Pharmaceutics Basic Principles

PHARMACEUTICS

We dedicate this text to our teachers and professors who inculcated in us a love of scientific principles and pharmaceutics.

PHARMACEUTICSBasic Principles and

Application to Pharmacy Practice

Edited by

ALEKHA K. DASH, RPH, PHDCreighton University, Nebraska, USA

SOMNATH SINGH, PHDCreighton University, Nebraska, USA

JUSTIN TOLMAN, PHARMD, PHDCreighton University, Nebraska, USA

AMSTERDAM BOSTON HEIDELBERG LONDON

NEW YORK OXFORD PARIS SAN DIEGO

SAN FRANCISCO SINGAPORE SYDNEY TOKYO

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14 15 16 17 10 9 8 7 6 5 4 3 2 1

Contents

Preface ixAcknowledgments xiList of Contributors xii

IPHYSICAL PRINCIPLES AND

PROPERTIES OF PHARMACEUTICS

1. Introduction: Terminology, BasicMathematical Skills, and Calculations

EMAN ATEF AND SOMNATH SINGH

1.1. Introduction 31.2. Review of Basic Mathematical Skills 31.3. Graphical Representation 111.4. Dimensions and Units 121.5. Conclusions 14Case Studies 14References 15

2. Physical States and ThermodynamicPrinciples in Pharmaceutics

VIVEK S. DAVE, SEON HEPBURN

AND STEPHEN W. HOAG

2.1. Introduction 172.2. Composition of Matter 172.3. Forces of Attraction and Repulsion 182.4. States of Matter 222.5. Thermodynamics 332.6. Basic Concepts and Definitions 352.7. The First Law of Thermodynamics 372.8. Enthalpy and Heat Capacity 392.9. The Second Law of Thermodynamics 412.10. The Third Law of Thermodynamics 442.11. Free Energy and Thermodynamic Functions 442.12. Chemical Equilibrium 462.13. Open Systems 462.14. Conclusions 47Case Studies 47Appendix 2.1 Calculus Review 48References 49

3. Physical Properties, Their Determination,and Importance in PharmaceuticsSOMNATH SINGH AND ALEKHA K. DASH

3.1. Introduction 513.2. Surface and Interfacial Tension 523.3. Adsorption 533.4. Solubilization 593.5. Rheology 613.6. Colligative Properties 653.7. Osmolarity and Osmolality 683.8. Solubility and Solutions of Nonelectrolytes 713.9. Spectroscopy 733.10. Conclusions 83Case Studies 82References 84

4. Equilibrium Processes in PharmaceuticsSUNIL S. JAMBHEKAR

4.1. Introduction 854.2. Gastrointestinal Physiology 874.3. Ionization 904.4. Partition Coefficient: Lipophilicity 934.5. Equilibrium Solubility 964.6. The Effect of pH 984.7. Use of Co-solvents 994.8. Drug Dissolution and Dissolution Process 1004.9. Factors Influencing the Dissolution Rate 1024.10. Passive Diffusion 1044.11. Biopharmaceutics Classification System (BCS) 1054.12. Conclusions 106Case Studies 106References 107Suggested Readings 108

5. Kinetic Processes in PharmaceuticsRAMPRAKASH GOVINDARAJAN

5.1. Introduction 1095.2. Thermodynamics vs. Kinetics 1095.3. Chemical Reaction Kinetics and Drug Stability 1105.4. Diffusion 1275.5. Dissolution 1305.6. Conclusions 136

v

Case Studies 136References 137

6. BiopolymersSOMNATH SINGH AND JUSTIN TOLMAN

6.1. Introduction to Polymers 1396.2. Introduction to Peptides and Proteins 1426.3. Introduction to Oligonucleotides 1466.4. Conclusions 149Case Studies 149References 150Suggested Readings 150

IIPRACTICAL ASPECTS OF

PHARMACEUTICS

7. Drug, Dosage Form, andDrug Delivery Systems

ALEKHA K. DASH

7.1. Introduction 1537.2. Pharmaceutical Ingredients 1547.3. Preformulation Studies 1547.4. Physical Description 1547.5. Liquid Dosage Forms 1547.6. Solid Dosage Forms 1567.7. Partition Coefficient and pKa 1577.8. Solubility 1577.9. Dissolution 1577.10. Polymorphism 1577.11. Stability 1587.12. Conclusions 158Case Studies 158References 159

8. Solid Dosage FormsALEKHA K. DASH

8.1. Introduction 1618.2. Powders 1618.3. Capsules 1658.4. Tablets 1688.5. Manufacture of Compressed Tablets 1708.6. Methods Used for Manufacture of Compressed

Tablets 1728.7. Tablet Compression and Basic Functional Units

of a Tablet Press 1738.8. Quality Control of Tablet Dosage Form 1758.9. Tablet Coating 1788.10. Conclusions 179Case Studies 179References 180

9. Liquid Dosage FormsHARI R. DESU, AJIT S. NARANG, LAURA A. THOMA

AND RAM I. MAHATO

9.1. Introduction 1819.2. Selection of Liquid Dosage Forms 1819.3. Types of Liquid Dosage Forms 1839.4. General Aspects of Liquid Dosage Forms 2079.5. Manufacturing Processes and Conditions 2109.6. Packaging 2149.7. Labeling 2179.8. Quality Assurance and Quality Control 2179.9. Regulatory Considerations 2199.10. Conclusions 221List of Abbreviations 221Case Studies 221References 222

10. Aerosol Dosage FormsJUSTIN A. TOLMAN AND MEGAN HUSLIG

10.1. Introduction 22510.2. Lung Anatomy 22610.3. Lung Physiology 22710.4. Pulmonary Drug Targets 22810.5. Pulmonary Drug Deposition 22810.6. Therapeutic Gases 23010.7. Inhaled Aerosols 23210.8. Conclusions 237Case Studies 238Acknowledgments 238References 238

11. Semisolid Dosage FormsSHAILENDRA KUMAR SINGH, KALPANA NAGPAL

AND SANGITA SAINI

11.1. Introduction 24111.2. Classification of Semisolid Dosage Forms 24311.3. Percutaneous Absorption 24911.4. Theory of Semisolid Dosage Forms 25311.5. Methods of Enhancement of Percutaneous

Absorption 25511.6. Characterization and Evaluation of Semisolid

Dosage Forms 26011.7. Procedure and Apparatus for Diffusion Experiment 26311.8. Conclusions 269Case Studies 269References 270

12. Special Dosage Forms andDrug Delivery Systems

SARAT K. MOHAPATRA AND ALEKHA K. DASH

12.1. Introduction 27312.2. Special Dosage Forms 274

vi CONTENTS

12.3. Parenteral Drug Delivery 27512.4. Osmotic Delivery 27712.5. Nanotechnology for Drug Delivery 28012.6. Implantable Drug Delivery 29212.7. Prodrugs 29912.8. Conclusions 301Case Studies 302References 303

IIIBIOLOGICAL APPLICATIONS

OF PHARMACEUTICS

13. Membrane Transport and PermeationJUSTIN A. TOLMAN AND MARIA P. LAMBROS

13.1. Introduction 30713.2. Cell Membranes 30713.3. Membrane Transport 30913.4. Pharmacologically Relevant Membrane

Transport Processes 31113.5. Conclusions 315Case Studies 315References 315

14. Factors Affecting Drug Absorptionand Disposition

CHONG-HUI GU, ANUJ KULDIPKUMAR

AND HARSH CHAUHAN

14.1. Introduction 31714.2. Drug Absorption 31714.3. Oral Drug Absorption Processes 31814.4. Food Effects on Oral Drug Absorption 323

14.5. Evaluation of Oral Absorption 32414.6. Drug Disposition 32514.7. Conclusions 327Case Studies 327References 329

15. Routes of Drug AdministrationMOHSEN A. HEDAYA AND JUSTIN A. TOLMAN

15.1. Introduction 33315.2. Parenteral Drug Administration 33415.3. Transdermal Drug Administration 33715.4. Ophthalmic Drug Administration 33815.5. Auricular (Otic) Drug Administration 34015.6. Nasal Drug Administration 34115.7. Pulmonary Drug Administration 34115.8. Oral Drug Administration 34215.9. Rectal Drug Administration 34615.10. Vaginal Drug Administration 34715.11. Conclusions 347References 348

16. Bioavailability and BioequivalenceAJIT S. NARANG AND RAM I. MAHATO

16.1. Introduction 34916.2. Bioavailability 34916.3. Factors Affecting Bioavailability 35216.4. Bioequivalence 35916.5. Conclusions 362Case Studies 362References 362

Index 365

viiCONTENTS

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Preface

Pharmaceutical education in the United States ofAmerica has been undergoing substantial changesover the past several decades to address changes in apharmacists role in the provision of pharmaceuticalcare. Pharmacy education has had a historical per-spective that prepared student pharmacists to engagein pharmaceutical dispensing or pursue graduatepharmaceutical education focused on research. Anyclinically-focused education was then obtained throughpost- baccalaureate training and experience. The cur-rently evolving perspective of pharmacy education isfocused on preparing student pharmacists as providersof clinical pharmaceutical care and as the medicationexpert in the healthcare system.

These evolutions have increased the need forpharmacy education to be solidly-grounded in scien-tific principles. Key domains of pharmaceutical knowl-edge include: medicinal chemistry and pharmacologyfor an understanding of drug molecule properties andmechanisms of action; pharmaceutics and biopharma-ceutics to utilize physicochemical properties of drugsto develop a safe, effective and reliable drug productand their interactions with human physiology; phar-macokinetics and pharmacodynamics to explain drugmovement and pharmacologic effects within systems;pharmacy practice to interpret the role of medicationsin the diagnosis, treatment, and prevention of disease;

and social and administrative studies to evaluatehealth services and patient safety. Pharmaceuticaleducation should substantively address all of thesedomains to provide scientific foundations for rationalclinical decision making. Additionally, only pharmacyeducation can provide the scientific depth and breadthacross these various levels of knowledge domains.

