Preview pharmaceutical physical chemistry theory and practices by s k bhasin (2016)

Preview Pharmaceutical Physical Chemistry Theory and Practices by S. K. Bhasin (2016) Preview Pharmaceutical Physical Chemistry Theory and Practices by S. K. Bhasin (2016) Preview Pharmaceutical Physical Chemistry Theory and Practices by S. K. Bhasin (2016) Preview Pharmaceutical Physical Chemistry Theory and Practices by S. K. Bhasin (2016) Preview Pharmaceutical Physical Chemistry Theory and Practices by S. K. Bhasin (2016)

Pharmaceutical Physical Chemistry

Theory and Practices

S K Bhasin

No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent.

This eBook may or may not include all assets that were part of the print version The publisher reserves the right to remove any material present in this eBook at any time

ISBN 9788131765272

eISBN 9788131775981

Head Office: A-8(A), Sector 62, Knowledge Boulevard, 7th Floor, NOIDA 201 309, India

All Those Who Toiled in Shaping Me into What I Am Today

Preface xxi

1.1 Introduction 2 1.2 Gas Laws 2

1.2.1 Boyle’s Law 2 1.2.2 Charles Law 3 1.2.3 Avogadro’s Law 4 1.2.4 The Combined Gas Law Equation or the Gas Equation 4 1.2.5 Graham’s Law of Diffusion 6

1.2.6 Dalton’s Law of Partial Pressure 6

1.3 Kinetic Theory of Gases 6

1.3.1 Postulates (Assumptions) of Kinetic Theory 6

1.4 Derivation of Kinetic Gas Equation 7 1.5 Derivation of Gas Laws from Kinetic Equation 9

1.5.1 Some Useful Deductions from Kinetic Theory of Gases 12

1.6 Ideal and Real Gases 17

1.6.1 Ideal Gases 17 1.6.2 Real Gas 17

1.7 Deviations of Real Gases from Gas Laws 18

1.7.1 Deviations from Boyle’s Law 18

1.8 Causes of the Derivations from Ideal Behaviour 20 1.9 van der Waals’ Equation (Reduced Equation of State) (Equation of State for Real Gases) 20

1.9.1 Units of van der Waals’ Constants 23 1.9.2 Signifi cance of van der Waals’ Constant 24

1.10 Explanation of Behaviour of Real Gases on the Basis of van der Waals’ Equation 24

1.11 Isotherms of Carbon Dioxide—Critical Phenomenon 281.12 Principle of Continuity of States 30

1.13 Critical Constants 31

1.13.1 Relations Between van der Waals’ Constants and Critical Constants 31

3

1.13.3 Calculation of van der Waals’ Constants in terms of T c and P c 34

1.14 Law of Corresponding States 34

1.14.1 Signifi cance of Law of Corresponding States 35

1.15 Limitations of van der Waals’ Equation 36

2.4.1 Some Important Results 47 2.4.2 Effect of Temperature on Surface Tension 48 2.4.3 Measurement of Surface Tension 48 2.4.4 Surface Tension in Everyday Life 51 2.4.5 Surface Tension and Chemical Constitution (Parachor) 52

2.5 Viscosity 57

2.5.3 Effect of Temperature on Viscosity 60

2.5.5 Viscosity and Chemical Constitution 62

Multiple Choice Questions 81

3.5 Colligative Properties of Dilute Solution 95 3.6 Lowering of Vapour Pressure 95

3.6.1 Determination of Molecular Masses of Non-volatile Solute 96

3.7 Elevation in Boiling Point 99

3.7.1 Expression for the Elevation in Boiling Point 99 3.7.2 Calculation of Molecular Masses 101

3.8 Depression of Freezing Point 102

3.8.1 Expression for the Depression in Freezing Point 103 3.8.2 Calculation of Molecular Masses 104

3.9.6 Calculation of Molecular Masses from Osmotic Pressure 108

3.10 Abnormal Molecular Masses 112

3.10.1 Modifi ed Equation for Colligative Properties in Case of Abnormal Molecular Masses 114

4.2 Some Common Thermodynamics Terms 121

4.2.3 Reversible and Irreversible Processes 123

4.3 Zeroth Law of Thermodynamics 126

4.3.1 Absolute Scale of Temperature 126

4.4 Work, Heat and Energy Changes 127

4.4.3 Equivalence Between Mechanical Work and Heat 130

4.5 First Law of Thermodynamics 131

4.5.1 Mathematical Formulation of First Law of Thermodynamics 132 4.5.2 Some Special Forms of First Law of Thermodynamics 132 4.5.3 Limitations of the First Law of Thermodynamics 133

