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Questions
Answer the following in brief.
Obtain the relationship between the rate constant and half-life of a first-order reaction.
Derive a relationship between the half-life and the rate constant for a first order reaction.
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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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