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

Derivation

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 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 = t1/2

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
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#### APPEARS IN

Balbharati Chemistry 12th Standard HSC for Maharashtra State Board
Chapter 6 Chemical Kinetics
Exercises | Q 3.04 | Page 136