4.6 The Heat Content or Enthalpy of a System 135 4.7 Heat Capacities at Constant Pressure and at Constant Volume 136

4.7.1 Heat Capacity at Constant Volume 137 4.7.2 Heat Capacity at Constant Pressure 137 4.7.3 Relationship Between C p and C v 138

4.8 Joule-Thomson Effect 138

4.9 Reversible-Isothermal Expansion of an Ideal Gas 140

4.10 Second Law of Thermodynamics 142

4.10.1 Spontaneous Processes and Reactions (Basis of Second Law) 143

4.11 Entropy 145

4.11.1 Mathematical Explanation of Entropy 145 4.11.2 Entropy Change in Chemical Reaction 147 4.11.3 Units of Entropy 147

4.11.4 Physical Signifi cance of Entropy 147 4.11.5 Entropy Change Accompanying Change of Phase 147 4.11.6 Entropy Changes in Reversible Processes 148 4.11.7 Entropy Changes in Irreversible Processes 149 4.11.8 Entropy as Criterion of Spontaneity 150 4.11.9 Entropy Changes for an Ideal Gas 150

Multiple Choice Questions 156

5 Adsorption and Catalysis 159

5.1 Adsorption 1605.2 Types of Adsorption 1615.3 Factors Affecting Adsorption of Gases on Solids 1625.4 Adsorption Isobar (Effect of Temperature on Adsorption) 1635.5 Adsorption Isotherm (Effect of Pressure) 163

5.5.1 Explanation of Type I Isotherm 164 5.5.2 Freundlich Adsorption Isotherm 164

5.5.3 The Langmuir Adsorption Isotherm 165

5.5.5 Explanation of Type II and III Isotherms 167 5.5.6 Explanation of Type IV and V Isotherms 167

5.6 Theory of Adsorption 168 5.7 Gibbs’ Adsorption Equation 169 5.8 Applications of Gibbs’ Adsorption Equation 173 5.9 Equation for Multi-Layer Adsorption (B.E.T Equation) 176

5.9.1 Determination of Surface Area of the Adsorbent 178

5.10 Catalysis 179

5.10.1 Positive and Negative Catalyses 179

5.11 Homogeneous and Heterogeneous Catalyses 1795.12 How Does a Catalyst Work? 180

5.12.1 Characteristics of Catalytic Reactions 181

5.13 Mechanism of Homogeneous and Heterogeneous Catalyses 184

5.13.1 Signifi cant Characteristics of Heterogeneous Catalysis 186 5.13.2 Facts Explained by Adsorption Theory 187

6.4.1 Grotthus–Drapper Principle of Photochemical Activation:

(First Law of Photochemistry) 201 6.4.2 Stark–Einstein’s Law of Photochemical Equivalence—

6.5 Quantum Effi ciency 204

6.5.1 Explanation of the Unexpected Behaviour 204 6.5.2 Classifi cation of Photochemical Reactions (Based on their Quantum Effi ciencies) 205

6.6 Study of Some Photochemical Reactions 2086.7 Fluorescence and Phosphorescence 211

7.3 Rate Constant and Rate Equation 225

7.3.1 Differences Between Rate of a Reaction and Rate Constant 226

7.6.1 Characteristics of a Zero-order Reaction 232

7.7 Intergrated Rate Law Equation for First-order Reactions 234

7.7.1 Characteristics or Signifi cance of First-order Reaction 235 7.7.2 Examples of the Reactions of First Order 236

7.7.3 Pseudo First-order Reaction 239

7.10 Reactions of Higher Order 2517.11 Determination of Rate Law, Rate Constant and Order of Reaction 2517.12 Some Complications in Determination of Order of a Reaction 256

7.13 Temperature Dependence of Reaction Rates 258

7.13.1 Explanation of Effect of Temperature 258

8.3 Origin of Quantum Mechanics 275

8.3.1 Classical Mechanics versus Quantum (or wave) Mechanics 275

8.4 Black Body Radiations 276 8.5 Kirchoff’s Law 277

8.5.1 Spectral Distribution of Black Body Radiation 278

8.6 Stefan-Boltzmann Fourth Power Law 279 8.7 Wien’s Displacement Law 279

8.8 Planck’s Radiation Law 281 8.9 Postulates of Quantum Mechanics 2838.10 Operators in Quantum Mechanics 285

8.10.1 Types of Operators 286

8.11 Schrödinger Wave Equation 288

8.11.1 Derivation of Schrödinger Wave Equation 288

8.12 Eigenvalues and Eigenfunctions (or Wave Functions) 290

8.12.1 Physical Signifi cance of the Wave Function 290

8.13 Normalized and Orthogonal Eigenfunctions 2908.14 Concept of Atomic Orbital 291

8.15 Probability Distribution Curves 2928.16 Radial Probability Distribution Curves 292

8.16.1 Radial Probability Distribution Curve for 1s Orbital 293 8.16.2 Radial Probability Distribution Curves for other s Orbitals 293 8.16.3 Comparison of Radial Probability, Distribution Curves for 1s with Other s Atomic Orbitals 294

