Physical chemistry by r l madan

TheStateof Chemical Equilibrium LawofMass Action Applications of EquilibriumConstant Partial MolarQuantitiesFree-Energy Change as aCriterionofSpontaneity ThermodynamicDerivationof theLaw

Also available as

PHYSICAL CHEMISTRY

Me

Graw

Hill

Physical Chemistry

R L Madan

FormerPrincipal ,GovernmentCollege , Panchkula

& FormerHead , Chemistry DepartmentGovernment PostGraduate College Faridabad , Haryana

McGraw Hill Education Offices

New Delhi New York StLouisSanFranciscoAuckland Bogota Caracas Kuala Lumpur Lisbon London MadridMexicoCity Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto

About the Author

R L Madan has about 35 years of experience in teaching chemistryatthe

undergraduate level After obtaining a PhD in Chemistry, he began his teaching career

atGovernment Postgraduate College, Gurgaon Thereafter, he has held positions of Head of Chemistry DepartmentatGovernment Postgraduate College, Faridabad, and Principal, Government College Panchkula and Tigaon (Faridabad) Prof Madan has doneextensiveresearch work in the areas of adsorption and surfacescienceatthe Chemistry Department of IndianInstituteof Technology, New Delhi He did

postdoctoral work for studies on polymersatPrague University, Czech Republic, under

a UNESCO fellowship He has also been awarded a fellowship by the SwedishInstitute

for Research Work on SuperconductivityatStockholm University, Sweden Prof

Madan has authored several best - selling books on chemistry

Physical Chemistry

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Physical Chemistry

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Dedicated to the Memory of

MyLovingMother

Late MrsjankiDevi

Who made me what I am andtowhom I owe a lot

Chemistry has made a significant impact on society andisintimately linkedto the well

-being of humankind The rate of advancement of basic sciences is so high that academicians look forward for strategies to cope with those advancements Physical chemistry provides a decent bridge between mathematical sciences and experimental

sciences The present textbookisasincereeffort in this direction This book provides a unified approach to the study of chemistry for undergraduate (Pass and Honours) students The text is well illustrated with examples from surrounding environments

which will help students grasp the concepts easily

There are several highlights of this book The chapter dealing with mathematical concepts and computers will provide a powerful tool to students in understanding the basic concepts and solve numerical problems in shortertime Partial differentiation and integration needs special mention in this regard Incorporation of computers and introducing analog and digital devices will provide a good basic platform to

understanding physical chemistry Biographical sketches of some scientists relevant to

particular topics will surely enthuse students Liquid state, colloidal state chemical kinetics and thermodynamics have been written especially in a systematic, lucid and comprehensible style Learning objectives have been given exhaustively in each chapter

of the book Sufficient examples have been givenatthe end of each chapter, along with summary for quick revision, key relations, question bank including multiple - choice questions and general questions so that students can evaluate their learning potential themselves

The level of text material in each chapter is sufficient to escalate this book to

international standard I am sure this book will procure the author and publishers a good name in the market The hard work done by the authoriscommendable and deservesto

be congratulated I wish huge successtothis book

N K Sandle

Prof Department of Chemistry, IndianInstituteof Technology (IIT), Delhi

Mathematical Concepts and Computers

IntroductiontoMathematical Concepts 2.2 Logarithmic Relations

KineticTheory of Gases

Maxwell Distribution of Velocities

3.6 Principle of Equipartition of Energy

3.7 CollisionDiameter, Collision Number,Collision Frequency and

Mean FreePath

4.2 VapourPressureof a Liquid

4.3 SurfaceTensionof a Liquid

4.4 Determinationof SurfaceTension

SpaceLatticeandUnitCell of a Crystal

Packing of Particles in a Crystal

Isomorphism and Polymorphism

PointGroups and Space Groups Investigation of Internal Structure of a Solid by X - Ray

