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INTERNATIONAL UNION OF PURE AND APPLIED CHEMISTRY

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Physical and Biophysical Chemistry Division

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Quantities, Units and Symbols in Physical Chemistry

Third Edition

Prepared for publication by Tomislav Cvita²

Jeremy G. Frey

Bertil Holmström

Kozo Kuchitsu

Roberto Marquardt

Ian Mills Jürgen Stohner

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E. Richard Cohen

Franco Pavese

Martin Quack

Herbert L. Strauss

Michio Takami

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Anders J Thor

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The rst and second editions were prepared for publication by

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Ian Mills Tomislav Cvita² Klaus Homann Nikola Kallay Kozo Kuchitsu

IUPAC 2007

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E.R. Cohen, T. Cvitas, J.G. Frey, B. Holmström, K. Kuchitsu, R. Marquardt, I. Mills, F. Pavese, M. Quack, J. Stohner, H.L. Strauss, M. Takami, and A.J. Thor, "Quantities, Units and Symbols in Physical Chemistry", IUPAC Green Book, 3rd Edition, 2nd Printing, IUPAC & RSC Publishing, Cambridge (2008)

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Please cite this document as follows:

ISBN: 978-0-85404-433-7

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A catalogue record for this book is available from the British Library © International Union of Pure and Applied Chemistry 2007

All rights reserved

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Reprinted 2008

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Apart from fair dealing for the purposes of research for non-commercial purposes or for private study, criticism or review, as permitted under the Copyright, Designs and Patents Act 1988 and the Copyright and Related Rights Regulations 2003, this publication may not be reproduced, stored or transmitted, in any form or by any means, without the prior permission in writing of The Royal Society of Chemistry, or in the case of reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to The Royal Society of Chemistry at the address printed on this page.

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Published by The Royal Society of Chemistry, Thomas Graham House, Science Park, Milton Road, Cambridge CB4 0WF, UK Registered Charity Number 207890

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For further information see our web site at www.rsc.org

Professor Tom Cvita²

17735, Corinthian Drive

University of Zagreb

Encino, CA 91316-3704

Department of Chemistry

USA

Horvatovac 102a

email: [email protected]

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Professor E. Richard Cohen

HR-10000 Zagreb Croatia

email: [email protected]

Professor Jeremy G. Frey

Professor Bertil Holmström

University of Southampton

Ulveliden 15

Department of Chemistry

SE-41674 Göteborg

Southampton, SO 17 1BJ

Sweden

United Kingdom

email: [email protected]

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email: [email protected]

Professor Kozo Kuchitsu

Professor Roberto Marquardt

Tokyo University of Agriculture and Technology

Laboratoire de Chimie Quantique

Graduate School of BASE

Institut de Chimie

Naka-cho, Koganei

Université Louis Pasteur

Tokyo 184-8588

4, Rue Blaise Pascal

Japan

F-67000 Strasbourg

email: [email protected]

France

email: [email protected]

Professor Ian Mills

Professor Franco Pavese

University of Reading

Instituto Nazionale di Ricerca Metrologica (INRIM)

Department of Chemistry

strada delle Cacce 73-91

Reading, RG6 6AD

I-10135 Torino Italia

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United Kingdom email: [email protected]

email: [email protected]

Professor Martin Quack

Professor Jürgen Stohner

ETH Zürich

ZHAW Zürich University of Applied Sciences ICBC Institute of Chemistry & Biological Chemistry

CH-8093 Zürich Switzerland email: [email protected] or

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[email protected]

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Physical Chemistry

Campus Reidbach T, Einsiedlerstr. 31 CH-8820 Wädenswil Switzerland email: [email protected] or [email protected]

Professor Michio Takami

University of California

3-10-8 Atago

Berkeley, CA 94720-1460

Niiza, Saitama 352-0021

USA

Japan

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Professor Herbert L. Strauss

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email: [email protected]

Dr. Anders J Thor

Secretariat of ISO/TC 12

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SIS Swedish Standards Institute Sankt Paulsgatan 6 SE-11880 Stockholm Sweden

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email: [email protected]

email: [email protected]

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PREFACE HISTORICAL INTRODUCTION PHYSICAL QUANTITIES AND UNITS

1.4 1.5 1.6

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TABLES OF PHYSICAL QUANTITIES

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2.8 2.9 2.10 2.11 2.12 2.13

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Space and time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electricity and magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum mechanics and quantum chemistry . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Ab initio Hartree-Fock self-consistent eld theory (ab initio SCF) . . . . . . . 2.4.2 Hartree-Fock-Roothaan SCF theory, using molecular orbitals expanded as linear combinations of atomic-orbital basis functions (LCAO-MO theory) . . Atoms and molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Symbols for angular momentum operators and quantum numbers . . . . . . 2.6.2 Symbols for symmetry operators and labels for symmetry species . . . . . . . 2.6.3 Other symbols and conventions in optical spectroscopy . . . . . . . . . . . . . Electromagnetic radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Quantities and symbols concerned with the measurement of absorption intensity 2.7.2 Conventions for absorption intensities in condensed phases . . . . . . . . . . . Solid state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Symbols for planes and directions in crystals . . . . . . . . . . . . . . . . . . Statistical thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10.1 Other symbols and conventions in chemistry . . . . . . . . . . . . . . . . . . Chemical thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11.1 Other symbols and conventions in chemical thermodynamics . . . . . . . . . . Chemical kinetics and photochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12.1 Other symbols, terms, and conventions used in chemical kinetics . . . . . . . Electrochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13.1 Sign and notation conventions in electrochemistry . . . . . . . . . . . . . . . . Colloid and surface chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14.1 Surface structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.15.1 Transport characteristic numbers: Quantities of dimension one . . . . . . . .

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Physical quantities and quantity calculus . . . . . . . . . . . . . . . . . Base quantities and derived quantities . . . . . . . . . . . . . . . . . . Symbols for physical quantities and units . . . . . . . . . . . . . . . . 1.3.1 General rules for symbols for quantities . . . . . . . . . . . . . 1.3.2 General rules for symbols for units . . . . . . . . . . . . . . . . Use of the words extensive, intensive, specic, and molar . . . . Products and quotients of physical quantities and units . . . . . . . . . The use of italic and Roman fonts for symbols in scientic publications

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1.1 1.2 1.3

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CONTENTS

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21 22 25 30 31 32 34 38 40 42 44 45 47 49 56 59 63 68 70 73 77 79 81 82

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DEFINITIONS AND SYMBOLS FOR UNITS

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3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9

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The International System of Units (SI) . . . . . . . . . . . . . . . . . . . . . . . . . . Names and symbols for the SI base units . . . . . . . . . . . . . . . . . . . . . . . . . Denitions of the SI base units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SI derived units with special names and symbols . . . . . . . . . . . . . . . . . . . . SI derived units for other quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . SI prexes and prexes for binary multiples . . . . . . . . . . . . . . . . . . . . . . . Non-SI units accepted for use with the SI . . . . . . . . . . . . . . . . . . . . . . . . Coherent units and checking dimensions . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental physical constants used as units . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Atomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2 The equations of quantum chemistry expressed in terms of reduced quantities using atomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Dimensionless quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Fractions (relative values, yields, and eciencies) . . . . . . . . . . . . . . . . 3.10.2 Deprecated usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.3 Units for logarithmic quantities: neper, bel, and decibel . . . . . . . . . . . . RECOMMENDED MATHEMATICAL SYMBOLS

