Differential and Integrated Rate Equation

Differential and Integrated Rate Equation Zero and First Order Kinetics
Differential and Integrated Rate Equation Zero and First Order Kinetics

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Differential and Integrated Rate Equation of Zero Order Kinetics

A reaction is considered to be of zero order if its rate is independent of the concentration of the reactants.

Let us consider a general zero-order reaction.

image 53
Initially concentration At time t=0  a0
At time, t(a-x)       x

The rate of the zero-order reaction is;

Rate = K0 [A]0

image 54

The differential Rate Equation for the zero-order reaction is;

image 55

Now, integrating the above equation we get,

image 56

Where, I = integration constant

When t=0, x=0, then I=O

Substituting the value of I in equation (i), we get

x = K0 t

K0 = x/tโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆ. (ii)

Equation ii is the integrated rate equation of the zero-order reaction.

Graph of Zero-order Reaction

image 57
The concentration of reactant Vs Time
Rate of reaction vs Concentration
Rate of reaction vs Concentration
[Rate is independent of concentration]

Characteristics of Zero-Order Reactions

  • The concentration of the reactant has no effect on the rate of a zero-order reaction.
  • The concentration of product increases linearly with time.
  • The rate constant is equal to the rate of reaction at all concentrations.
  • The unit of zero-order reaction is mol L-1sec-1

The half-life period of Zero-order Reaction

It is the time required to decrease the concentration of reactant to half of its initial concentration. In other words, it is the time taken to complete half of the reaction to be completed. It is denoted by (t1/2).

For zero-order reactions,

During half-life, t = t1/2 and x =a/2

Now,

image 59

Hence, the half-life period of the zero-order reaction is directly proportional to the initial concentration of reactants.

Examples of Zero-Order Reaction

Examples of zero-order reactions that are independent of reactant concentration include the following. Zero-order reactions are those that require a catalyst and are saturated by the reactants.

  • The reaction of hydrogen with chlorine (Photochemical reaction).
image 60
  • Decomposition of nitrous oxide over a hot platinum surface.
image 61
  • Iodization of Acetone (In H+ ion-rich medium)
image 62

Differential and Integrated Rate Equation of First Order Reaction

A first-order reaction can be defined as a chemical reaction in which the reaction rate is linearly dependent on the concentration of only one reactant. In other words, a first-order reaction is a chemical reaction in which the rate varies based on the changes in the concentration of only one of the reactants. Thus, the order of these reactions is equal to 1.

Let us consider a general reaction;

image 63
Initially concentration At time t=0  a0
At time, t(a-x)       X

The rate of the zero-order reaction is;

image 64

Integrating both sides we get,

image 65

When t=0, x=0 then;

-In a = C โ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆโ€ฆ (iii)

Substitute Value of c in eqn (ii)

-In (a-x) = K1t – In a

In a – -In (a-x) = K1

image 66

This is the integrated equation of the first-order reaction.

Graphical Representation of First-order Reaction

Equation iv can be written as;

image 67

Equation (v) is in the form of y = mx + c

image 68

Now, plotting a graph between log(a-x) against t, we obtained a straight line with the slope K1/2.303. Similarly, a graph of log a/(a-x) against โ€˜tโ€™ gives a straight line passing through the origin.

image 69

Characteristics of First-order Reaction

  • The unit of the rate constant of a first-order reaction is secโˆ’1 or timeโˆ’1
  • The units of rate constant do not depend on the units of concentration expressed.
  • The rate constant remains unchanged throughout the reaction.
  • The time required to complete a definite fraction of the reaction is independent of the initial concentration.

The half-life of First-order Reaction

The half-life of a reaction is the time frame in which a reactantโ€™s concentration is reduced to one-half of its initial concentration. It can be calculated as follows

From the integral form, we have the following equation

image 70

At half-life, t = t1/2 and [A] = [A]o/2. Substituting these values in the equation for concentration, we get

image 71

Thus, the half-life of a first-order reaction is equal to 0.693/k (where โ€˜kโ€™ denotes the rate constant, whose units are s-1). It is independent of the initial concentration of reactants.

image 72

Examples of First-order Reaction

Here are some examples of first-order chemical reactions.

