Half life of zero order reaction: Definition, derivation, graphical representation and numericals

Half life of zero order reaction is used to represent the rate of reaction. The definition of the half-life period and the half-life period of zero-order reaction has discussed here.

Half life of a chemical reaction

Half life period of a chemical reaction is defined as the time required to decrease the concentration of reactant to half of its initial concentration. Half-lifetime is the time required for one-half of the reaction to be completed. It is denoted by t_1/2.

Half life of zero order reaction/how to find half life of zero order reaction?

The half-life of zero-order reaction can be derived by using integrated rate law expression of zero-order reaction.

Consider a general zero-order reaction

A → Product

The rate law expression is given by,

Rate=dx/dt=k[A]^o

dx/dt=k

dx=kdt…………………………………………………………….(1)

on integration this equation (1), we get the integrated rate law expression as:

x=kt …………………………………………………..(2)

This is integrated rate law expression for zero order reaction.

Now, At half-life time, t becomes t_1/2 and the concentration of reactant becomes half of its original value. Therefore,

t=t_1/2 and x=a/2. Putting these values, equation (2) becomes,

a/2= k t_1/2

t_1/2= a/2k ……………………………………………………………….(3)

This is the expression of the halflife period of zero-order reaction. This equation indicates that the half-life period of the zero-order reaction is directly proportional to the initial concentration of the reactant. On increasing the concentration of reactant twofold, the half-life period will be doubled. In general,

t_1/2 ∝ a ……………………………………………………………………..(4).

where a= initial concentration of reactant.

Graphical representation of half life time vs concentration

For zero-order reaction, the half-life time can be plotted against the concentration of reactant, a straight line passing through the origin and having a slope equal to 1/2k is obtained as shown in the figure.

Numerical problems related to Zero-order reaction

Question No.1: If a zero-order reaction completed 50% in 30 minutes, how much time will take it to complete 90%?

Let a=100 mole L^-1

First case: Reaction is 50% completed which means x=50 mole L^-1 so, for zero-order reaction,

x=kt

k=x/t = 50/30= 5/3

Second case: Reaction is 90% completed which means x=90 mol L^-1 so,

k=90/t

Putting the value of k,

5/3=90/t

t= 54 min

Hence, it takes 54 minutes to complete the 90% reaction.

Question No. 2: If a zero-order reaction completes 50% in 20 minutes, how much time will it take to complete a 75% reaction?

Solve yours.

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