# Chemical Kinetics

Contents

• Introduction to chemical kinetics

• Concept of molecularity and order of a reaction

• Different types of order of a reaction including rate constants, half-life, shelf life, apparent or pseudo order

• First order reaction kinetics- examples, derivation, half-life and shelf life

• Pseudo first order reaction kinetics

• Second order reaction kinetics- examples, derivation, half-life and shelf life

• Determination of order of a reaction

Learning Objectives

At the end of this lecture, student will be able to:

• Define rate, order of a reaction and molecularity

• Explain the use of   apparent zero-order kinetics to the practice of pharmacy

• Describe the concept and applications of half-life and shelf life in the formulation and production of different pharmaceutical products and drugs

• Describe the principles and concepts of first order reaction kinetics

• Explain the importance of apparent first-order kinetics to the practice of pharmacy

• Describe half-life and shelf life of pharmaceutical products and drugs

• Describe the principles and concepts of second order reaction kinetics

• Calculate half-life and shelf life of pharmaceutical products and drugs

• Determine the order of a reaction

## Chemical Kinetics

• Chemical kinetics involve the study of the rate of a chemical process

• The rate, velocity or speed of a reaction is given by ± (dc/dt)

• dc is the small change in the concentration within a given time interval

• Negative sign indicates the decrease in concentration over a period of time

• Law of mass action explains the rate of a chemical reaction is proportional to the product of the molar concentration of the reactants each raised to a power equal to the number of molecules of the substance undergoing reaction

## Molecularity of a Reaction

• Molecularity is defined in terms of a number which is equal to the   number   of   molecules   or   atoms   that   must   collide simultaneously to give the products

• Unimolecular reactions- one type of molecule stoichiometrically participate in the reaction

Example-Isomerisation of trans-stilbene to cis-stilbene

• Bimolecular reaction- Two types of molecules stoichiometrically involved in the reaction

Example- Oxidation of hydrogen peroxide

• Termolecular reaction- Termolecular and other higher molecularity are seldom observed

• Three or more molecules having sufficient kinetic energy meeting simultaneously in the same region of space is unlikely

## Order of a Reaction

• Order of a reaction is defined as the number of concentration terms on which the rate of a reaction depends

• The overall order of reaction is equal to the powers of the concentration terms affecting the experimentally determined rate

• In contrast to molecularity, it is possible for the order of a reaction to assume fractional or zero values

## Zero Order Reaction

• Zero order reaction is defined as a reaction in which the rate does not depend on the concentration terms of the reactants

• Mathematically expressed as:

−dc / dt   =k0

Where, k0 is the specific rate constant

• Examples: - Colour loss of liquid multisulfonamide preparation

- Oxidation of vitamin A in an oily solution

- Photochemical degradation of chlorpromazine in aqueous solution

• Mechanism: The rate must depend upon some factor other than the concentration term

Derivation:

The rate equation for zero order can be written as

−dA / dt   =k0…………..(1)

Where A is the absorbance (optical density) of the preparation

• The concentration is measured in terms of optical density

• Integrating equation (1) between initial absorbance, A0 at t=0 time, and absorbance, At at t=t

or,                                                    k0= A0−At / t    …………..(2)

• The initial concentration is expressed as ‘a’ and the concentration at any time t, is ‘c’, then equation (2) becomes

k0=(a− c) / t   ………………(3)

• Equation (3) may be written as

c= a - k0t………….(4)

• The units for k0 are conc/time, if the conc. Is expressed in moles/liter, then k0 will be moles/liter.sec

Half life

• It is the time required for the concentration of the reactant to reduce to half of its initial concentration

• The half-life can be derived as follows:

c= a/2      and              t=t1/2

Substituting the values in the equation (3) gives:

• The unit for half-life period is sec/conc., min/conc. hr/conc. etc.

• Shelf life

• It is the time required for the concentration of the reactant to reduce 90% of its initial concentration

• Terms in equation (3) change to

C = 90a / 100 and  t= t90

• Substituting the values in equation (3) gives:

t90=(a− 0.9a) / k0 =  0.1a / k0 ………(6)

• Units- time/conc

## Apparent Zero Order Reaction

• Pseudo zero order is a reaction, which may be a first order, but behaves like a zero order

• In suspensions, drug degradation is a chemical reaction and follows an apparent zero order

• The rate equation can be written as:

d[A] / dt   =k1[A]…………..(7)

Where [A]is concentration of undecomposed drug at time t, and k1 is the first order rate constant

• When [A] is maintained constant due to reservoir of solids in the suspension, the rate equation (7) changes to

d[A]

-          ------ = k1 X constant = k0 …………..(8)

dt

## First Order Reaction

• First order reaction is defined as a reaction in which the rate of reaction depends on the concentration of one reactant

• The first order rate equation can be written as:

−dc / dt   α c , therefore, −dc / dt  = k1c

Where  c  is  the  concentration  of  the  reactant  and  k1   is  the specific rate constant for first order

Examples

• Decomposition of hydrogen peroxide catalysed by 0.02 M potassium iodide

• Acid hydrolysis of ethyl acetate and methyl acetate

• Diffusion of drugs across biological membranes

Derivation

• Rate expression is written as:

