From Deep Sea to Laboratory 3 from Tait's Work on the Compressibility

From Deep Sea to Laboratory 3 Illustration representative of the book: the Challenger expedition (route, vol. 1), physical measurements (samples, vol. 2) and the compressibility of liquids (globes, vol.3) From Deep Sea to Laboratory 3 From Tait's Work on the Compressibility of Seawater to Equations-of-State for Liquids Frédéric Aitken Jean-Numa Foulc First published 2019 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com Cover image © John Steven Dews (b.1949), H.M.S. Challenger in Royal Sound, Kerguelen Island, in the Southern Ocean (oil on canvas). © ISTE Ltd 2019 The rights of Frédéric Aitken and Jean-Numa Foulc to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2019943766 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-376-9 Contents Foreword ........................................... vii Preface ............................................ xi Notations ........................................... xv Chapter 1. The Compressibility of Liquids and Tait’s Equation-of-State ................................. 1 1.1. Introduction ..................................... 2 1.2. Concepts of compressibility ............................ 3 1.3. The first instruments to measure the compressibility of liquids ......... 5 1.4. The piezometers used onboard the Challenger .................. 21 1.5. Sources of pressure measurement errors ...................... 24 1.5.1. Apparent compressibility of water and mercury ............... 24 1.5.2. Apparent compressibility of liquid and piezometer ............. 27 1.6. Compressibility of fresh and salt water ...................... 32 1.6.1. Results on fresh water compressibility .................... 34 1.6.2. Results on seawater compressibility ...................... 38 1.6.3. Results on the compressibility of saline solutions .............. 40 1.6.4. Equilibrium of a water column ........................ 42 Chapter 2. Interpretations of the Parameters of Tait’s Equation ..................................... 45 2.1. Introduction ..................................... 46 2.2. Comparison and analogy with the Boyle–Mariotte equation-of-state ..................................... 46 2.3. Comparison and analogy with the Hirn equation-of-state ............ 54 vi From Deep Sea to Laboratory 3 2.4. Comparison and analogy with the van der Waals equation-of-state ..................................... 84 2.4.1. The molecular motion model ......................... 88 2.4.2. Establishing the van der Waals equation ................... 94 2.4.3. The different expressions and interpretations of covolume ......... 111 Chapter 3. Tait–Tammann–Gibson Equations-of-State ............. 147 3.1. Introduction ..................................... 148 3.2. Examples of compressibility equations-of-state .................. 150 3.3. Evolution of the parameters of the mixed modulus ................ 155 3.3.1. Application in the case of fresh water ..................... 160 3.3.2. Application in the case of standard seawater ................. 168 3.3.3. Application in the case of helium-4 ...................... 179 3.3.4. Application in the case of helium-3 ...................... 192 3.3.5. Density anomalies ............................... 199 3.3.6. Compressibility anomalies ........................... 201 3.4. Discussion and conclusion ............................. 207 Chapter 4. The Modified Tait Equation ........................ 245 4.1. Introduction ..................................... 246 4.2. Development of a complete equation-of-state ................... 249 4.3. Study of the adiabatic elastic modulus ....................... 255 4.3.1. Application in the case of fresh water ..................... 255 4.3.2. Application in the case of helium-3 ...................... 264 4.3.3. Application in the case of helium-4 ...................... 271 Conclusion ......................................... 279 Appendices ......................................... 283 Appendix A. Compressibility of a Straight Tube ................. 285 Appendix B. Virial Theorem ............................... 291 References .......................................... 335 Index .............................................. 343 Summary of Volume 1 .................................. 347 Summary of Volume 2 .................................. 351 Foreword It is a beautiful adventure that Frédéric Aitken and Jean-Numa Foulc have undertaken, using physical data from the Challenger expedition, the first major oceanographic expedition, sponsored by the British Admiralty in the 1870s. Indeed, this data, temperature and pressure readings at various depths and at multiple points of the world, was relatively little used at the time despite the visionary intuition of one of the initiators of the expedition, Professor Carpenter, that this data would allow for the reconstruction of ocean circulation. The authors attribute this relative lack of interest to the fact that most scientists on the expedition were naturalists, and that from the point of view of biology, the total benefits were already huge, with, for example, the discovery of life at a great depth. Exploiting data is not the least interesting of the physicist’s tasks. To deal with the problem, we simplify the situation and try not to delete anything essential. The terms of the equations are evaluated, keeping only the most important, and then two situations may arise. Let us say that the discrepancy with the data is clear: we are generally convinced that it has been oversimplified, but where? We are tempted in bad faith to defend our idea, even if it means becoming the Devil’s advocate and destroying what we have built. We go back to the overlooked terms one by one, and, with some luck, this may lead to a new effect. We make do with what we know; the battle is tough, and this is its appeal. Let us say that the similarity is acceptable. This is when a good physicist is suspicious: is it not a coincidence that two important effects are not offset by any chance? It would be necessary to make a prediction, and to repeat the experiment in different conditions, but it is not always possible. Another boat was not sent out with 200 people around the world for three years! The rigor with which experiments have been conducted, and the confidence that can be placed in the measures, are essential. The experimenters have had to multiply the situations blindly, without knowing viii From Deep Sea to Laboratory 3 which ones would be used as a test, with the sole aim of doing their best every time, by describing their protocol for future use. The development of the measurement protocol is part of the experiment’s design, as was instrument construction. At that time, a physicist worth his salt would never have used an instrument that he did not know how to build. How can one measure a temperature in a place that one cannot reach oneself (2,000 m below the surface of the sea, for example)? We can record the maximum and minimum temperatures reached during the descent (I found, with much emotion, the description of the maximum and minimum thermometer used by my grandfather in his garden). But what to do for intermediate temperatures? How to make sure that the line does not break in bad weather under the boat’s blows? How to decide the real depth despite currents, and the fact that the line continues to run under its own weight once the sensor is at the bottom? The design phase of the experiment can be exciting: I knew a physicist who was ready to sabotage a barely built experience (under the pretext, of course, of improving it) to be able to move more quickly to the design of the following experiment. Despite all the attention given to the design, sometimes an error is suspected in the measurements. This is the case here. Having reached unexpected depths (they discovered the Mariana Trench), the Challenger scientists wondered if their measurements had not been distorted by contraction of the glass envelopes. After their return, they assigned Peter Tait, a physicist from Edinburgh, the task of assessing these errors. One thing leading to another, he raised questions about the compressibility of seawater, and other liquids, and so about their equation-of-state, connecting pressure, temperature and density (and even salinity). The result of his studies left a lasting mark on the physics of liquids. Estimating errors, a task hated and despised by the typical physics student, yielded new knowledge. From the same period as the van der Waals equation, Tait’s efforts were part of the first trials to represent the equation-of-state of dense, liquid and solid bodies by continuous functions. The goal was twofold: metrological, to interpolate between

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