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Introduction to Modern Liquid Chromatography
Introduction to Modern Liquid Chromatography
Introduction to Modern Liquid Chromatography
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Introduction to Modern Liquid Chromatography

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The latest edition of the authoritative reference to HPLC

High-performance liquid chromatography (HPLC) is today the leading technique for chemical analysis and related applications, with an ability to separate, analyze, and/or purify virtually any sample. Snyder and Kirkland's Introduction to Modern Liquid Chromatography has long represented the premier reference to HPLC. This Third Edition, with John Dolan as added coauthor, addresses important improvements in columns and equipment, as well as major advances in our understanding of HPLC separation, our ability to solve problems that were troublesome in the past, and the application of HPLC for new kinds of samples.

This carefully considered Third Edition maintains the strengths of the previous edition while significantly modifying its organization in light of recent research and experience. The text begins by introducing the reader to HPLC, its use in relation to other modern separation techniques, and its history, then leads into such specific topics as:

  • The basis of HPLC separation and the general effects of different experimental conditions
  • Equipment and detection
  • The column—the "heart" of the HPLC system
  • Reversed-phase separation, normal-phase chromatography, gradient elution, two-dimensional separation, and other techniques
  • Computer simulation, qualitative and quantitative analysis, and method validation and quality control
  • The separation of large molecules, including both biological and synthetic polymers
  • Chiral separations, preparative separations, and sample preparation
  • Systematic development of HPLC separations—new to this edition
  • Troubleshooting tricks, techniques, and case studies for both equipment and chromatograms
Designed to fulfill the needs of the full range of HPLC users, from novices to experts, Introduction to Modern Liquid Chromatography, Third Edition offers the most up-to-date, comprehensive, and accessible survey of HPLC methods and applications available.
LanguageEnglish
PublisherWiley
Release dateSep 20, 2011
ISBN9781118210390
Introduction to Modern Liquid Chromatography

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    Introduction to Modern Liquid Chromatography - Lloyd R. Snyder

