Two first order reactions proceed at `25^(@)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 these reactions at `75^(@)C`.

Explanation

For fisrt order, `r_(1) = k[A]`
`(r_(1))/(r_(2)) = (k_(1))/(k_(2)) =` temperature coefficient
Let the rate of the reaction for first reaction at `25^(@)C` be `r_(1)` and rate of reaction for second reaction at `25^(@)C` be `r_(2)`.
`r_(1) = r_(2)`
`{:(,,"Rates of reaction",,),(At,25^(@)C,r_(1),r_(2),),(,35^(@)C,2r_(1),3r_(2),),(,45^(@)C,(2)^(2)r_(1),(3)^(2)r_(2),),(,55^(@)C,(2)^(3)r_(1),(3)^(3)r_(2),),(,65^(@)C,(2)^(4)r_(1),(3)^(4)r_(2),),(,75^(@)C,(2)^(5)r_(1),(3)^(5)r_(2),):}`
Temperature coefficient for first reaction `= (k_(35))/(k_(25)) = (r_(35))/(r_(25)) = 2` i.e., for each `10^(@)C` rise in temperature, rate becomes `2` times. ismilarly, for second reaction, it become `3` times.
At `75^(@)C = ("Rate of reaction for second reaction")/("Rate of reaction for first reaction")`
`= ((3)^(5)r_(2))/((2)^(5)r_(1)) = 7.5937`
`(because r_(1) = r_(2))`