The reaction $\mathrm{A}$ 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?

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 reaction of formation of phosgene from $\mathrm{CO}$ and ${\mathrm{Cl}}_2$ is $\mathrm{CO}+{\mathrm{Cl}}_2\to {\mathrm{COCl}}_2$. The proposed mechanism is

Shows that the above leads to the following rate law $\frac{\mathrm{d}\left[{\mathrm{COCl}}_2\right]}{\mathrm{dt}}=\mathrm{K}\left[\mathrm{CO}\right]{\left[{\mathrm{Cl}}_2\right]}^{\frac{3}{2}}$

Where $\mathrm{K}={\mathrm{k}}_3·\frac{{\mathrm{k}}_2}{{\mathrm{k}}_{-2}}{\left(\frac{{\mathrm{k}}_1}{{\mathrm{k}}_{-1}}\right)}^{\frac{1}{2}}$

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 $\mathrm{T}=32.5 \mathrm{mmHg}$.

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 $12.0 \mathrm{k} \mathrm{cal} {\mathrm{mole}}^{-1}$ respectively and for reaction (b) are $1.52\times {10}^{-4} {\min }^{-1}$ at $237 °\mathrm{C}$ and $24.5 \mathrm{k} \mathrm{cal} {\mathrm{mole}}^{-1}$ 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 $V$ litters is $1 \mathrm{atm}$. Pressure at time $t=10 \sec$ is $1.4 \mathrm{atm}$ and after infinite time it becomes $1.5$ atmosphere. Find the rate constant $k_1$ and $k_2$ for the appropriate reactions.