    CONTENTS

    PREFACE

    GLOSSARY OF SYMBOLS AND ABBREVIATIONS

    1 INTRODUCTION

    1.1 Background Information

    1.2 A Short History of HPLC

    1.3 Some Alternatives to HPLC

    1.4 Other Sources of HPLC Information

    References

    2 BASIC CONCEPTS AND THE CONTROL OF SEPARATION

    2.1 Introduction

    2.2 The Chromatographic Process

    2.3 Retention

    2.4 Peak Width and the Column Plate Number N

    2.5 Resolution and Method Development

    2.6 Sample Size Effects

    2.7 RELATED TOPICS

    References

    3 EQUIPMENT

    3.1 Introduction

    3.2 Reservoirs and Solvent Filtration

    3.3 Mobile-Phase Degassing

    3.4 Tubing and Fittings

    3.5 Pumping Systems

    3.6 Autosamplers

    3.7 Column Ovens

    3.8 Data Systems

    3.9 Extra-Column Effects

    3.10 Maintenance

    References

    4 DETECTION

    4.1 Introduction

    4.2 Detector Characteristics

    4.3 Introduction to Individual Detectors

    4.4 UV-Visible Detectors

    4.5 Fluorescence Detectors

    4.6 Electrochemical (Amperometric) Detectors

    4.7 Radioactivity Detectors

    4.8 Conductivity Detectors

    4.9 Chemiluminescent Nitrogen Detector

    4.10 Chiral Detectors

    4.11 Refractive Index Detectors

    4.12 Light-Scattering Detectors

    4.13 Corona-Discharge Detector (CAD)

    4.14 Mass Spectral Detectors (MS)

    4.15 Other Hyphenated Detectors

    4.16 Sample Derivatization and Reaction Detectors

    References

    5 THE COLUMN

    5.1 Introduction

    5.2 Column Supports

    5.3 Stationary Phases

    5.4 Column Selectivity

    5.5 Column Hardware

    5.6 Column-Packing Methods

    5.7 Column Specifications

    5.8 Column Handling

    References

    6 REVERSED-PHASE CHROMATOGRAPHY FOR NEUTRAL SAMPLES

    6.1 Introduction

    6.2 Retention

    6.3 Selectivity

    6.4 Method Development and Strategies for Optimizing Selectivity

    6.5 Nonaqueous Reversed-Phase Chromatography (NARP)

    6.6 Special Problems

    References

    7 IONIC SAMPLES: REVERSED-PHASE, ION-PAIR, AND IONEXCHANGE CHROMATOGRAPHY

    7.1 Introduction

    7.2 Acid-Base Equilibria and Reversed-Phase Retention

    7.3 Separation of Ionic Samples by Reversed-Phase Chromatography (RPC)

    7.4 Ion-Pair Chromatography (IPC)

    7.5 Ion-Exchange Chromatography (IEC)

    References

    8 NORMAL-PHASE CHROMATOGRAPHY

    8.1 Introduction

    8.2 Retention

    8.3 Selectivity

    8.4 Method-Development Summary

    8.5 Problems in the Use of NPC

    8.6 Hydrophilic Interaction Chromatography (HILIC)

    References

    9 GRADIENT ELUTION

    9.1 INTRODUCTION

    9.2 Experimental Conditions and Their Effects on Separation

    9.3 Method Development

    9.4 Large-Molecule Separations

    9.5 Other Separation Modes

    9.6 Problems

    References

    10 COMPUTER-ASSISTED METHOD DEVELOPMENT

    10.1 Introduction

    10.2 Computer-Simulation Software

    10.3 Other Method-Development Software

    10.4 Computer Simulation and Method Development

    References

    11 QUALITATIVE AND QUANTITATIVE ANALYSIS

    11.1 Introduction

    11.2 Signal Measurement

    11.3 Qualitative Analysis

    11.4 Quantitative Analysis

    11.5 Summary

    References

    12 METHOD VALIDATION with Michael Swartz

    12.1 Introduction

    12.2 Terms and Definitions

    12.3 System Suitability

    12.4 Documentation

    12.5 Validation for Different Pharmaceutical-Method Types

    12.6 Bioanalytical Methods

    12.7 Analytical Method Transfer (AMT)

    12.8 Method Adjustment or Method Modification

    12.9 quality Control and Quality Assurance

    12.10 Summary

    References

    13 BIOCHEMICAL AND SYNTHETIC POLYMER SEPARATIONS with Timothy Wehr, Carl Scandella, and Peter Schoenmakers

    13.1 Biomacromolecules

    13.2 Molecular Structure and Conformation

    13.3 Special Considerations for Biomolecule HPLC

    13.4 Separation of Peptides and Proteins

    13.5 Separation of Nucleic Acids

    13.6 Separation of Carbohydrates

    13.7 Separation of Viruses

    13.8 Size-Exclusion Chromatography (SEC)

    13.9 Large-Scale Purification of Large Biomolecules

    13.10 Synthetic Polymers

    References

    14 ENANTIOMER SEPARATIONS with Michael Lämmerhofer, Norbert M. Maier and Wolfgang Lindner

    14.1 Introduction

    14.2 Background and Definitions

    14.3 Indirect Method

    14.4 Direct Method

    14.5 Peak Dispersion and Tailing

    14.6 Chiral Stationary Phases and Their Characteristics

    14.7 Thermodynamic Considerations

    References

    15 PREPARATIVE SEPARATIONS with Geoff Cox

    15.1 Introduction

    15.2 Equipment for Prep-LC Separation

    15.3 Isocratic Elution

    15.4 Severely Overloaded Separation

    15.5 Gradient Elution

    15.6 Production-Scale Separation

    References

    16 SAMPLE PREPARATION with Ronald Majors

    16.1 Introduction

    16.2 Types of Samples

    16.3 Preliminary Processing of Solid and Semi-Solid Samples

    16.4 Sample Preparation for Liquid Samples

    16.5 Liquid-Liquid Extraction

    16.6 Solid-Phase Extraction (SPE)

    16.7 Membrane Techniques in Sample Preparation

    16.8 Sample Preparation Methods for Solid Samples

    16.9 Column-Switching

    16.10 Sample Preparation for Biochromatography

    16.11 Sample Preparation for LC-MS

    16.12 Derivatization in HPLC

    References

    17 TROUBLESHOOTING Quick Fix

    17.1 Introduction

    17.2 Prevention of Problems

    17.3 Problem-Isolation Strategies

    17.4 Common Symptoms of HPLC Problems

    17.5 Troubleshooting Tables

    References

    APPENDIX I PROPERTIES OF HPLC SOLVENTS

    I.1 Solvent-Detector Compatibility

    I.2 Solvent Polarity and Selectivity

    I.3 Solvent Safety

    Reference

    APPENDIX II. PREPARINGBUFFERED MOBILE PHASES

    II.1 Sequence of Operations

    II.2 Recipes for Some Commonly Used Buffers

    Reference

    Index

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    Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved.

    Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

    Published simultaneously in Canada.

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    Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

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    Library of Congress Cataloging-in-Publication Data:

    Snyder, Lloyd R.

    Introduction to modern liquid chromatography / Lloyd R. Snyder, Joseph J. Kirkland. – 3rd ed. / John W. Dolan.

    p. cm.

    Includes index.

    ISBN 978-0-470-16754-0 (cloth)

    1. Liquid chromatography. I. Kirkland, J. J. (Joseph Jack), 1925- II. Dolan, John W. III. Title.

    QD79.C454S58 2009

    543'.84–dc22

    2009005626

    PREFACE

    High-performance liquid chromatography (HPLC) is today the premier technique for chemical analysis and related applications, with an ability to separate, analyze, and/or purify virtually any sample. The second edition of this book appeared in 1979, and for tens of thousands of readers it eventually became their choice of an HPLC reference book. The remarkable staying power of the second edition (with significant sales into the first decade of the present century) can be attributed to certain features which continue to be true for the present book. First, all three editions have been closely tied to short courses presented by the three authors over the past four decades, to an audience of more than 10,000 industrial, governmental, and academic chromatographers. Teaching allows different approaches to a subject to be tried and evaluated, and a pragmatic emphasis is essential when dealing with practicing chromatographers as students. Second, all three editions have tried to combine practical suggestions (''how to?'') with a theoretical background (''why?''). Both theory and practice continue to be emphasized so that the reader can better understand and evaluate the various recommendations presented here. Finally, each of the three authors has been an active participant in HPLC research, development, and/or routine application throughout most of their careers.

    Since the preparation of the second edition in 1979, there have been major improvements in columns and equipment, as well as numerous advances in (1) our understanding of HPLC separation, (2) our ability to solve problems that were troublesome in the past, and (3) the application of HPLC for new kinds of samples. Whereas six different HPLC procedures received comparable attention in the second edition, today reversed-phase chromatography (RPC) accounts for about 80% of all HPLC applications—and therefore receives major (but not exclusive) attention in the present edition. Over the past three decades the use of HPLC for biological samples, enantiomeric (chiral) separations, and sample purification has expanded enormously, accompanied by a much better understanding of these and other HPLC applications.

    Commercial HPLC columns continue to be improved, and many new kinds of columns have been introduced for specific applications, as well as for faster, trouble-free operation. Prior to 1990, HPLC method development was an uncertain process—often requiring several months for the acceptable separation of a sample. Since then it has become possible to greatly accelerate method development, especially with the help of appropriate software. At the same time HPLC practice is increasingly carried out in a regulatory environment that can slow the release of a final method. These various advances and changes in the way HPLC is carried out have mandated major changes in the present edition.

    The organization of the present book, while similar to that of the second edition, has been significantly modified in light of subsequent research and experience. Chapter 1 provides a general background for HPLC, with a summary of how its use compares with other modern separation techniques. Chapter 1 also reviews some of the history of HPLC. Chapter 2 develops the basis of HPLC separation and the general effects of different experimental conditions. Chapters 3 and 4 deal with equipment and detection, respectively. In 1979 the detector was still the weak link in the use of HPLC, but today the widespread use of diode-array UV and mass-spectrometric detection—as well as the availability of several special-purpose detectors—has largely addressed this problem. Chapter 5 deals with the column: the ''heart'' of the HPLC system. In 1979, numerous problems were associated with the column: peak tailing—especially for basic samples, column instability at elevated temperatures or extremes in mobile-phase pH, and batch-to-batch column variability; today these problems are much less common. We also now know a good deal about how performance varies among different columns, allowing a better choice of column for specific applications. Finally, improvements in the column are largely responsible for our current ability to carry out ultra-fast separations (run times of a few minutes or less) and to better separate mixtures that contain hundreds or even thousands of components.

