$ is the rate constant.

Explanation of Solution

Adsorption of Isobutene is limited.

The rate of adsorption of isobutene has the rate law as shown below:

${\text{r}}_{\text{ADI}}\text{=}{\text{k}}_{\text{AI}}\left[{\text{C}}_{\text{I}}\text{-}\frac{{\text{C}}_{\text{TBA}}}{{\text{C}}_{\text{W}}{\text{K}}_{\text{p}}}\right]{\text{C}}_{\text{V}}$

The following assumptions can be made on considering the adsorption of isobutene:

$\frac{{\text{r}}_{\text{ADW}}}{{\text{k}}_{\text{AW}}}\text{=}0$

$\frac{{\text{r}}_{\text{D}}}{{\text{k}}_{\text{D}}}\text{=}0$

$\frac{{\text{r}}_{\text{s}}}{{\text{k}}_{\text{s}}}\text{=0}$

${\text{C}}_{\text{I}\text{.S}}=\frac{{\text{C}}_{\text{V}}{\text{C}}_{\text{TBA}\text{.S}}}{{\text{C}}_{\text{W}\text{.S}}{\text{K}}_{\text{S}}}$

$\begin{array}{l}\because {\text{C}}_{\text{TBA}\text{.S}}={\text{C}}_{\text{TBA}}{\text{C}}_{\text{V}}{\text{K}}_{\text{TBA}}\text{and}\\ \because {\text{C}}_{\text{W}\text{.S}}={\text{C}}_{\text{W}}{\text{C}}_{\text{V}}{\text{K}}_{\text{W}}\end{array}$

Therefore, on substituting these expressions the expression of ${\text{C}}_{\text{I}\text{.S}}$ will be obtained as follows:

$\begin{array}{c}{\text{C}}_{\text{I}\text{.S}}=\frac{{\text{C}}_{\text{TBA}}{\text{K}}_{\text{TBA}}{\text{C}}_{\text{V}}{\bcancel{\text{C}}}_{\text{V}}}{{\text{C}}_{\text{W}}{\bcancel{\text{C}}}_{\text{V}}{\text{K}}_{\text{W}}{\text{K}}_{\text{S}}}\\ =\frac{{\text{C}}_{\text{TBA}}{\text{K}}_{\text{TBA}}{\text{C}}_{\text{V}}}{{\text{C}}_{\text{W}}{\text{K}}_{\text{W}}{\text{K}}_{\text{S}}}\end{array}$

It is known that ${\text{r}}_{\text{ADI}}\text{=}{\text{k}}_{\text{AI}}\left[{\text{C}}_{\text{I}}\text{-}\frac{{\text{C}}_{\text{TBA}}}{{\text{C}}_{\text{W}}{\text{K}}_{\text{p}}}\right]{\text{C}}_{\text{V}}$

${\text{K}}_{\text{p}}=\frac{{\text{K}}_{\text{w}}{\text{K}}_{\text{s}}}{{\text{K}}_{\text{TBA}}}$

${\text{-r}}_{\text{I}}^{\text{'}}\text{=}\frac{{\text{k}}_{\text{AI}}{\text{C}}_{\text{I}}\left[{\text{C}}_{\text{I}}\text{-}\frac{{\text{C}}_{\text{TBA}}}{{\text{C}}_{\text{W}}{\text{K}}_{\text{p}}}\right]}{\text{1}\text{+}{\text{C}}_{\text{W}}{\text{K}}_{\text{W}}\text{+}{\text{K}}_{\text{TBA}}{\text{C}}_{\text{TBA}}\text{+}\frac{{\text{C}}_{\text{TBA}}}{{\text{C}}_{\text{W}}{\text{K}}_{\text{p}}}}$

Interpretation Introduction

Interpretation: The rate laws corresponding to the Eley-Rideal kinetics for the given reaction has to be determined.

Concept Introduction:

Rate law:

The rate law of the chemical reaction states that the rate of reaction is the function of the concentration of the reactants and the products present in that specific reaction. The rate is actually predicted by the slowest step of the reaction.