6.4 Preparationof Colloidal Solutions

6.5 Propertiesof Sols (Colloidal Solutions)

6.6 ProtectiveColloids and Gold Number

7.4 RateLaw andRate Constant

7.5 Zero-OrderReaction

7.6 First- OrderReaction

7.7 Radioactive Decay as aFirst- Order Phenomenon

7.8 Second - OrderReaction

7.9 Third - OrderReaction

7.10 Relations between Half - Life Period and Order of aReaction

7.11 Formulation of Mechanism of aReaction

7.12 Methods for theDeterminationof Order of aReaction

7.13 Complications in theDeterminationof Order ofReaction

7.14 SurfaceReactions

7.15 ChainReactions

7.16 Temperature Dependence and Arrhenius Equation

7.17 Collision Theory ofReactionRates

7.18 TransitionState Theory or Theory of AbsoluteReactionRates

or Theory of Activated ComplexFormation

7.19 Experimental Methods of ChemicalKinetics

Important Features of Solid Catalysts

Shape - Selective Catalysis by Zeolites

9.3 Calculation of Values of w, q , AH and AH in Adiabatic

Expansionof an IdealGas

Latest SignConventionsforQor AH,AH andW

Heat ofReaction(or Enthalpy of Reaction) StandardState

Different Kinds of Heat (or Enthalpy) ofReaction

Hess ’ s Law of ConstantHeatSummation

Bond Enthalpy or Bond Energy

Variationof Heat ofReactionwith Temperature — Kirchhoff ’ s

Short - AnswerQuestions

Entropy Change for an Ideal Gas Under Different Conditions Entropy Change on Mixing of Ideal Gases

Helmholtz Free Energy Gibbs Free Energy (or Gibbs Energy)

Criteriafor Feasibility or Spontaneity of a Process Maxwell Relationships

Gibbs - Helmholtz Equation

12.2 Statements of Third Law of Thermodynamics

12.3 Residual Entropy

12.4 Partial MolarProperties(Open Systems)

12.5 Clausius-ClapeyronEquation

12.6 FugacityandActivity

13.8 VanT Hoff ReactionIsotherm

13.9 Relation between Kp and Kc

13.10 VanT Hoff Equation for Temperature Dependence of

EquilibriumConstant(Van’t HoffReactionIsochore)

13.11

13.12

13.13

13.

TheStateof Chemical Equilibrium

LawofMass Action

Applications of EquilibriumConstant

Partial MolarQuantitiesFree-Energy Change as aCriterionofSpontaneity

ThermodynamicDerivationof theLawof Chemical

Le - Chatelier’s Principle Clausius - Clapeyron Equation Hammett Equation

Phase Diagram of Carbon Dioxide System Phase Diagram of Sulphur System Two - Component Systems

Types of Two - Component Systems Involving Solid - Liquid

14.15

14.16

14.17

PhaseDiagramof Na2S04-H2OSystem

Sodium Chloride-WaterSystem

PhaseDiagramof Copper Sulphate-Water System

ThermodynamicDerivationof DistributionLaw

Modification in DistributionLawinCaseof Changein

Applications of DistributionLaw

Studying Chemical Equilibrium InvolvingFormationof Complex Compounds

15.6

15.7

and Volume of Extracting Liquid used EachTime

15.8 Determinationof Degree of Hydrolysis from DistributionLaw

16.2 ElectricalResistanceand Electrical Conductance

16.3 Measurement of Electrolytic Conductance, Specific

Conductivity, Equivalent Conductivity and Molar Conductivity

16.4 Numerical Problems on Specific, Equivalent and Molar Conductivities

16.5 Effect of Dilution on Specific Conductance and Equivalent Conductance

Short - AnswerQuestions

Utility of DistributionLawin theProcessofExtraction

General Formula forAmountof Substance Left Unextracted

16

Arrhenius Theory ofIonisation

Ostwald Dilution Law

Migration of Ions and Transport Number Hittorf’s TheoreticalDevice— Change inConcentration

Transport Number Moving Boundary Method for theDeterminationof Transport

17.6 HydrolysisConstantand Degree of Hydrolysis

17.7 Expressions for the HydrolysisConstant,Degree of Hydrolysis and pH of Solutions of Salts ofStrongAcid and WeakBase

17.8 Expressions for the HydrolysisConstant,Degree of Hydrolysis and pH of Solution of a Salt of Weak Acid andStrong Base

17.9 Expressions for HydrolysisConstant,Degree of Hydrolysis and

pH of Solutions of Salts of Weak Acid and WeakBase

Coefficient of the Emf

18.11 Measurement of Electrode Potentials of ZincElectrode and Copper Electrode

HendersonEquationfor the pH of a BufferMixture

BufferCapacity and Buffer Index Neutralization Phenomenon

18.

Electrolytic Cell Schematic Representation of an Electrochemical Cell Emf of a Cell

Weston Standard Cell Reversible Cell Irreversible Cell Types of Electrodes Relationship between Electrical Energy and Chemical Energy Expressions for AG,AH and AS in Terms of Temperature

ElectrochemicalSeries

Activity and Activity Coefficient of Electrolytes Nemst Equation for Measuring Emf of a Cell Nemst Equation for Single Electrode Potential Electrode Potential of Reduction - Oxidation Electrode Calculation of Equilibrium Constant (K) from Nernst Equation Irreversible Electrode Processes

ConcentrationCells Emf of ElectrodeConcentrationCell without Transference Emf of ElectrolyteConcentrationCells without Transference Emf of an ElectrolyteConcentrationCell with Transference LiquidJunctionPotential