    Chapter 6, which deals with the reversed-phase separation of non-ionic samples, extends the discussion of Chapter 2 for these important HPLC applications. A similar treatment for normal-phase chromatography (NPC) is given in Chapter 8, including special attention to hydrophilic interaction liquid chromatography (HILIC). In Chapter 7 the separation of ionized or ionizable samples is treated, whether by RPC, ion-pair chromatography, or ion-exchange chromatography. Gradient elution is introduced in Chapter 9 for small-molecule samples, and as an essential prerequisite for the separation of large biomolecules in Chapter 13; two-dimensional separation—another technique of growing importance—is also discussed. Chapter 10 covers the use of computer-facilitated method development (computer simulation). Other important, general topics are covered in Chapters 11 (Qualitative and Quantitative Analysis) and 12 (Method Validation).

    Chapter 13 introduces the separation of large molecules, including both biological and synthetic polymers. HPLC procedures that are uniquely useful for these separations are emphasized: reversed-phase, ion-exchange, and size-exclusion, as well as related two-dimensional separations. Chapter 14 (Enantiomer Separations) marks a decisive shift in approach, as the resolution of enantiomers requires columns and conditions that are sample-specific—unlike most of the HPLC applications described in earlier chapters.

    Chapter 15 deals with preparative separations (prep-LC), where much larger sample weights are introduced to the column. The big change since 1979 for prep-LC is that we now have a much better understanding of how such separations vary with conditions, in turn making method development much more systematic and efficient. Chapter 16 (Sample Preparation) provides a comprehensive coverage of this important supplement to HPLC separation. As in the case of other HPLC-related topics, the past 30 years have seen numerous developments that today make sample preparation a routine addition to many HPLC procedures. Finally, Chapter 17 deals with HPLC troubleshooting. Despite all our advances in equipment, columns, materials, technique, and understanding, trouble-free HPLC operation is still not guaranteed. Fortunately, our ability to anticipate, diagnose, and solve HPLC problems is now more informed and systematic. One of our three authors (JWD) has been especially active in this area.

    Different readers will use this book in different ways. An experienced worker may wish to explore topics of his or her choice, or find an answer to specific problems. For this audience, the Index may be the best starting place. Beginning readers might first skim Chapters 1 through 7, followed by 9 through 10, all of which emphasize reversed-phase HPLC. The latter sequence is similar to the core of the basic HPLC short courses developed by the authors. After this introduction, the reader can jump to chapters or sections of special interest. Other readers may wish to begin with topics of interest from the Contents pages at the front of the book or at the beginning of individual chapters. The present book has been organized with these various options in mind.

    This third edition is highly cross-referenced, so as to allow the reader to follow up on topics of special interest, or to clarify questions that may arise during reading. Because extensive cross-referencing represents a potential distraction, in most cases it is recommended that the reader simply ignore (or defer) these invitations to jump to other parts of the book. Some chapters include sections that are more advanced, detailed, and of less immediate interest; these sections are in each case clearly identified by an introductory advisory in italics, so that they can be bypassed at the option of the reader. We have also taken pains to provide definitions for all symbols used in this book (Glossary section), along with a comprehensive and detailed index. Finally, attention should be drawn to a ''best practices'' entry in the Index, which summarizes various recommendations for both method development and routine use.

    We very much appreciate the participation of eight collaborators in the preparation of the present book: Peter Schoenmakers (Sections 9.3.10, 13.10), Mike Swartz (Chapter 12), Tim Wehr (Sections 13.1-13.8), Carl Scandella (Section 13.9), Wolfgang Lindner, Michael Lämmerhofer, and Norbert Maier (Chapter 14), Geoff Cox (Chapter 15), and Ron Majors (Chapter 16). Their affiliations are as follows:

    We also are indebted to the following reviewers of various parts of the book: Peter Carr, Tom Chambers, Geoff Cox, Roy Eksteen, John Fetzer, Dick Henry, Vladimir Ioffe, Pavel Jandera, Peter Johnson, Tom Jupille, Ron Majors, Dan Marchand, David McCalley, Imre Molnar, Tom Mourey, Uwe Neue, Ravi Ravichandran, Karen Russo, Carl Scandella, Peter Schoenmakers, and Loren Wrisley. However, the authors accept responsibility for any errors or other shortcomings in this book.

    LLOYD R. SNYDER

    J. J. (JACK) KIRKLAND

    JOHN W. DOLAN

    Orinda, CA

    Wilmington, DE

    Amity, OR

    GLOSSARY OF SYMBOLS AND ABBREVIATIONS

    This section is divided into ''frequently used''and "less-frequently used'' symbols.'' Most symbols of interest will be included in ''frequently used symbols''. Equations that define a particular symbol are listed with that symbol; for example, ''Equation 2.18'' refers to Equation (2.18) in Chapter 2. The units for all symbols used in this book are indicated. Where IUPAC definitions or symbols differ from those used in this book, we have indicated the corresponding IUPAC term (from ASDLID 009921), for example, tM instead of t0.

    FREQUENTLY USED SYMBOLS AND ABBREVIATIONS

    LESS-FREQUENTLY USED (OR LESS-COMMONLY UNDERSTOOD) SYMBOLS AND ABBREVIATIONS

    CHAPTER ONE

    INTRODUCTION

    High-performance liquid chromatography (HPLC) is one of several chromatographic methods for the separation and analysis of chemical mixtures (Section 1.3). Compared to these other separation procedures, HPLC is exceptional in terms of the following characteristics:

    almost universal applicability; few samples are excluded from the possibility of HPLC separation

    remarkable assay precision (±0.5% or better in many cases)

    a wide range of equipment, columns, and other materials is commercially available, allowing the use of HPLC for almost every application

    most laboratories that deal with a need for analyzing chemical mixtures are equipped for HPLC; it is often the first choice of technique

    As a result, HPLC is today one of the most useful and widely applied analytical techniques. Mass spectrometry rivals and complements HPLC in many respects; the use of these two techniques in combination (LC-MS) is already substantial (Section 4.14), and will continue to grow in importance.