4.1 4.2

95 97 97 97 98 101

Printing of numbers and mathematical symbols . . . . . . . . . . . . . . . . . . . . . 103 Symbols, operators, and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

FUNDAMENTAL PHYSICAL CONSTANTS

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PROPERTIES OF PARTICLES, ELEMENTS, AND NUCLIDES

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6.1 6.2 6.3

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The use of quantity calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conversion tables for units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The esu, emu, Gaussian, and atomic unit systems in relation to the SI . . . . . . . . Transformation of equations of electromagnetic theory between the ISQ(SI) and Gaussian forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

UNCERTAINTY

8.1 8.2 8.3

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131 135 143 146 149

Reporting uncertainty for a single measured quantity . . . . . . . . . . . . . . . . . . 151 Propagation of uncertainty for uncorrelated measurements . . . . . . . . . . . . . . . 153 Reporting uncertainties in terms of condence intervals . . . . . . . . . . . . . . . . . 154

ABBREVIATIONS AND ACRONYMS

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CONVERSION OF UNITS

7.1 7.2 7.3 7.4

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Properties of selected particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Standard atomic weights of the elements 2005 . . . . . . . . . . . . . . . . . . . . . . 117 Properties of nuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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10 REFERENCES

155 165

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10.1 Primary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 10.2 Secondary sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 vi

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12 INDEX OF SYMBOLS 13 SUBJECT INDEX NOTES PRESSURE CONVERSION FACTORS NUMERICAL ENERGY CONVERSION FACTORS

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IUPAC PERIODIC TABLE OF THE ELEMENTS

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11 GREEK ALPHABET

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181 195 231 233

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PREFACE

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The purpose of this manual is to improve the exchange of scientic information among the readers in dierent disciplines and across dierent nations. As the volume of scientic literature expands, each discipline has a tendency to retreat into its own jargon. This book attempts to provide a readable compilation of widely used terms and symbols from many sources together with brief understandable denitions. This Third Edition reects the experience of the contributors with the previous editions and we are grateful for the many thoughtful comments we have received. Most of the material in this book is standard, but a few denitions and symbols are not universally accepted. In such cases, we have attempted to list acceptable alternatives. The references list the reports from IUPAC and other sources in which some of these notational problems are discussed further. IUPAC is the acronym for International Union of Pure and Applied Chemistry. A spectacular example of the consequences of confusion of units is provided by the loss of the United States NASA satellite, the Mars Climate Orbiter (MCO). The Mishap Investigation Board (Phase I Report, November 10, 1999)1 found that the root cause for the loss of the MCO was the failure to use metric units in the coding of the ground (based) software le. The impulse was reported in Imperial units of pounds (force)-seconds (lbf-s) rather than in the metric units of Newton (force)-seconds (N-s). This caused an error of a factor of 4.45 and threw the satellite o course.2 We urge the users of this book always to dene explicitly the terms, the units, and the symbols that they use. This edition has been compiled in machine-readable form by Martin Quack and Jürgen Stohner. The entire text of the manual will be available on the Internet some time after the publication of the book and will be accessible via the IUPAC web site, http://www.iupac.org. Suggestions and comments are welcome and may be addressed in care of the

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IUPAC Secretariat PO Box 13757 Research Triangle Park, NC 27709-3757, USA email: [email protected]

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The MCO report can be found at ftp://ftp.hq.nasa.gov/pub/pao/reports/1999/MCO_report.pdf. Impulse (change of momentum) means here the time-integral of the force.

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Corrections to the manual will be listed periodically. The book has been systematically brought up to date and new sections have been added. As in previous editions, the rst chapter describes the use of quantity calculus for handling physical quantities and the general rules for the symbolism of quantities and units and includes an expanded description on the use of roman and italic fonts in scientic printing. The second chapter lists the symbols for quantities in a wide range of topics used in physical chemistry. New parts of this chapter include a section on surface structure. The third chapter describes the use of the International System of units (SI) and of a few other systems such as atomic units. Chapter 4 outlines mathematical symbols and their use in print. Chapter 5 presents the 2006 revision of the fundamental physical constants, and Chapter 6 the properties of elementary particles, elements and nuclides. Conversion of units follows in Chapter 7, together with the equations of electricity and magnetism in their various forms. Chapter 8 is entirely new and outlines the treatment of uncertainty in physical measurements. Chapter 9 lists abbreviations and acronyms. Chapter 10 provides the references, and Chapter 11, the Greek alphabet. In Chapters 12 and 13, we end with indexes. The IUPAC Periodic Table of the Elements is shown on the inside back cover and the preceding pages list frequently used conversion factors for pressure and energy units.

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Many people have contributed to this volume. The people most directly responsible are acknowledged in the Historical Introduction. Many of the members of IUPAC I.1 have continued to make active contributions long after their terms on the Commission expired. We also wish to acknowledge the members of the other Commissions of the Physical Chemistry Division: Chemical Kinetics, Colloid and Surface Chemistry, Electrochemistry, Spectroscopy, and Thermodynamics, who have each contributed to the sections of the book that concern their various interests. We also thank all those who have contributed whom we have inadvertently missed out of these lists.

Jeremy G. Frey and Herbert L. Strauss

Commission on Physicochemical Symbols,

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Terminology and Units

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HISTORICAL INTRODUCTION

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The Manual of Symbols and Terminology for Physicochemical Quantities and Units [1.a], to which this is a direct successor, was rst prepared for publication on behalf of the Physical Chemistry Division of IUPAC by M. L. McGlashan in 1969, when he was chairman of the Commission on Physicochemical Symbols, Terminology and Units (I.1). He made a substantial contribution towards the objective which he described in the preface to that rst edition as being to secure clarity and precision, and wider agreement in the use of symbols, by chemists in dierent countries, among physicists, chemists and engineers, and by editors of scientic journals. The second edition of that manual prepared for publication by M. A. Paul in 1973 [1.b], and the third edition prepared by D. H. Whien in 1976 [1.c], were revisions to take account of various developments in the Système International d'Unités (International System of Units, international abbreviation SI), and other developments in terminology. The rst edition of Quantities, Units and Symbols in Physical Chemistry published in 1988 [2.a] was a substantially revised and extended version of the earlier manuals. The decision to embark on this project originally proposed by N. Kallay was taken at the IUPAC General Assembly at Leuven in 1981, when D. R. Lide was chairman of the Commission. The working party was established at the 1983 meeting in Lyngby, when K. Kuchitsu was chairman, and the project has received strong support throughout from all present and past members of the Commission I.1 and other Physical Chemistry Commissions, particularly D. R. Lide, D. H. Whien, and N. Sheppard. The extensions included some of the material previously published in appendices [1.d−1.k]; all the newer resolutions and recommendations on units by the Conférence Générale des Poids et Mesures (CGPM); and the recommendations of the International Union of Pure and Applied Physics (IUPAP) of 1978 and of Technical Committee 12 of the International Organization for Standardization, Quantities, units, symbols, conversion factors (ISO/TC 12). The tables of physical quantities (Chapter 2) were extended to include dening equations and SI units for each quantity. The style was also slightly changed from being a book of rules towards a manual of advice and assistance for the day-to-day use of practicing scientists. Examples of this are the inclusion of extensive notes and explanatory text inserts in Chapter 2, the introduction to quantity calculus, and the tables of conversion factors between SI and non-SI units and equations in Chapter 7. The second edition (1993) was a revised and extended version of the previous edition. The revisions were based on the recent resolutions of the CGPM [3]; the new recommendations by IUPAP [4]; the new international standards ISO 31 [5,6]; some recommendations published by other IUPAC commissions; and numerous comments we have received from chemists throughout the world. The revisions in the second edition were mainly carried out by Ian Mills and Tom Cvita² with substantial input from Robert Alberty, Kozo Kuchitsu, Martin Quack as well as from other members of the IUPAC Commission on Physicochemical Symbols, Terminology and Units. The manual has found wide acceptance in the chemical community, and various editions have been translated into Russian [2.c], Hungarian [2.d], Japanese [2.e], German [2.f], Romanian [2.g], Spanish [2.h], and Catalan [2.i]. Large parts of it have been reproduced in the 71st and subsequent editions of the Handbook of Chemistry and Physics published by CRC Press. The work on revisions of the second edition started immediately after its publication and between 1995 and 1997 it was discussed to change the title to Physical-Chemical Quantities, Units and Symbols and to apply rather complete revisions in various parts. It was emphasized that the book covers as much the eld generally called physical chemistry as the eld called chemical physics. Indeed we consider the larger interdisciplinary eld where the boundary between physics and chemistry has largely disappeared [10]. At the same time it was decided to produce the whole book as a text le in computer readable form to allow for future access directly by computer, some xi