  • Trimethyl bromomethane ((CH3)3CBr) reacts with sodium hydroxide (NaOH) to give trimethyl methanol ((CH3)3COH) and sodium bromide (NaBr).

(CH3)3CBr + NaOH โ†’ (CH3)3COH + NaBr

  • Decomposition of hydrogen peroxide (H2O2) into water (H2O) and oxygen (O2)

2 H2O2 (l) โ†’ H2O (l) + O2 (g)

  • Hydrolysis of methyl acetate (CH3COOCH3) in the presence of mineral acids

CH3COOCH3 (aq.) + H2O (l) โ†’ CH3COOH (aq.)  + CH3OH (aq.)

  • Decomposition of ammonium nitrite (NH4NO2) in aqueous solution

NH4NO2 (aq.) โ†’ N2 (g) + 2 H2O (l)

C2H4ย (g) + H2ย (g) โ†’ C2H6ย (g)

Differential and Integrated Rate Equation of Second-Order Kinetics

A chemical reaction whose rate depends upon two concentration terms is called second-order
reaction. These chemical reactions depend on either the concentrations of two first-order
reactants or the concentration of one-second-order reactants.

CASE I: Reaction involving one reactant

Let us consider a general reaction with one reactant.

A + A โ†’ Products

Initially concentration At time t=0  a0
At the time, t(a-x)       x

The rate law for the second-order reaction is given by;

image 74

This is the differential rate law for second-order reactions.
Rearranging equation (ii) we get;

image 75

Integrating both sides we get;

image 76

Applying Boundary condition; t=0, x=0, we get;

image 77

This is the required integrated rate law for second-order reactions when the concentration of
reactant molecules are the same.

Graphical Representation

Taking Eqn (iii) and comparing with y =mx +c

image 78

Now plotting 1/aโˆ’x vs t we get, the following graph

image 79

Examples

image 80

CASE II: Reactant involving two reactants with different concentrations

Let us consider a general reaction with one reactant.

A + B โ†’ Products

Initially concentration At time t=0  ab0
At the time, t(a-x)       (b-x)x

The rate law for the second-order reaction is given by;

image 81

This is the differential rate law for second-order reactions.
Rearranging equation i) we get;

image 82

Applying boundary condition, x=0, t = 0 we get;

image 83

Substitute value of C in eqn (iii)

image 84

This is the required integrated equation for second-order reaction with different reaction
concentrations.

Graphical Representation

Rearranging eqn (iv)

image 85
image 86
image 87

Examples

image 91

Characteristics of a second-order reaction

  • The concentration(s) of one or two second-order reactants or one first-order reactant will determine the outcome of a second-order reaction.
  • A second-order reaction is classified as being simple if its rate is proportional to the square of the concentration of a single component.
  • The units for concentration and reaction rate are respectively moles per liter per second (mol L-1 s-1)

The half-life of Second-order Reaction


The half-life of a reaction is the time frame in which a reactantโ€™s concentration is reduced
to one-half of its initial concentration. It can be calculated as follows
From the integral form, we have the following equation

image 88

At half-life, t = t1/2 and [x] = [a]/2. Substituting these values in the equation for
concentration, we get;

image 89
image 90

Thus, the half-life of the second-order reaction is inversely proportional to initial
the concentration of reactant.

References

  • https://testbook.com/chemistry/second-order-reaction
  • https://www.aakash.ac.in/important-concepts/chemistry/second-order-reaction
  • https://collegedunia.com/exams/second-order-reactions-derivation-rate-law-and-graphs-chemistry-articleid-1578
  • https://byjus.com/chemistry/rate-law-laws-of-motion/
  • https://byjus.com/chemistry/integrated-rate-equation/

About Author

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Jyoti Bashyal

Jyoti Bashyal, a graduate of the Central Department of Chemistry, is an avid explorer of the molecular realm. Fueled by her fascination with chemical reactions and natural compounds, she navigates her field's complexities with precision and passion. Outside the lab, Jyoti is dedicated to making science accessible to all. She aspires to deepen audiences' understanding of the wonders of various scientific subjects and their impact on the world by sharing them with a wide range of readers through herย writing.

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