This textbook is intended to provide a basicscientific introduction to the fields of pharmaceuticsand biopharmaceutics specifically tailored to meet theneed of practice of Pharmacy. Current educationalresources in these fields are principally focused ona historical perspective of pharmaceutical education.They either provide a mathematically rigorous andtheoretical introduction to these fields or are brieflyintegrated into larger resources focused on otherknowledge domains. Pharmaceutics: Basic Principles andApplication to Pharmacy Practice will help pharmacystudents gain the scientific foundation to understanddrug physicochemical properties, practical aspects ofdosage forms and drug delivery systems, and thebiological applications of drug administration.

Alexha H. DashJustin TolmanSomnath Singh

Pharmaceutics: Basic Principles and Application to Pharmacy Practice includes a companion website with a full colorimage bank and flip videos featuring difficult processes and procedures, as well as sample questions for studentsto test their knowledge. To access these resources, please visit booksite.elsevier.com/9780123868909.

ix

Acknowledgments

We would like to thank all our family members for their unstinted support during the preparation of this text.We would also like to acknowledge the following individuals who have contributed to this book or supplementalmaterials:

Daniel MuntBarbara BittnerDawn TrojanowskiMegan HusligRoger LiuKatherine Smith

xi

List of Contributors

Eman Atef School of Pharmacy-Boston, MassachusettsCollege of Pharmacy and Health Sciences, Boston, MA,USA

Harsh Chauhan College of Pharmacy, CreightonUniversity, Omaha, NE, USA

Alekha K. Dash Department of Pharmacy Sciences, Schoolof Pharmacy and Health Professions, CreightonUniversity, Omaha, NE, USA

Vivek S. Dave St. John Fisher College, Wegmans School ofPharmacy, Rochester, NY USA

Hari R. Desu Department of Pharmaceutical Sciences,University of Tennessee Health Science Center, Memphis,TN, USA

Ramprakash Govindarajan Research and Development,GlaxoSmithKline, Research Triangle Park, North Carolina,USA

Chong-Hui Gu Vertex Pharmaceuticals, Inc., Cambridge,MA, USA

Mohsen A. Hedaya Department of Pharmaceutics, Facultyof Pharmacy, Kuwait University, Safat, Kuwait

Seon Hepburn University of Marweyland, School ofPharmacy, Baltimore, MD, USA

Stephen W. Hoag University of Marweyland, School ofPharmacy, Baltimore, MD, USA

Megan Huslig Affiliation to come

Sunil S. Jambhekar LECOM Bradenton, School of Pharmacy,Bradenton, FL, USA

Anuj Kuldipkumar Vertex Pharmaceuticals, Inc., Cambridge,MA, USA

Maria P. Lambros Department of Pharmaceutical Sciences,College of Pharmacy, Western University of HealthSciences, Pomona, CA, USA

Ram I. Mahato Department of Pharmaceutical Sciences,University of Tennessee Health Science Center, Memphis,TN, USA

Sarat K. Mohapatra Department of Pharmacy Sciences,School of Pharmacy and Health Professions, CreightonUniversity, Omaha, NE, USA

Kalpana Nagpal Department of Pharmaceutical Sciences,G. J. University of Science and Technology, Hisar, India

Ajit S. Narang Drug Product Science and Technology,Bristol-Myers Squibb, Co., New Brunswick, NJ, USA

Sangita Saini PDM College of Pharmacy, SaraiAurangabad, Bahadurgarh, Haryana, India

Shailendra Kumar Singh Department of PharmaceuticalSciences, G. J. University of Science and Technology,Hisar, India

Somnath Singh Pharmacy Sciences, School of Pharmacyand Health Professions, Creighton University, Omaha, NE,USA

Laura A. Thoma Department of Pharmaceutical Sciences,University of Tennessee Health Science Center, Memphis,TN, USA; Creighton University School of Pharmacy andHealth Professions, Omaha, NE, USA

xiii

P A R T I

PHYSICAL PRINCIPLES ANDPROPERTIES OF PHARMACEUTICS

C H A P T E R

1

Introduction: Terminology, Basic MathematicalSkills, and CalculationsEman Atef1 and Somnath Singh2

1School of Pharmacy-Boston, Massachusetts College of Pharmacy and Health Sciences, Boston, MA, USA2Pharmacy Sciences, School of Pharmacy and Health Professions, Creighton University, Omaha, NE, USA

CHAPTER OBJECTIVES

Review the basic mathematics applicable inpharmacy.

Apply the concept of significant figures inpharmacy.

Apply basic calculus, logarithms, andantilogarithms to solve pharmaceutical problems.

Apply basic statistics (mean, mode, median, andstandard deviation) to interpret pharmaceuticaldata.

Interpret a graph and straight-line trend of datato derive useful information.

Review frequently used units and dimensions inpharmacy.

Keywords Basic mathematics review

Basic statistics

Dimensional analysis

Graphical representations

Logarithmic calculations

Significant figures

Units and dimensions

1.1. INTRODUCTION

How much drug should be prescribed to a newbornbaby compared to an adult? How do different patho-logical conditions affect the prescribed dose? How isthe drug therapeutic dose determined? How long is a

drug stable and can be used without compromising itstherapeutic efficacy? Why do some drugs expirewithin 1 month, whereas others expire after a coupleof years? How do you interpret data reported in theliterature to derive some useful and clinically signifi-cant information about the therapeutic outcomes of adrug that can be used to counsel a patient and answersome of the pertinent questions a pharmacist encoun-ters daily? To answer such questions and more, thepharmacist must have adequate mathematical and sta-tistical skills. Therefore, this chapter provides a basicintroduction to pharmaceutical calculations, units, andbasic statistics terms.

1.2. REVIEW OF BASICMATHEMATICAL SKILLS

1.2.1 Integers

The numbers 0, 1, 2, 3, 21, 22, 23, and so on, arecalled integers or whole numbers, which can be eitherpositive or negative and can be arranged in ascendingorder, as shown in Figure 1.1, where they increase asyou move from left to right on the line. Therefore, anegative integer such as 23 is smaller than 22.

1.2.2 Zero and Infinity

Mathematical operations involving zero and infinitydo not work in the usual way, which sometimes isthe reason for errors in pharmaceutical calculations.The following examples and key concepts illustrate the

3Pharmaceutics. DOI: http://dx.doi.org/10.1016/B978-0-12-386890-9.00001-7 2014 Elsevier Inc. All rights reserved.

special rules governing the role of zero and infinity inmathematical operations:

Any number multiplied by zero equals zero, e.g.,123 05 0. This result is unusual because generallymultiplication of any number x by y results in anumber that is different from either x or y, exceptwhen y is equal to 1, which results in no change inx. Otherwise, x increases if y is a positive integer(i.e., a whole number) greater than 1 and decreasesif y is a fraction or an integer lower than 1. In thefollowing examples, x is always 12:

123 15 12 (i.e., no change in the value of x if y5 1).123 35 36 (i.e., the value of x increases from 12

to 36 if y5 3).123 2 352 36 (i.e., the value of x decreases

from 12 to 236 if y523)123 2 13 52 4 (i.e., the value of x decreases from

12 to 24 if y52 13which is a negative fraction123 13 5 4 (i.e., the value of x decreases from 12

to 4 if y5 13which is a positive fraction) Any number multiplied by infinity (N) equals

infinity, e.g., 123N5N. This is also unusualfollowing the discussion provided formultiplication by zero.

Any number divided by zero is mathematicallyundefined; e.g.,12=05Undefined. This result isunusual because generally division of any number xby y results in a number z, which provides x whenmultiplied by y. For example, dividing 12 by 4results in 3, which is correct because 3 multiplied by4 provides the original number 12. However, 12divided by 0 cannot result in a specific number thatcan provide 12 when multiplied by 0. Therefore, theoutcome of 12 divided by 0 is undefined.

Any number divided by infinity is mathematicallyundefined; e.g.,12=N5Undefined. This result isalso unusual following the discussion provided fordivision by zero because any number multipliedbyN would result inN; it cannot ever provide theoriginal number, 12.

1.2.3 Rule of Indices

A number with a power or exponent such as 127 iscalled an indice, where 12 is called the base and 7 isthe exponent. Mathematical problems involving indi-ces with a common base are solved easily by applyingthe following rules:

Exponents are added when multiplying indices, e.g.,

1273 1255 127155 1212

1273 1253 12235 12715235 129

The exponent of the divisor is subtracted from theexponent of the dividend when dividing one indiceby another, e.g.,

1254 1235 125235 122

1234 1255 123255 1222

1293 1234 1243 1225 1291324125 126

Multiple exponents of a base are multiplied, e.g.,

12535 1253 35 1215

122535 12253 35 12215126

p5 12

63125 123

1263p

5 1263

135 122

An indice having a negative exponent is equal to itsinverse with a positive exponent, e.g.,

122351

123

5

12

0@

1A235 12

5

0@

1A35 123 123 12

53 53 5

0@

1A5 1728

1255 13:82

An indice having a fraction as its exponent is equalto its root with a power equal to the denominator ofthe fraction followed by an exponent equal to thenumerator of the fraction, e.g.,

6425564

5p

2523 23 23 23 2

5p

25 225 4

Any indice having zero as an exponent is equal to1, e.g.,

1205 1

10005 1

All the rules governing mathematical operationsinvolving indices can be summarized as shownhere, assuming x as a base:

xy3 xz 5 xy1z

xy

xz5 xy2z

xyz 5 xyzx2y 5

1

xy

xyz 5 xzp y

x0 5 1

0 321123

FIGURE 1.1 Ascending order of integers from left to right.