    In the present chapter we will:

    examine some general features of HPLC

    summarize the history of HPLC

    very briefly consider some alternatives to HPLC, with their preferred use for certain applications

    list other sources of information about HPLC

    1.1 BACKGROUND INFORMATION

    1.1.1 What Is HPLC?

    Liquid chromatography began in the early 1900s, in the form illustrated in Figure 1.1a–e, known as ''classical column chromatography''. A glass cylinder was packed with a finely divided powder such as chalk (Fig. 1.1a), a sample was applied to the top of the column (Fig. 1.1b), and a solvent was poured onto the column (Fig. 1.1c). As the solvent flows down the column by gravity (Fig. 1.1d), the components of the sample (A, B, and C in this example) begins to move through the column at different speeds and became separated. In its initial form, colored samples were investigated so that the separation within the column could be observed visually. Then portions of the solvent leaving the column were collected, the solvent was evaporated, and the separated compounds were recovered for quantitative analysis or other use (Fig. 1.1e). In those days a new column was required for each sample, and the entire process was carried out manually (no automation). Consequently the effort required for each separation could be tedious and time-consuming. Still, even at this stage of development, chromatography provided a unique capability compared to other methods for the analysis of chemical mixtures.

    A simpler form of liquid chromatography was introduced in the 1940s, called paper chromatography (Fig. 1.1f). A strip of paper replaced the column of Figure 1.1a; after the sample was spotted near the bottom of the paper strip, the paper was placed in a container with solvent at the bottom. As the solvent migrated up the paper by capillary action, a similar separation as seen in Figure 1.1d took place, but in the opposite direction. This ''open bed'' form of chromatography was later modified by coating a thin layer of powdered silica onto a glass plate—as a replacement for the paper strip used in paper chromatography. The resulting procedure is referred to as thin-layer chromatography (TLC). The advantages of either paper or thin-layer chromatography included (1) greater convenience, (2) the ability to simultaneously separate several samples on the same paper strip or plate, and (3) easy detection of small amounts of separated compounds by the application of colorimetric reagents to the plate, after the separation was completed.

    HPLC (Fig. 1.1g, h) represents the modern culmination of the development of liquid chromatography. The user begins by placing samples on a tray for automatic injection into the column (Fig. 1.1g). Solvent is continually pumped through the column, and the separated compounds are continuously sensed by a detector as they leave the column. The resulting detector signal plotted against time is the chromatogram of Figure 1.1h, which can be compared with the result of Figure 1.1e — provided that the sample A + B + C and experimental conditions are the same. A computer controls the entire operation, so the only manual intervention required is the placement of samples on the tray. The computer can also generate a final analysis report for the sample. Apart from this automation of the entire process, HPLC is characterized by the use of high-pressure pumps for faster separation, re-usable and more effective columns for enhanced separation, and a better control of the overall process for more precise and reproducible results. More discussion of the history of HPLC can be found in Section 1.2.

    Figure 1.1 Different stages in the development of chromatography.

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    Figure 1.2 The expanding importance of HPLC research and application since 1966. (a) Number of HPLC-related publications per year [1]; (b) total sales of HPLC equipment and supplies per year (approximate data compiled from various sources).

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    The growth of HPLC, following its introduction in the late 1960s (Section 1.2), is illustrated in Figure 1.2. In (Fig. 1.2a) the annual number of HPLC publications is plotted against time. The first HPLC paper appeared in 1966 [2], and the number of publications grew each year exponentially, leveling off only after 1980. By 1990 the primary requirements of HPLC had largely been satisfied in terms of an understanding of the separation process, and the availability of suitable equipment and columns. At this time HPLC could be considered to have become a mature technique—one that is today practiced in every part of the world. While new, specialized applications of HPLC continued to emerge after 1990, and remaining gaps in our understanding receive ongoing attention, major future changes to our present understanding of HPLC seem unlikely.

    As the pace of HPLC research reached a plateau by 1990, a comparable flattening of the HPLC economy took a bit longer—as suggested by the plot in Figure 1.2b of annual expenditures against time for all HPLC products (not adjusted for inflation). The money spent annually on HPLC at the present time exceeds that for any other analytical technique.

    1.1.2 What Can HPLC Do?

    When the second edition of this book appeared in 1979, some examples of HPLC capability were presented, two of which are reproduced in Figure 1.3. Figure 1.3a shows a fast HPLC separation where 15 compounds are separated in just one minute. Figure 1.3b shows the separation power of HPLC by the partial separation of more than 100 recognizable peaks in just 30 minutes. In Figure 1.4 are illustrated comparable separations that were carried out 25 years later. Notice that in Figure 1.4a, six proteins are separated in 7 seconds, while in Figure 1.4b, c, about 1000 peptides plus proteins are separated in a total time of 1.5 hours. The improvement in Figure 1.4a compared with Figure 1.3a can be ascribed to several factors, some of which are discussed in Section 1.2. The separation of 1000 compounds in Figure 1.4b, c is the result of so-called two-dimensional separation (Section 9.3.10): a first column (Fig. 1.4b) provides fractions for further separation by a second column (Fig. 1.4c). In this example 4-minute fractions were collected from the first column and further separated with the second column; Figure 1.4c shows the separation of fraction 7. The total number of (recognizable) peaks in the sample is then obtained by adding the unique peaks present in each of the fractions. The enormous progress made in HPLC performance (Fig. 1.4 vs. Fig. 1.3) suggests that comparable major improvements in speed or separation power in the coming years are not so likely.

    Figure 1.3 Examples of HPLC capability during the mid-1970s. (a) Fast separation of a mixture of small molecules [3]; (b) high-resolution separation of a urine sample [4]. (a) is adapted from [3], and (b) is adapted from [4].

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    Some other improvements in HPLC since 1979 have been equally significant. Beginning in the 1980s, the introduction of suitable columns for the separation of proteins and other large biomolecules [7, 8] has opened up an entirely new field of application and facilitated major advances in biochemistry. Similarly the development of chiral columns for the separation of enantiomeric mixtures by Pirkle [9] and others enabled comparable advances in the areas of pharmaceuticals and related life sciences. The use of HPLC for large-scale purification is also increasing, as a result of the availability of appropriate equipment, an increase in our understanding of how such separations should best be carried out, and regulatory pressures for higher purity pharmaceutical products.

    Figure 1.4 Recent examples of HPLC capability. (a) Fast separation of six proteins, using gradient elution with a 150 × 4.6-mm column packed with 1.5-µm-diameter pellicular particles [5]; (b) initial separation of peptides and proteins from human fetal fibroblast cell by gradient cation-exchange chromatography; (c) further separation of fraction 7 (collected between 24–28 min) on a second column by gradient reversed-phase chromatography [6]. Figures adapted from original publications [5, 6].