8.16.4 Radial Probability Distribution Curves for p Orbitals 295 8.16.5 Comparison of Radial Probability Distribution Curves for 2s and 2p Orbitals 295

8.16.6 Comparison of Radial Probability Distribution Curves for 3s, 3p and 3d Orbitals 295

9.2 Arrhenius Theory of Ionization 302

9.2.1 Degree of Dissociation or Ionization 303

9.3 Ionisation of Weak Electrolytes—Ostwald’s Dilution Law 304

9.3.1 Verifi cation of Ostwald’s Dilution Law 305

9.4 Arrhenius Concept of Acids and Bases 306

9.4.1 Limitation of Arrhenius Theory 306

9.5 Ionisation Constant of Weak Acids and Bases (Arrhenius Concept) 307 9.6 Bronsted–Lowry Concept of Acids and Bases 308

9.6.1 Conjugate Acid Base Pairs 309 9.6.2 Relative Strength of Acids and Bases 310 9.6.3 Limitation of Bronsted—Lowry Theory 311

9.7 Lewis Concept of Acids and Bases 311

9.7.1 Limitations of Lewis Concept 312

9.8 Ionic Product of Water 313

9.8.1 Concentrations of H 3 O + and OH – ions in Aqueous Solutions of Acids and Bases 314

10.4 Thermodynamic Derivation of Distribution Law 329

10.5 Distribution Law and Molecular State of Solute 331

10.5.1 Case I: When the Solute Undergoes Association in One of the Solvents 331 10.5.2 Case II: When the Solute Undergoes Dissociation in One of the Solvents 334 10.5.3 Case III: When the Solute Enters into Chemical Combination with One of the Solvents 335

10.6 Applications of Distribution Law 336

10.6.1 Determination of Solubility of a Solute in a Solvent 337 10.6.2 Determination of Molecular State of Solute in Different Solvents 337 10.6.3 Determination of Distribution Indicators 337

10.6.4 Study of Complex Ions 338 10.6.5 In the Process of Extraction 339 10.6.6 Application of Principle of Extraction To Desilverization of Lead 342 10.6.7 Determination of Degree of Hydrolysis 343

11.3.7 Experimental Measurement of Conductance 359 11.3.8 Effect of Dilution on Conductance 363

11.6 Transport Number 369

11.6.1 Important Relations Concerning Transport Number 369 11.6.2 Factors Controlling Transport Number 370

11.6.3 Determination of Transport Numbers 371

11.7 Limitations of Arrhenius Theory 37511.8 Modern Theory of Strong Electrolytes 376

12.6 Electrochemical Series 393

12.6.1 Applications of Electrochemical Series 394

12.7 Cell Potential or EMF of a Cell 396

12.7.1 Calculation of EMF of a Cell 396

12.8 Derivation of Nernst Equation (Concentration Dependence of Electrode Potential) 399

12.8.1 Application of Nernst Equation 400

13.5 Elements of Symmetry 419

13.5.1 Plane of Symmetry and Refl ections 419 13.5.2 Axis of Symmetry or Axis of Rotation 419 13.5.3 Centre of Symmetry or Inversion Centre 421 13.5.4 Improper Axis or Rotation Refl ector Axis and Improper Rotation 421 13.5.5 Axis of Rotation Inversion 421

13.5.6 Total Elements of Symmetry 421

13.11 Types of Crystalline Solids 433

14.3.5 Bonding and Anti-bonding Molecular Orbitals in Terms of Wave Functions 447

14.3.6 Characteristics of Bonding and Anti-bonding Molecular Orbitals 448 14.3.7 Combination of Atomic Orbitals—Sigma (s) and Pi (p) Molecular Orbitals 448

14.4 Relative Energies of Molecular Orbitals and Filling of Electrons 45114.5 Stability of Molecules 453

14.5.1 Stability of Molecules in Terms of Bonding and Anti-bonding Electrons 453

14.5.2 Stability of Molecules in Terms of Bond Order 453

14.6 Molecular Orbital Confi gurations 454

14.6.1 Bonding in Some Homonuclear Diatomic Molecules and Ions – Electronic Confi gurations 454 14.6.2 Helium Ion, He 2 + 457

14.6.3 Nitrogen Molecule, N 2 459 14.6.4 Oxygen Molecule, O 2 460 14.6.5 The Fluorine Molecule, F 2 462 14.6.6 Hypothetical Neon Molecule, Ne 2 462 14.6.7 Molecular Orbital Electronic Confi guration of Some Common Heteronuclear Molecules 462