Determinationof Activity and Activity Coefficient from Emf Applications of emf Measurements

Different Types of Electrodes in Practical Use

18.28 Determinationof pH or H Concentrationof a Solution by emf Measurement

Electrochemical Theory ofCorrosion

Methods ofPreventionofCorrosion

Postulates ofQuantumMechanics

19.21 Particle in a One - Dimensional Box

Photoelectric Effect Heat Capacity of Solids

Normalised and Orthogonal WaveFunctions

Operators Postulates ofQuantumMechanics

Derivationof Schrodinger Wave Equation Based on the

Calculation of Expectation Values using WaveFunction

Particle in a Three - Dimensional Box Concept of Degeneracy

Schrodinger Wave Equation in Terms of Polar (Spherical) Separation of Variables

Expressions for Angular Spherical WaveFunctions

Expression for the Radial WaveFunctionQuantumNumbers from Schrodinger Wave Equation Concept of Orbital

QuantumMechanics and Chemical Bonding

Axis inBonding and Antibonding Molecular Orbitals

20.6 Formationof Molecular Orbitals fromAtomicOrbitals

ConceptofO, a *, TT,TT*Orbitals and their Characteristics

Formationof H2Molecule Hybridisation

QuantumMechanical Principles of Hybridisation Calculation of the Coefficients of AtomicOrbitals in different

Absorption andEmission Spectroscopy

Types of Molecular Energies and Born - Oppenheimer Approximation

21.6 Types of Molecular Spectra

21.7 Selection Rules for Rotational and Vibrational Spectra

21.8 Width andIntensitiesof the SpectralLines

21.9 Degrees of Freedom

Pure Rotational Spectra of DiatomicMolecules Nonrigid Rotor (Qualitative Description) Isotopic Effect

Vibrational Spectra of DiatomicMolecules (Infrared Spectra) Vibrational Energy of an Anharmonic Oscillator (Morse

21.15 Vibrational - Rotational Spectra Obtained for aDiatomic

Molecule Takingitas Anharmonic Oscillator

21.16 Applications of Study of Infrared Spectrum

Vibrational Frequencies of Functional Groups Introduction

21.22 Potential Energy Curve and Frank - Condon Principle

21.23 ElectronicTransitionin o,TT, and n Molecular Orbitals

Exercises

Rotational - Vibrational Raman Spectra ofDiatomicMolecules Introduction

Short - AnswerQuestions

QuantumEfficiency orQuantumYield

SomeExamples of PhotochemicalReactionLuminescence

Excitationof Electrons (Jablonski Diagram) Photosensitisation

23.

Optical Activity

Measurementof Optical Activity

Causeof Optical Activity in a Compound Dipole Moment

Polarisation of Molecules (Induced Polarisation) Dipole Moment and Structure of Molecules Magnetic Susceptibility

Explanation of Diamagnetism and Paramagnetism

Determinationof the Magnetic Moment of a Paramagnetic

Short - AnswerQuestions

24.9

24.10

Principle of Distillation of Binary Miscible Liquids Distillation of Binary Miscible Solutions

Immiscible Liquids

Steam Distillation Partially Miscible Liquids Effect ofImpuritieson Consolute Temperature (Critical Solution Temperature)

Mass of the Solute

25.14 Van’t Hoff Factor

Elevation in BoilingPoint

Relation between Elevation in BoilingPointof Solution and Depression in FreezingPoint

Relation between Depression in FreezingPointand Molecular

Short - AnswerQuestions

Adsorption by Solids from Solutions Gibbs Adsorption Equation

Applications of Gibbs Adsorption Equation Applications of Adsorption

Summary

Key Relations

Methods for theDeterminationof MolecularMasses27.6 StepPolymerisation

Artificial Disintegration orTransmutationof Elements

QValue of NuclearReactions

Soddy’s Group - Displacement Law

I feel immensely pleasedto present Physical Chemistrytothe teaching fraternity and the students Physical chemistry is the study of macroscopic, atomic, sub-atomic and particulate phenomena in chemical systems It applies the principles, practices and concepts of motion, energy , force,time, thermodynamics, quantum chemistry,statistical mechanics and equilibrium

Physical chemistry, a branch of chemistry,isa core subjectatundergraduatescience

and technology course curricula of different universities and autonomous institutes in the country In the study of this subject for the above courses, a number of books by foreign authors are widely used as reference books The teaching fraternity in India was feeling the need of a textbookwritten in a simple language and an interesting manner which could be grasped even by an average student The present book has been written

in responsetothe aspirations of the teaching community and the students

This book covers the syllabus of Physical Chemistry for the students of BSc I, II and III (Pass and Honours or Major and Minor) of different Indian Universities Students of other undergraduate professional courses having physical chemistry as a major subject may also use this book The only prerequisiteisthat the students mustpossess a basic knowledge of chemistry of 11th and 12th standard to utilize this book A basic knowledge of mathematicsisalso expected

