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 of zero order reaction is,

$t_{1/2}=\frac{[A]_0}{2K}$

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,

A  →  products

Rate expression for zero order reaction,

Rate = $-\frac{d[A]}{dt}=K[A]^0$ = K = constant

By rearranging the terms,

d[A] = - K dt     ..........(1)

Integrating equation (1), we get

$\int_{[A]_0}^{[A]}d[A]=\int_{0}^{t}-Kdt$

At t = 0, the concentration of reactant [A] = $[A]_0$ which is the initial concentration.

$[A]=[A]_0-Kt$       ........(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.

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

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

$\frac{1}{2} [A]=[A]_0-Kt_{1/2}$

$t_{1/2}=\frac{[A]_0}{2K}$

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

$t_{1/2}=\frac{[A]_0}{2K}$

Answer 2

The relationship between rate constant and half-life period of zero-order reaction is given by:

t=k[A0]−[A]

At t=t1/2, [A]=21[A0]

t1/2=2k[A0]