14.7 Comparison of Valence Bond (VB) Theory and Molecular Orbital (MO) Theory 463

14.7.1 Points of Similarly 463 14.7.2 Points of Difference 463

15.2.1 True and Metastable Equilibrium 470

15.2.4 Degrees of Freedom or Variance 474

15.3 Mathematical Statement of Phase Rule 474 15.4 Phase Diagrams 476

15.9.1 Characteristics of Eutectic Point 485 15.9.2 Use of Eutectic Systems 486

15.9.4 Pattinson’s Process for Desilverization of Lead 488

15.10 Type B—System in Which Two Components form a Stable Compound (Zinc–Magnesium Alloy System) 488

15.10.1 Eutectic Points and Congruent Melting Point 490

15.11 Type C—The Two-component Form: A Compound With Incongruent Melting Point 490

15.11.1 Sodium – Potassium System 490

15.12 Thermal Analysis (Cooling Curve) 492

Object 535, Theory 535, Materials 535, Description of the Apparatus 535, Procedure 536, Observations and Calculations 536, Calculations 537, Result 537, Precautions 537, Viva-voce 538

Object 540, Theory 540, Apparatus/Reagents Required 540, Procedure 540, Observations 541, Calculations 541, Calculate Values of K for Each Set 542, Results 542, Precautions 542, Viva-voce 542

Object 543, Theory 543, Apparatus Reagents Required 543, Method 543, Observations 544, Burette Readings 544, Calculations 544, Result 545, Precautions 545, Viva-voce 545

It gives me immense pleasure to place before a large community of pharmacy students, my humble work

on Pharmaceutical Physical Chemistry, written in accordance with the recent syllabus prescribed for the

B.Pharma (II Semester) students of all Indian universities My aim in writing this book is to present the fundamental principles of physical chemistry for the pharmacy students on modern lines No single book on pharmacy that covers the revised syllabus exclusively is available in the market Keeping in view the requirement of the students and the teachers, this book has been written to cover all the topics with the desired limits of the prescribed syllabus I hope the book will be useful and meets the requirements

of students at large

Based on my vast teaching experience, I have prepared the text in a simple, lucid and comprehensive style, keeping in view the diffi culties of the students; in addition, the manuscript has been presented as if the teacher is talking to the students in the class Th roughout the text, special care has been taken to add

‘Review Problems for Tutorials’ at relevant stages for the students to assess their grasp of the topic

cov-ered Students are advised to solve the problems themselves and look for their solution only aft erwards

An added feature of this book is that it contains the ‘Laboratory Manual’, which contains 20

experi-ments covering the syllabus in the practical of all Indian universities along with the ‘Viva-Voce’ at the end of each experiment In this section, we have described the theory of the experiment in details before giving the procedural details

A large number of solved and unsolved ‘Numerical Problems’ have also been included wherever required Students should solve the unsolved numerical problems in the tutorial classes under the guid-

ance of their learned teachers

Here is a book that resolves all your queries and doubts Th is book discusses the fundamentals of pharmaceutical physical chemistry required for a pharmacist Th e book will not only help you to score better in the examination but would also develop your skills to apply your knowledge of pharmaceutical physical chemistry in solving the problems that you may face during your pharmaceutical career

I sincerely express my thanks to the authors and the publishers whose works I have consulted in

pro-ducing this book I am grateful to the editors of Pearson Education for the sustained interest shown by them during the publication of this book

Although care has been taken while preparing and typing the manuscript, yet errors and misprints might have crept in; I shall be grateful to the students and teachers who would be kind enough to send their suggestions for the further improvement of the book

I shall consider my eff orts amply rewarded if all those for whom the book is intended are benefi ted

by it

Dr S K Bhasin

S.K Bhasin is presently Director and Professor in Chemistry in the Global Research Institute of

Management and Technology, Radaur A matured academician, experienced teacher, established author and devoted researcher, Dr Bhasin has a teaching experience of over 50 years; 35 years of which relates

to teaching of undergraduate and postgraduate students in chemistry and 15 years of teaching in

profes-sional institutes

He is a ‘life fellow’ of four professional bodies, namely, the International Congress of Chemistry and Environment (FICCE), Fellow of Indian Council of Chemists (FICC), Fellow of Indian Association of

Environmental Management (FIAEM) and Fellow of Indian Journal of Environmental Protection (FIJEP)

He is a member on the Editorial Board of International Journal of Environmental Research He has

pub-lished 64 research papers in reputed international and national journals and has pubpub-lished 14 articles in national-level magazines and newspapers He has also convened a number of national-level conferences and workshops and has acted as resource person in many conferences

He has presented his research papers in international conferences abroad—two research papers in Kuwait in 2007 and one paper in Th ailand in 2009 Recently, he has presented his research paper in an international conference held at Malaysia, May 2011 organized by the Indian Congress of Chemistry and Environment (ICCE) and yet another international conference organized by the Indian Council of Chemists (ICC), June 2011, at Bangkok (Th ailand)

