**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__

• 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 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__

• 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__

• 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, A_{0 }at t=0 time, and absorbance, At at t=t

or, ** ****k0= A _{0}−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 – k _{0}t………….(4)**

• The units for k_{0} 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) / k _{0 }=
0.1a / k_{0 }………(6)**

• Units- time/conc

__Apparent Zero Order Reaction__

__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__

• 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 c_{0} = -k_{1}(t-0)**

**ln c _{t} = ln c_{0}-k_{1}t………………(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}= c_{0}/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) c_{0 } and t=t_{90}**

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

__Pseudo First Order Reaction Reaction__

__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 = k _{2}[A][B]…………..(10)**

• Where A and B are the reactants in the reaction and k_{2 }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 = k _{2}[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__

__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

k_{2} 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__

__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 / a^{n−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

**t _{90} =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 = k _{2}[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 / ak_{2}**

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

– Graphical method

– Half-life method

– Substitution method

**Chemical Kinetics FAQs**

**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.

**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.

**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.

**What distinguishes chemical kinetics from thermodynamics?**- Chemical kinetics deals with reaction rates and mechanisms, while thermodynamics focuses on the energy changes during reactions.

**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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