Question

Two $1^{st}$ order reactions having same reactant concentrations proceed at ${25}^{o}C$ at the same rate. The temperature coefficient of the rate of the first is 2 and that of second reaction is 3. Find the ratio of the rates of these reactions at ${75}^o$C:

Answer 1

For the first-order reaction,
$R=k{\left[C\right]}^1$

The effect of temperature on reaction rate​ can be expressed with​ temperature coefficient.

Temperature coefficient is the ratio of the rate of reaction/rate constant at two different temperature differing by ​${10}^{o}C$.
$\text{Temperature coefficient (T.C.)}=\frac{K_{t+10}}{K_t}$

$\therefore \frac{R_1}{R_2}=\frac{K_1}{K_2}=\text{Temperature coefficient}$

Let consider temperature co-efficient be 'a'

$\therefore \frac{R_{35}}{R_{25}}=a$

$\frac{R_{45}}{R_{35}}=a$

$\frac{R_{45}}{R_{35}}\times \frac{R_{35}}{R_{25}}=a\times \frac{R_{35}}{R_{25}}$

$\frac{R_{45}}{R_{25}}=a\times a=a^2$

Similarly,
$\frac{R_{75}}{R_{25}}=a^5$

For the $1^{st}$ reaction,
${\left(R_{75}\right)}_1=2^5\times {\left(R_{25}\right)}_1$
For the $2^{nd}$ reaction,
${\left(R_{75}\right)}_2=3^5\times {\left(R_{25}\right)}_2$
$\therefore \frac{{\left(R_{75}\right)}_{II}}{{\left(R_{75}\right)}_I}=\frac{3^5}{2^5}=7.59[\therefore {\left(R_{25}\right)}_1={\left(R_{25}\right)}_2]$