Question

Derive the relation between rate constant and half life period of zero order reaction

Answer 1

Answer : The relation between the rate constant and the half life period of zero order reaction is,

$t_{1/2}=\frac{[x]_0}{2k}$

Solution:

Zero order reaction are those in which the rate is independent on the concentration of the reactant. It is directly proportional to the rate constant.

For a reaction,

x  →  products

Rate expression for zero order reaction,

$Rate=-\frac{d[x]}{dt}=k[x]_0=k=Constant$

By rearranging the terms,

d[x] = - k dt     ..........(1)

Integrating equation (1), we get

$\int_{[x]_0}^{[x]}d[x]=\int_{o}^{t}-k\times dt$

At t = 0, the concentration of reactant [x] = $[x]_{0}$ which is the initial concentration of reactant.

$[x]=[x]_0-kt$     ........(2)

Half life of the reaction : It is the time required to reduce the concentration of the reactant to half of its initial value. It is represented by $t_{1/2}$.

Now for half life reaction, the concentration of reactant become half.

$[x]=\frac{1}{2}[x]_0$    and     $t=t_{1/2}$

By Using these conditions in equation (2), we get the relation between half life and the rate constant.

$\frac{1}{2}[x]_0=[x]_0-kt_{1/2}$

$t_{1/2}=\frac{[x]_0}{2k}$

The relation between the rate constant and the half life of zero order reaction is,

$t_{1/2}=\frac{[x]_0}{2k}$