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time after the printed version would be available. Support for this decision came from the IUPAC secretariat in the Research Triangle Park, NC (USA) (John W. Jost). The practical work on the revisions was carried out at the ETH Zürich, while the major input on this edition came from the group of editors listed now in full on the cover. It ts with the new structure of IUPAC that these are dened as project members and not only through membership in the commission. The basic structure of this edition was nally established at a working meeting of the project members in Engelberg, Switzerland in March 1999, while further revisions were discussed at the Berlin meeting (August 1999) and thereafter. In 2001 it was decided nally to use the old title. In this edition the whole text and all tables have been revised, many chapters substantially. This work was carried out mainly at ETH Zürich, where Jürgen Stohner coordinated the various contributions and corrections from the current project group members and prepared the print-ready electronic document. Larger changes compared to previous editions concern a complete and substantial update of recently available improved constants, sections on uncertainty in physical quantities, dimensionless quantities, mathematical symbols and numerous other sections. At the end of this historical survey we might refer also to what might be called the tradition of this manual. It is not the aim to present a list of recommendations in form of commandments. Rather we have always followed the principle that this manual should help the user in what may be called good practice of scientic language. If there are several well established uses or conventions, these have been mentioned, giving preference to one, when this is useful, but making allowance for variety, if such variety is not harmful to clarity. In a few cases possible improvements of conventions or language are mentioned with appropriate reference, even if uncommon, but without specic recommendation. In those cases where certain common uses are deprecated, there are very strong reasons for this and the reader should follow the corresponding advice.

Martin Quack

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Titular members

1963−1967 G. Waddington (USA); 1967−1971 M.L. McGlashan (UK); 1971−1973 M.A. Paul (USA); 1973−1977 D.H. Whien (UK); 1977−1981 D.R. Lide Jr (USA); 1981−1985 K. Kuchitsu (Japan); 1985−1989 I.M. Mills (UK); 1989−1993 T. Cvita² (Croatia); 1993−1999 H.L. Strauss (USA); 2000−2007 J.G. Frey (UK).

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Chairman:

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The membership of the Commission during the period 1963 to 2006, during which the successive editions of this manual were prepared, was as follows:

1963−1967 H. Brusset (France); 1967−1971 M.A. Paul (USA); 1971−1975 M. Fayard (France); 1975−1979 K.G. Weil (Germany); 1979−1983 I. Ansara (France); 1983−1985 N. Kallay (Croatia); 1985−1987 K.H. Homann (Germany); 1987−1989 T. Cvita² (Croatia); 1989−1991 I.M. Mills (UK); 1991−1997, 2001−2005 M. Quack (Switzerland); 1997−2001 B. Holmström (Sweden).

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Secretary:

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Other titular members

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1975−1983 I. Ansara (France); 1965−1969 K.V. Astachov (Russia); 1963−1971 R.G. Bates (USA); 1963−1967 H. Brusset (France); 1985−1997 T. Cvita² (Croatia); 1963 F. Daniels (USA); 1979−1981 D.H.W. den Boer (Netherlands); 1981−1989 E.T. Denisov (Russia); 1967−1975 M. Fayard (France); 1997−2005 J. Frey (UK); 1963−1965 J.I. Gerassimov (Russia); 1991−2001 B. Holmström (Sweden); 1979−1987 K.H. Homann (Germany); 1963−1971 W. Jaenicke (Germany); 1967−1971 F. Jellinek (Netherlands); 1977−1985 N. Kallay (Croatia); 1973−1981 V. Kellö (Czechoslovakia); 1989−1997 I.V. Khudyakov (Russia); 1985−1987 W.H. Kirchho (USA); 1971−1979 J. Koefoed (Denmark); 1979−1987 K. Kuchitsu (Japan); 1971−1981 D.R. Lide Jr (USA); 1997−2001, 2006− R. Marquardt (France); 1963−1971 M.L. McGlashan (UK); 1983−1991 I.M. Mills (UK); 1963−1967 M. Milone (Italy); 1967−1973 M.A. Paul (USA); 1991−1999, 2006− F. Pavese (Italy); 1963−1967 K.J. Pedersen (Denmark); 1967−1975 A. Perez-Masiá (Spain); 1987−1997 and 2002−2005 M. Quack (Switzerland); 1971−1979 A. Schuy (Netherlands); 1967−1970 L.G. Sillén (Sweden); 1989−1999 and 2002−2005 H.L. Strauss (USA); 1995−2001 M. Takami (Japan); 1987−1991 M. Tasumi (Japan); 1963−1967 G. Waddington (USA); 1981−1985 D.D. Wagman (USA); 1971−1979 K.G. Weil (Germany); 1971−1977 D.H. Whien (UK); 1963−1967 E.H. Wiebenga (Netherlands). Associate members

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1983−1991 R.A. Alberty (USA); 1983−1987 I. Ansara (France); 1979−1991 E.R. Cohen (USA); 1979−1981 E.T. Denisov (Russia); 1987−1991 G.H. Findenegg (Germany); 1987−1991 K.H. Homann (Germany); 1971−1973 W. Jaenicke (Germany); 1985−1989 N. Kallay (Croatia); 1987−1989 and 1998−1999 I.V. Khudyakov (Russia); 1979−1980 J. Koefoed (Denmark); 1987−1991 K. Kuchitsu (Japan); 1981−1983 D.R. Lide Jr (USA); 1971−1979 M.L. McGlashan (UK); 1991−1993 I.M. Mills (UK); 1973−1981 M.A. Paul (USA); 1999−2005 F. Pavese (Italy); 1975−1983 A. Perez-Masiá (Spain); 1997−1999 M. Quack (Switzerland); 1979−1987 A. Schuy (Netherlands); 1963−1971 S. Seki (Japan); 2000−2001 H.L. Strauss (USA); 1991−1995 M. Tasumi (Japan); 1969−1977 J. Terrien (France); 1994−2001 A J Thor (Sweden); 1975−1979 L. Villena (Spain); 1967−1969 G. Waddington (USA); 1979−1983 K.G. Weil (Germany); 1977−1985 D.H. Whien (UK). National representatives

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Numerous national representatives have served on the commission over many years. We do not provide this long list here.

