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First order reaction, AtoB requires acti...

First order reaction, `AtoB` requires activation energy of 70 kJ `"mol"^(-1)`. When a `20%` solution of A was kept at `25^(@)C` for 20 min. `25%` decomposition in the same time in a `30%` solution maintained at `40^(@)C` ? Assume that activation energy remains constant in this range of temperature.

Text Solution

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We know that the fraction of reactants reacted in a first order reaction is independent of initial concentration. So, we need not worry about the terms `20%` solution and `30%` solution.
For first order reaction, `k=(2.303)/(t)"log"(a)/(a-x)`.
Given, `t=20` min., `a=100,(a-x)=100-25=75`
`:." "k=(2.303)/(20)"log"(100)/(75)=1.44xx10^(-2)"min"^(-1)`
Let, `k_(2)` be the rate constant of this reaction at `40^(@)C`
According to Arrhenius equation,
`"log"(k_(2))/(k_(1))=(E_(a))/(2.303R)[(1)/(T_(1))-(1)/(T_(2))]`
`"log "k_(2)=(70xx10^(3)" J mol"^(-1))/(2.3030xx8.314" JK"^(-1)"mol"^(-1))[(1)/(298K)-(1)/(313K)]`
`k_(2)=5.56xx10^(-2)"min"^(-1)`
Now, again using first order reaction
`"log"(a)/(a-x)=(k)/(2.303),t=(5.56xx10^(-2)"min"^(-1))/(2.303)xx20min=0.33`
`:.` Percentage of A remaining `=33%`
Percentage of A reacted `=100-33=67%`
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