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    1.2 A SHORT HISTORY OF HPLC

    We have noted the development of liquid chromatography prior to the advent of HPLC (Section 1.1). For a more complete account of this pre-1965 period, several review articles have been written by Leslie Ettre, our ''historian of chromatography'':

    precursors to chromatography; developments prior to 1900 [10, 11]

    invention of chromatography by M. S. Tswett in the early 1900s [12]

    rediscovery of chromatography in the early 1930s [13]

    A. J. P. Martin's invention of partition and paper chromatography in the early 1940s [14]

    development of the amino-acid analyzer by S. Moore and W. S. Stein in the late 1950s [15]

    development of the gel-permeation chromatograph by Waters Associates in the early 1960s [16]

    Carl Runge, a German dye-chemist born in 1856, first reported crude dye separations by means of a technique similar to paper chromatography [10], but neither he nor others pursued the practical possibilities of this work. In the late 1890s David Day at the US Geological survey carried out separations of petroleum by a technique that resembles classical column chromatography [11]; however, his goal was not the development of a separation technique, but rather the demonstration that petroleum deposits of different quality result from their separation during migration through the ground. As in the case of Runge's work, Day's investigations did not proceed further. In the early 1900s, Mikhail Tswett invented classical column chromatography and demonstrated its ability to separate different plant extracts [12]. This was certainly the beginning of chromatography, but the value of his work was not appreciated for another two decades. In the early 1930s, Tswett's work was rediscovered [13], leading to an explosive subsequent growth of chromatography. The invention of paper chromatography by A.J.P. Martin followed in 1943 [14], accompanied by the development of thin-layer chromatography between the late 1930s and the mid-1950s [17]. This short summary necessarily omits numerous other contributions to the development of chromatography before 1955.

    The amino-acid analyzer, introduced in the late 1950s [15], was an important precursor to HPLC; it was an automated means for analyzing mixtures of amino acids by use of ion-exchange chromatography (Section 7.5). This was followed by the invention of gel permeation chromatography (Section 13.7) by Moore [18] and the introduction in the early 1960s of a gel-permeation chromatograph by Waters Associates [16]. Each of these latter techniques was close in concept to what later became HPLC, differing little from the schematic of Figure 1.1g. In each case the solvent was pumped at high pressure through a reusable, small-particle column, the column effluent was continuously monitored by a detector, and the output of the device was a chromatogram as in Figure 1.1h. What each of these two systems lacked, however, was an ability to separate and analyze other kinds of samples. The amino-acid analyzer was restricted to the analysis of mixtures of amino acids, while the gel-permeation chromatograph was used exclusively for determining the molecular weight distribution of synthetic polymers. In neither case were these devices readily adaptable for the separation of other samples.

    During the early 1960s, two different groups embarked on the development of a general-purpose HPLC system, under the leadership of Csaba Horváth in the United States and Josef Huber in Europe. Each of these two men have described their early work on HPLC in a collection of personal recollections [19], and Ettre has provided additional detail on early work in Horváth's laboratory [20]. The immediate results of these two groups, plus related work by others that was carried out a few years later, are described in publications that appeared in 1966 to 1968 [2, 21–24]. The introduction of commercial equipment for HPLC followed in the late 1960s, with systems from Waters Associates and DuPont initially dominating the market. Other companies soon offered competing equipment, and research on HPLC began to accelerate (as seen from Fig. 1.2a). By 1971, the first HPLC book had been published [25], and an HPLC short course was offered by the American Chemical Society (Modern Liquid Chromatography), with J. J. Kirkland and L. R. Snyder as course instructors).

    Progressive improvements in HPLC from 1960 to 2010 are illustrated by the representative separations of Figure 1.5a–f, which show separation times decreasing by several orders of magnitude during this 50-year interval. Figure 1.5g shows how this reduction in separation time (°,—) was related to increases in the pressure drop across the column (---) and a reduction in the size of particles (•)that were used to pack the column. In the early days of HPLC the technique was sometimes referred to as ''high-pressure liquid chromatography'' or ''high-speed liquid chromatography,'' for reasons suggested by Figure 1.5g. Figure 1.5h shows corresponding changes in column length (•) and flow rate (°) for the separations of Figure 1.5a–e.

    A theoretical foundation for the eventual development of HPLC was established well before the 1960s. In 1941, Martin reported [27] that ''the most efficient columns… should be obtainable by using very small particles and high-pressure differences across the length of the column;'' this summarized the requirements for HPLC separation in a nutshell (as demonstrated by Fig. 1.5g). In the early 1950s, the related technique of gas chromatography was invented by Martin [28]; its rapid acceptance by the world [29] led to a number of theoretical studies that would prove relevant to the later development of HPLC. Giddings summarized and extended this work for specific application to HPLC in the early 1960s [30], work that was later to prove important for both column design and the selection of preferred experimental conditions.

    For a further background on the early days of HPLC, see [19, 31–33]. Additional historical details on the progress of HPLC after 1980 are provided by the collected biographies of several HPLC practitioners [34].

    1.3 SOME ALTERNATIVES TO HPLC

    Two, still-important techniques, each of which can substitute for HPLC in certain applications, existed prior to 1965: gas chromatography (GC) and thin-layer chromatography (TLC). Countercurrent chromatography (CCC) is another pre-1965 technique that, in principle, might compete with HPLC in many applications but falls considerably short of the speed and separation power of HPLC. Several additional, potentially competitive, techniques were introduced after HPLC: supercritical fluid chromatography (SFC) in the 1970s, capillary electrophoresis (CE) in the 1980s, and capillary electrochromatography (CEC) in the 1990s.

    1.3.1 Gas Chromatography (GC)

    Because GC [35] is limited to samples that are volatile below 300°C, this technique is not applicable for very-high-boiling or nonvolatile materials. Thus about 75% of all known compounds cannot be separated by GC. On the other hand, GC is considerably more efficient than HPLC (higher values of the plate number N), which means faster and/or better separations are possible. GC is therefore preferred to HPLC for gases, most low-boiling samples, and many higher boiling samples that are thermally stable under the conditions of separation. GC also has available several very sensitive and/or element-specific detectors that permit considerably lower detection limits.