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1 PHYSICAL QUANTITIES AND UNITS

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PHYSICAL QUANTITIES AND QUANTITY CALCULUS

The value of a a unit [Q]

physical quantity

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1.1

Q can be expressed as the product of a Q = {Q} [Q]

numerical value

{Q} and (1)

Neither the name of the physical quantity, nor the symbol used to denote it, implies a particular choice of unit (see footnote 1 , p. 4). Physical quantities, numerical values, and units may all be manipulated by the ordinary rules of algebra. Thus we may write, for example, for the wavelength λ of one of the yellow sodium lines

λ = 5.896 × 10−7 m = 589.6 nm

(2)

20 08 )C op yri

where m is the symbol for the unit of length called the metre (or meter, see Sections 3.2 and 3.3, p. 86 and 87), nm is the symbol for the nanometre, and the units metre and nanometre are related by 1 nm = 10−9 m or nm = 10−9 m (3) The equivalence of the two expressions for λ in Equation (2) follows at once when we treat the units by the rules of algebra and recognize the identity of 1 nm and 10−9 m in Equation (3). The wavelength may equally well be expressed in the form

λ/m = 5.896 × 10−7

(4)

λ/nm = 589.6

(5)

or

Example

tin g(

It can be useful to work with variables that are dened by dividing the quantity by a particular unit. For instance, in tabulating the numerical values of physical quantities or labeling the axes of graphs, it is particularly convenient to use the quotient of a physical quantity and a unit in such a form that the values to be tabulated are numerical values, as in Equations (4) and (5).

2.5

ln(p/MPa) = a + b/T = 3

a + b (10 K/T )

ln(p/MPa) −0.6578 1.2486 1.9990

rin

p/MPa 0.5180 3.4853 7.3815

dP

103 K/T 4.6179 3.6610 3.2874

2.0 1.5 1.0 0.5 0.0 –0.5 –1.0 3.0

2n

T /K 216.55 273.15 304.19

(6)

ln (p/MPa)

0

3.5

4.0 103 K/T

4.5

5.0

GB

3R

SC

Algebraically equivalent forms may be used in place of 103 K/T , such as kK/T or 103 (T /K)−1 . Equations between numerical values depend on the choice of units, whereas equations between quantities have the advantage of being independent of this choice. Therefore the use of equations between quantities should generally be preferred. The method described here for handling physical quantities and their units is known as quantity calculus [1113]. It is recommended for use throughout science and technology. The use of quantity calculus does not imply any particular choice of units; indeed one of the advantages of quantity calculus is that it makes changes between units particularly easy to follow. Further examples of the use of quantity calculus are given in Section 7.1, p. 131, which is concerned with the problems of transforming from one set of units to another. 3

BASE QUANTITIES AND DERIVED QUANTITIES

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1.2

By convention physical quantities are organized in a dimensional system built upon seven base quantities, each of which is regarded as having its own dimension. These base quantities in the International System of Quantities (ISQ) on which the International System of units (SI) is based, and the principal symbols used to denote them and their dimensions are as follows: Symbol for quantity

length mass time electric current thermodynamic temperature amount of substance luminous intensity

l m t I T n

Symbol for dimension

L M T I Θ

N J

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Base quantity

Iv

All other quantities are called derived quantities and are regarded as having dimensions derived algebraically from the seven base quantities by multiplication and division. Example

dimension of energy is equal to dimension of M L2 T−2 This can be written with the symbol dim for dimension (see footnote 1 , below) dim(E) = dim(m · l2 · t−2 ) = M L2 T−2

dP

rin

tin g(

The quantity amount of substance is of special importance to chemists. Amount of substance is proportional to the number of specied elementary entities of the substance considered. The proportionality factor is the same for all substances; its reciprocal is the Avogadro constant (see Section 2.10, p. 47, Section 3.3, p. 88, and Chapter 5, p. 111). The SI unit of amount of substance is the mole, dened in Section 3.3, p. 88. The physical quantity amount of substance should no longer be called number of moles, just as the physical quantity mass should not be called number of kilograms. The name amount of substance, sometimes also called chemical amount, may often be usefully abbreviated to the single word amount, particularly in such phrases as amount concentration (see footnote 2 , below), and amount of N2 . A possible name for international usage has been suggested: enplethy [10] (from Greek, similar to enthalpy and entropy). The number and choice of base quantities is pure convention. Other quantities could be considered to be more fundamental, such as electric charge Q instead of electric current I.

Zt2 Q=

I dt

(7)

t1

SC

2n

However, in the ISQ, electric current is chosen as base quantity and ampere is the SI base unit. In atomic and molecular physics, the so-called atomic units are useful (see Section 3.9, p. 94).

3R

The symbol [Q] was formerly used for dimension of Q, but this symbol is used and preferred for of Q. The Clinical Chemistry Division of IUPAC recommended that amount-of-substance concentration be abbreviated substance concentration [14].

1

GB

unit 2

4

SYMBOLS FOR PHYSICAL QUANTITIES AND UNITS [5.a]

gh t IU PA C

1.3

A clear distinction should be drawn between the names and symbols for physical quantities, and the names and symbols for units. Names and symbols for many quantities are given in Chapter 2, p. 11; the symbols given there are recommendations. If other symbols are used they should be clearly dened. Names and symbols for units are given in Chapter 3, p. 83; the symbols for units listed there are quoted from the Bureau International des Poids et Mesures (BIPM) and are mandatory. 1.3.1

General rules for symbols for quantities

Cp pi CB µB a Ek µr ◦ ∆r H − Vm A10

Examples

but

for for for for for for for for for

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The symbol for a physical quantity should be a single letter (see footnote 1 , below) of the Latin or Greek alphabet (see Section 1.6, p. 7). Capital or lower case letters may both be used. The letter should be printed in italic (sloping) type. When necessary the symbol may be modied by subscripts and superscripts of specied meaning. Subscripts and superscripts that are themselves symbols for physical quantities or for numbers should be printed in italic type; other subscripts and superscripts should be printed in Roman (upright) type. heat capacity at constant pressure partial pressure of the i th substance heat capacity of substance B chemical potential of substance B in phase kinetic energy relative permeability standard reaction enthalpy molar volume decadic absorbance

a

Examples

tin g(

The meaning of symbols for physical quantities may be further qualied by the use of one or more subscripts, or by information contained in parentheses. ◦ (HgCl , cr, 25 ◦ C) = −154.3 J K−1 mol−1 ∆f S − 2 µi = (∂G/∂ni )T,p,...,nj ,...; j6=i or µi = (∂G/∂ni )T,p,nj6=i

General rules for symbols for units

dP

1.3.2

rin

Vectors and matrices may be printed in bold-face italic type, e.g. A , a . Tensors may be printed in bold-face italic sans serif type, e.g. S , T . Vectors may alternatively be characterized by an arrow, → → ~~ ~~ A, a and second-rank tensors by a double arrow, S, T.

Symbols for units should be printed in Roman (upright) type. They should remain unaltered in the plural, and should not be followed by a full stop except at the end of a sentence. r

= 10 cm, not cm. or cms.