P Covers all important topics Gaseous State, Liquid State, Solid State, Colloidal State, ChemicalKinetics, Catalysis, Thermodynamics, Chemical Equilibrium, Phase Equilibria, Distribution Law, Electrochemistry,AtomicStructure and Wave

Mechanics, Chemical Bonding and Wave Mechanics, Spectroscopy,

Photochemistry, Physical Properties and Molecular Structure, Liquid Solutions, Dilute Solutions, Adsorption, Polymers and Nuclear Chemistry

P Latest IUPACnotationsand SIunitsused

P Principles and laws explained in the simplest possible way

P Analogies from daily life usedtoclarify difficult points

P Conceptexplanation through more than 1200step-wisesolved problems

P Numerical problems for practice

P Applications and case studies like applications of equilibriumconstant,casesto

understandstructureof liquid crystals and how they flow

P Morethan500line,phase, and real-timeillustrations

• Prologue on mathematicalconcepts

• Key words and Key relations provided in the summary

• Conceptboxestohighlightimportantfacts that are asked inexaminations

• Special features such as interesting facts, common pitfalls,key terms

interspersed

• Subject index provided for quick location ofterms, laws, theories,etc.

• Over450solved examples covering all types of problems in different chapters

• Morethan800 reviewquestions

• Over400objective-type questions

About the Book

The style of the book is such that it encourages students to understand physical chemistry rather than learn it by rote memorization Simple and comprehensible language has been used to explain the principles involved in different topics The opening chapter,Basic Concepts, acquaints the students with different systems andunits

of physical quantities As recommended by IUPAC and CGPM, SI system of

measurementandunitsof physical quantities have been followed by and large, although CGS system has been retained in some problems This is because questions in some university papers still appear in CGS units IUPAC has recommended changes in the names of certainterms, for example, conductivity in place of specific conductance, and Gibbs energy in place of Gibbs free energy This has been highlighted in the chapters on electrochemistry and thermodynamics Readers can refer to the Table of Content to

know more about the topics covered in this book

Biographical sketches of scientists are given to inspire and stimulate the minds of students towards the subject Analogies from daily life have been chosen to explain

intricate points Historical developments leading to the establishment of different theories and laws have found due place in the book How the different principles of physical chemistry come to the rescue of people in difficult situations has been explained with the help of interesting pictures Although this partis non - evaluative,it

makes the subject exciting and easily understandable The text is supported by a large number of solved examples and problems for practice

Learning Objectives given at the beginning of each chapter give the detailed

contents of the chapter Summary provided at the end of a chapter lists the important information in brief for last -minute revision before the examination Also given in the summary are Key Points and Key Relations, which are required in solving numerical problems Exercisesconsistoftwoparts, objective evaluation and subjective evaluation

As per the latest recommendation on the setting of question papers , multiple - choice questions, fill - in - the - blanks questions, short - answer questions and general (long -

answer) questions have been provided to prepare the students to face examinations

boldly

The text is accompanied by an Online Learning Center, which can be accessed at

https://www mhhe com/madan/pcl This includes objective-type questions, concept

boxes, and answerstounsolved problems

Acknowledgements

I express my thanks and regards to my mentor and Ph D guide Prof N K Sandle,

Department of chemistry, IndianInstituteof Technology (IIT), NewDelhi for providing

me inspiration and support during the preparation of the book

I thank all the reviewers who read the initial manuscript and gave commendable suggestions, some of which have been incorporated in the text Their names are given below:

Readers are welcome to send constructive criticism and feedback to me at

rattanlal madan@gmail com

RL Madan

McGraw Hill Education (India) invites suggestions and comments from you , all of which can besent toinfo india@mheducation com (kindly mentionthe title and author name in the subject line) Piracy - relatedissuesmay also be reported

Acquaint with different systems of measurement of different quantities

Understand howwegradually shifted from the British system to the metric systemKnow thesevenbasic physicalquantities and their units

Learn in detail about the units of thesevenphysicalquantities

Differentiatebetweenmassandweight

Write very large or verysmall numbers in scientific notation

Carry outarithmetical calculations usingexponentialnotation

Differentiate between precision and accuracy

Define significant figures andlearn the rules for deciding significant figuresindifferentkinds of numbers

Learnaddition,subtraction,multiplicationanddivision ofsignificantfigures

Learnconversion of units from one system to another through dimensional analysis

A substanceisassociated with physicalpropertiesand chemical properties. Some examples ofphysical properties are colour, odour, smell, density, melting point and boiling point. Theseproperties can be measured without the substance undergoing decomposition or association.

