Archive for the ‘history of science’ Category

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The postmark on this card is Tuesday 25th February 1908 – the date Ronald Ross left Mauritius for England, having spent three months on the island to prepare an official report on measures for the prevention of malaria, while privately thinking about how epidemics can be explained in terms of mathematical principle.

CarnotCycle is a thermodynamics blog but occasionally it ventures into new areas. This post concerns the modeling of disease transmission.

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Calamity

In 1867, a violent epidemic of malaria broke out on the island of Mauritius in the Indian Ocean. In the coastal town of Port Louis 6,224 inhabitants out of a local population of 87,000 perished in just one month. Across the island as a whole there were 43,000 deaths out of a total population of 330,000. It was the worst calamity that Mauritius has ever suffered, and it had a serious impact on the island’s economy which in those days was principally generated by sugar cane plantations.

At the time, Mauritius was ruled by the British. The island had little in the way of natural resources, but perhaps because of its strategic position for Britain’s armed forces, the government was keen to keep the malaria problem under observation. Medical statistics show that following the great epidemic of 1867, deaths from malaria dropped to zero by the end of the century.

In the first years of the 20th century however, a small but significant rise in deaths from malarial fever was observed. And in May 1907 the British Secretary of State for the Colonies requested Ronald Ross, Professor of Tropical Medicine at Liverpool University, to visit Mauritius in order to report on measures for the prevention of malaria there. Ross sailed from England in October 1907 and arrived in Mauritius a month later.

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Genesis

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Just south of Port Louis, on the west coast of Mauritius, lies the township of Albion. Today it is home to a Club Med beach resort, but in 1907 when Ronald Ross visited the island, there were sugar plantations here – Albion Estate and Gros Cailloux estate, employing considerable numbers of Indian laborers. This part of the sea-coast was known for its marshy localities and it was here that the first sporadic cases of malaria were observed in 1865, two years before the great epidemic broke out.

Ronald Ross no doubt toured this area, his mind occupied with the genesis of the outbreak. Just a handful of cases in 1865, then in 1866 there were 207 cases on Albion Estate and 517 cases on Gros Cailloux Estate. From these estates the disease spread north and south, and during 1867 the epidemic broke out with such severity along sixty miles of coastline that those who survived were scarcely able to bury the dead.

How could this rapid increase in cases be explained? Ronald Ross was probably better placed than anyone to furnish an answer. Not only was he the discoverer of the role of the marsh-breeding Anopheline mosquito in spreading malaria (for which he received a Nobel Prize in 1902), he was also a thinker with a mathematical turn of mind.

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Statistics

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Soon after the malaria epidemic broke out on Mauritius, the British government appointed a commission of enquiry, which published a bulky report in 1868. This was followed by numerous other publications, giving Ronald Ross an abundance of statistical data with which to chart the course of the epidemic.

I can picture Ross studying the monthly totals of malaria cases as the epidemic unfolded, and noting how they followed an exponential curve. And I can imagine him seeing the list of figures as the terms of a mathematical sequence, with the question forming in his mind “What is the formula that generates the numbers in this sequence?”.

Although trained in medicine rather than mathematics, Ross nevertheless knew that one route to finding the formula was to construct a first-order difference equation which expresses the next term in a sequence as a function of the previous term. In his 1908 report he adopts this approach, finds a formula, and demonstrates some remarkable results with it. Although at times loosely worded, his pioneering elaboration of what he calls the ‘malaria function’ displays original thinking of a high order.

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Solution

Ronald Ross was a mathematician by nature but not by training, which explains the absence of formal rigor in his mathematical argument. The style of exposition is somewhat saltatory; in fact he never actually states the difference equation, but instead leaps straight to its general solution (the malaria function) without showing the intermediate steps.

Ross begins with the argumentation leading to his famous ‘fsbaimp’ expression (familiarity is assumed; otherwise see Appendix 1), but it is not particularly conducive to understanding his overall scheme since he presents it as an algebraic thing-in-itself rather than a component variable in a first-order difference equation.

To apprehend the architecture of Ross’s thinking, one has to work backwards from the malaria function to obtain the difference equation, which can be expressed in words as

Infections (month n+1) = Infections (month n) – number of recoveries + number of new cases

Now although Ross did not address the matter of dimensions at any point in his argumentation, it was nonetheless a crucial consideration in formulating the above equation. Equality is symmetric, so the dimensions of each RHS term must be the same as the LHS term, which according to Ross’s terminology for infected people is mp. Since Ross is seeking to obtain a difference equation of the form

where α is the growth/decay constant, each of the three RHS terms must be the product of mp and a dimensionless coefficient k:

Clearly k1 is a dimensionless 1 since the total infections in month n is simply m(n)p. The coefficient k2 is the dimensionless recovery constant for the infected population (Ross uses the symbol r), whose value lies in the range 0–1. The real difficulty is with k3 – how to transform fsbai into a dimensionless quantity. Ross achieved this (see Appendix 1) by introducing a one-to-one correspondence constraint which had the effect of changing the units of a from mosquitoes to people, thereby cancelling out the units of b (1/people) and rendering fsbai dimensionless. This could with some justification be regarded as an exercise in artifice, but Ross really had no alternative to employing facilitated convenience if he was to solve this equation.