2n

Example

SC

An exception is made for certain characteristic numbers or dimensionless quantities used in the study of transport processes for which the internationally agreed symbols consist of two letters (see Section 2.15.1, p. 82). 1

Reynolds number, p. 70 and 75).

3R

Example

Re ;

another example is pH (see Sections 2.13 and 2.13.1 (viii),

GB

When such symbols appear as factors in a product, they should be separated from other symbols by a space, multiplication sign, or parentheses. 5

Examples

m (metre),

s (second),

gh t IU PA C

Symbols for units shall be printed in lower case letters, unless they are derived from a personal name when they shall begin with a capital letter. An exception is the symbol for the litre which may be either L or l, i.e. either capital or lower case (see footnote 2 , below). but J (joule), Hz (hertz)

Decimal multiples and submultiples of units may be indicated by the use of prexes as dened in Section 3.6, p. 91. Examples

1.4

nm (nanometre), MHz (megahertz),

kV (kilovolt)

USE OF THE WORDS EXTENSIVE, INTENSIVE, SPECIFIC, AND MOLAR

Examples

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A quantity that is additive for independent, noninteracting subsystems is called extensive ; examples are mass m, volume V, Gibbs energy G. A quantity that is independent of the extent of the system is called intensive ; examples are temperature T, pressure p, chemical potential (partial molar Gibbs energy) µ. The adjective specic before the name of an extensive quantity is used to mean divided by mass. When the symbol for the extensive quantity is a capital letter, the symbol used for the specic quantity is often the corresponding lower case letter. volume, V, and specic volume, v = V /m = l/ρ (where ρ is mass density); heat capacity at constant pressure, C p , and specic heat capacity at constant pressure, c p = C p /m

volume, V enthalpy, H

molar volume, Vm = V /n (Section 2.10, p. 47) molar enthalpy, Hm = H/n

rin

Examples

tin g(

ISO [5.a] and the Clinical Chemistry Division of IUPAC recommend systematic naming of physical quantities derived by division with mass, volume, area, and length by using the attributes massic or specic, volumic, areic, and lineic, respectively. In addition the Clinical Chemistry Division of IUPAC recommends the use of the attribute entitic for quantities derived by division with the number of entities [14]. Thus, for example, the specic volume could be called massic volume and the surface charge density would be areic charge. The adjective molar before the name of an extensive quantity generally means divided by amount of substance. The subscript m on the symbol for the extensive quantity denotes the corresponding molar quantity.

SC

2n

dP

If the name enplethy (see Section 1.2, p. 4) is accepted for amount of substance one can use enplethic volume instead of molar volume, for instance. The word molar violates the principle that the name of the quantity should not be mixed with the name of the unit (mole in this case). The use of enplethic resolves this problem. It is sometimes convenient to divide all extensive quantities by amount of substance, so that all quantities become intensive; the subscript m may then be omitted if this convention is stated and there is no risk of ambiguity. (See also the symbols recommended for partial molar quantities in Section 2.11, p. 57, and in Section 2.11.1 (iii), p. 60.) There are a few cases where the adjective molar has a dierent meaning, namely divided by amount-of-substance concentration. absorption coecient, a molar absorption coecient, ε = a/c (see Section 2.7, note 22, p. 37) conductivity, κ molar conductivity, = κ/c (see Section 2.13, p. 73)

3R

Examples

L

However, only the lower case l is used by ISO and the International Electrotechnical Commission (IEC).

GB

2

6

PRODUCTS AND QUOTIENTS OF PHYSICAL QUANTITIES AND UNITS

Products of physical quantities may be written in any of the ways a b

or

ab

or

a

·b

or

and similarly quotients may be written

a/b or F = ma,

Examples

a b

a

gh t IU PA C

1.5

×b

or by writing the product of a and b−1 , e.g. ab−1

p = nRT /V

Example

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Not more than one solidus (/) shall be used in the same expression unless parentheses are used to eliminate ambiguity.

(a/b)/c or a/(b/c) (in general dierent),

not a/b/c

In evaluating combinations of many factors, multiplication written without a multiplication sign takes precedence over division in the sense that a/bc is interpreted as a/(bc) and not as (a/b)c; however, it is necessary to use parentheses to eliminate ambiguity under all circumstances, thus avoiding expressions of the kind a/bcd etc. Furthermore, a/b + c is interpreted as (a/b) + c and not as a/(b + c). Again, the use of parentheses is recommended (required for a/(b + c)). Products and quotients of units may be written in a similar way, except that the cross (×) is not used as a multiplication sign between units. When a product of units is written without any multiplication sign a space shall be left between the unit symbols. Example

not 1 mkgs−2

THE USE OF ITALIC AND ROMAN FONTS FOR SYMBOLS IN SCIENTIFIC PUBLICATIONS

tin g(

1.6

1 N = 1 m kg s−2 = 1 m·kg·s−2 = 1 m kg/s2 ,

rin

Scientic manuscripts should follow the accepted conventions concerning the use of italic and Roman fonts for symbols. An italic font is generally used for emphasis in running text, but it has a quite specic meaning when used for symbols in scientic text and equations. The following summary is intended to help in the correct use of italic fonts in preparing manuscript material.

2n

dP

1. The general rules concerning the use of italic (sloping) font or Roman (upright) font are presented in Section 1.3.2, p. 5 and in Section 4.1, p. 103 in relation to mathematical symbols and operators. These rules are also presented in the International Standards ISO 31 (successively being replaced by ISO/IEC 80000) [5], ISO 1000 [6], and in the SI Brochure [3].

3R

SC

2. The overall rule is that symbols representing physical quantities or variables are italic, but symbols representing units, mathematical constants, or labels, are roman. Sometimes there may seem to be doubt as to whether a symbol represents a quantity or has some other meaning (such as label): a good rule is that quantities, or variables, may have a range of numerical values, but labels cannot. Vectors, tensors and matrices are denoted using a bold-face (heavy) font, but they shall be italic since they are quantities.

GB

Examples

The Planck constant h = 6.626 068 96(33)×10−34 J s. The electric eld strength has components Ex , Ey , and Ez . The mass of my pen is m = 24 g = 0.024 kg.

E

7

gh t IU PA C

3. The above rule applies equally to all letter symbols from both the Greek and the Latin alphabet, although some authors resist putting Greek letters into italic. Example When the symbol µ is used to denote a physical quantity (such as permeability or reduced mass) it should be italic, but when it is used as a prex in a unit such as microgram, mg, or when it is used as the symbol for the muon, m (see paragraph 5 below), it should be roman.