Observationofa chemicalpropertyrequiresa chemical changetooccur.Chemicalpropertiesof

a substance include acidic or basicnature,combustibility,etc.

Werequiresomeparameters toassesscertain properties.For example,propertieslike length,

width,area, volume,etc ,arequantitativein nature.That is, theyare associated with definitevalues.Anyquantitative measurement isrepresented bya number followed byunitsin whichit

ismeasured Itisabsurd tosay thata particular substance measures 6—in length we needtospecify whetherit is6cm or 6m.Whenwe sayit is6m long,we meanitmeasures 6unitsonthe scaleofmetre.

1.2

In scientific studies and research,twodifferentsystemsofmeasurementhave been in use.These

are the English System and the Metric System.The metric system originated in France in thelate eighteenthcenturyandismoreconvenient touse becauseit isbasedon the decimalsystem

of numbers Before 1960, both thesystemsof measurementwerebeing usedby the scientificcommunity.Somepartsof theglobe preferredtheEnglish systembecausetheywere usedtoit.

Greaterproportionof the scientific population advocatedaswitchtothemetric system because

it wasmoreconvenientand hencemore scientific

The metric system is called the International System of Units (called Le Systems

established by the 11thGeneralConference on Weights and Measures (Conference General desPoides atMeasures in French and abbreviated as CGPM).The CGPM is an intergovernmentaltreaty organization createdby a diplomatic treaty known as Meter Conventionwhich was signed

in Paris in 1875.

Asa consequence of serial meetings of the international scientific community, it was agreed

in 1960 to adopt SI units for scientific experimentation and studies, although both the unitscontinue to be used in textbooks.We have been rather slow in adopting SI units in our works.The SI system has seven base units pertaining to seven fundamental quantities as listed in Table

1.1.

Table 1.1Seven basic physical quantities and their units

Symbol for Quantity

Name of SI Unit

Symbol for SI Unit Physical Quantity

SIbaseunitsare separately definedas under:

1 Metre The metre is thelength ofthepath travelled by lightin vacuum during atime interval of 1/299792458 of a second

2 Kilogram Thekilogram is equal to the mass ofthe international prototype ofthe kilogram

3 Second The second is the duration of 9192631770periods of radiationcorrespondingto the transition between the two hyperfine levels ofthe groundstate ofthe caesium-133atom

4 Ampere Theampere is that constant current which,if maintainedin twostraightparallel conductors of infinite length,ofnegligible circularcross section

and placed 1 metre apartin vacuum would produce between these conductorsaforceequal to 2 *10“7newton per metreoflength.

5. Kelvin The Kelvin is the1/273.16 fraction ofthethermodynamictemperature

ofthetriple pointofwater

6 Mole The mole is theamountofsubstance which containsasmany

elementaryentitiesare there areatoms in0.012kilogramofcarbon-12.Itssymbolis mol Theelementaryentitiesmustbespecifiedandmaybeone ofatoms, molecules, ions, electrons or otherspecifiedparticles.

7 Candela The candela is the luminousintensity,inagivendirection, of asource that emits monochromatic radiations of 540* 1012hertz frequency andthat hasa radiantintensity inthat direction of1/683watt persteradian.

The SI system allows the use of prefixes to indicate the multiples or submultiples of a unit asgiven below in Table 1.2. Infact, this is the strength of the SI system because it is based ondecimal system (multiples or submultiples of 10)

Table 1.2Prefixes in SI system

Massofa substanceistheamountofmatterpresentinitwhile weightistheforce exertedbygravityon anobject.Themass of a substance is constantwhereas theweight may varyfromplace to placedueto changeingravitationalforce.Themass ofasubstancecanbedeterminedaccuratelyin thelaboratory by usingan analytical balance.Presently, wehave the electronictop-

weighing balancewhich directly gives the mass of the substance correct up to three or fourdecimal places dependingupon the precision of thebalanceused.TheSIunitofmassisonekg.

However, its 1CT3multiple (1 kg =1000g), that isgram, is used morefrequently for weighingsmall amountsof chemicals that are used in chemical reactions.

Physical quantity measured Base unit SI abbreviation

volumesare often denoted in cm or dm units.

Litre (L)whichis notan SIunit isusedfor themeasurementof volumes of liquids.

1L=1000mL and 1000 cm3=1 dm3

1L=1 dm3=1000mL=1000 cm3Volumesof liquidsor solutions are measured with the help or graduated cylinders,burettes,

pipettes,etc.

