Chemical Kinetics

Learning Objectives

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

• Define rate, order of a reaction and molecularity

• Explain the use of apparent zero-order kinetics in 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 the half-life and shelf life of pharmaceutical products and drugs

• Determine the order of a reaction

Chemical Kinetics

• Chemical kinetics involves 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 that is equal to the number of molecules or atoms that must collide simultaneously to give the products

• Unimolecular reactions- one type of molecule stoichiometrically participates 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 the 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 multi sulfonamide 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

• Negative sign indicates color fading

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

or,                                                                                   k0= A₀−A_t / 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 the 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 the concentration of undecomposed drug at time t, and k1 is the first order rate constant

• When [A] is maintained constant due to the 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 catalyzed 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 c_t – ln c₀ = -k₁(t-0)

ln c_t = 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 represents 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

c_t= c₀/2
and t=t_1/2

• Substituting the terms in equation (4) gives:

Shelf life

The term in equation (4) can be changed to

c_t= (90/100) c₀       and t=t₉₀

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

Pseudo First Order Reaction Reaction

• It is a reaction that 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 the second order rate constant

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

−dc / dt  = k₂[A][constant]…………..(11)

Examples

• Hydrolysis of esters catalyzed by H+ ions

• Base catalyzed oxidative degradation of prednisolone in an aqueous solution

• Acid catalyzed 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 concentrations 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 the 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 experimentally

• The methods employed to determine the order of a reaction are:

Graphical method

• The kinetic experiment is conducted and the data are collected

• The data are plotted on 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 data are collected

• The data 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 t_1/2 is calculated by using the equation for each order

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

T_1/2 = 1 / aⁿ⁻¹  ………….(6)

Where n is the order of the reaction

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

• The half-life t_1/2(1) and t_1/2 (2) are related as follows

Where n is the order of the reaction

Chemical Kinetics Summary

• Rate- The 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 that is equal to the number of molecules or atoms that must collide simultaneously to give the products

• Order – The 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 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 behavior

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

t_1/2 =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 that 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 catalyzed 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 of 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

t_1/2 = 1 / ak₂

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

– Graphical method

– Half-life method

– Substitution method

Chemical Kinetics FAQs

1. What is the Arrhenius equation, and how does it relate to chemical kinetics?
   • The Arrhenius equation relates temperature, activation energy, and the rate constant of a reaction. It helps us understand how temperature influences reaction rates.
2. What is the role of catalysts in chemical kinetics?
   • Catalysts increase reaction rates by providing an alternative reaction pathway with lower activation energy, allowing reactions to occur more quickly.
3. How do concentration and temperature affect reaction rates?
   • Higher concentrations and temperatures generally lead to faster reaction rates by increasing the frequency of collisions and the energy of reactant molecules.
4. What distinguishes chemical kinetics from thermodynamics?
   • Chemical kinetics deals with reaction rates and mechanisms, while thermodynamics focuses on the energy changes during reactions.
5. What are some real-world applications of chemical kinetics?
   • Chemical kinetics is applied in pharmaceuticals, environmental science, and materials science to predict and control reaction behavior for practical purposes.

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