4 1. INTRODUCTION: TERMINOLOGY, BASIC MATHEMATICAL SKILLS, AND CALCULATIONS

I. PHYSICAL PRINCIPLES AND PROPERTIES OF PHARMACEUTICS

1.2.4 Scientific or Exponential Notation

Pharmacists often encounter extremely large or smallnumbers, which creates a challenge when doing simplemathematical operations involving such numbers. Forexample, the normal range of testosterone level in men(1630 years old) is 72148 pg/mL (i.e.,0.000,000,000,0720.000,000,000,148 g/mL) [1], and thenumber of skin cells in humans is 110,000,000,000 [2].Therefore, scientific notation is used to handle suchlarge or small numbers, using exponential notationor the power of 10. Thus, the testosterone level canbe conveniently expressed as 7.23 10211 to1.483 10210 g/mL. Similarly, the number of skin cellscan be represented by 1.13 1011. The number expressedby scientific notation is called the scientific number.

Generally, only one figure appears before the deci-mal point in the first part of scientific notation; it iscalled the coefficient. When multiplying or dividingtwo scientific numbers, the exponents are added orsubtracted respectively, as shown below:

Multiplication of scientific numbers:

(1.13 10211)3 (7.23 1010)5 7.923 1021; where theexponents, 211 and 10, have been added.(1.13 1011)3 (7.23 1010)5 7.923 1021; where theexponents, 11 and 10, are added.

Division of scientific numbers:

(1.13 1011)4 (7.23 10211)5 0.153 1022 or 1.53 1021;where exponent 211 is subtracted from exponent 11.(1.13 1011)4 (7.23 107)5 0.153 104 or 1.53 103;where exponent 7 is subtracted from exponent 11.

Addition or subtraction of scientific numbers can beeasily carried by following the two steps shown below:

Step 1: The exponent of each number must be sameas shown in example below where (7.23 109) hasbeen converted to (0.0723 1011).Step 2: The coefficients are added or subtracteddepending on the problem.

Addition of scientific numbers:

(1.13 1011)1 (7.23 109)5 (1.13 1011)1(.0723 1011)5 (1.11 .072)3 1011 or 1.173 1011;where the decimal point in coefficient 7.2 is movedleft by two positions to make exponents in both thescientific numbers equal to 11.

Alternatively, 1.13 1011 can be converted to110.03 109 to make the exponents in both the scientificnumbers equal to 9 as shown below:

(1.13 1011)1 (7.23 109)5 (110.03 109)1(7.23 109)5 (110.01 7.2)3 109 or117.23 1095 1.173 1011

Subtraction of scientific numbers:

(1.13 1011)2 (7.23 109)5 (1.13 1011)2(.0723 1011)5 (1.12 .072)3 1011 or1.0283 10115 1.033 1011

1.2.5 Logarithms and Antilogarithms

Exponential data often are used in pharmacy calcula-tions; e.g., the acidity constant, Ka, of acetaminophen is3.093 10210, [3] which is used for developing itsstable formulation. Performing mathematical calcula-tions using such exponentials is not convenient.Furthermore, in many instances such as accelerated sta-bility studies of drugs, it is difficult to find any correla-tion between exponential data. Another example is ofdata generated out of first-order rate kinetic studies. Insuch situations, using logarithms is helpful because itlinearizes the data. Using logarithms makes calcula-tions such as multiplication or division involving expo-nentials easy because it converts them into easy-to-handle simple addition or subtraction problems. A log-arithm is the power to which a base must be raised toobtain a number. Therefore, there are two kinds oflogarithms on the basis of differences in the base: thecommon logarithm (log), where the base is 10, andnatural logarithm (ln), where the base is e (wheree5 2.7182818. . .). The following examples clarify thisconcept:

Using log10(log to the base 10):log1010005 3 (i.e., log of 1000 to the base 10 is 3) isequivalent to 1035 1000 where 10 is the base, 3 isthe logarithm (i.e., the exponent or power), and 1000is the number.

Using natural log (loge or ln):ln 1005 4.6052 (i.e., log of 100 to the base e is 4.6052)is equivalent to e4.60525 100 or 2.71834.60525 100where e or 2.7183 is the base, 4.6052 is the logarithm(i.e., the exponent or power), and 100 is the number.

Anytime something, c, changes at a rate propor-tional to c, it is represented by a natural logarithmicequation, e.g., the equation representing the first-orderrate kinetics as shown next.

The first-order reaction is represented by dc=c52 k1dt, where c is the concentration of the reactant atany time, t and k1 is the proportionality constant.Integration of this equation between concentration C0at time t5 0 and concentration Ct at time t5 t resultsin the following equation using natural log:

lnCt5 lnC02 k1t

Therefore, it is essential to know the interconversionfrom a common logarithm to a natural logarithm andvice versa, which can be derived as shown next.

51.2. REVIEW OF BASIC MATHEMATICAL SKILLS


Assume that the ratio of a natural and common logof the same number is x, i.e.

ln10

log105 x

Since ln 105 2.303 and log 105 1, the ratiox5 2.303.

Therefore, for any number y,

ln y5 2:303 log y

Sometimes the logarithm (or ln) of a number isavailable, but you need to find the number itself,which can be done by finding the antilogarithm of alogarithmic number. Therefore, the antilogarithm isalso called the inverse logarithm. The following exam-ples illustrate this concept:

log x5 2; x5 antilog of 25 100because 1025 100

log x52 2; x5 antilog 2 25 0:01because 10225 0:01

The natural logarithm also works in the same way:

ln x5 2:303; so; x5 antiln 2:3035 10The rules governing logarithmic calculation are

shown in Table 1.1.

1.2.6 Accepted Errors and Significant Figures

All numbers can be categorized as either exact orinexact numbers:

Exact numbers: Any numbers that can bedetermined with complete certainty; e.g., there are110 students in a class, 12 eggs in one dozen eggs, 7days in a week, 12 months in a year, etc. All thesenumbers can be figured out without any doubt.

Inexact numbers: Numbers associated with anymeasurement are not exact because accuracydepends on the sensitivity of the instrument used insaid measurement. You can increase the precision ofthe measurement by carefully following thestandard operating procedure or by selecting amore sensitive instrument.

Lets start by defining and differentiating two termsthat often are interchanged mistakenly: accuracy andprecision.

Accuracy refers to how closely measured valuesagree with the correct value, whereas precision refers tohow closely an individual measurement agrees withanother. Precision is correlated to reproducibility of ameasurement and is indicated by standard deviationof multiple repeated measurements. Obviously, a high-er standard deviation indicates a lower precision ofmeasurements. Therefore, a measurement can be ofhigh precision but of low accuracy. For example, 100grams of a drug are weighed using a balance having110% errors due to a manufacturing defect. Thus, youcan weigh out 100 grams multiple times with 99.99%precision (i.e., each 100 grams weighed out does notdiffer from another by more than 0.01 gram), but theweight accuracy is 90% due to the systematic error.

In another example, if an assay method reports495 mg of ampicillin in a 500 mg capsule of ampicillin,the measurement accuracy is 99% [(495/500)3 100],i.e., a 1% error. If the assay is repeated 5 times forthe same sample of ampicillin capsule and each timethe result is 495 mg of ampicillin, the precision of theexperimental method is 100%. Thus, precision indi-cates the repeatability of an experimental method. Ifthe same error or mistake is repeated during eachexperiment, the result may be precise but inaccurate.

1.2.6.1 Measurement Accuracy

All measurements have a degree of uncertaintybecause no device can provide absolutely perfect mea-surement with absolute zero error. The error can bepredicted from but not limited to, for example, theprocess used to prepare the dosage form, the sensitiv-ity of the utilized balance or measuring devices, or thenumber of significant figures.

1.2.6.1.1 BASED ON THE OFFICIAL COMPENDIA

The U.S. Pharmacopeia [6] states that

Unless otherwise specified, when a substance is weighedfor an assay, the uncertainty should not exceed 0.1% of thereading.

Also according to the USP,

Measurement uncertainty is satisfactory if 3 times the stan-dard deviation of not less than 10 replicates weighingsdivided by the amount weighed, does not exceed 0.001.

Another commonly used parameter is the relativestandard deviation (RSD), which equals to Standarddeviation=Mean3 100.

TABLE 1.1 Rules for Logarithmic Mathematical Operations

Common Logarithm Natural Logarithm

log xy5 log x1 log y ln xy5 ln x1 ln y

logx

y5 log x2 log y ln

x

y5 ln x2 ln y

log xy5 y log x ln xy5 y ln x

logxy

p5 log x

1y5

1

ylog x ln

xy

p5 ln x

1y5

1

yln x



For example, if the weight of 100 mg of active phar-maceutical ingredient (API) is taken 10 times (n5 10)and the following weights were recorded

0.1001 g, 0.1002 g, 0.0999 g, 0.1003 g, 0.1003 g,0.1002 g, 0.1001 g, 0.1001 g, 0.1003 g, 0.1003 g

you can determine the average weight, standard devia-tion, and relative standard deviations for these mea-surements as follows:

Average weight Mean weight

50:1001 g1 0:1002 g1 0:0999 g1 0:1003 g1 0:1003 g1 0:1002 g1 0:1001 g1 0:1001 g1 0:1003 g1 0:1003 g

10

51:0018 g

105 0:10018 g

Standard deviation SD5Variance

p5

The average of the squared differences from the Mean

p

Difference from the Mean Squared Differences

(0.10012 0.10018) (2 0.00008)2

(0.10022 0.10018) (0.00002)2

(0.09992 0.10018) (2 0.00028)2

(0.10032 0.10018) (0.00012)2

(0.10032 0.10018) (0.00012)2

(0.10022 0.10018) (0.00002)2

(0.10012 0.10018) (2 0.00008)2

(0.10012 0.10018) (2 0.00008)2

(0.10032 0.10018) (0.00012)2

(0.10032 0.10018) (0.00012)2

Average of the squared differences

520:0000821?1 20:000122

105 1:563 1028

The standard deviation SD51:563 1028

p5 0:000125

Relative standard deviation RSD

SDMean

3 10050:000125

0:100183 1005 0:125%

Since 33 0.000125 (i.e., 33 SD)5 0.000375 and0.000375, 0.001, the preceding measurement uncer-tainty is acceptable according to the USP.