    Figure 1.5 Representative chromatograms that illustrate the improvement in HPLC performance over time. Sample: five herbicides. Conditions: 50% methanol-water, ambient temperature. Chromatograms a–f are DryLabR computer simulations (Section 10.2), based on data of [26]; g and h provide details for the separations of a–f . Column-packings of identical selectivity and 4.6-mm-diameter columns are assumed.

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    1.3.2 Thin-Layer Chromatography (TLC)

    The strong points of TLC [36] are its ability to separate several samples simultaneously on a single plate, combined with the fact that every component in the sample is visible on the final plate (strongly retained compounds may be missed in HPLC). With the advent of specialized equipment for the pressurized flow of solvent across the plate, so-called high-performance TLC (HP-TLC) has become possible. Regardless of how it is carried out, however, TLC lacks the separation efficiency of HPLC (as measured by values of N), and quantitation is less convenient and less precise. At the time of publication of the present book, TLC was used relatively infrequently in the United States for quantitative analysis, although it is a convenient means for semi-quantitative analysis and for the detection of sample impurities. It is widely used for screening large numbers of samples, with little need for sample cleanup (e.g., plasma drug screening). In Europe HP-TLC is more popular than in the United States but much less popular than HPLC.

    1.3.3 Supercritical Fluid Chromatography (SFC)

    SFC [37] is carried out with equipment and columns that are similar to HPLC. The solvent is, by definition, a supercritical fluid, usually a gas such as CO2, under conditions of elevated pressure and temperature. SFC can be regarded as an extension of GC, in that supercritical fluids can dissolve and separate samples that are normally considered to be nonvolatile. SFC may be considered as a hybrid of GC and HPLC, as it is characterized by greater separation efficiency than for HPLC (higher N) but lower efficiency than GC. Similarly the solvent in SFC plays a greater role in determining separation than in GC, but less so than in HPLC. Detection sensitivity is also intermediate between what is possible with HPLC compared to GC. A major application of SFC is for the analysis of natural or synthetic polymeric mixtures, for example, the separation of polyphenols as described in [38]. Whereas HPLC may be unable to resolve individual polymeric species with molecular weights above some maximum value, SFC can usually extend this upper molecular-weight limit considerably. SFC has also been used for separating enantiomers, whose very similar retention may require greater separation efficiency (larger value of N).

    1.3.4 Capillary Electrophoresis (CE)

    CE [1, 39] is not a form of chromatography, but it competes effectively with HPLC for the separation of certain classes of compounds. The principle of separation is the differential migration of sample compounds in a capillary, under the influence of an electric field, with the result that compounds are separated on the basis of their mass-to-charge ratio (m/z); compounds with smaller m/z migrate faster. Consequently compounds that are to be separated by CE must carry an ionic charge. CE is characterized by a greater separation efficiency than for HPLC (higher value of N), especially for the separation of compounds of high molecular weight. However, detection sensitivity is usually much poorer than for HPLC. CE is heavily used for the genomic analysis of various species, based on the fractionation of DNA fragments. CE has also proved popular for analytical separations of enantiomeric samples, where its performance may exceed that of HPLC for two reasons. First, these separations are often difficult and therefore are facilitated by the larger values of N available from CE. Second, HPLC separations of enantiomers usually rely on chiral columns. The separation of a particular enantiomeric sample may require the trial-and-error testing of several different (and expensive) columns before a successful separation is achieved. CE allows the use of small amounts of different chiral complexing agents—instead of different columns, allowing for a faster, cheaper, and more versatile alternative to HPLC. The required flow rates for HPLC compared with CE (e.g., mL/min vs. µL/min) make the use of costly chiral complexing reagents impractical for HPLC. Several variations of CE exist, which allow its extension to other sample types; for example, non-ionized compounds can be separated by micellar electrokinetic chromatography [40].

    1.3.5 Countercurrent Chromatography

    CCC [41, 42] is an older form of liquid-liquid partition chromatography that was later improved in various ways. HPLC with a liquid stationary phase was since replaced by bonded-phase HPLC, the use of CCC as an alternative to HPLC has become relatively less frequent. An often-cited feature of CCC is its freedom from problems caused by irreversible attachment of the sample to the large internal surface present in HPLC columns. However, the improved HPLC columns used today are largely free from this problem. CCC may possess certain advantages for the preparative separation of enantiomers [43]; otherwise, the technique is used mainly for the isolation of labile natural products.

    1.3.6 Special Forms of HPLC

    The five separation techniques mentioned above (Sections 1.3.1–l.3.5) differ in essential ways from HPLC. Four other procedures, which will not be discussed in this book, can be regarded as HPLC variants. However, much of the information in following chapters can be adapted for use with the following procedures.

    Capillary electrochromatography [44, 45] (CEC) is generally similar to HPLC, except that the flow of solvent is achieved by means of an electrical potential across the column (endoosmotic flow), rather than by use of a pump. Because solvent flow is not affected by the size of particles within the column (and column efficiency can be greater for small particles), much larger values of N are, in principle, possible by means of CEC. Higher values of N also result from endoosmotic flow per se. Because of these potentially greater values of N in CEC than in HPLC, considerable effort has been invested since 1995 into making this technique practical. However, major technical problems remain to be solved, and CEC had not become a routine alternative to HPLC at the time this book went to press.

    HPLC on a chip [46] is a recently introduced technology for the convenient separation of very small samples. A micro-column (e.g., 43 × 0.06 mm) forms part of the chip, which can be interfaced between a micro pump and a mass spectrometer. The principles of separation are the same as for HPLC with conventional columns and equipment, but a chip offers advantages in terms of separation power and convenience for very small samples.

    Ion chromatography [47, 48] is widely used for the analysis of mixtures that contain inorganic anions and cations; for example, Cl- and Na+, respectively. While the principles of separation are the same as for ion-exchange HPLC (Section 7.5), ion chromatography involves special equipment and is used mainly for inorganic analysis.

    Micellar liquid chromatography is a variant of reversed-phase chromatography in which the usual aqueous-organic solvent is replaced by an aqueous surfactant solution [49]. It is little used at present because of the lower efficiency of these separations.