20 08 )C op yri

4. Numbers, and labels, are roman (upright). Examples The ground and rst excited electronic state of the CH2 molecule are denoted e 3 B1 , and . . .(2a1 )2 (1b2 )2 (3a1 )2 , e . . .(2a1 )2 (1b2 )2 (3a1 )1 (1b1 )1 , X a 1 A1 , respectively. The p-electron conguration and symmetry of the benzene molecule in e 1 A1g . All these symbols are its ground state are denoted: . . .(a2u )2 (e1g )4 , X labels and are roman. 5. Symbols for elements in the periodic system shall be roman. Similarly the symbols used to represent elementary particles are always roman. (See, however, paragraph 9 below for use of italic font in chemical-compound names.) Examples H, He, Li, Be, B, C, N, O, F, Ne, ... for atoms; e for the electron, p for the proton, n for the neutron, m for the muon, a for the alpha particle, etc.

tin g(

6. Symbols for physical quantities are single, or exceptionally two letters of the Latin or Greek alphabet, but they are frequently supplemented with subscripts, superscripts or information in parentheses to specify further the quantity. Further symbols used in this way are either italic or roman depending on what they represent. Examples H denotes enthalpy, but Hm denotes molar enthalpy (m is a mnemonic label for molar, and is therefore roman). Cp and CV denote the heat capacity at constant pressure p and volume V , respectively (note the roman m but italic p and V ). The chemical potential of argon might be denoted µAr or µ(Ar), but the chemical potential of the ith component in a mixture would be denoted µi , where i is italic because it is a variable subscript.

rin

7. Symbols for mathematical operators are always roman. This applies to the symbol ∆ for a dierence, d for an innitesimal variation, d for an innitesimal dierence (in calculus), and to capital Σ and Π for summation and product signs, respectively. The symbols p (3.141 592. . . ), e (2.718 281. . . , base of natural logarithms), i (square root of minus one), etc. are always roman, as are the symbols for specied functions such as log (lg for log10 , ln for loge , or lb for log2 ), exp, sin, cos, tan, erf, , , rot , etc. The particular operators and rot and the corresponding symbols ∇ for , ∇ × for rot , and ∇ · for are printed in bold-face to indicate the vector or tensor character following [5.k]. Some of these letters, e.g. e for elementary charge, are also sometimes used to represent physical quantities; then of course they shall be italic, to distinguish them from the corresponding mathematical symbol. Examples ∆H = H(final) − H(initial); (dp/dt) used for the rate of change of pressure; dx used to denote an innitesimal variation of x. But for a damped linear oscillator the amplitude F as a function of time t might be expressed by the equation F = F0 exp(−δt) sin(ωt) where δ is the decay coecient (SI unit: Np/s) and ω is the angular frequency (SI unit: rad/s). Note the use of roman d for the operator in an innitesimal variation of x, dx, but italic δ for the decay coecient in the product δt. Note that the products δt and ωt are both dimensionless, but are described as having the unit neper (Np = 1) and radian (rad = 1), respectively.

GB

3R

SC

2n

dP

grad

div grad grad

8

div

gh t IU PA C

8. The fundamental physical constants are always regarded as quantities subject to measurement (even though they are not considered to be variables) and they should accordingly always be italic. Sometimes fundamental physical constants are used as though they were units, but they are still given italic symbols. An example is the hartree, Eh (see Section 3.9.1, p. 95). However, the electronvolt, eV, the dalton, Da, or the unied atomic mass unit, u, and the astronomical unit, ua, have been recognized as units by the Comité International des Poids et Mesures (CIPM) of the BIPM and they are accordingly given Roman symbols. Examples c0 for the speed of light in vacuum, me for the electron mass, h for the Planck constant, NA or L for the Avogadro constant, e for the elementary charge, a0 for the Bohr radius, etc. The electronvolt 1 eV = e·1 V = 1.602 176 487(40)×10−19 J.

20 08 )C op yri

9. Greek letters are used in systematic organic, inorganic, macromolecular, and biochemical nomenclature. These should be roman (upright), since they are not symbols for physical quantities. They designate the position of substitution in side chains, ligating-atom attachment and bridging mode in coordination compounds, end groups in structure-based names for macromolecules, and stereochemistry in carbohydrates and natural products. Letter symbols for elements are italic when they are locants in chemical-compound names indicating attachments to heteroatoms, e.g. O-, N -, S -, and P -. The italic symbol H denotes indicated or added hydrogen (see reference [15]). Examples a-ethylcyclopentaneacetic acid b-methyl-4-propylcyclohexaneethanol tetracarbonyl(η4 -2-methylidenepropane-1,3-diyl)chromium a-(trichloromethyl)-ω-chloropoly(1,4-phenylenemethylene) a- -glucopyranose 5a-androstan-3b-ol N -methylbenzamide O-ethyl hexanethioate 3H -pyrrole naphthalen-2(1H )-one

GB

3R

SC

2n

dP

rin

tin g(

d

9

3R

GB

10

SC 20 08 )C op yri

tin g(

rin

dP

2n

gh t IU PA C

TABLES OF PHYSICAL QUANTITIES

gh t IU PA C

2

GB

3R

SC

2n

dP

rin

tin g(

20 08 )C op yri

The following tables contain the internationally recommended names and symbols for the physical quantities most likely to be used by chemists. Further quantities and symbols may be found in recommendations by IUPAP [4] and ISO [5]. Although authors are free to choose any symbols they wish for the quantities they discuss, provided that they dene their notation and conform to the general rules indicated in Chapter 1, it is clearly an aid to scientic communication if we all generally follow a standard notation. The symbols below have been chosen to conform with current usage and to minimize conict so far as possible. Small variations from the recommended symbols may often be desirable in particular situations, perhaps by adding or modifying subscripts or superscripts, or by the alternative use of upper or lower case. Within a limited subject area it may also be possible to simplify notation, for example by omitting qualifying subscripts or superscripts, without introducing ambiguity. The notation adopted should in any case always be dened. Major deviations from the recommended symbols should be particularly carefully dened. The tables are arranged by subject. The ve columns in each table give the name of the quantity, the recommended symbol(s), a brief denition, the symbol for the coherent SI unit (without multiple or submultiple prexes, see Section 3.6, p. 91), and note references. When two or more symbols are recommended, commas are used to separate symbols that are equally acceptable, and symbols of second choice are put in parentheses. A semicolon is used to separate symbols of slightly dierent quantities. The denitions are given primarily for identication purposes and are not necessarily complete; they should be regarded as useful relations rather than formal denitions. For some of the quantities listed in this chapter, the denitions given in various IUPAC documents are collected in [16]. Useful denitions of physical quantities in physical organic chemistry can be found in [17] and those in polymer chemistry in [18]. For dimensionless quantities, a 1 is entered in the SI unit column (see Section 3.10, p. 97). Further information is added in notes, and in text inserts between the tables, as appropriate. Other symbols used are dened within the same table (not necessarily in the order of appearance) and in the notes.

11

3R

GB

12

SC 20 08 )C op yri

tin g(

rin

dP

2n

gh t IU PA C

SPACE AND TIME

gh t IU PA C

2.1

The names and symbols recommended here are in agreement with those recommended by IUPAP [4] and ISO [5.b]. Name

Symbols

cartesian space coordinates cylindrical coordinates spherical polar coordinates generalized coordinates position vector length special symbols: height breadth thickness distance radius diameter path length length of arc area volume plane angle solid angle time, duration period frequency angular frequency characteristic time interval, relaxation time, time constant angular velocity velocity speed acceleration

x; y; z

Denition

m, 1, m m, 1, 1

q, q i

r

r = x e x+y e y +z e z

h b

δ

d r d s s A, As , S V,

(v) α, β , γ , ϑ, ϕ, ... , (ω )

ω = dϕ/dt = d /dt v = | | = d /dt

rad s−1 , s−1 m s−1 m s−1 m s−2

2, 5

α = s/r = A/r2

W T

f T

rin

ν, ω τ,

1

= t /N ν = 1/T ω = 2pν τ = |dt /d ln x |

m2 m3 rad, 1 sr, 1 s s Hz, s−1 rad s−1 , s−1 s

tin g(

W T

ω , , , , v, u, w, c

(varies) m m

20 08 )C op yri

l

t

Notes

m

ρ; ϑ; z r ; ϑ; ϕ

d,

SI unit

3 2, 4

v r v 6 a v 7 (1) An innitesimal area may be regarded as a vector e n dA, where e n is the unit vector normal to .