1.2.6.1.2 COMPOUNDING PRESCRIPTIONS AND

INDUSTRIAL MANUFACTURING

Based on the USP, a maximum error of 6 5% isacceptable in compounding prescriptions [4]. Unless

otherwise indicated, especially relatively potent pre-scriptions may require higher accuracy.

On the other hand, a maximum error of 6 1% isacceptable in pharmaceutical industrial measurements.

1.2.6.1.3 THE SENSITIVITY OF THE UTILIZED

BALANCE

The sensitivity (i.e., the lowest weight detected) of abalance is an important and crucial parameter,

enabling pharmacists to decide on which balance touse to fulfill the needed accuracy. The following exam-ples illustrate this concept.

A powder weight is found to be 13.2 g using a bal-ance with sensitivity5 0.1 g. In other words, you maybe somewhat uncertain about that last digit; it couldbe a 2, 1, or 3.

On the other hand, a measurement done to the clos-est hundredth of a gram indicates the following:13.21 g can be 13.22 g or 13.20 g.

Thus, the former balance should be used if accuracyof a tenth of a gram is required. However, if a drug ishighly potent and has a narrow therapeutic window,such as digoxin or warfarin, a higher order of accuracycould be needed. Therefore, a balance with a sensitiv-ity of a thousandth of a gram should be preferred tousing a balance with sensitivity only up to a tenth orhundredth of a gram.

The sensitivity is also called resolution, whichdepends on types of balances, which include thefollowing:

Precision top pan balances have 0.001 g resolution. Analytical balances have 0.1 mg or 0.01 mg

resolution. Semi-micro balances have 0.001 mg or 0.002 mg

resolution. Micro balances have at least 0.0001 mg resolution.

1.2.6.1.4 SIGNIFICANT FIGURES

The number of significant figures is simply thenumber of figures that are known with some degree ofreliability. The number 13.2 is said to have three signif-icant figures. The number 13.20 is said to have foursignificant figures.



Therefore, the number of significant figures in ameasurement is the number of digits that are knownwith certainty, plus the last one that is not absolutelycertain but rather an approximate (inexact) number.For example, 13.2 g has three significant figures: 0.2 isthe last and thus an inexact number; it could be a 0.249or 0.15. Both are rounded to 0.2. For simplicity, youcan indicate the tolerance as 13.26 0.05. A mass of13.20 g indicates an uncertainty of 0.00 g, so theexpected weight would be any value in the range of13.20 g6 0.005 g. Thus, any measuring instrumentsuch as a balance with greater sensitivity would pro-vide measured value having a greater number of sig-nificant figures. Table 1.2 summarizes the rules helpfulin deciding the number of significant figures in a mea-sured value.

The potential ambiguity in the last rule can beavoided through the use of standard exponential, orscientific, notation. For example, depending onwhether two or three significant figures are correct,you could write 120 g as follows:

1:23 102 g two significant figuresor

12:03 10 g three significant figures

1.2.6.1.5 DETERMINING SIGNIFICANT FIGURES IN

MATHEMATICAL OPERATIONS

1.2.6.1.5.1 ADDITION AND SUBTRACTION Whenmeasured quantities are used in addition or subtrac-tion, the uncertainty is determined by the absoluteuncertainty in the least precise measurement (not bythe number of significant figures). Sometimes this isconsidered to be the number of digits after the decimalpoint.

Now consider these numbers:

78956.23 m11.875 m

Addition of the preceding two numbers provides78968.105 m, but the sum should be reported as78968.10 m because there are two digits after the deci-mal point in 78956.23 m, which is less precise than11.875 m.

1.2.6.1.5.2 MULTIPLICATION AND DIVISION Whenexperimental quantities are multiplied or divided, thenumber of significant figures in the result is deter-mined by the quantity with the least number of signifi-cant figures. If, for example, a density calculation ismade in which 27.124 grams is divided by 2.1 mL, thedensity should be reported as 13 g/mL, not as13.562 g/mL because the number of significantfigures in 2.1 is two.

1.2.6.1.5.3 LOGARITHMIC CALCULATIONS In loga-rithmic calculations, the same number of significantfigures is retained in the mantissa as there are in theoriginal number; e.g., a 10-digit calculator would showthat log 5795 2.762678564. Since the original number(579) contains three significant figures, the resultshould be reported as 2.763; i.e., the mantissa alsoshould contain three significant figures. Likewise,when you are taking antilogarithms, the resultingnumber should have as many significant figures as themantissa in the logarithm (so the antilog of1.5795 37.9, not 37.931). For any log, the number tothe left of the decimal point is called the characteristic,and the number to the right of the decimal point iscalled the mantissa.

Caution: The concept of significant figure should beapplied with caution while dispensing a pharmaceuti-cal prescription. The minimum weight or volume ofeach ingredient in a pharmaceutical formula or pre-scription should be large enough that the error intro-duced is not greater than 5% (5 in 100); i.e.,pharmaceutical calculations incorporating three digitsafter decimal point are of acceptable precision [4].While applying the concept of significant figures, youshould know that some of the values could never be

TABLE 1.2 Rules for Determining Number of Significant Figures

Measured

Values

Number of

Significant Figures

Rules

12.786 g 5 All nonzero digits are significant.

12.078 g 5 Zero is significant if flanked by nonzero digits.

0.02 g 1 Zero immediately after a decimal point but before a nonzero digit is not significant where it merelyindicates its position.

0.20 g 2 Zero after a decimal point is significant if preceded by a nonzero digit.

120 g 2 or 3 Zero at the end of a number and not preceded by a decimal point is not necessarily significant. Ifsensitivity of the balance is 10 g, the number of significant figures would be two. Similarly, the numberof significant figures would be three if sensitivity of the balance is 1 g.



approximate because they are exact. The question ofsignificant figures arises only when there is approxi-mation in a measurement. For example, if a pharmacistcombines five unit dose packages of a liquid that are4.5 ml each, the total volume obtained would be(53 4.55 22.5) 22.5 mL, which should be rounded offto two significant figures, not one. The reason is thatthe only inexact number, 4.5 mL, contains only twosignificant figures; and 5, which contains one signifi-cant figure, is an exact number. So, now the questionis whether 22.5 should be reported as 22 or 23.Actually, it should be reported as 23 mL. The follow-ing rules should help when deciding to round up ordown.

1.2.6.1.5.4 RULES FOR ROUNDING OFF

NUMBERS The rules for rounding off are based onwhether the digit to be dropped is equal to or greaterthan 5, as well as the digits flanking that digit. The fol-lowing examples illustrate rules for rounding off num-bers up to the required significant figures such as two:

The last digit to be retained is increased by one ifthe digit to be dropped is greater than 5; e.g., 17.9 isrounded up to18 because the digit to be dropped, 9,is greater than 5.

The last digit to be retained is left unaltered if thedigit to be dropped is less than 5; e.g., 17.4 isrounded down to 17 because the digit to bedropped, 4, is less than 5.

If 5 is the digit to be dropped but is followed bynonzero digit(s), the last remaining digit isincreased by one; e.g., 17.512 is rounded up to 18because digits 1 and 2 follow 5 and are not zero.

If 5 is the digit to be dropped and is followed byzero only or no other digits, the last remaining digitshould be rounded up or down depending onwhether it is an even number or odd number. Thelast remaining digit is increased by one if it is anodd digit or left unaltered if it is an even digit; e.g.,17.5 is rounded up to 18 because the last remainingdigit, 7, is an odd digit, but 18.5 is rounded down to18 because the last remaining digit is an even digit.

1.2.7 Significant Difference

Conclusions can frequently be drawn about signifi-cant differences by looking at the standard deviationor standard error bars in case of clear overlapping. Butsometimes the following points should be considered:

Clinical significance versus statistical significance Comparison between treatments and treatments

versus control

The effect of three antipyretic drugs in Figure 1.2 isused to clarify the preceding two points. Compared tothe control group, the three tested antipyretic drugsresulted in a statistically significant drop in thepatients temperature, p, 0:05. But Antipyretic 1,although statistically significant from the control, maybe insignificant because a drop of less than 1C is notenough to create a clinically important effect.Although Antipyretics 2 and 3 are not significantly dif-ferent, they are both statistically different from the con-trol group; i.e., both are effective medications.

1.2.8 Samples and Measure of Centrality

Collecting, managing, and interpreting sample dataare important responsibilities for pharmacists. Samplesare generally small numbers of observations or datataken from a comparatively large population withclearly defined parameters [5]. For example, all thehypertensive patients having systolic blood pressuregreater than 170 mmHg are a population, but 150 suchpatients selected for a clinical trial study of a hyperten-sive drug constitute a sample. Such a study generates alarge amount of data based on experimental design ofthe study. One hundred twenty patients selected for thestudy may be divided in different groups being admin-istered different dosages of the hypertensive drug understudy. One group may be administered placebo,whereas another group may get the hypertensive drugunder current clinical practice. The blood pressurechange could be different in different groups or even indifferent subjects in the same group. Thus, reporting theconclusion of the study requires a summary number(s)because the original raw data are not communicable.

3

2.5

2

1.5

1

0.5

0Control Antipyretic 1 Antipyretic 2

Treatment group

Ave

rage

dro

p in

the

patie

nts

tem

pera

ture

Antipyretic 3

FIGURE 1.2 Average drop in the patients body temperaturefollowing treatment with antipyretic.



One of the most useful approaches is using somesort of summary number or numbers, which aregood indicators of centrality of sampling observa-tions or data. The three measures of central tendencyused in pharmacy are mean, median, and mode.Calculating mean is discussed in the Section 1.2.6.1.1.Following sections describe how to calculate medianand mode using the same example used for calculat-ing mean.

