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Answer in Brief

Derivation

**Answer the following in brief.**

Obtain the relationship between the rate constant and half-life of 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 a reactant at t = 0. It falls to [A]_{t} at time t after the start of the reaction. T

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 = t_{1/2}

Putting this condition in the integrated rate law we write

k = `2.303/"t"_(1/2) "log"_10 ["A"]_"t"/(["A"]_0/2) = 2.303/"t"_(1/2) "log"_10 2`

Substituting 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"`

Concept: Integrated Rate Law

Is there an error in this question or solution?

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