He has acted as Chairman of ICC chapter at M.L.N College Yamuna Nagar for consecutive three years He has been honoured with appreciation awards twice at international conferences organized

by the ICCE in the year 2009 and in Malaysia in 2011 Has also been decorated with Dronacharya Award in recognition of his services in the fi eld of education and research He has over 24 graduate-level books in chemistry, engineering and pharmacy to his credit

A vacuum is nothing and what is nothing cannot exist

1.3 Kinetic Theory of Gases

1.4 Derivation of Kinetic Gas Equation

1.5 Derivation of Gas laws from Kinetic Equation

1.6 Ideal and Real Gases

1.7 Deviations of Real Gases from Gas laws

1.8 Causes of the Derivations from

Ideal Behaviour

1.9 van der Waals’ Equation (Reduced Equation

of State) (Equation of State for Real Gases)1.10 Explanation of Behaviour of Real Gases

on the Basis of van der Waals’ Equation1.11 Isotherms of Carbon

Dioxide-Critical Phenomenon1.12 Principle of Continuity of States1.13 Critical Constants

1.14 law of Corresponding States1.15 limitations of van der Waals’ Equation

1.1 INTRODUCTION

All matter exists in one of the three states of aggregation: solids, liquids or gases A solid has definite shape and volume due to strong intermolecular attractive forces A substance will be a solid if it melts at

a temperature higher than room temperature under atmospheric pressure

In the liquid state, the attractive forces are relatively weak and it has a definite volume but no definite shape A substance will be a liquid if its freezing point (i.e., melting point) is below the room temperature under atmospheric pressure

In the gaseous state, the molecular forces are much weaker Hence, due to possible random motion in all directions, gases have no bounding surface and lend to fill completely any available space The gases thus have neither definite shape nor definite volume A substance will be a gas if its boiling point is below room temperature under atmospheric pressure

Out of the three states of matter, i.e., solid, liquid and gases in which the different substances exist, the gases show the most uniform behaviour irrespective of the nature of the gas Some common properties

of gases are given below:

(vi) exert pressure on the walls of the vessel in which they are contained.

In addition to the above general properties of gases, another important feature of all gases is that they

obey certain gas laws such as Boyle’s law and Charles’ law which are briefly outlines below A gas which obeys the gas laws under all conditions of temperature and pressure is known as an ideal gas However,

gases deviate from behaviour especially at low temperature and high pressure The concept of ideal gases

is only hypothetical Gases obey gas laws at high temperature and low pressure usually and are called real gases In this chapter, we will discuss the deal gas behaviour in terms of kinetic theory of gases and also the deviation of real gases from ideal behaviour

1.2 GAS LAWS

1.2.1 Boyle’s Law

This law was proposed by Robert Boyle in 1662 based on his experimental study of the variation of

volume of a gas with change of pressure at constant temperature The law states as follow:

Temperature remaining constant, the volume of a given mass of a gas is inversely proportional to its pressure.

Mathematically, if V is the volume of a gas at pressure P, then

V P

∝ 1, if temperature is kept constant

or PV = constant, if temperature is kept constant (1.1)

In other words, if at constant temperature, V1 is the volume of a gas at pressure P1 and on changing

the pressure to P2, the volume changes to V2, then

1.2.2 Charles Law

Charles in 1787 observed that at constant pressure, the volume of every gas increased by the same

amount when heated from 0 °C to a particular temperature Gay-Lussac in 1802 found the exact increase

in volume for every degree rise of temperature Based on their study, they put forward the following generalization

Pressure remaining constant, the volume of a given mass of a gas increases or decreases by 1/273 of its volume at 0 °C for one degree rise or fall of temperature.

Thus, if V0 is the volume of a gas at 0 °C and V t is the volume at t °C, then mathematically, the law may

This means that at −273 °C, the volume of the gas is reduced to zero i.e., the gas ceases to exist; in fact,

the gas gets liquefied This temperature (−273 °C) at which the gas ceases to exist is called absolute zero

It is represented by 0 °A or 0 K (K stands for kelvin Kelvin is the SI unit of temperature According to SI unit, it should be written without putting the symbol of degree i.e., we should write 0 K; 273 K etc.) Thus,

−273 °C = 0 K or 0 °C = 273 K

This means that to convert t °C into K, we should add 273

T ° C = (t + 273) K = T K Substituting 273 + t °C = T K in Eq (1.1), we get

273

This implies that V T t∝ if pressure remains constant or simply V T∝ if pressure constant pressure

Thus, the Charles Gay-Lussac’s law may also be defined as follows:

Pressure remaining constant, the volume of a given mass of a gas is directly proportional to its absolute temperature

Mathematically, the law may also written as: V ∝ T (at constant pressure)

1 1

2

=2

under constant pressure (1.3)

where V1, is the volume of the gas at pressure P1 and V2 is the volume at pressure P2

Absolute scale/Kelvin scale of temperature: Taking −273 °C as 0 K, the scale of temperature obtained

is called absolute scale or Kelvin scale On this scale, the temperature in degrees kelvin is obtained by adding 273 to the temperature in degrees centigrade

1.2.3 Avogadro’s Law

Avogadro, in 1811, put forward the following hypothesis.