1.2.8.1 Calculating Median

The median is the middle data of observationsarranged in ascending or descending order. Thus, halfof the data or observation would be greater than themedian, but the other half would be less than it.

If the weights of 100 mg of active pharmaceuticalingredient (API) are taken 10 times (n5 10) and theseweightings are

0.1001 g, 0.1002 g, 0.0999 g, 0.1003 g, 0.1003 g,0.1002 g, 0.1001 g, 0.1001 g, 0.1003 g, 0.1003 g

you can arrange the preceding data in ascending orderas follows:

0.0999 g, 0.1001 g, 0.1001 g, 0.1001 g, 0.1002 g,0.1002 g, 0.1003 g, 0.1003 g, 0.1003 g, 0.1003 g

Therefore, the median is the mean of the fifth (i.e.,0.1002) and sixth (i.e., 0.1002) data, which is equal to0.1002. Obviously, median is the mean of two middledata in case of an even number of observations ordata, but in the case of an odd number of observations,it would be the middle datum.

It is evident that median is not influenced by anyextreme data because it would be the same in the pre-ceding example whether the first datum is any numberless than 0.0999 g or the tenth datum is greater than0.1003. On contrast, mean or average is significantlyinfluenced by any extreme data.

1.2.8.2 Calculating Mode

Mode is simply the data that occur most of the timeand, therefore, generally is used for a large set ofobservations or data. In the preceding example, modeis equal to 0.1003. Sometimes, the frequency of twonumbers could be equal in a data set, in which casethe data are termed bimodal.

1.2.9 Dimensional Analysis

Dimensional analysis is a mathematical method,also known as the unit factor method, that utilizes theunits and ratios between them in calculating a desiredquantity with the required unit.

To use the dimensional analysis method, you haveto know the relations between different units, as inthese examples: 1 kg5 2.2 lb, 1 ft5 12 inches, 1 g5 1000milligrams, 1 day5 24 hours.

Examples:

1. The recommended amoxicillin dose for severeinfection is 25 mg/kg/day in divided doses every12 hours. How many milliliters of amoxicillin canbe given to a 66 lb patient per 12 hours, knowingthat the oral suspension has 125 mg per 5 mL?

To solve this problem using dimensionalanalysis, you should recognize the given parameterswith the correct units and the unit required in thefinal answer.

You need the final answer unit to be in mL/12hours. And you have to utilize the given patientand drug information as well as your knowledge ofratios between different units to solve this problem.

The following relations are needed to solve theproblem:

1 kg5 2.2 lb; 1 day is 24 hours.

The number of mL=12 hours

525 mg

kg3 day3

1 kg

2:2 lb3 66 lb3

5 mL

125 mg

31 day

212 h 5mL

12 h

In the preceding formula, all the units that areundesired in the final answer will cancel each other,and you end up with a final answer in the desiredunit, i.e., the number of milliliters every 12 hours.(The correct answer is 15 mL/12 hours.)

It is worth mentioning that the use ofdimensional analysis is not the only way to solvethis problem. The problem can be solved usingmultiple sets of proportions and can be performedstepwise.

2. The digoxin dose of a premature baby is 20microgram/kg once a day. The available elixir is0.05 mg/mL. How many milliliters should be givento a 5.5 lb baby per day?

Remember that1 kg5 2.2 lb; 1 mg5 1000 microgram.

The number of milliliters per day520 microgram

kg=day

31 kg

2:2 lb3 5:5 lb3

1 mg

1000 micrograms

31 mL

0:05 mg5 mL=day

(The correct answer is 1 mL/12 hours.)



3. A drug provides 10,000 units/250 mg tablet. Howmany total units does the patient get byadministering 4 tablets per day for 10 days?

Key: The 250 mg is the total tablet weight. This isa distracting number and has nothing to do withyour calculation.

(The correct answer is 400,000 units/10 days.)4. The recommended dose of a drug is 10 mg/kg/day

(the drug is given every 6 hours). How many mL/6hours should be prescribed to 60 lb child? Theavailable suspension is 150 mg/tsp.

(The correct answer is 2.3 mL/6 hours.)

1.3. GRAPHICAL REPRESENTATION

A graph is simply a visual representation showingthe relationship between two or more variables. Itshows how one variable (a dependent variable)changes with alteration in another variable (an inde-pendent variable). A graph consists of four quadrantsin which the abscissa or ordinate is negative or posi-tive, as shown in Table 1.3.

Looking at the theoretical and measured valuechanges with time in Table 1.4, you might find it diffi-cult to observe the relationship between the two vari-ables. However, when you look at the graph inFigure 1.3, the relationship becomes quite apparent.Thus, the graph is a better tool to present data in aclear, visual manner.

1.3.1 Interpreting Graphs

When you attend a lecture, your initial comprehensionis high. However, as the lecture progresses, your compre-hension typically decreases with time. This informationmay be presented graphically as a two-dimensional graph

consisting of a dependent variable (comprehension) andan independent variable (time).

The magnitude of the independent variable is usu-ally measured along the x-axis, or the horizontal scale.The dependent variable is measured along the y-axis,or the vertical scale. See Figure 1.4.

The graph in this figure enables you to see quicklythat for the initial time of about 30 minutes, the com-prehension is high at approximately 9.5, and it gradu-ally falls to about 2.0 after about 60 minutes, or 1 hour.It is evident that graphs are useful in providing avisual representation of data.

1.3.2 Straight-Line Graphs (Simple LinearRegression)

A graph is a straight line (linear) only if the equa-tion from which it is derived has the following form:

y5mx1 b

TABLE 1.4 Theoretical and Measured Value Changes with Time

Time Measured Value Theoretical Value

0 2 4.6 25

1 2 3.4 23

2 2 0.6 21

3 0.8 1

4 3.4 3

5 4.4 5

TABLE 1.3 Quadrants on a Cartesian Graph

Quadrant II (2 x, 1 y) Quadrant I (1 x, 1 y)

Quadrant III (2 x, 2 y) Quadrant IV (1 x, 2 y)

6

4

2

0

2

4

6Time (hours)

Measured values Theoretical values

210Val

ues

3 4 5 6

FIGURE 1.3 Graphical representation of the theoretical andmeasured values changes with time.

10

9

8

7

6

5

4

3

2

1

00 50

Time (min)

Com

preh

ensi

on (

Sca

le o

f 1-1

0)

100 150

FIGURE 1.4 Graph representation of the comprehension of thestudents versus time.

111.3. GRAPHICAL REPRESENTATION


where y is the dependent variable, x is the indepen-dent variable, m is the slope of the straightline5y=x, and b is the y intercept (when x5 0).

Example:The data in Table 1.5 represent the osmolality and

molality of solution of nicotinamide.

1. Find the linear relationship between the molalityand osmolality of nicotinamide solutions.

2. What is the predicted value of osmolality when themolality of the solution is 255 mmol/kg?

3. Calculate the correlation coefficient for the linearrelationship that exists between the osmolality andmolality of solutions of nicotinamide.

Answers:

1. Microsoft Excel was used to plot a graph using thefollowing data (see Figure 1.5).

Using Excel functions, you can find out that thelinear relationship between molality and osmolalitycan be described using the straight-line equation asy5 (0.9483x)1 0.8045; i.e., Osmolality5 (0.94833Molality)1 0.8045 at low concentrations.

2. Substitute the molality term with 255 in thepreceding equation:

Osmolality5 (0.94833 255)1 0.80455242.6 mOsmol/kg.

3. The correlation coefficient, r, is the square root of R2

shown in the plot of molality and osmolality inFigure 1.5. So,

Correlation coefficient,r5OR25O0.999515 0.9995.Note: The coefficient of correlation value, r, points

to the strength of the relationship between the x and y

variables, which can range between 21 and 11. If thevalue of r is zero, this means there is no relationshipbetween the two variables. If r521 or 11, there isperfect negative or positive linear correlation, respec-tively. In science generally, an acceptable value for rmust be at least 0.70. Values below 0.70 reflect weakcorrelation.

The square of r is known as the coefficient of deter-mination (represented by R2 in Figure 1.5), which tellshow much of the variability in the dependent variabley is explained by x, the independent variable. An R2 of0.60 means that 60% of the variability in y is explainedby x and 40% of the variability in y could be due toother factors.

1.4. DIMENSIONS AND UNITS

Matter is anything that has weight and occupiesspace. Based on this broad definition, anything yousee, feel, or interact withsuch as computers, tables,coffee mugs, tablets, capsules, and solutionsconstitu-tes matter. To define the properties of matteramount,composition, position in space and time, and moreyou need quantitative tools called dimensions and units.How heavy is your laptop? If you said 5 pounds,you have just used the unit of weight to define theheaviness of a substance that constitutes matter.

Why do you need to study dimensions and units?The pharmacist, being the drug expert on the health-care team, is responsible for formulating, dispensing,and evaluating drugs and dosage forms for optimaltherapeutic efficacy. A few examples in which thisknowledge will help practicing pharmacists include (1)effectively formulating tablets, capsules, powders, solu-tions, ointments, or other dosage forms to meet thera-peutic objectives; (2) performing dosage adjustments forpatients based on the patients weight, age, or body sur-face area; (3) determining the amount of active ingredi-ent in a dosage form; and (4) determining the rate ofinfusion of a parenteral dosage form.

TABLE 1.5 Osmolality and Molality of Solution of Nicotinamide

Molality, mmol/kg Osmolality, mOsmol/kg

25 23.9

50 48.2

75 71.9

100 93.2

125 122.5

150 143.7

175 168.1

200 191.2

225 215.2

250 231.1

275 262.9

300 286.9

400

300

Osm

olal

ity (

mO

smol

/kg)

200

100

0 50 150100 200 250

y = 0.9483x + 0.8045R2 = 0.9991

3000

350

Molality (mmo1/kg)

Relationship between molality andosmolality of nicotinamide

FIGURE 1.5 Graphical representation of the relationship betweenosmolality and the molality of solution of nicotinamide.