Thus,

Derive I

L

Volume Force Pressure

Energy

liter newton pascal

joule

N Pa

J

Fig 1.2 Derived SI units

1 DensityThe density of a substance is its amount per unit mass.The SI unitof density can becomputed as under:

SI unit of mass Density SI units) =

SI unit of volume

k§

- or kg m 3

m

Achemistoftenexpresses the densityas g cm-3becausethe SIunit giveslarge values.

Fig 1.3 Anders Celsius (1701-1744 ) was a Swedish astronomer The scale of temperature

named after him was devised by him

2 Temperature Three scales have been in use for the measurement of temperature—degreeCelsius (°C), degree Fahrenheit (°F) and Kelvin (K). Kelvin is the SI unit of temperature.

However,thermometersare manufactured basedon Celsius and Fahrenheit degrees.Generally,

Celsius scale thermometersare calibrated from 0°to100°C basedon the freezing and boilingpointsof water.However, thermometersfrom 0°C toabout 250°C are available when highertemperaturesare requiredtobe measured.The Celsius scaleoftemperaturewas earlier known ascentigrade scale.The Fahrenheit scaleisrepresented between 32°Fto212°F.Thetemperatures

on Fahrenheit and Celsius scales are relatedtoeach other by the following relationship.

Fig.1.4 Temperature conversion

The relation between the Kelvin scale and Celsius scaleisas under:

K = °C+273.16

It may be of interest to note that negative valuesof temperature on the Celsius scale ispossible while thisis notso on the Kelvin scale.

( 1.2 )

Revamping Reference Standards

The mass standard has been the kilogram since 1889. Ithas been defined as the mass ofplatinum-iridium cylinder that is stored in an airtight jar at the International Bureau ofWeights and Measures in Sevres, France Pt-Irwaschosen for this standard because it ishighly resistant to chemical attack and atmospheric action and its mass is not likely tochange for an extremely long time Scientists are in search of a new standard for mass This

is being done though accurate determination of Avogadro constant Work on this newstandard follows methods tomeasureaccurately the number of atoms in awell-defined mass

of the sample This will provide a precise value of the Avogadro number and a newreference standard formass

3 Velocity Velocityisexpressedasm/s in SIunits.In the CGSsystemit isexpressedas cm/s.

4 Acceleration Acceleration is expressed as m/s in SI units In the CGS system, it isexpressedas cm/s.

5 ForceForceisobtainedbymultiplyingmass with acceleration.Theunitof forceis newton.

A massof1kg movingwithanaccelerationof1m/s hasaforceofonenewton.In theCGSsystem,the unit offorce is dyne.Dyne is theforce exerted by a massof 1 gmovingwith anaccelerationof 1 cm/s2,1 Newton=1kgms-2

6 Work Work is obtained by multiplying force with displacement.The SI unitof work isjoule

When a force of one newton makes a displacement of one metre, a work of one joule is said to

be done.Inthe CGS system, the unit of work iserg Anergof work is done when a force of one

It offers a real challenge in chemical arithmetic.This problem is solved by using scientific

quantity can berepresentedas

wheren isthe exponenthaving positive or negative value and N variesfrom 1to 9.Thisisillustratedas under:

The number4297.32can bewritten as 4.29732x103inscientific notation.

How didwe do that?

Wewantthe decimalpoint tobeplaced after 4(remember Nvariesbetween 1to9).Thus,

we havetomove three placestothe left.Theexponentnisequalto the numberof placeswehave moved the decimal point.Whenwe move the decimalpoint tothe left,the exponentnis

Letus takean example of a smallquantity,say 0.0000029.Thiscan bewrittenas 2.9x10

in scientific notation. Remember we have to convertit into the form N x 10n, where N lies

between 1 and 9.To obtain the number in this form,we needtoshift the decimalpoint sixplaces

to the right. The exponent will again be equal to the number of places we are shifting thedecimal point, i.e. 6, but with a negative sign, i.e -6. Hence, the scientific notation for thisnumber will be 2.9x10-6.

1.5 1 Addition and Subtraction

For the purpose of addition or subtraction of numbers in scientific notation, the numbersarewrittenin sucha way that they have the sameexponent.After that the coefficients are added orsubtractedas the case may be.