Under similar conditions of temperature and pressure, equal volumes of all gases contain the same number of molecules.

In other words, if temperature and pressure are kept constant, the volume (V ) of a gas is directly proportional to the number of molecules (N) present in the gas Mathematically,

V ∝ Ν, if T and P are kept constant (1.4)However, the number of molecules present in 1 mole of the substance (solid, liquid or gas) has been determined experimentally It has been found that one mole of every gas (i.e., 22.4 litre at N.T.P.) contains 6.023 × 1023 molecules This number is known as Avogadro’s number and is usu-

ally represented by N0

Thus, what was originally known as hypothesis is now known as Avogadro’s law Obviously, the

number of moles (n) of a gas may be calculated as follows:

1.2.4 The Combined Gas Law Equation or the Gas Equation

The above three gas laws may be combined to give a general equation, called the gas equation This may

According to Avogadro’s law

V n∝ if T and P are kept constant (1.7)Combining Eqs (1.5), (1.6) and (1.7), we get

1

or V R

where R is constant of proportionality and is called gas constant.

This equation is known as gas equation (or more correctly, ideal gas equation), which will be explained later.

Standard Temperature and Pressure or Normal Temperature and Pressure (N.T.P.)

As discussed above, volume of a given mass of gas changes with temperature and pressure Hence,

to compare the volume of different gases, they have to be taken or converted to same condition of

temperature and pressure using gas equation The conditions most commonly chosen are P = 1 atm and T = 0 °C or 273 K.

Nature/significance of gas constant, R: From the gas equation, PV = nRT, we have

×

ForceArea (Length)Moles Degrees

Force(Length) (Length)

WorkMoles Degrees

R = Work done per degree per mole.

Numerical values of the gas constant, R

At N.T.P conditions, for 1 mole of the gas

For expressing in SI units, put 10

1.2.5 Graham’s Law of Diffusion

It states that: Under similar conditions of temperature and pressure, the rates of diffusion of different gases are inversely proportional to the square root of their densities Mathematically,

r r

d d

1 2

2 1

1.2.6 Dalton’s Law of Partial Pressure

If states that: If a number of gases which do not react chemically with one another are enclosed together in the same vessel, the total pressure exerted by the mixture of gases is the sum of their par-

tial pressures, i.e., the pressures which each gas would exert if present alone in the same vessel

Mathematically,

where P is the total pressure and p1, p2, p3 etc are partial pressures

1.3 KINETIC THEORY OF GASES

It was observed that the gas laws were based on experimental observations The theoretical foundation

or mathematical representation was missing However, several workers studied the properties of gases and found that the gases are essentially composed of freely moving molecules To explain the behaviour

of gases, they put forward a theory called kinetic theory of gases This theory succeeded to attain a rigid mathematical form The assumptions made in this theory are given below.

1.3.1 Postulates (Assumptions) of Kinetic Theory

A gas consists of a large number of very small spherical tiny particles called molecules

(i)

The volume occupied by the molecules is negligible in comparison to the total volume of the gas

(ii)

The molecules are in a state of constant rapid motion which is completely random During their (iii)

motion, they collide into one another and with the walls of the vessel

The pressure of a gas is due to the collisions of the molecules with the walls of the vessel

(iv)

The molecules are perfectly elastic, i.e., there is no loss of energy when they collide with one (v)

another or with the walls of the vessel

There is no force of attraction or repulsion amongst the molecules

1.4 DERIVATION OF KINETIC GAS EQUATION

Based on the above postulates of the kinetic theory, we can calculate the

pressure of a gas in terms of molecular quantities Suppose a volume

of gas is enclosed in a cubical vessel (Fig 1.1) of side of l centimetres

Suppose the number of molecules present in it be n and the mass of

each molecule be m.