    1.4 OTHER SOURCES OF HPLC INFORMATION

    A wide variety of resources is available that can be consulted to supplement the use of the present book. These include various other publications (Sections 1.4.1–1.4.3), short courses (Section 1.4.4), and the Internet (Section 1.4.5).

    1.4.1 Books

    Literally hundreds of books on chromatography have now been published, as reference to Amazon.com and other internet sources can readily verify. Books on HPLC can be divided into two groups: (1) specialized texts that address the HPLC separation of a certain kind of sample (e.g., proteins, carbohydrates, enantiomers), or by means of special detection (e.g., mass spectrometer, chemical derivatization), and (2) more general books, such as the present book, that cover all aspects of HPLC. Specialized HPLC books are referenced in later chapters that address different HPLC topics. Table 1.1 provides a partial listing of more general HPLC books published after 1995 that might serve as useful supplements to the present book.

    1.4.2 Journals

    Technical articles that involve HPLC can appear in most journals that deal with the chemical or biochemical sciences. However, the journals below are of special value to those readers wishing to keep abreast of new developments in the field.

    Analytical Chemistry, American Chemical Society

    Chromatographia,Springer

    Journal of Chromatographic Science, Preston

    Journal of Chromatography A, Elsevier

    Journal of Chromatography B, Elsevier

    Journal of Liquid Chromatography, Wiley

    Journal of Separation Science, Wiley

    LCGC, Advanstar (separate issues for North America and Europe)

    1.4.3 Reviews

    Review articles that deal with HPLC can be found in the journals listed above and in other journals. Additionally there are series of publications that are devoted in part to HPLC, either as collections of review articles

    Advances in Chromatography, Dekker

    High-Performance Liquid Chromatography. Advances and Perspectives, Academic Press (published only between 1980 and 1986)

    or as individual books:

    Journal of Chromatography Library, Elsevier

    1.4.4 Short Courses

    There are numerous short courses offered either live or on the Internet (see Section 1.4.5). For a current listing of short courses, see the back pages of LCGC magazine or search the Internet for ''HPLC training.''

    1.4.5 The Internet

    The dynamic nature of the Internet ensures that any listing in a book will soon be obsolete. A number of sites are links to other sites and, as such, presumably will be continuously updated:

    http://www.lcresources.com/

    http://matematicas.udea.edu.co/~carlopez/index7.xhtml

    http://lchromatography.com/hplcfind/index.xhtml

    http://tech.groups.yahoo.com/group/chrom-L/links

    http://userpages.umbc.edu/~dfrey1/Freylink

    http://wwwJnfochembioxthzxh/links/en/analytchem_chromat.xhtml

    http://www.chromatographyonline.com/

    Table 1.1 Some HPLC Books of General Interest Published since 1995

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    REFERENCES

    1. R. L. Cunico, K. M. Gooding, and T. Wehr, Basic HPLC and CE of Biomolecules,Bay Bioanalytical Laboratory, Richmond, CA, 1998, p. 4. C.

    2. C. Horváth and S. R. Lipsky, Nature, 211 (1966) 748.

    3. I. Halasz, R. Endele, and J. Asshauer, J. Chromatogr., 112 (1975) 37.

    4. I. Molnar and C. Horváth, J. Chromatogr. (Biomed App.), 143 (1977) 391.

    5. T. Issaeva, A. Kourganov, and K. Unger, J. Chromatogr. A, 846 (1999) 13.

    6. K. Wagner, T. Miliotis, G. Marko-Varga, R. Biscoff, and K. Unger, Anal. Chem., 74 (2002) 809.

    7. S. H. Chang, K. M. Gooding, and F. E. Regnier, J. Chromatogr., 125 (1976) 103.

    8. W. W. Hancock, C. A. Bishop, and M. T. W. Hearn, Science, 153 (1978) 1168.

    9. W. H. Pirkle, D. W. House, and J. M. Finn, J. Chromatogr., 192 (1980) 143.

    10. H. H. Bussemas and L. S. Ettre, LCGC, 22 (2004) 262.

    11. L. S. Ettre, LCGC, 23 (2005) 1274.

    12. L. S. Ettre, LCGC, 21 (2003) 458.

    13. L. S. Ettre, LCGC, 25 (2007) 640.

    14. L. S. Ettre, LCGC, 19 (2001) 506.

    15. L. S. Ettre, LCGC, 243 (2006) 390.

    16. L. S. Ettre, LCGC, 23 (2005) 752.

    17. J. G. Kirchner, Thin-layer Chromatography, Wiley-Interscience, New York, 1978, pp. 5–8.

    18. J. C. Moore, J. Polymer Sci. Part A, 2 (1964) 835.

    19. L. S. Ettre and A. Zlatkis, eds., 75 years of Chromatography—A Historical Dialog, Elsevier, Amsterdam, 1979.

    20. L. S. Ettre, LCGC, 23 (2005) 486.

    21. J. F. K. Huber and J. A. R. Hulsman, J. Anal. Chim. Acta, 38 (1967) 305.

    22. J. J. Kirkland, Anal. Chem., 40 (1968) 218.

    23. L. R. Snyder, Anal. Chem., 39 (1967) 698, 705.

    24. R. P. W. Scott, W. J. Blackburn, and T. J. Wilkens, J. Gas Chrommatogr., 5 (1967) 183.

    25. J. J. Kirkland, ed., Modern Practice of Liquid Chromatography, Wiley-Interscience, New York, 1971.

    26. T. Braumann, G. Weber, and L. H. Grimme, J. Chromatogr., 261 (1983) 329.

    27. A. J. P. Martin and R. L. M. Synge, Biochem. J., 35 (1941) 1358.

    28. A. T. James and A. J. P. Martin, Biochem. J., 50 (1952) 679.

    29. L. S. Ettre, LCGC, 19 (2001) 120.

    30 J. C. Giddings, Dynamics of Chromatography. Principles and Theory, Dekker, New York, 1965.

    31. L. R. Snyder, J. Chem. Ed., 74 (1997) 37.

    32. L. S. Ettre, LCGC Europe, 1 (2001) 314.

    33. L. R. Snyder, Anal. Chem., 72 (2000) 412A.

    34. C. W. Gehrke, ed., Chromatography—A Century of Discovery 1900–2000, Elsevier, Amsterdam, 2001.

    35. R. L. Grob and E. F. Barry, Modern Practice of Gas Chromatography, 4th ed., Wiley-Interscience, NewYork, 2004.

    36. B. Fried and J. Sherma, Thin-Layer Chromatography (Chromatographic Science, Vol. 81), Dekker, New York, 1999.

    37. R. M. Smith and S. M. Hawthorne, eds., Supercritical Fluids in Chromatography and Extraction, Elsevier, Amsterdam, 1997.