1.4.1 The Three Fundamental Dimensions

The properties of matter are usually expressedthrough the use of three fundamental dimensions:length, mass, and time. Each of these properties isassigned a definite unit and a reference standard. Inthe metric system, these units are assigned the centi-meter (cm), the gram (g), and the second (sec); accord-ingly, it is called the cgs system.

The International Union of Pure and AppliedChemistry (IUPAC) introduced a System International,or SI, unit system to establish an internationally uni-form set of units. Although physical pharmacy usescgs units for most calculations, SI units are appearingwith increasing frequency in textbooks.

1.4.2 Units Based on Length

Length and area: The SI unit for length is the meter.Other commonly used prefixes are listed in Table 1.6.In addition to the units here, many textbooks preferusing angstrom units (A, equal to 10210 meters or1028 cm) to express microscopic distances. The prefixesshown in the table may also be used to represent otherdimensions such as mass and time. For example, 1029

seconds is termed as a nanosecond.The units of area are cm2 or m2 in cgs and SI sys-

tems, respectively. Therefore, area is represented asthe square of length.

Volume: Volume is also derived from units basedon length, and uses units in cm3, also represented ascubic centimeter or cc (or m3 in the SI system).Volume is also frequently defined in terms of the liter,with 1 liter or 1 L being equal to 1000.027 cm3. Thefrequently used unit for volume in physical pharmacyis the milliliter, or mL, which is roughly equal to1 cm3 or mL.

1.4.3 Units Based On Mass

Mass and weight: The SI unit of mass is the kilogram,or the Kg. The cgs unit of mass is the gram, which is1/1000 of the kilogram. Mass is often expressed as theweight of a substance, which is actually a force, andis discussed under Derived Dimensions inSection 1.4.4.

Example:The concentration of a drug in a patients blood was

reported to be 15 mcg(microgram)/mL. Total volumeof the blood in the same patient was 5 liters. Answerthe following questions based on the information pro-vided in this case study.

1. Identify the amount, volume, and concentrationterms from this example.

2. What is the total amount of drug in the patientsblood?

3. Is there any relationship among concentration,volume, and amount? If so, identify it.

Answers:

1. In this example, 15 mcg is an amount term, 1 mland 5 liters are volume terms, and 15 mcg/mL is aconcentration unit.

2. Total amount of drug in theblood5Concentration3Total Volume:5 15 mcg/mL3 5000 mL5 75000 mcg5 75 mg

3. Yes, a relationship exists:Amount5Concentration3Volume.

1.4.4 Derived Dimensions

Four derived dimensions are usually discussed inpharmacy calculations. They include (1) density andspecific gravity; (2) force; (3) pressure; and (4) work,energy, and heat.

1.4.4.1 Density and Specific Gravity

The pharmacist uses the quantities density and spe-cific gravity for interconversions between mass andvolume. Density is a derived quantity and combinesthe units of mass and volume:

Density5mass/volume

The units of a derived quantity can be obtained bysubstituting the units for the individual fundamentalunits. This process is called dimensional analysis. Forexample, the units of mass and volume in the cgs sys-tem are g and cm3. So the units of density in the cgssystem are g/cm3.

The specific gravity of a substance is the ratio of itsdensity to that of water, at a constant temperature.

TABLE 1.6 Common Multiples and Their Prefixes and Symbols

Multiple Prefix Symbol

1012 Tera T

109 Giga G

106 Mega M

103 Kilo K

1022 Centi c

1023 Milli m

1026 Micro m

1029 Nano n

10212 Pico p

131.4. DIMENSIONS AND UNITS


Note that, being a ratio of two similar quantities, thespecific gravity is not described by a unit:

Specific gravity5density of a substance/density ofwater

For the same reason, any quantity expressed as aratio is always dimensionless.

The density of the drug, excipients, and dosageform are important for the following reasons:

1. During manufacturing, mixing solids with similardensities ensures complete mixing and minimizesthe solid segregation (i.e., demixing).

2. Knowing the density of a dosage form helps inpredicting the final volume occupied by theprescription.

3. Knowing the density of a substance can allow theconversion of percentage (w/w) to % (w/v) andvice versa.

Example:Knowing that concentrated hydrochloric acid has

36% (w/w), (specific gravity 1.179), can you calculatethe percentage w/v?

Answer:Here, 36% (w/w)5 36 grams in 100 grams, based

on the density the 100 grams occupies:

100 grams

1:179 grams=mL5 84:8 mL

To calculate the % (w/v), you set a proportion:

36 grams

84:8mL5x grams

100mL.x542:5-36%w=w542:5%w=v

Example 2:Calculate the volume occupied by the container vol-

ume of 21.2 grams of a toothpaste if the density5 0.94.Answer:

21:2 grams

0:94 grammL5 22:6 mL

1.4.4.2 Force

The force exerted on a body is equal to its massmultiplied by the acceleration achieved as a result ofthat force:

Force5mass3 acceleration

Now can you derive the units of force in the cgssystem, given that the units of acceleration are cm/s2

in the cgs system? Also, can you derive the units in theSI system?

The weight of a body is equal to the force exertedon that body due to gravity. The weight of a substancewith a mass of 1 g is therefore equal to

Weight5 1 g3 981 cm/sec25 981 g cm/sec2, or 981dynes

However, it is a common practice to express weightin the units of mass (g) for convenience.

1.4.4.3 Pressure

Pressure is the force applied per unit area and isexpressed as dynes per cm2. Its cgs units are

(g cm/sec2)/cm25 g/(cm sec2).

1.4.4.4 Work, Energy, and Heat

When you apply a force on a body and move it fora certain distance, you do work. Work is defined asforce3distance, and its cgs units are dynes cm, orergs. Another commonly used unit for work is joules(J), which is equal to 107 ergs, and is the SI unit ofwork. Energy is the capacity to do work and has thesame unit as work.

Heat and work are equivalent forms of energy, andtheir units are interchangeable. The cgs unit of heat isthe calorie and is equal to 4.184 J.

1.5. CONCLUSIONS

This chapter reviewed the basic mathematical con-cepts frequently used in the practice of pharmacy andintroduced the basic concepts of graphical data repre-sentation, interpretation, and analysis for finding linearregression. Moreover, the system of units, their inter-conversion, and dimensional analysis are invaluable inpharmacy calculations. We hope that the concepts pre-sented in this chapter will help students in interpretingliterature data more efficiently and that they will findit a handy tool while doing calculations for dispensingprescriptions.

CASE STUDIES

Case 1.1

The USP monograph states, Pravastatin sodiumcontains not less than 90.0 percent and not more than110.0 percent of the labeled amount of pravastatinsodium (C23H35NaO7). The chemical analysis of apravastatin sodium 80 mg tablet found it to contain71.9 mg of the chemical. Does it comply with the USPstandard?

Approach: No, it does not comply with the USPstandard because 71.9 mg is 89.9% of the labeledamount. The USP standard is not less than 90.0%,which means 90.1% or 90.2% is an acceptable amount,but not 89.9%.



Case 1.2

A physician in a hospital wrote a prescription for342.45 mg of theophylline in 250 ml of 5% dextrosesolution. The pharmacy department of the hospitalcompounded an intravenous solution containing340 mg of theophylline per 250 ml 5% dextrose solu-tion. The prescribing physician thought that his pre-scription was not accurately dispensed and returnedthe compounded theophylline admixture. How wouldyou resolve this situation?

Approach: The computer program used in calcula-tion of dose generates data consisting of three or moredigits after the decimal point, which happened in thisinstance. If the pharmacy department dispensed340 mg of the drug instead of 342.45 mg, that amountis acceptable because the error introduced is wellbelow the acceptable limit of 5%. The compoundingpharmacist should resolve this issue by patiently andprofessionally explaining the concept of significantfigures and the realistic precision expected duringmeasurement of ingredient(s) for intravenous fluids.

Case 1.3

A prescription with a dose of 2 mg/kg was writtenfor a 66 lb patient. The pharmacy technician calculatedthe dose and forgot the right conversion (1 kg5 2.2 lb).Instead, the technician used a wrong conversion factor

of 1 kg5 2 lb. As the pharmacist in charge, you aresupposed to inform the technician regarding her mis-take and find out whether the error is within theacceptable limit of 6 5%.

Approach: You know that 66 lb should be66/2.25 30 kg.

The actual dose5 303 25 60 kg3mg/kg5 60 mg.The wrong dose calculated by the technician5

66/2.05 33 kg. The calculated dose5 333 25 66 mg.% Error5 (662 60)/603 1005 (6/60)3 1005 10%This error is higher than 5% of the allowed limit

and is not an acceptable calculation.

References

[1] ,http://www.hgh-pro.com/hormones.html. Normal hormonelevel, [accessed on 01.06.2013].

[2] ,http://bionumbers.hms.harvard.edu/search.aspx?task5searchbyamaz. Amazing BioNumbers, [accessed on 01.06.2013].

[3] Ma JKH, Hadzija BW. Basic physical pharmacy. Burlington, MA,USA: Jones & Bartlett Learning; 2013. p. 53

[4] Brecht EA. Pharmaceutical measurements. In: Sprowls JB, DittertLW, editors. Sprowls American pharmacy: an introduction topharmaceutical techniques and dosage forms. 7th ed.Philadelphia: Lippincott; 2008. p. 511. 1974, digitized 2008

[5] Bolton S. Pharmaceutical statistics: practical and clinical applications.2nd ed. New York: Marcel Dekker, Inc.; 1990.

[6] United States Pharmacopenia 36 General Chapter 41: Weightsand Balances, p. 53.

15REFERENCES


C H A P T E R

2

Physical States and Thermodynamic Principlesin Pharmaceutics

Vivek S. Dave1, Seon Hepburn2 and Stephen W. Hoag21St. John Fisher College, Wegmans School of Pharmacy, Rochester, NY USA

2University of Maryland, School of Pharmacy, Baltimore, MD, USA

CHAPTER OBJECTIVES

Define atoms, molecules, elements, andcompounds, and discuss their roles in thecomposition of matter.

