Chapter 13, Problem 13.129CP

Interpretation Introduction

Interpretation:

General equation for the half − life of a zeroth − order reaction of the type A ( Products, comparison of length of each half-life with previous one and compare first and second − order reactions in same manner should be determined.

Concept Introduction:

Consider a general reaction of the type A ( Products.
The rate of the reaction w. r. t. zero order can be expressed as, Rate = $-\frac{d[A]}{dt}=k{[A]}^0=k$ or

  $\text{d}\left[\text{A}\right]=-\text{kdt}.$

On integrating both sides, we get
$\left[\text{A}\right]=-\text{kt }+ \text{I}$. Where ‘I’ is an integration constant. If t = 0 means, the above equation changes as
[A] = [A]₀.
Hence the above equation can be written as $\left[\text{A}\right]=-\text{kt }+{\left[\text{A}\right]}_0$ or $\text{kt }={\left[\text{A}\right]}_0–\left[\text{A}\right]$ or
$\text{k }=\text{ 1}/\text{t }\left[{\left(\text{A}\right)}_0–\left(\text{A}\right)\right].$

Half-life period is the time in which half of the substance has reacted.
If [ $\text{A}\left]=\right[\text{A}{]}_0/\text{2},\text{ t }={\text{ t}}_{\text{1}/\text{2}}$.
The half-life can be expressed as ${\text{t}}_{\text{1}/\text{2}}={\left[\text{A}\right]}_0/\text{2k}$. Hence half − life period of a zero-order reaction is directly proportional to initial concentration, i.e., ${\text{t}}_{\text{1}/\text{2}}\alpha {\left[\text{A}\right]}_0$

The half − life expressions for zero − and second − order processes include the initial concentration, whereas that for a first − order process does not require, because it is independent of the initial concentration. This concludes that half − lives for zero − and second − order processes vary with initial concentration; each successive half − life requires a different amount of time than the one before it.