The molecules move with different velocities in different directions

colliding with each other and with the walls of the containing vessel

Suppose u is the velocity of a molecule at any instant This velocity may

be resolved in three rectangular components, u x , u y and u z, along the

three axes x, y and z of the cube Then, we have

Y

X Z

  1 Air is dense at sea level, because it is compressed by the mass of air above it However, the density

and pressure decrease with altitude The atmospheric pressure at Mount Everest is only 0.5 atm

  2 Pressure of a pure gas is measured by manometer while that of a mixture of gases by a barometer

  3 1 Bar = 0.9862 atm = 1.0 atm

    1 atm = 1.03125 × 105 Pa

  4 At 4.58 mm Hg pressure and at 0.0098 °C, ice (solid), water (liquid) and vapour (gas) may be

present simultaneously and all are stable

  Ice/water/vapour

    The corresponding temperature at which all the three states coexist is called triple point

  5 Centigrade scale and Fahrenheit scale of temperature were related as C = 5/9 (F−32)

  6 Charle’s law is not applicable to liquids

  7 Gas constant per molecule is known as Boltzmann constant (K)

K = R/N = 1.38 × 10−10 erg/K/molecule or 1.38 × 10−23 JK−1 mol−1

8 Numerical value of R = 0.0821 litre atm K−1 mol−1 = 8.314 JK−1 mol−1

= 1.987 cal K−1 mol−1

Now, consider the molecule moving with a velocity ux along x-axis striking the walls A and A′ Since the collisions are perfectly elastic, the molecule after colliding with the wall A would rebound with

exactly the same velocity but in opposite direction

The momentum of the molecule before it strikes

Therefore, the change of momentum per molecule along x-axis per second

=2mu ×u =2 2

l

mu l

Similarly, the change of momentum per molecule along y- and z-axes per second would be 2mu l2y/

 and 2mu l z2/, respectively

Therefore, the total change of momentum along all the three axes per second per molecule

l

mu l

mu l

=2m( 2+ 2+ 2)=2 2

mu l

mnu l

n

where u is the root mean square velocity.

Based on the kinetic theory of gases, we have three types of velocities, viz average velocity, root mean square velocity and most probable velocity Suppose in a gas, there are n1, n2, n3,… molecules possessing

velocities n1, n2, n3,…, respectively The average velocity (v) is then given by,

The root mean square velocity (u) is defined as the square root of the mean of squares of all the

velocities and is given by

Most probable velocity ( )u is defined as the velocity possessed by the maximum number of molecules

of the gas We have

v=09213×u and u= 23u

According to Newton’s second law of motion, the change of momentum per unit time is the force (F)

exerted by the molecules on the walls of the cube

Expressions (1.12), (1.13) and (1.14) are different forms of kinetic equation of gases

1.5 DERIVATION OF GAS LAWS FROM KINETIC EQUATION

(i) Boyle’s law: According to it, at constant temperature, the volume of a given mass of a gas is inversely

proportional to pressure, i.e.,

V

∝ 1 , at constant

The kinetic energy of the gas

∴

At constant temperature, the kinetic energy (E) of the gas is constant Therefore, at constant temperature

PV = constant [This is Boyle’s law.]

(ii) Charles’ law: According to it, at constant pressure, the volume of a given mass of a gas is directly

proportional to its absolute temperature, i.e.,

At constant pressure,

V= constant E or× V∝E

However, we know that E ∝T , where T is absolute temperature.

V T∝   [This is Charles’ law.]

(iii) Avogadro’s hypothesis: According to it, equal volumes of all gases under similar conditions of

temperature and pressure contain equal number of molecules For any two gases, the kinetic equation can be written as:

13

23

12

13

23

12

When the temperatures of the gases are the same, their mean kinetic energy will also be the same, i.e.,

  1

2

12

Dividing Eq (1.16) by Eq (1.17), we get, n1 = n2 [This is Avogadro’s hypothesis.]

(iv) Graham’s law of diffusion: According to it, the rate of diffusion (r) of a gas is inversely proportional

to the square root of the density (d) of the gas, at constant pressure, i.e.,

r d

=13 2 or = 3  = 3 

As Total mass of the gas

Volume Density of the gas,

  [This is Graham’s law of diffusion.]

(v) Dalton’s law of partial pressures: Suppose n1 molecules, each of mass m1, of a gas, A, are contained

in a vessel of volume V Then, according to the kinetic theory, the pressure, pA, of the gas A will be given by:

Now, suppose n2 molecules, each of mass m2, of another gas, B, are contained in the same vessel at

the same temperature and there is no other gas present at that time The pressure, pB, of the gas B is then given by,

m n u V

Similarly, if three, four or more gases are present in the same vessel, the total pressure is given by,

  P p= A+pB+pC+pD+ [This is Dalton’s law of partial pressures.]

(vi) Derivation of ideal gas equation:

23

PV = RT (for one mole of gas)

PV = n RT (for n moles of gas)

which is the required ideal gas equation

1.5.1 Some Useful Deductions from Kinetic Theory of Gases

1.5.1.1 calculation of Molecular velocities

According to kinetic gas equation

2

PV mn

2=3 =3 [∵ = is the mass of a gas]

If 1 mole of the gas is taken, M will be equal to the molecular weight of the gas.