−dc / dt  = k1c………(1)

Integrating equation (1) between concentration c0 at time t=0 and concentration ct at time t=t gives:

ln ct – ln c0 = -k1(t-0)

ln ct = ln c0-k1t………………(2)

Converting equation (2) to logarithm to the base 10 gives:

Rearranging the above equation,

• Graphically the equation may be represented as:

• The unit for k1 is reciprocal time, hours-1, minutes-1

• Equation (4) can also be written as

• The exponential form of the first order rate equation is

The equation in logarithms to base 10

• The exponential form of first order kinetics represent that the curve will be asymptotic

Half life

• Time required to reduce the concentration of the reactant to half of its initial concentration the term in equation (4) can be changed to

ct= c0/2    and t=t1/2

• Substituting the terms in equation (4) gives:

Shelf life

The term in equation (4) can be changed to

ct= (90/100) c0       and t=t90

Substituting the terms in equation (4) and rearranging gives:

## Pseudo First Order Reaction Reaction

• It is a reaction which is originally a second order but made to behave like a first order

• In second order reaction, the rate depends on the concentration terms of two reactants, the rate equation would be:

−dc / dt = k2[A][B]…………..(10)

• Where A and B are the reactants in the reaction and k2 is second order rate constant

• In pseudo first order the concentration of one of the reactant is in large excess, and considered to be constant

−dc / dt  = k2[A][constant]…………..(11)

Examples

• Hydrolysis of esters catalysed by H+ ions

• Base catalysed oxidative degradation of prednisolone in aqueous solution

• Acid catalysed hydrolysis of digoxin

## Second Order Reaction Reaction

• It is defined as a reaction in which the rate depends on the concentration terms of two reactants each raised to the power one

• The rate equation can be written as

Where [A] and [B] are the concentration of A and B

k2 is the specific rate constant for second order

Examples

• Alkaline hydrolysis of esters such as methyl acetate or ethyl acetate

• Hydrolysis of chlorobutanol in presence of sodium hydroxide

Derivation: As per the definition the rate equation is

• Let ‘a’ and ‘b’ be the initial concentration of A and B, respectively, and ‘x’ be the concentration of each species reacting in time t

• Substituting the above terms in equation (1) gives:

• Considering a=b, the above equation changes to

• Integrating equation (2) employing the conditions x=0 at t=0 and x = x at t=t

• Equation (3) is the integral equation for second order reaction kinetics when a=b

• When a ‡ b, the integral equation is :

• Graphical representation of second order kinetics

Half life

• As per the definition the terms in equation (3) can be changed to

• Unit for half-life- time/ conc.

## Determination of Order of a Reaction

• The order of a reaction can be determined by experimentally

• The method employed to determine order of a reaction are:

### Graphical method

• The kinetic experiment is conducted and the datas are collected

• The datas are plotted on a graph paper

• The graph which gives a better fit for the straight line, the reaction is considered to be of that order

### Substitution method

• The kinetic experiment is conducted and the datas are collected

• The datas are substituted in the integral equation of zero, first and second order to get the k values

• The order in which the k values at different time period remain constant, the reaction is considered to be of that order

### Half-life method

• Initially the t1/2 is calculated by using the equation for each order

• The relationship between half-life and initial concentration is as follows:

T1/2 = 1 / an−1  ………….(6)

Where, n is the order of the reaction

• Alternatively an experiment is conducted at two different initial concentration a1 and a2

• The half-life t1/2(1) and t1/2 (2) are related as follows

Where n is the order of the reaction

Summary

Rate- Rate of a reaction is given by ± (dc/dt) dc is the small change in the concentration within a given time interval

Molecularity - It is defined in terms of a number which is equal to the number of molecules or atoms that must collide simultaneously to give the products

Order – Order of a reaction is defined as the number of concentration terms on which the rate of a reaction depends

Zero order reaction- It is defined as a reaction in which the rate does not depend on the concentration terms of the reactants

Pseudo zero order reaction- It is a reaction, which may be a first order, but behaves like a zero order

First order reaction- It is defined as a reaction in which the rate of reaction depends on the concentration of one reactant

Example- Acid hydrolysis of ethyl acetate and methyl acetate

First order reaction kinetics equation-

• First order reaction kinetics is monoexponential in nature

• The curve of a first order reaction kinetics shows asymptotic behaviour

The half-life of first order reaction kinetics is given by

t1/2 =0.693 / 1

• Shelf life of first order reaction kinetics is given by

t90 =0.105 / 1

• Pseudo first order reaction- It is a reaction which is originally a second order but made to behave like a first order

• Pseudo first order reaction is represented by

−dc / dt  = k2[A][constant]

Example for pseudo first order reaction kinetics- Hydrolysis of esters catalysed by H+ ions

Second order reaction - It is defined as a reaction in which the rate depends on the concentration terms of two reactants each raised to the power one

Example- Alkaline hydrolysis of esters such as methyl acetate or ethyl acetate

• Rate equation when a= b

• Rate equation when a ‡ b

Half-life equation for second order

t1/2 = 1 / ak2

• There are three methods of determination of order of a reaction:

- Graphical method

- Half-life method

- Substitution method