2n

dP

v u w c r a

2 2

GB

3R

SC

the plane. (2) The units radian (rad) and steradian (sr) for plane angle and solid angle are derived. Since they are of dimension one (i.e. dimensionless), they may be included in expressions for derived SI units if appropriate, or omitted if clarity and meaning is not lost thereby. (3) N is the number of identical (periodic) events during the time t. (4) The unit Hz is not to be used for angular frequency. (5) Angular velocity can be treated as a vector, ω , perpendicular to the plane of rotation dened by v = ω × r . (6) For the speeds of light and sound the symbol c is customary. (7) For acceleration of free fall the symbol g is used. 13

CLASSICAL MECHANICS

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2.2

The names and symbols recommended here are in agreement with those recommended by IUPAP [4] and ISO [5.c]. Additional quantities and symbols used in acoustics can be found in [4,5.g]. Name

Symbol

mass reduced mass density, mass density relative density surface density specic volume momentum angular momentum moment of inertia

m

Denition

µ ρ d ρA , ρS v

µ = m1 m2 / (m1 + m2 ) ρ = m/V ◦ d = ρ/ρ− ρA = m/A v = V /m = 1/ρ =m = P× I = mi ri 2

p v L r p

F M ,(T )

F = dp/dt = ma M=r×F

I, J

i

force moment of force, torque energy potential energy kinetic energy work power generalized coordinate generalized momentum Lagrange function Hamilton function

E Ep , V , Ek , T , K W , A, w P q p L H

action pressure surface tension weight gravitational constant

S p, (P ) γ, σ G, (W, P ) G

F r F r F v

R Ep = − ·d Ek = R(1/2)mv 2 W = ·d P = · = dW /dt

L(q, q) ˙ = TP (q, q) ˙ − V (q) ˙ H(q, p) = pi q˙i − L(q, q) i R S = L dt p = F/A γ = dW /dA G = mg F = Gm1 m2 /r2

rin

tin g(

F

Notes

kg kg kg m−3 1 kg m−2 m3 kg−1 kg m s−1 Js kg m2

1

2 3

20 08 )C op yri

p L

SI unit

N Nm

J J J J W (varies) (varies) J J

4

Js Pa, N m−2 N m−1 , J m−2 N N m2 kg−2

5

SC

2n

dP

◦ = ρ(H O, 4 ◦ C). (1) Usually ρ− 2 (2) Other symbols are customary in atomic and molecular spectroscopy (see Section 2.6, p. 25).  P P (3) In general I is a tensor quantity: Iαα = i mi βi2 + γi2 , and Iαβ = − i mi αi βi if α 6= β , where α, β , γ is a permutation of x, y , z . For a continuous mass distribution the sums are replaced by integrals. (4) Strictly speaking, only potential energy dierences have physical signicance, thus the integral is to be interpreted as a denite integral, for instance Z r2 Ep (r1 , r2 ) = − ·d r1

F r

3R

or possibly with the upper limit innity Z ∞ Ep (r) = − ·d r

F r

GB

(5) Action is the time integral over the Lagrange function L, which is equivalent to (see [19]). 14

R

pdq −

R

H dt

Symbol

Denition

SI unit

Notes

normal stress shear stress linear strain, relative elongation modulus of elasticity, Young's modulus shear strain shear modulus, Coulomb's modulus volume strain, bulk strain bulk modulus, compression modulus viscosity, dynamic viscosity uidity kinematic viscosity dynamic friction factor sound energy ux acoustic factors, reection absorption transmission dissipation

σ τ ε, e

σ = F/A τ = F/A ε = ∆l/l

Pa Pa 1

6 6

E

E = σ/ε

Pa

6

γ G

γ = ∆x/d G = τ /γ

1 Pa

6, 7 6

ϑ

ϑ = ∆V /V0

1

6

K

K = −V0 (dp/dV )

Pa

6

ϕ ν µ, (f ) P, Pa ρ αa , (α) τ δ

20 08 )C op yri

η, (µ)

gh t IU PA C

Name

τxz = η (dvx /dz)

Pa s

ϕ = 1/η ν = η/ρ Ffrict = µFnorm P = dE /dt

m kg−1 s m2 s−1 1 W

ρ = Pr /P0 αa = 1 − ρ τ = Ptr /P0 δ = αa − τ

1 1 1 1

8 9 8

GB

3R

SC

2n

dP

rin

tin g(

(6) In general these can be tensor quantities. (7) d is the distance between the layers displaced by ∆x. (8) P0 is the incident sound energy ux, Pr the reected ux and Ptr the transmitted ux. (9) This denition is special to acoustics and is dierent from the usage in radiation, where the absorption factor corresponds to the acoustic dissipation factor.

15

ELECTRICITY AND MAGNETISM

gh t IU PA C

2.3

The names and symbols recommended here are in agreement with those recommended by IUPAP [4] and ISO [5.e]. Name

Symbol

Denition

electric current electric current density electric charge, quantity of electricity charge density surface density of charge electric potential electric potential dierence, electric tension electromotive force

I, i , Q

R I = R · n dA Q = I dt

ρ σ V, φ U ,∆V, ∆φ

ρ = Q/V σ = Q/A V = dW/dQ U = V2 − V1

E Y D C ε ε0 εr

P

F

r

H E = ( /Q)·d

E F Y D D D E

χe χe (2) χe (3)

V

∇V = R /Q = −∇ = · e n dA ∇· =ρ C = Q/U =ε ε0 = µ0 −1 c0 −2

V C C m−2 F, C V−1 F m−1 F m−1

εr = ε/ε0 = − ε0

1 C m−2

P D

Notes

1 2 1

C m−3 C m−2 V, J C−1 V

20 08 )C op yri

E, (Emf , EMK )

A A m−2 C

j e

E

tin g(

electric eld strength electric ux electric displacement capacitance permittivity electric constant, permittivity of vacuum relative permittivity dielectric polarization, electric polarization (electric dipole moment per volume) electric susceptibility 1st hyper-susceptibility 2nd hyper-susceptibility

jJ

SI unit

χe = εr − 1 χe (2) = ε0 −1 (∂2 P/ ∂E 2 ) χe (3) = ε0 −1 (∂3 P/∂E 3 )

m−1

1 C m J−1 , m V−1 C2 m2 J−2 , m2 V−2

3 2 4 5 6

7 7

SC

2n

dP

rin

(1) The electric current I is a base quantity in ISQ. (2) e n dA is a vector element of area (see Section 2.1, note 1, p. 13). (3) The name electromotive force and the symbol emf are no longer recommended, since an electric potential dierence is not a force (see Section 2.13, note 14, p. 71). (4) ε can be a second-rank tensor. (5) c0 is the speed of light in vacuum. (6) This quantity was formerly called dielectric constant. (7) The hyper-susceptibilities are the coecients of the non-linear terms in the expansion of the magnitude P of the dielectric polarization P in powers of the electric eld strength E , quite related to the expansion of the dipole moment vector described in Section 2.5, note 17, p. 24. In isotropic media, the expansion of the component i of the dielectric polarization is given by