PV M

Further, for one mole of the gas (assuming it is ideal)

PV = RT

= 3 can also be written as

Example 1 Calculate the root mean square velocity of CO 2 molecule at 1,000 °C.

Solution: We know that u PV

M

RT M

R = 8.314 × 107 erg deg−1 mol−1

= 0.88314 × 108 erg deg−1 mol−1

u = 84,985 cm sec−1 or 849.85 m sec -1

Example 2 Calculate the RMS velocity of chlorine molecules at 12 °C and 78 cm pressure.

Solution: At given condition: At STP

V2 = ? V1 = 22,400 ml

T2 = 12 + 273 = 285 K T1 = 273 K

P2 = 78 cm P1 = 76 cm Applying P V

T

P V T

u = 31,652 cm sec−1 or 316.52 m sec -1

Example 3 Calculate the average velocity of nitrogen molecule at STP.

Solution: Substituting the given values in this equation

= 49.330 cm sec−1

∴ Average velocity = 0.9213 × 49,330 cm sec−1 = 45,447 cm sec -1

Example 4 Oxygen at 1 atmosphere pressure and 0 °C has density of 1.4290 grams per litre Find the RMS velocity of oxygen molecules.

Solution: We have P = 1 atom = 76 × 13.6 × 981 dynes cm−2

1.5.1.2 calculation of the Kinetic Energy of the Molecules

For 1 mole of the gas containing n molecules, each of the mass m, the kinetic energy will be

From the equation, we observe that the kinetic energy of an ideal gas is independent of the nature

or pressure of the gas and it depends only on the absolute temperature, being directly proportional to it The average kinetic energy per molecule can be obtained from the above equation by dividing with the

Avogadro’s number N Thus, average K.E per molecule

32

where k R

N

= is called Boltzmann constant

Example 5 Calculate the total and average kinetic energy of 32 g methane molecules at 27 °C

JK molmolecules mol K = 6.21 × 10−21 J mol−1

2 RT mol−1= 32 × 8.314 JK−1 mol−1 × 300 K = 3741.3 J mol−1

Example 6 Calculate the root mean square speed of methane molecules at 27 °C.

Solution:

ur.m.s = 3RT

M

Using C.G.S units, put R = 8.314 × 107 erg K−1 mol−1, T = 27 + 273 = 300 K,

M (for CH4) = 16 g mol−1, we get ur.m.s = 3 8 314 10 300

23

Thus, the molecular velocity of any gas is directly proportional to the square root of its absolute

temperature Hence, when T = 0, u also becomes zero This temperature is called absolute zero Hence,

absolute zero is that temperature at which the molecular motion creases

Remember 0 °A or 0 K = −273 °C

1.5.1.4 Slow Diffusion of gases

In spite of the fact that the gas molecules move with high speeds, yet their motion is not so fast For example, it takes quite some time for a smell to spread from one corner of a room to the other This is obviously due to the fact the molecules of a gas undergo an extremely large number of colli-

sions, per second In every collision, the path of the molecule is deflected, so much so that sometimes

it is even deflected back Hence, the net linear movement of the molecules in a particular direction

is quite slow

1.6 IDEAL AND REAL GASES

1.6.1 Ideal Gases

Gases which obey gas equation PV = RT rigidly for all values of temperature T and pressure P are called

ideal or perfect gases Thus, an ideal or perfect gas is one which rigidly obeys Boyle’s and Charles law for

all values of temperature and pressure.

1.6.1.1 characteristics of an Ideal gas

The product of pressure and volume of a given mass of an ideal gas at constant temperature (i)

should be constant, i.e., if product PV is plotted against pressure P at constant temperature, the

curve obtained should be a straight line parallel to the pressure axis (Boyle’s law)

The compressibility factor

become zero at −273 °C or 0 K (Charles law)

If a gas is allowed to expand without doing any external work, it should show no thermal effect (iv)

(no force of attraction between the molecules)

At constant temperature, if pressure is doubled, the volume should reduce to half (Boyle’s law)

Thus, an ideal or perfect gas is only hypothetical It has been observed that at high pressures and low

temperatures, gases do not obey the gas laws However, at low pressures and high temperatures, they obey the gas laws to a fair degree of approximation Such gases are, therefore, known as real gases

1.6.2 Real Gas

Real gas is one which obeys the gas laws fairly well under low pressure and high temperature It means all gases are real gases, but they show deviation from ideal behaviour The deviations are more pronounced,

as the pressure is increased or the temperature is decreased Further, in case of real gases, it is not

pos-sible to reduce uniformly the volume of a gas to zero at −273 °C, as all real gases liquefy when sufficiently cooled Real gases such as O2, H2 and N2 which are difficult to be liquefied differ from an ideal gas to a