    38. T. Bamba, E. Fukusaki, Y. Nakazawa, H. Sato, K. Ute, T. Kitayama, and A. Kobayashi, J. Chromatogr. A, 995 (2003) 203.

    39. K. D. Altria and D. Elder, J. Chromatogr. A, 1023 (2004) 1.

    40. A. Berthod and C. Gárcia-Alvarez-Coque, Micellar Liquid Chromatography, Dekker, New York, 2000.

    41. Y. Ito and W. D. Conway, eds., High-Speed Countercurrent Chromatography, Wiley, New York, 1996.

    42. J.-M. Menet and D. Thiebaut, eds., Countercurrent Chromatography, Dekker, New York, 1999.

    43. E. Gavioli, N. M. Maier, C. Minguillón, and W. Lindner, Anal. Chem., 76 (2004) 5837.

    44. K. D. Bartle and P. Meyers, Capillary Electrochromatography (Chromatography Monographs), 2001.

    45. F. Svec, ed., J. Chromatogr. A, 1044 (2004).

    46. H. Yin and K. Killeen, J. Sep. Sci., 30 (2007) 1427.

    47. J. S. Fritz and D. T. Gjerde, Ion Chromatography, 3rd ed., Wiley-VCH, Weinheim, 2000.

    48. J. Weiss, Handbook of Ion Chromatography, 3rd ed., Wiley, 2005.

    49. A. Berthod and M. C. Gárcia-Alvarez-Coque, Micellar Liquid Chromatography, Dekker, New York, 2000.

    CHAPTER TWO

    BASIC CONCEPTS AND THE CONTROL OF SEPARATION

    2.1 INTRODUCTION

    The successful use of HPLC requires an understanding of how separation is affected by experimental conditions: the column, solvent, temperature, flow rate and so forth. In this chapter we review some general features of HPLC for use in the laboratory, in order to develop an adequate separation (method development), to carry out a routine HPLC procedure for sample analysis, or to solve problems as they arise. A descriptive or qualitative approach is usually best suited for understanding both method development and the routine application of HPLC. For this reason the reader may wish to skim or skip any of the following derivations — at least initially. Important equations that are useful in practice are enclosed within a box; for example, Equation (2.5).

    2.2 THE CHROMATOGRAPHIC PROCESS

    A schematic of an HPLC system is shown in Figure 2.1, with emphasis on the flow path of the solvent (solid arrows) as it proceeds from the solvent reservoir to the detector (the solvent is usually referred to as the mobile phase or eluent). A detailed discussion of each part of the system (HPLC equipment) is given in Chapter 3. After injection of the sample, a separation takes place within the column, and separated sample components leave (are eluted or washed from) the column—with detection in most cases by either ultraviolet absorption (UV) or mass spectrometry (MS); see Chapter 4 for details on the use of these and other HPLC detectors. The fundamental nature or mode of the separation is determined mainly by the choice of column, as summarized in Table 2.1. For sample analysis, the predominant HPLC mode in use today is reversed-phase chromatography (RPC), which features a nonpolar column in combination with a (polar) mixture of water plus an organic solvent as mobile phase. Unless noted otherwise, RPC separation will be assumed in this book. Other HPLC modes are described in later sections of the book, as noted in Table 2.1. In Chapters 2 through 8 we will assume that the composition of the solvent remains the same throughout separation, which is called isocratic elution, as opposed to gradient elution where the solvent composition is deliberately changed during the separation (Section 2.7.2, Chapter 9).

    Figure 2.1 Schematic of an HPLC system.

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    The column consists of a cylindrical tube that is typically filled with small (usually 1.5- to 5-μm diameter) spherical particles (Fig. 2.2a). These particles are in most cases porous silica, with an individual pore portrayed in Figure 2.2b as a cylinder of some specified diameter (typically about 10 nm for use with small-molecule samples i.e., molecular weights <1000 Da). The inside of each pore is covered with the stationary phase — in this example, C18 groups that are attached to the silica particle. Figure 2.2c shows a more realistic representation of present-day porous particles for HPLC. The particle is formed by aggregating small, spherical, subparticles as shown. The actual pores are formed by the spaces between the subparticles. Because almost all of the surface of the particle is contained within these pores, most sample molecules are held inside the particle rather than on the surface of the particle. That is, the internal surfaces of the pores account for J » 99% of the total surface area of the particle; the external surface area (and its effect on separation) is in most cases negligible. The mobile phase surrounds each particle as it flows through the column, and sample molecules can enter the particle pores by diffusion (there is normally no significant flow of mobile phase through the particle).

    Figure 2.2 The HPLC column. (a) Column packed with spherical particles; (b) schematic of an individual particle, showing an idealized pore with attached C18 groups; (c) more realistic picture of a spherical, porous particle, showing detail (10× expansion).

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    Table 2.1 HPLC Separation Modes.

    Figure 2.3 illustrates a hypothetical separation of a sample that contains three sample compounds (or solutes), with individual sample molecules represented by • for solute X, □ for solute Y, and ▲ for solute Z. For clarity, molecules of the mobile phase are not shown, and molecules of the solvent that the sample is dissolved in are portrayed by +. The sample is applied to the column in (Fig. 2.3a) is carried through the column by the flowing mobile phase in successive stages (Fig. 2.3b–d), and eventually the sample leaves the column (Fig. 2.3e)to provide a plot of detector response versus time (a chromatogram, or record of the separation). As the separation proceeds in Figure 2.3a–d, molecules of sample components X, Y, and Z exhibit two characteristic behaviors: differential migration and molecular spreading. By Figure 2.3d, solutes X, Y, and Z have become separated from each other within the column.

    Figure 2.3 Illustration of the separation process in HPLC. (a–d) Sequential separation within the column (i.e., as a function of time); (e) the final chromatogram; (f) estimating values of k from the chromatogram (e). Solute molecules X, Y, and Z are represented by ●, □ and ▲, respectively; sample solvent molecules are shown by +.

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    Differential migration (different average speeds at which solute molecules of X, Y, and Z move,

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