If there is a chemical reaction which has reactants A and B that reacts to form products then their rate law is given as follows.

    $r=k{\left[A\right]}^a{\left[B\right]}^b$

Here, $\left[A\right]$ is the concentration of the reactant A, $\left[B\right]$ is the concentration of the reactant B and $k$ is the rate constant.

Eley-Rideal model is one of the models of rate law for the surface-catalyzed reactions. It suits best for the surface reactions in which the reactant A is adsorbed while the second reactant is not adsorbed on the catalyst surface.

Explanation of Solution

The given reaction is:

$\text{I}\text{.}\text{S}\text{.}\text{W}\stackrel{}{\to }\text{TBA}\text{.S}$

Eley-Rideal Kinetics for the given reaction can be derived as follows:

${\text{r}}_{\text{S}}\text{=}{\text{k}}_{\text{S}}\left[{\text{C}}_{\text{W}}{\text{C}}_{\text{I}\text{.S}}-\frac{{\text{C}}_{\text{TBA}\text{.S}}}{{\text{k}}_{\text{S}}}\right]$

$\begin{array}{l}\because {\text{C}}_{\text{I}\text{.S}}\text{=}{\text{k}}_{\text{I}}{\text{C}}_{\text{I}}{\text{C}}_{\text{v}}\text{and}\\ \because {\text{C}}_{\text{TBA}\text{.S}}={\text{C}}_{\text{TBA}}{\text{C}}_{\text{V}}{\text{K}}_{\text{TBA}}\end{array}$

Therefore,

${\text{-r}}_{\text{I}}^{\text{'}}\text{=}\frac{\text{k}\left[{\text{C}}_{\text{W}}{\text{C}}_{\text{I}}\text{-}\frac{{\text{C}}_{\text{TBA}}}{{\text{K}}_{\text{H}}}\right]}{\text{1}\text{+}{\text{k}}_{\text{I}}{\text{C}}_{\text{I}}\text{+}{\text{K}}_{\text{TBA}}{\text{C}}_{\text{TBA}}}$

Where,

$\begin{array}{l}\text{k}\text{=}{\text{k}}_{\text{s}}{\text{k}}_{\text{I}}{\text{C}}_{\text{I}}\\ {\text{K}}_{\text{H}}=\frac{{\text{k}}_{\text{I}}{\text{k}}_{\text{s}}}{{\text{K}}_{\text{TBA}}}\\ \end{array}$

Interpretation Introduction

Interpretation: The rate law corresponding to the surface reactions as the rate-limiting step when isobutene is adsorbed on different sites.

Concept Introduction:

Rate law:

The rate law of the chemical reaction states that the rate of reaction is the function of the concentration of the reactants and the products present in that specific reaction. The rate is actually predicted by the slowest step of the reaction.

If there is a chemical reaction which has reactants A and B that reacts to form products then their rate law is given as follows.

    $r=k{\left[A\right]}^a{\left[B\right]}^b$

Here, $\left[A\right]$ is the concentration of the reactant A, $\left[B\right]$ is the concentration of the reactant B and $k$ is the rate constant.

Explanation of Solution

The adsorption reactions of isobutene on different sites are given as shown below:

$\begin{array}{l}\text{I}\text{+}{\text{S}}_{\text{1}}\stackrel{}{\rightleftarrows }\text{I}\text{.}{\text{S}}_{\text{1}}\\ \text{W}\text{+}{\text{S}}_{\text{2}}\stackrel{}{\rightleftarrows }\text{W}\text{.}{\text{S}}_{\text{2}}\end{array}$

Here

$\text{I}$-Isobutene

$\text{W}$- Liquid-phase hydration of Isobutene $\text{(I)}$

${\text{S}}_1=\text{Surface}\text{1}$.

${\text{S}}_2=\text{Surface}2$.