Explain the binding forces between molecules.

Define gaseous state and describe the kinetictheory of gas.

Analyze various gas laws and interpret theliquefaction of gases.

Discuss supercritical fluids and apply thisdiscussion in explaining aerosols and theimplantable infusion pump.

Define the liquid state and explain vaporpressure and boiling of a liquid.

Apply the ClausiusClapeyron equation,Raoults law, and Henrys law for explaining thebehavior and properties of a liquid state.

Define solid state and discuss amorphous andcrystalline solids.

Interpret the significance of polymorphism,dissolution, wetting, and solid dispersion inpharmacy.

Define the basic terminology used inthermodynamics.

Discuss laws of thermodynamics and theirapplication in explaining protein stability andspontaneity of the transport phenomenon.

Keywords

First law of thermodynamics Gas Liquid Physical states of matter Second law of thermodynamics Solid Thermodynamics

2.1. INTRODUCTION

This chapter is divided into two parts: the first partdeals with the nature of matter, and the second partdeals with the thermodynamics of pharmaceutical sys-tems. The goal of this chapter is to introduce the scien-tific principles you need to understand how and whypharmaceutical dosage forms work and what kinds ofproblems a dispensing pharmacist can encounter whenworking with pharmaceutical products and how tosolve these problems.

2.2. COMPOSITION OF MATTER

Matter can be defined as anything that has a massand a volume. The mass of matter is generally deter-mined by its inertia or its resistance to change in accel-eration when in motion or at rest. One common way

17Pharmaceutics. DOI: http://dx.doi.org/10.1016/B978-0-12-386890-9.00002-9 2014 Elsevier Inc. All rights reserved.

of defining this is to consider the acceleration when anexternal force is applied to a mass. The acceleration ofmatter is described by Newtons second law of motionand expressed by

F5ma 2:1Thus, a greater mass will have a slower acceleration

for the same applied force. The volume of matter isdetermined by the space it occupies in three dimen-sions. Almost all matter is composed of atoms, alsocalled atomic matter. There are forces between theatoms and molecules that make up matter, and thenature of these forces dictate some of the importantproperties of matter.

One of the most important concepts is to under-stand the state of matter and the properties associatedwith each state. When dispensing tablets or capsules,the things you have to worry about are very differentfrom that when you are dispensing a solution, emul-sion, or suspension. Because every prescription shouldhave storage conditions listed on the packaging, thisknowledge will affect how you label every prescrip-tion. For example, everyone knows tablets and cap-sules should not be stored in the same bathroomwhere the patient likes to take hot and steamyshowers, but how about cough syrup? The goal of thefollowing sections is to give you the scientific princi-ples to answer these questions so that you can bettercounsel patients and advise physicians. The key con-cepts you need to understand are the states of matter,the properties associated with each state, and wherethese properties come from.

2.3. FORCES OF ATTRACTIONAND REPULSION

Molecules interact with each other via the forces ofattraction and repulsion. Attractive forces are of twotypes: cohesive forces and adhesive forces. The forces ofattraction between molecules of the same substanceare known as cohesive forces. The forces of attractionbetween the molecules of different substances areknown as adhesive forces. The forces that act on mole-cules to push them apart are known as repulsive forces.

Consider two atoms that start far apart and cometogether. As they approach each other, a combinationof attractive and repulsive forces act on the two atoms.The attractive forces act to pull the molecules closer.Attractive forces (FA) are inversely proportional to thedistance separating the molecules (r), as shown by therelationship

FA ~ 1rn 2:2

where n varies with the type of atoms/molecules [1].For example, n typically equals 6, but for some gasessuch as nitrogen n 7. These forces arise from theVan der Waals or dispersion forces, which aredescribed later. Using Eq. 2.2, one can represent theforce of attraction between atoms/molecules as a func-tion of the distance between them using a potentialenergy diagram, as shown in Figure 2.1. As the attrac-tive forces increase, the potential energy becomesincreasingly negative. From this curve, you can seeseveral important characteristics. First, as the atoms ormolecules get close together, the attractive forcesincrease very rapidly; and second, the magnitude ofthe attractive forces act over a range of atomic dis-tances, and it requires close proximity for the forces toaffect molecular behavior.

If the overlap of the electron cloud is small, thelong-range component of attractive forces is signifi-cant. Conversely, when the molecules come closeenough that their electron clouds interact, the short-range component of the attractive forces dominate(see Figure 2.2).

However, as you bring the atoms or molecules veryclose together, the electron clouds start to overlap,which leads to very strong repulsive forces. The repul-sive forces FR are proportional to an exponential rela-tionship with the reciprocal of the distance separatingthe molecules r as follows:

FR ~ e1=r 2:3

For repulsive forces, an exponential functionchanges more rapidly on the potential energy diagram(see Figure 2.3). As the repulsive forces increase, thepotential energy becomes increasingly positive.Compared to repulsive forces, attractive forces act overa longer distance.

The total force on two atoms or molecules as a func-tion of distance is given by the sum of the attractive

0

+

Distance of separation

Pot

entia

l ene

rgy

FIGURE 2.1 Potential energy diagram as a function of separationdistance for attractive forces.

18 2. PHYSICAL STATES AND THERMODYNAMIC PRINCIPLES IN PHARMACEUTICS


and repulsive forces; this sum is given in Figure 2.4. Astwo distant molecules approach each other, the energychanges are gradual and attractive to a point of mini-mum energy; this minimum in potential is the equilib-rium or average bond length, which is the balancepoint between attractive and repulsive forces. After theminimum as the molecules come closer together, theenergy starts rising rapidly, and repulsive forces domi-nate. The distance where the attractive and repulsiveforces balance each other is the collision diameter.

It is important to distinguish between intramolecularand intermolecular bonds. Intramolecular bonds are forcesof attraction between the atoms that hold an individualmolecule together (e.g., covalent or ionic bonds).Intermolecular bonds are forces of attractions between amolecule and its neighboring molecule. All moleculesexhibit intermolecular bonding to a certain degree.Most of these attractions are relatively weak in nature.The common types of intermolecular attractive forcescan be divided in several classes. They include electro-static forces, polarization forces, dispersion forces, and

hydrogen bonding [2]. The sum of the electrostatic,polarization, and dispersion forces is often called theVan der Waals forces. Each of these classes is describedin the following text.

Before discussing these forces, we need to introducethe concept of a dipole. A dipole is a charge separatedover a range. For example, HCl has a permanentdipole:

Because H is much less electronegative than Cl, theelectrons are predominantly around the Cl atom,which creates a permanent negative charge. Also, theH is electron deficient, so it has a permanent positivecharge, which is separated by the bond length. Dipolescan be permanent or transient. The degree of chargeseparation can be quantified by calculating the dipolemoment; interested readers can check out references[3] and [4].

2.3.1 Electrostatic Forces

The class of electrostatic forces includes the interac-tions between charged atoms and molecules such asion-ion, ion-permanent dipole, and permanentdipolepermanent dipole. These interaction forces canbe intra- and intermolecular.

One example of electrostatic interactions is an ionicbond, which is a type of chemical bond formed throughan electrostatic attraction between two oppositelycharged ions. Ionic bonds are formed between a cation,which is usually a metal, and an anion, which is usuallya nonmetal. The larger the difference in electronegativ-ity between the two atoms involved in a bond, the more

Bond energy

+

0

Distance of separationBond length

Pot

entia

l ene

rgy

(kJ/

mol

)

FIGURE 2.4 The total potential energy diagram of two atoms ormolecules.FIGURE 2.2 Overlapping electron clouds.

0

+

Distance of separation

Pot

entia

l ene

rgy

FIGURE 2.3 Potential energy diagram for repulsive forces.

192.3. FORCES OF ATTRACTION AND REPULSION


ionic (polar) the bond is. An ionic bond is formed whenthe atom of an element (metal), whose ionizationenergy is low, loses an electron(s) to become a cation,and the other atom (nonmetal), with a higher electronaffinity, accepts the electron(s) and becomes an anion.An ionic bond is a relatively strong bond with the bondenergy. 5 kcal/mole, e.g., sodium chloride.

Ion-dipole bonds are forces that originate fromthe electrostatic interactions between an ion and a neu-tral molecule containing a permanent dipole. Theseinteractions commonly occur when solutions of ioniccompounds are dissolved in polar liquidsforexample, NaCl dissolving in water. The interactionsoccur when a positive ion attracts the partially nega-tive end of a neutral polar molecule or vice versa. Ion-dipole attractions become stronger as either the chargeon the ion increases or as the magnitude of the dipoleof the polar molecule increases. Ion-dipole interactionsare relatively strong and relatively insensitive to tem-perature and distance. When an organic base is added

to an acidic medium, an ionic salt may be formed that,if dissociable, will have increased water solubilityowing to ion-dipole bonding.

Dipole-dipole forces are the forces that originate fromthe interaction of permanent dipoles. For example, theinteraction of a Cl atom with the H of an adjacent HClmolecule looks like this:

These are also known as Keesom forces, namedafter Willem Hendrik Keesom. Of the Van der Waalsforces, these are relatively strong forces with theenergy of attractionB17 kcal/mole.

2.3.2 Polarization and Dispersion Forces

The polarization class includes the interactionsbetween dipoles induced in a molecule by an electricfield from a nearby permanent dipole, ionized mole-cule, or ion. The dipole-induced dipole interaction,

also known as Debye Forces, is named after PeterJ. W. Debye. Dispersion forces include the interactionsbetween atoms and molecules even if they are chargeneutral and dont have permanent dipoles. Dispersionforces are electrodynamic in nature and occur whencharge separation occurs in a molecule due to the ran-dom motion of elections, and this transient chargeinduces a dipole in an adjacent molecule. These forces,called Van der Waals forces (dispersion forces), arealso known as London forces, named after aGermanAmerican physicist Fritz London, who cameup with this theory.