Pi = ε0 [χe (1) Ei + (1/2)χe (2) Ei2 + (1/6)χe (3) Ei3 + · · · ]

GB

3R

where Ei is the i-th component of the electric eld strength, and χe (1) is the usual electric susceptibility χe , equal to εr − 1 in the absence of higher terms. In anisotropic media, χ e (1) , χ e (2) , and χ e (3) are tensors of rank 2, 3, and 4. For an isotropic medium (such as a liquid), or for a crystal with a centrosymmetric unit cell, χe (2) is zero by symmetry. These quantities are macroscopic

16

µ µ0 µr

M χ, κ, (χm ) χm ,µ R G δ X Z

m

Y B ρ κ, γ, σ L M, L12

Notes

Cm

8

T

9

Wb A m−1

2

µ0 = 4p × 10−7 H m−1

N A−2 , H m−1 H m−1

10

µr = µ/µ0 = /µ0 −

1 A m−1

p = Qiri i F = Qv×B F = R B · en dA ∇×H=j B = µH M B

A S

H

χ = µr − 1 χm = Vm χ E=− · R = U/I G = 1/R δ = ϕU − ϕI X = (U/I) sin δ Z = R + iX

1 m3 mol−1 A m2 , J T−1 Ω S rad Ω Ω

Y = 1/Z

S

Y = G + iB =ρ =κ E = −L(dI/dt) E1 = −L12 (dI2 /dt) =∇× = ×

S Ωm S m−1 H, V s A−1 H, V s A−1 Wb m−1 W m−2

m B

E j j E

B A S E H

rin

magnetic ux density, magnetic induction magnetic ux magnetic eld strength, magnetizing eld strength permeability magnetic constant, permeability of vacuum relative permeability magnetization (magnetic dipole moment per volume) magnetic susceptibility molar magnetic susceptibility magnetic dipole moment electric resistance conductance loss angle reactance impedance, (complex impedance) admittance, (complex admittance) susceptance resistivity conductivity self-inductance mutual inductance magnetic vector potential Poynting vector

p, µ B F H

SI unit

P

gh t IU PA C

electric dipole moment

Denition

20 08 )C op yri

Symbol

tin g(

Name

11 12 12 13

14 14, 15

16

GB

3R

SC

2n

dP

(7) (continued) properties and characterize a dielectric medium in the same way that the microscopic quantities polarizability (α) and hyper-polarizabilities (β, γ) characterize a molecule. For a homogeneous, saturated, isotropic dielectric medium of molar volume Vm one has αm = ε0 χe Vm , where αm = NA α is the molar polarizability (Section 2.5, note 17, p. 24 and Section 2.7.2, p. 40). (8) When a dipole is composed of two point charges Q and −Q separated by a distance r, the direction of the dipole vector is taken to be from the negative to the positive charge. The opposite convention is sometimes used, but is to be discouraged. The dipole moment of an ion depends on the choice of the origin. (9) This quantity should not be called magnetic eld. (10) µ is a second-rank tensor in anisotropic materials. (11) The symbol χm is sometimes used for magnetic susceptibility, but it should be reserved for molar magnetic susceptibility. (12) In a material with reactance R = (U/I) cos δ , and G = R/(R2 + X 2 ). (13) ϕI and ϕU are the phases of current and potential dierence. (14) This quantity is a tensor in anisotropic materials. (15) ISO gives only γ and σ , but not κ. (16) This quantity is also called the Poynting-Umov vector. 17

QUANTUM MECHANICS AND QUANTUM CHEMISTRY

gh t IU PA C

2.4

The names and symbols for quantities used in quantum mechanics and recommended here are in agreement with those recommended by IUPAP [4]. The names and symbols for quantities used mainly in the eld of quantum chemistry have been chosen on the basis of the current practice in the eld. Guidelines for the presentation of methodological choices in publications of computational results have been presented [20]. A list of acronyms used in theoretical chemistry has been published by IUPAC [21]; see also Chapter 9, p. 155. Symbol

p

b momentum operator kinetic energy operator Tb b hamiltonian operator, H hamiltonian wavefunction, , ψ, φ state function hydrogen-like ψnlm (r, θ, φ) wavefunction spherical harmonic Ylm (θ, φ) function probability density P charge density ρ of electrons probability current density, probability ux electric current density of electrons integration element dτ matrix element Aij , hi |A| ji b of operator A expectation value hAi, A b of operator A

Y

S

SI unit

pb = −i~ ∇

∇2 Tb = −(~2 /2m)∇ b = Tb + Vb H

Js J J

Notes

m−1

1 1 1

b = Eψ Hψ

(m−3/2 )

2, 3

ψnlm = Rnl (r)Ylm (θ, φ)

(m−3/2 )

3

Ylm = Nl|m| Pl |m| (cos θ)eimφ

1

4

P = ψ∗ψ ρ = −eP

(m−3 ) (C m−3 )

3, 5 3, 5, 6

S = −(i~/2m)×

(m−2 s−1 )

3

j = −eS

(A m−2 )

3, 6

dτ = dx dy dz R b j dτ Aij = ψi ∗Aψ

(varies) (varies)

7

(varies)

7

(ψ ∗∇ ψ

tin g(

j

Denition

20 08 )C op yri

Name

R

b dτ ψ ∗Aψ

rin

hAi =



∇ψ ∗ ) ψ∇

dP

(1) The circumex (or hat), b , serves to distinguish an operator from an algebraic quantity. This denition applies to a coordinate representation, where ∇ denotes the nabla operator (see Section 4.2, p. 107). (2) Capital and lower case ψ are commonly used for the time-dependent function (x, t) and the amplitude function ψ(x) respectively. Thus for a stationary state (x, t) = ψ(x)exp(−iEt/}). (3) For the normalized wavefunction of a single particle in three-dimensional space the appropriate SI unit is given in parentheses. Results in quantum chemistry, however, are commonly expressed in terms of atomic units (see Section 3.9.2, p. 95 and Section 7.3 (iv), p. 145; and reference [22]). If distances, energies, angular momenta, charges and masses are all expressed as dimensionless ratios r/a0, E/Eh , etc., then all quantities are dimensionless. (4) Pl |m| denotes the associated Legendre function of degree l and order |m|. Nl|m| is a normalization factor. (5) ψ ∗ is the complex conjugate of ψ . For an anti-symmetrized n electron wavefunction (r1 , · · · , rn ), R R the total probability density of electrons is 2 · · · n ∗ dτ2 · · · dτn , where the integration extends over the coordinates of all electrons but one. (6) −e is the charge of an electron. (7) The unit is the same as for the physical quantity A that the operator represents.

Y

SC

2n

Y

GB

3R

YY

18

Y

Symbol

Denition



= (Aji )∗

hermitian conjugate b of operator A

b A

commutator b and B b of A

b B] b , [A, b B] b− [A,

b B] b =A bB b −B bA b [A,

anticommutator b and B b of A angular momentum operators spin wavefunction

b B] b+ [A,

b B] b + =A bB b +B bA b [A,

A

 † ij

SI unit

Notes

(varies)

7

(varies)

8

(varies)

8

1

9

gh t IU PA C

Name

see SPECTROSCOPY, Section 2.6.1, p. 30

α; β

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