The adsorption reaction of Isobutene is:

$\text{W}\text{.}\text{S}\text{+}\text{I}\text{.}\text{S}\stackrel{}{\rightleftarrows }\text{TBA}\text{.}\text{S}\text{+}\text{S}$

It can be modified with respect to surfaces 1 and 2 as shown below:

$\text{W}\text{.}{\text{S}}_2\text{+}\text{I}\text{.}{\text{S}}_1\stackrel{}{\rightleftarrows }\text{TBA+}{\text{S}}_2\text{+}{\text{S}}_1$

${\text{r}}_{\text{S}}\text{=}{\text{k}}_{\text{S}}\left[{\text{C}}_{\text{W}{\text{.S}}_2}{\text{C}}_{\text{I}{\text{.S}}_1}-\frac{{\text{C}}_{\text{TBA}\text{.S}}{\text{C}}_{{\text{V}}_{\text{1}}}{\text{C}}_{{\text{V}}_{\text{2}}}}{{\text{k}}_{\text{S}}}\right]$

${\text{C}}_{{\text{T}}_{\text{1}}}\text{=}{\text{C}}_{{\text{V}}_{\text{1}}}\text{+}{\text{C}}_{\text{I}{\text{.S}}_{\text{1}}}$

${\text{C}}_{{\text{T}}_{\text{2}}}\text{=}{\text{C}}_{{\text{V}}_{\text{2}}}\text{+}{\text{C}}_{\text{W}{\text{.S}}_{\text{2}}}$

${\text{C}}_{\text{I}{\text{.S}}_{\text{1}}}\text{=}{\text{K}}_{\text{I}}{\text{C}}_{\text{I}}{\text{C}}_{{\text{V}}_{\text{1}}}$

${\text{C}}_{{\text{T}}_{\text{1}}}\text{=}{\text{C}}_{{\text{V}}_{\text{1}}}\left(\text{1+}{\text{K}}_{\text{I}}{\text{C}}_{\text{I}}\right)$

${\text{C}}_{\text{W}{\text{.S}}_{\text{2}}}\text{=}{\text{K}}_{\text{w}}{\text{C}}_{\text{W}}{\text{C}}_{{\text{V}}_{\text{2}}}$

${\text{C}}_{{\text{T}}_{\text{2}}}\text{=}{\text{C}}_{{\text{V}}_{\text{2}}}\text{+}\left(\text{1+}{\text{K}}_{\text{W}}{\text{C}}_{\text{W}}\right)$

${\text{r}}_{\text{I}}^{\text{'}}\text{=}\frac{\text{k}\left[{\text{C}}_{\text{W}}{\text{C}}_{\text{I}}\text{-}\frac{{\text{C}}_{\text{TBA}}}{{\text{K}}_{\text{S}}}\right]}{\left(\text{1}\text{+}{\text{k}}_{\text{I}}{\text{C}}_{\text{I}}\right)\left(\text{1}\text{+}{\text{k}}_{\text{W}}{\text{C}}_{\text{W}}\right)}$

$\text{k}\text{=}{\text{k}}_{\text{S}}{\text{k}}_{\text{W}}{\text{k}}_{\text{t}}{\text{C}}_{{\text{r}}_{\text{1}}}{\text{C}}_{{\text{r}}_{\text{2}}}$

Interpretation Introduction

Interpretation: The generalization of rate laws of the previous sub-parts have to be discussed.

Concept Introduction:

Rate law:

The rate law of the chemical reaction states that the rate of reaction is the function of the concentration of the reactants and the products present in that specific reaction. The rate is actually predicted by the slowest step of the reaction.

If there is a chemical reaction which has reactants A and B that reacts to form products then their rate law is given as follows.

    $r=k{\left[A\right]}^a{\left[B\right]}^b$

Here, $\left[A\right]$ is the concentration of the reactant A, $\left[B\right]$ is the concentration of the reactant B and $k$ is the rate constant.

Explanation of Solution

The rate laws make to realize that the adsorption of isobutene is the rate-limiting step.