# Half-life Of A Reaction

The half-life of a reaction is generally denoted by t1/2. The half-life of reactions depends on the order of reaction and takes different forms for different reaction orders. From the integrated rate equations, concentration of reactants and products at any moment can be determined with the knowledge of time, initial concentration and rate constant of the reaction. Similarly, we can determine time too, with the knowledge of other two variables. The time in which the concentration of the reactant is reduced to one-half of the initial value is known as the half-life of a reaction.

## Zero order reaction

In zero order reaction, the rate of reaction depends upon the zeroth power of concentration of reactants. From the integrated rate equation for a zero order reaction with rate constant, k, concentration at any time, t is expressed as,

A $\rightarrow$ B

$[A]$ = $-kt ~+~ [A]_0$

From the definition of half-life of a reaction, at $t$ = $t_{\frac{1}{2}};~ [A]$ = $\frac{[A]_0}{2}$

$\Rightarrow$ = $-kt_{\frac{1}{2}}~+~ [A]_0$

$\Rightarrow~-kt_{\frac{1}{2}}$ = $-\frac{[A]_0}{2}$

$\Rightarrow~ t_{\frac{1}{2}}$ = $\frac{[A]_0}{2k}$

Hence, from the above equation we can conclude that the half life of a zero order reaction depends on initial concentration of reacting species and the rate constant, k. It is directly proportional to initial concentration of the reactant whereas it is inversely proportional to the rate constant, k.

## First order reaction

In first order reaction, the rate of reaction depends upon the first power of concentration of reactants. From the integrated rate equation for a first order reaction with rate constant, k, concentration at any time, t is expressed as,

$A \rightarrow B$

$ln ~[A]$ = $-kt~ + ~ln ~[A]_0$

From the definition of half-life of a reaction, at $t$ = $t_{\frac{1}{2}}; [A]$ = $\frac{[A]_0}{2}$

$\Rightarrow~ln \frac{[A]_0}{2} – ln ~[A]_0$ = $-~kt_{\frac{1}{2}}$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\Rightarrow~ln~\frac{\frac{[A]_0}{2}}{[A]_0}$ = $-kt_{\frac{1}{2}}$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\Rightarrow~ln~\frac{1}{2}$  = $-kt_{\frac{1}{2}}$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\Rightarrow~ t_{\frac{1}{2}}$  = $\frac{ln~2}{k}$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\Rightarrow~ t_{\frac{1}{2}}$  = $2.303\frac{log_{10}2}{k}$

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\Rightarrow~ t_{\frac{1}{2}}$  = $\frac{0.693}{k}$

From the above equation, we observe that the half life of a first order reaction is independent of initial concentration. It only depends on the value of rate constant. It is inversely proportional to the rate constant of the reaction. From the half life formula for the first order reaction and zero order reaction it can be concluded that half life of a reaction depends on order of reaction.

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#### Practise This Question

The half-life of a first order reaction is 24 hrs. If we start with 10 M initial concentration of the reactant then conc. after 96 hrs will be