Embibe Experts Solutions for Chapter: Chemical Kinetics, Exercise 4: EXERCISE-4

For the two parallel reactions $\mathrm{A}\stackrel{{\mathrm{k}}_1}{⟶}\mathrm{B}$ and $\mathrm{A}\stackrel{{\mathrm{k}}_2}{⟶}\mathrm{C}$, show that the activation energy $\mathrm{E}$ ' for the disappearance of $\mathrm{A}$ is given in terms of activation energies ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ for the two paths by ${\mathrm{E}}^{\text{'}}=\frac{{\mathrm{k}}_1{\mathrm{E}}_1+{\mathrm{k}}_2{\mathrm{E}}_2}{{\mathrm{k}}_1+{\mathrm{k}}_2}$

For the mechanism $\mathrm{A}+\mathrm{B}\underset{{\mathrm{k}}_2}{\stackrel{{\mathrm{k}}_1}{\rightleftharpoons }}\mathrm{C}$     $\mathrm{C}\stackrel{{\mathrm{k}}_3}{⟶}\mathrm{D}$

Derive the rate law using the steady-state approximation to eliminate the concentration of $\mathrm{C}$. 

Assuming that ${\mathrm{k}}_3<<{\mathrm{k}}_2$, express the pre-exponential factor $\mathrm{A}$ and ${\mathrm{E}}_{\mathrm{a}}$ for the apparent second-order rate constant in terms of ${\mathrm{A}}_1,{\mathrm{A}}_2$ and ${\mathrm{A}}_3$ and ${\mathrm{E}}_{{\mathrm{a}}_1},{\mathrm{E}}_{{\mathrm{a}}_2}$ and ${\mathrm{E}}_{{\mathrm{a}}_3}$ for the three steps.

The decomposition of a compound $\mathrm{P}$, at temperature $\mathrm{T}$ according to the equation

$2{\mathrm{P}}_{\left(\mathrm{g}\right) }⟶ 4{\mathrm{Q}}_{\left(\mathrm{g}\right)}+ {\mathrm{R}}_{\left(\mathrm{g}\right)}+ {\mathrm{S}}_{\left(\mathrm{t}\right)}$

is the first order reaction. After $30$ minutes from the start of decomposition in a closed vessel, the total pressure developed is found to be $317 \mathrm{mm} \mathrm{Hg}$ and after a long period of time the total pressure observed to be $617 \mathrm{mmHg}$. Calculate the total pressure of the vessel after $75$ minute, if volume of liquid $\mathrm{S}$ is supposed to be negligible. Also calculate the time fraction  ${\mathrm{t}}_{7/8}$

Given : Vapour pressure of $\mathrm{S}\left(\mathrm{l}\right)$ at temperature $\mathrm{T}=32.5 \mathrm{mmHg}$.

A certain reactant  ${\mathrm{B}}^{\mathrm{n}+}$ is getting converted to ${\mathrm{B}}^{(\mathrm{n}+4)+}$  in solution. The rate constant of this reaction is measured by titrating a volume of the solution with a reducing reagent which only reacts with ${\mathrm{B}}^{\mathrm{nt}}$and ${\mathrm{B}}^{(\mathrm{n}+4)+}$. In this process, it converts ${\mathrm{B}}^{\mathrm{n}+}$ to ${\mathrm{B}}^{(\mathrm{n}-2)+}$ and ${\mathrm{B}}^{(\mathrm{n}+4)+}$ to ${\mathrm{B}}^{(\mathrm{n}-1)+}$. At  $\mathrm{t}=0$, the volume of the reagent consumed is $25 \mathrm{mL}$ and at $\mathrm{t}=10 \min$, the volume used up is $32 \mathrm{mL}$. Calculate the rate constant of the conversion of ${\mathrm{B}}^{\mathrm{nt}}$ to  ${\mathrm{B}}^{(\mathrm{n}+4)+}$ assuming it to be a first order reaction.

The reaction $\mathrm{A}\left(\mathrm{aq}\right)⟶\mathrm{B}$ (aq) is monitored by measuring optical rotation of reaction mixture at different time interval. The species $A, B$ and $C$ are optically active with specific rotations $20°,30°$ and $-40°$ Respectively. Starting with pure A if the value of optical rotations was found to be $2.5°$ after $6.93$ Minutes and optical rotations was $-5°$ After infinite time. Find the rate constant for order conversion of A into B and C.

The reaction A proceeds in parallel channels         Suppose the half life values for the two branches are $60$ minutes and $90$ minutes, what is the overall half-life value?

The catalytic decomposition of formic acid may take place in two ways :

(a) $\mathrm{HCOOH}={\mathrm{H}}_2\mathrm{O}+\mathrm{CO}$

(b) $\mathrm{HCOOH}={\mathrm{H}}_2+{\mathrm{CO}}_2$

The rate constant and activation energy for reaction (a) are $2.79\times {10}^{-3}{ \min }^{-1}$ at $236°\mathrm{C}$ and $12.0$ kcal ${\mathrm{mole}}^{-1}$ . respectively and for reaction (b) are $1.52\times {10}^{-4}{ \min }^{-1}$ at $237°\mathrm{C}$ and $24.5{\mathrm{kcalmole}}^{-1}$ respectively. Find the temperature which will give a product made up of equimolar quantities of water vapour, carbon monoxide, hydrogen and carbon dioxide.

For the following first order gaseous reaction

The initial pressure in a container of capacity $\mathrm{V}$ litres is $1 \mathrm{atm}$. Pressure at time $\mathrm{t}=10\sec$ is $1.4 \mathrm{atm}$ and after infinite time it becomes $1.5$ atmosphere. Find the rate constant ${\mathrm{k}}_1$ and ${\mathrm{k}}_2$ for the appropriate reactions.