Question

Two $1$ order reaction having same reactant concentration proceed at ${25}^{\circ }C$ at the same rate. The temperature coefficient of the rate of the first reaction is $2$ and that of second reaction is $3$. Find the ratio of the rates of there reactions at ${75}^{\circ }C$.

Answer 1

For first order reaction, $r_1=k{\left[A\right]}^1$
$\therefore r_1/r_2=k_1/k_2=$ temperature coefficient
Let the rate of reaction for Ist at ${25}^{o}C$ be $R_1$ and the rate of reaction for IInd at ${25}^{o}C$ be $R_2$.
Also, $R_1=R_2$
Rates of reaction
$\therefore$ At ${25}^{o}C$ $R_1$ $R_2$
${35}^{o}C$ $2R_1$ $3R_2$
${45}^{o}C$ ${\left(2\right)}^2{R}_1$ ${\left(3\right)}^2{R}_2$
${55}^{o}C$ ${\left(2\right)}^3{R}_1$ ${\left(3\right)}^3{R}_2$
${65}^{o}C$ ${\left(2\right)}^4{R}_1$ ${\left(3\right)}^4{R}_2$
${75}^{o}C$ ${\left(2\right)}^5{R}_1$ ${\left(3\right)}^5{R}_2$
$\therefore$ Temperature coefficient for 1st reaction $=\frac{k_{35}}{k_{25}}=\frac{R_{35}}{R_{25}}=2$
i.e., for each ${10}^{o}C$ rise in temperature, rate becomes $2$ times.
Similarly, for $2$nd reaction it becomes $3$ times,
$\therefore$ At ${75}^{o}C$
$\frac{rateofreactionfor2nd}{rateofreactionfor1st}=\frac{{\left(3\right)}^5{R}_2}{{\left(2\right)}^5{R}_1}=7.5937(\because R_1=R_2)$