Show that for a first order reaction, the time taken to complete half of the change is Independent of the initial concentration of the reactant.

Explanation

Half Life of a Reaction : It is defined as the time required for the completion of half of the reaction or the time during which the concentration of the reactants is reduced to half of the initial concentration. It is denoted by `t_(1//2)`.
Let us consider the half life period of a first order reaction. For a first order reaction,
`k=(2.303)/(t)log([R]_0)/([R])`
or `t=(2.303)/(t)log([R]_)/([R])`
When, `[R]=([R]_0)/(2), then t=t_(1//2)`
`t_(1//2)=(2.303)/(k)log([R]_0)/(([R]_0)/(2))`
`t_(1//2)=(2.303)/(k)log2=(2.303xx0.3010)/(k)`
`t_(1//2)=(0.693)/(k)`
Thus, it is quite clear from the above expression that the half life period for a first order reaction does not depend upon initial concentration of the reactant.