Polarization forces originate as a result of temporarydipoles induced in a molecule by a permanent dipolein a neighboring polar or charged molecule. As adipole approaches a molecule, the charge attracts theopposite charge and repels the same charge, whichresults in the polarization of the adjacent molecule,and this polarization leads to an electrostatic interac-tion between the two molecules:

These interactions are relatively weak with theenergy of attractionB13 kcal/mole.

Van der Waals forces originate from temporary dipolefluctuations, which affect electron distributions in adja-cent molecules. The attraction between the moleculesis electrical in nature. In an electrically symmetricalmolecule like hydrogen, there doesnt seem to be anyelectrical distortion to produce positive or negativeparts, but thats only true on average.

The preceding diagram represents a small symmet-rical molecule of hydrogen (H2). The even shadingshows that on average there is no electrical distortionor polarization. But the electrons are mobile, and atany given moment, they might position themselvestoward one end of the molecule, making that end 2.The other end will be temporarily devoid of electronsand become 1. A moment later the electrons maywell move to the other end, reversing the polarity ofthe molecule, as illustrated here:



This constant motion of the electrons in the mole-cule results in rapidly fluctuating dipoles even in themost symmetrical molecule. Imagine a molecule thathas a temporary polarity being approached by a mole-cule that happens to be entirely nonpolar just at thatmoment:

As the molecule on the right approaches, its elec-trons tend to be attracted by the slightly positive endof the molecule on the left. This sets up an induceddipole in the neighboring molecule, which is orientedin such a way that the 1 end of one is attracted to the2 end of the other:

A moment later the electrons in the left moleculemay move up the other end. In doing so, they repelthe electrons in the molecule at the right:

For groups of molecules, these random fluctuationsresult in attractive forces that hold the moleculestogether:

These forces between molecules are much weakerthan the covalent bonds within molecules. It is difficultto give an exact value, because the extent of the

attraction varies considerably with the size of the mol-ecule and its shape. Some of the main characteristics ofthese forces are as follows:

Van der Waals forces are extremely weak; i.e.,the typical bond energies range from 0.5 to1.0 kcal/mole for each atom involved.

They are temperature dependent; i.e., withincreasing temperatures, the attractive forcesdiminish significantly.

They occur at very short distances; i.e., they requiretight packing of molecules.

Steric factors influence the attraction; e.g., branchingin molecules significantly decreases attraction.

These forces commonly occur in lipophilic materialsand are relatively less significant in aqueoussystems.

Despite being relatively weak in nature, Van derWaals forces may play an important role in pharmaceu-tical systems. An important implication of these forces isobserved in the flocculation and deflocculation phe-nomena commonly observed in pharmaceutical suspen-sions [5]. The presence of Van der Waals forces betweenthe suspended particles results in the formation of looseagglomerates, or floccules, which rapidly settle downupon standing but are easily redispersible upon shaking.Conversely, if the repulsive forces predominate, the sus-pended particles do not flocculate but remain as discreteentities. These particles are slower to settle on standing;however, once settled, they form a relatively densermass in a process commonly known as caking, whichis difficult to redisperse.

2.3.3 Hydrogen Bonds

Hydrogen bonds are stronger and an important formof dipole-dipole interactions. Hydrogen bonding origi-nates when at least one dipole contains electropositivehydrogen. The bond exhibits an electrostatic attraction

of a hydrogen atom for a strongly electronegative atomsuch as oxygen, nitrogen, fluoride. Because hydrogenatoms are so small, they can get very close to the

212.3. FORCES OF ATTRACTION AND REPULSION


electronegative atom; and in strong hydrogen bonds,the hydrogen bond is partly covalent in nature, as theelectron of the hydrogen atom is delocalized to theelectronegative atom. Hydrogen bonds can be inter- orintramolecular in nature. A common example of inter-molecular hydrogen bonding is that observed betweenwater molecules as discussed in the followingparagraphs.

An example of intramolecular hydrogen bonding isa DNA molecule, where the nitrogen bases from thetwo strands are joined by intramolecular hydrogenbonds (see Figure 2.5). The hydrogen bonding betweennitrogen bases is critical to DNA structure and isimportant to DNA translation and replication.

For an example of intermolecular hydrogen bonding,consider two or more water molecules coming together:

H H

H

H

H

H

O

O+ +

+ +

++

O

The 1 hydrogen of one molecule is stronglyattracted to the lone pair of electrons on the oxygen ofother molecule. It is not a covalent bond, but theattraction is significantly stronger than a typicaldipole-dipole interaction. Hydrogen bonds have abouta 1/10th of the strength of an average covalent bond,

and are being constantly broken and reformed inwater. The energy of hydrogen bonding is1.010 kcal/mole for each interaction. Each water mol-ecule can potentially form four hydrogen bonds withsurrounding water molecules. This is why the boilingpoint of water is high for its molecular size.

2.4. STATES OF MATTER

The three primary states or phases of matter aregases, liquids, and solids (see Figure 2.6). In the solidstate, molecules, atoms, and ions are held in closeproximity by intermolecular, interatomic, and ionicforces. Atoms exhibit restricted oscillations in a fixedposition within a solid. With an increase in tempera-ture, the atoms acquire sufficient energy to overcomethe forces that hold them in the solid lattice, whichleads to the disruption of the ordered arrangement ofthe lattice as the system moves into a liquid state. Thisprocess is called melting, and the temperature of thistransition is called the melting point of the substance.The further addition of energy to a liquid results in thetransition into a gaseous state. This process is calledboiling, and the temperature of this transition is calledthe boiling point.

Occasionally, some solids (particularly those withhigh vapor pressures, e.g., carbon dioxide) can passdirectly from the solid to the gaseous state withoutmelting. This process is called sublimation. The reverse

Sugar Sugar

Nitrogenbase

Nitrogenbase

Phosphate Phosphate

Sugar Sugar

Nitrogenbase

Nitrogenbase

Phosphate Phosphate

Sugar Sugar

Nitrogenbase

Nitrogenbase

Phosphate Phosphate

Sugar Sugar

Nitrogenbase

Nitrogenbase

Phosphate Phosphate

Intramolecular hydrogen bonding

FIGURE 2.5 Hydrogen bonding betweenDNA standards.



process, i.e., from gaseous to solid state, is called depo-sition. Note that a phase is defined as a homogeneousphysically distinct portion of a system that is separatedfrom the other portions of the system by bounding sur-faces. For example, ice in water is an example of twophasesa solid and a liquid. You can also have phasesof the same state; for example, you can have an oil andwater emulsion in which you have two phasesan oilphase and a water phaseboth in the liquid state.These concepts are discussed in more detail later inSection 2.5 on thermodynamics.

Under certain conditions, substances can exhibit anin-between phase known as mesophase (Greek:mesos5middle) as shown in the phase diagram inFigure 2.7. Commonly observed mesophase statesinclude liquid crystals and supercritical fluids. Liquid crys-tals are a state of matter that has properties betweenthose of a conventional liquid and those of a solidcrystal. Supercritical fluids occur when a substance is ata temperature and pressure above their critical point orat the triple point because there are three phases inequilibrium at this point. When a material is in thesupercritical fluid state, there is not a distinct liquid orgas phase, and the system has properties of both gasand liquid. One unique property of supercritical fluidsis that gases like CO2 can have properties similar to asolvent, and the solvent properties can be varied bychanging the pressure and temperature. Because ofthis unique solvent property that can be varied, super-critical fluids have found significant importance in thepharmaceutical industry. For example, the selectiveextraction of pharmaceutical actives from biologicalsources is efficiently carried out using this approach.

2.4.1 Gaseous State

In the gaseous state, the attractive forces betweenthe atoms or molecules are not sufficient to hold themodules in close contact, and the molecules are free to

randomly move about in three dimensions (seeFigure 2.6). Matter in gaseous state has the followinggeneral properties:

Molecules exhibiting rapid motion due to higherkinetic energy

Molecules having weaker intermolecular forces Devoid of regular shape Capable of filling all the space in an enclosed

system Compressible upon application of external pressure Mostly invisible to the human eye

The properties of matter in gaseous state can bedescribed by the ideal gas law. The ideal gas lawdescribes the behavior of an ideal gas as a function oftemperature, pressure, volume, and amount of gas.The ideal gas law is derived from a combination of gaslaws formulated by Boyle, Charles, and Gay-Lussac. Themain assumptions in the ideal gas law derivation are

The gas molecules are hard spheres with nointermolecular interactions between the molecules.

Collisions between the molecules are perfectlyelastic; i.e., there is no energy loss between themolecules of gas during collisions.

Boyles law states that for one mole of an ideal gas atfixed temperature, the product of pressure (P) and vol-ume (V) is a constant, which can be described by

PV5 k 2:4Gay-Lussacs and Charless laws state that the volume

and absolute temperature (T) of a given mass of gas at

Supercritical fluid region

LiquidSolid

Melting line

Critical point

Boiling point

Triple point

Vapor

Sublimation line

Pc

Tc

Temperature

Pre

ssur

e

FIGURE 2.7 A typical phase diagram of a closed system inequilibrium.

FIGURE 2.6 The different states of matter.

232.4. STATES OF MATTER


constant pressure are directly proportional, as givenby the following relationship:

V~T 2:5V5 kT 2:6

Combining both laws gives

P1V1T1

5P2V2T2

2:7

From the preceding equation, one can assume thatPV=T is constant and can be mathematically expressedas

PV

T5R or PV5RT 2:8

where R is the constant value for an ideal gas.However, this equation assumes there is only one moleof gas. For n moles of an ideal gas, the equationbecomes

PV5 nRT 2:9

This equation is known as the ideal gas law. The con-stant R in the equation of state is also known as the mo

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