Question

The gas phase decomposition of dimethyl ether follows first order kinetics,

$C{H}_{3}OC{H}_3\left(g\right)\to C{H}_4\left(g\right)+H_2\left(g\right)+CO\left(g\right)$

The reaction is carried out in a constant volume container at ${500}^{o}C$ and has a half-life of 14.5 minute. Initially only dimethyl ether is present at a pressure of 0.40 atmosphere. The total pressure of the system after 12 minutes assuming ideal gas behaviour in atm is__________. (write answer as the first integer after decimal, e.g., if answer is $2.56$ then answer should be $5$)

Answer 1

$C{H}_{3}OC{H}_3\left(g\right)\to C{H}_4\left(g\right)+H_2\left(g\right)+CO\left(g\right)$
At $t=0$ $0.4$ $0$ $0$ $0$

At $t=12min$ $(0.4-P)$ $P$ $P$ $P$

For ideal gas behaviour mole $\propto pressure$

$\therefore a\propto 0.40,(a-X)\propto (0.04-P)$

$\because K=\frac{2.303}{t}log\frac{a}{(a-X)}$

or $\frac{0.693}{14.5}=\frac{2.303}{12}log\frac{0.40}{(0.40-P)}$

$\therefore P=0.175atm$

Thus, total pressure after 12 minutes $=0.04-P+P+P+P$

$=0.40+2P$

$=0.40+2\times 0.175$

$=0.75atm$

Answer is 7.