Example 1Add 5.34 x 105and 7.97 x104

Solution: Make theexponent5 in both numbers(equaltohigher of thetwoexponents).Weobtain the numbersas

5.34x105and 0.797x105

Andnow we shall add the coefficient andrepresentthesum as

(5.34+0.797)x105 =6.137x105

Now considera caseof subtraction

Example 2 Subtract 8.2 x10 4 from 7.8 x10 3

Solution : Make the exponent-3in both cases to proceedfurther; thus, the numberscan bewrittenas 0.82 x10-3and 7.8x10-3.Therefore,

(7.8x10-3)-(0.82x10" 3)Subtract the coefficientsof two numbers and write the answer as

(7.8-0.82)x10"3=6.98x10"3

1.5 2 MultiplicationandDivision

Inthese two operations, the rules as applicable to exponential numbers are followed.Let the twonumbers that are multiplied be x10* andN 2 x10^ The multiplication result would beNi x

iV2 x10*+y.In the divisionof the above numbers, the result wouldbe

illustrated with thehelpof the following examples:

Example 3Multiply 6.7x 103and 8.9 *105

1 PrecisionPrecisionreferstothe closenessofvarious measurementsfor thesamequantity.

2 Accuracy Accuracyistheagreementofa particular valuetothetruevalueof the result.

We illustrate this with the help of the following table which records themeasurements inmetreby three studentsX,YandZ.Thetruevalueof the resultis3.00m.

Every student takes two measurements indicated under 1 and 2.

StudentX reports the measurements as 2.97mand 2.98m.These measurements are precise(because they are close to each other) but these are not accurate because they are far away fromthe true result which is 3.0m.

StudentYreports the measurements as 2.98mand 3.10.These results are neither precise noraccurate.

Student Z reports the measurements as 2.99 and 3.01.These resultsare both precise andaccurate.

1.6 1 Significant Figures

Theuncertainty inthemeasurementsor calculated valuesisindicatedbymentioningthe number

of significant figures.Significant figuresare meaningful digits which are known withcertainty.

The last digitinthe numberistaken asuncertainwhile other digitsarecertain.If wewritetheresult as 20.6 cm, we say that 20 is certain and 6 is uncertain. The uncertainty in themeasurementis ±1 in the last digit. An uncertainty of ±1isassumed in the last digit, unlessotherwise stated.

1 Rules for Number of Significant Figures

(a) All non-zero digits are significant.For example,in 375mL,thereare three significantfigures and in 0.65mL,therearetwosignificant figures.

(b) Zeros preceding the firstnon-zero digit arenotsignificant.Sucha zero indicates thepositionof decimalpoint.Thus,0.07hasone significant figure and 0.0075 hastwosignificant figures.Zeros betweentwonon-zero digits are significant.Thus,7.035hasfoursignificant figures.

(c) Zerosatthe endor right ofa number are significant provided they are on the right side ofthe decimalpoint.For example,0.440has three significant figures.If the above condition

is notsatisfied then thezeros arenotsignificant figures.For example,100 has onlyonesignificant figure.

(d) Exact numbers havean infinite number of significant figures.For example,in4 tables or

10 chairs,thereare infinite significant figuresas these can be represented bywritinginfinite numberofzeros after placing the decimalpoint.Thus,2=2,00000 and 10=

10,00000

In case a numberis writtenin scientificnotation (Nx10n),the numberof digits in the N part

of the numberare significant figures.

2 Addition and Subtraction of SignificantFigures The result of additionorsubtraction

of numbers cannot havemoredigits to the right of the decimal point than any of the numbers.

For example,

8.0 11.12

[5.214 34.334

The number 8.0 has only one digit after the decimal.Therefore, the result cannot have more thanonedigit after the decimal point.Hence, the result would be 34.3.

3 Multiplication and Division of Significant Figures In the multiplication and division

of numbers, the result must be reported with significant figures which are no more than in the number with least significant figures For example,

(a) It the digit on theextremerighttobe removedismore than 5,the preceding digitis

increased byone.For example,in the number4.678,if the last digitis tobe removed thenit

isroundedto4.68.

(b) If the digit on theextremerighttobe removedisless than 5,the preceding digitis notchanged.Thus,if in the number 7.884,the last digit 4is tobe removed thenitwill beroundedto7.88.

(c) It the digit on theextremerighttobe removedis5 then the preceding digitis notchanged if

it isan even number butit isincreased byone ifit isan odd number.For example,if 8.75is

tobe roundedoff byremoving5,we needtoincreasethe preceding digit by 1,thatis,theresult would be 8.8.However,if the numberis8.45andwe needtorounditbyremoving5then the preceding digit 4 willremainunchanged and the number will the roundedto8.4.

Example 5 What would be the SI unitfor the quantity P } L _ ?

