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Question
Answer the following in brief.
Obtain the relationship between the rate constant and half-life of a first-order reaction.
Solution
i. The integrated rate law for the first-order reaction is k = `2.303/"t" log_10 ["A"]_0/["A"]_"t"`
Where [A]0 is the initial concentration of the reactant at t = 0. The concentration falls to [A]t at time t after the start of the reaction. The concentration of the reactant falls to `["A"]_0/2` at time t1/2.
ii. The time required for [A]0 to become `["A"]_0/2` is denoted as `"t"_(1/2)` or `["A"]_"t" = ["A"]_0/2` at t = t1/2
Putting this condition in the integrated rate law we write
k = `2.303/"t"_(1/2) "log"_10 ["A"]_0/(["A"]_0/2)`
= `2.303/"t"_(1/2) "log"_10 2`
Substituting the value of `log_10 2`,
k = `2.303/"t"_(1/2) xx 0.3010`
∴ k = `0.693/"t"_(1/2)`
∴ `"t"_(1/2) = 0.693/"k"`
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