I inc

Example 7Convert into metre :

(a) 7 nm (diameter of a small virus )

(b) 40 Em (thickness of milky Way galaxy)

(c) 1.4 Gm (diameter of thesun)

(d) 41 Pm (distance of the nearest star)

( b ) 1 fg (massof human DNA )

(c) 500 Mg=( Mass of jumbo jet, loaded )

( d ) 3.34 x 10 ~ 24 g (massof hydrogen molecule )

Solution:

(a) 0.91x 10“ 27g=0.91x10“ 3

°kg=9.1x10“ 31kg(b) 1 fg=1x10" 15g=1x10-18kg

(c) 500 Mg=500x106g=500x103kg=5.0x105kg

(d) 3.34x10-24g=3.34x10-27kg

PROBLEMS FOR PRACTICE

1.Express the following in scientificnotation:

(i)0.0048

(ii)234,000

(iii) 8008

(iv) 500.0

(v)6.0012

[Ans.(i) 4.8x10"3(ii) 2.34x105(iii) 8.008x103(iv) 5.000xio2(v)6.0012x io°]

2.Match the following prefixes with their multiples:

10 " 6

io " 15

10

[Ans.Micro—>10 6,deca—>10,mega—>106, giga —>IO9,femto—>1015]

3.How many significant figures are present in the following?

[Ans.(i)2(ii)3(iii) 4 (iv) 4 (v)5)]

4.Roundup the following uptothree significant figures:

We often needto convertunitsfromonesystem toanother during calculations.The method that

is used to carry out the conversion is known as unit factor method. This is also called

Letus say wewant to convert5 inchesinto centimetres.This will be achievedas under.

We know that 1 inch=2.54cm

Example9 How many secondsarethere in 3 days ?

Solution :Weknow1 day=24hours

The unitfactorswillbe

Example 10 Pressureisdefined as force perunit areaof the surface.The SIunitof pressure,

pascalisshownasbelow

1 Pa = 1 NMm 2

o

If themassofairatsealevelis1034 gem ,calculate the pressureinpascals.

Solution:Pressureistheforceperunitarea and the force hereisthe weight

[000 mL

Idmx Idmx l dm

[0 cm x 10 cm x 10 cm

1.If the speed of lightis3.0x108ms 1,calculate the distance covered by light in 2.00 ns.

[Ans.0.6m]

2.Convertthe followingintobasicunits.

(i)28.7pm (ii)15.15pm (iii)25365 mg

2.Inscientific studies, two different systems of measurement have been in use.These are the

English System and the Metric System

3.The metric system is called the International System of Units (SI units).

4.The SI system has seven base units pertaining tosevenfundamental quantities, viz.length,mass, time, electric current,thermodynamic temperature, amount of the substance andluminous intensity.

5.The SI system allows the use of prefixes to indicate the multiples or submultiples of a unit.

6.Mass of a substance istheamountof matter present in it while weight istheforce exerted bygravity on anobject.

7.Density of asubstanceis its amountperunitvolume.

8.Avery largeor very smallnumber (or quantity)canbe represented usingscientifcnotation(N

x10n)wherenistheexponenthavingpositive or negativevalueandNvariesbetween 1 and

9.

9.Precisionreferstothe closenessofvarious measurementsfor thesamequantity.

10.Accuracy istheagreement ofa particular valuetothetruevalueof the result.

11.Theuncertaintyin themeasurementsor calculated valuesisindicated bymentioningthenumberof significant figures.

12.Significant figuresare meaningful digits which are known withcertainty.

13.The resultof additionor subtraction of numberscannothavemore digitstothe rightof thedecimalpointthan any of the numbers.

14.In the multiplication and division of numbers,the resultmustbe reported with significantfigures whichareno more than in the number with least significant figures.

15.Forconversionofunitsformonesystem toanother,unitfactor methodor dimensionalanalysisiscarriedout.

EXERCISES

Based on Different University Papers Objective Questions

Multiple - Choice Questions

1.Whichof the followingis nota baseunit?

(a) Length

(b)Area

8.The numberof significant figures in 1.20x102is

1.(b) 2.(a) 3.(c) 4.(b) 5.(a) 6.(b) 7.(a) 8.(a) 9.(c) 10.(b)

1.Whatisthe SIunitofmass? Howis itdefined?

2.Givethe prefixes for the multiples: 10-3,10-1,10+1,10+3

3.Whatdoyou mean by significant figures? Explain withan example.

4.Explain the following in scientificnotation:

(i)0.000123 (ii)5670000 (iii)74932

5.Roundoff the followingtofour significant figures:

(i)1.3785 (ii)11.175 (iii)0.069327

6.Fill in the blanks with the followingconversion:

7.Vanadiummetalisaddedtosteeltoimpartstrength.The densityof vanadiumis5.96g cm .

O

Express this in SIunitkgm .

8.Express the following numberstofour significant numbers:

(i) 5.607892 (ii) 1.78986x103 (iii) 5.608

9.Whatisthe significant figure in(i Avogadroa number(6.0x10 ),and(ii)Planck’sconstant