Putting all these pieces together, the difference equation Ross arrived at, but did not state, was:

where all terms except m (called the malaria rate) are considered constant. In his 1908 report, Ross skipped directly from the above equation, which is of the form

to its solution

which enabled him to compute his malaria function explicitly in terms of the initial value m(0)

or as Ross actually rendered it (by substituting f/p for b; see Appendix 1)

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Ratios

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Ronald Ross and a mosquito trap on Clairfond Marsh, Mauritius

In the above equation, Ross found an explanation not only of the outbreaks of malaria epidemics, but also of why malaria can diminish and even die out – as had happened for example in Europe – despite the continued presence of mosquitoes capable of carrying the disease.

Ross recognized that m(n) would increase or diminish indefinitely at an exponential rate as n increases, according to whether the contents of the parentheses were greater or less than unity, i.e.

Here was the riposte to those who claimed that malaria should persist wherever Anopheline mosquitoes continued to exist, and that anti-malarial strategies which merely reduced mosquito numbers would never eradicate the disease.

Ross could now show that it was the relation of the mosquito-human population ratio in a locality to its threshold value (a/p = r/f2si) that determined growth or decay of the malaria rate m(n), and that mosquito reduction measures, if sufficiently impactful, could indeed result in the disease diminishing and ultimately disappearing. He could even provide a rough estimate of the threshold value of a/p by assigning plausible values to s, i, f and r. In his 1908 report, Ross calculated this value to be 39.6, or about 40 mosquitoes per individual.

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Limitations

Ross’s malaria function was a remarkable result of some brilliantly original thinking, but as with most early forays into uncharted territory it had its limitations. Principal among them was that the equation was valid on the restrictive assumption that infected mosquitoes bit only uninfected human beings.

This clearly lacked credibility in the circumstances of a developed epidemic where a substantial proportion of the local population would be infected. So Ross was forced to preface his equation with the words ‘if … m is small’, which meant that the equation was strictly invalid for charting log phase growth or decay – thereby weakening support for his argument that total eradication of mosquitoes was unnecessary for disease control.

Another significant assumption in Ross’s equation was that the local population p was regarded as constant*, something wildly at variance with the actuality of the Mauritius epidemic of 1867, where a great many deaths occurred in the absence of any significant immigration.

*Although p cancels out from the mp term on both sides of the equation, it remains present in the third coefficient which is a component part of the growth/decay constant.

With limitations like these, it is evident that in his 1908 report Ross had not yet achieved a convincing mathematical argument to support his controversial views on how to control malaria. Ross was well aware of this, and over the next eight years he developed his ideas considerably – both in refining his model and advancing his mathematical approach.

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Extensions

The next phase of Ross’s mathematical thinking was published in a book entitled The Prevention of Malaria (1911) wherein Ross addresses the malaria rate issue using iterated difference equations, from which he computes a limiting value of m. In an addendum to the 2nd edition of this work, under the heading Theory of Happenings, Ross addresses the population variation issue using a systematized set of difference equations, and in the closing pages of the addendum makes the transition from the discrete time period of his difference equations to the infinitesimal time period of a corresponding set of differential equations. This allows him to address variations from the perspective of continuous functions.

Ross could have stopped there, but the instinctive mathematician in him had more to say. This resulted in a lengthy paper published in parts in the Proceedings of the Royal Society of London between July 1915 and October 1916. In this paper, Ross continues from where he left off in 1911, but in a more generalized form. He considers a population of whom a number are affected by something (such as a disease) and the remainder are non-affected; in an element of time dt a proportion of the non-affected become affected and a possibly different proportion of the affected revert to the non-affected group. He then supposes that both groups are subject to possibly different birth rates, death rates, immigration and emigration rates, and asks: What will be the number of affected individuals, the number of new cases, and the number of people living at time t?

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Hilda Hudson (1881-1965) and Ronald Ross (1857-1932)

To answer these questions, Ross attempts to integrate his differential equations; this forms the substance of Part I. For Parts II and III, Ross enlists the assistance of “Miss Hilda P. Hudson, MA, ScD”, a 34-year-old Cambridge mathematician, whom he acknowledges as co-author. In Part II they examine cases where the something that happens to the population (such as a disease) is not constant during the considered period. This propels them into the study of what they call hypometric happenings. In Part III they turn their attention to graphing some of the functions they have obtained, and note the steadily rising curve of a happening that gradually permeates the entire population, the symmetrical bell-shaped curve of an epidemic that dies away entirely, the unsymmetrical bell curve that begins with an epidemic and settles down to a steady endemic level, the periodic curve with its regular rise and fall due to seasonal disturbances, and the irregular curve where outbreaks of differing violence occur at unequal intervals. The conclusion they reach is that “the rise and fall of epidemics as far as we can see at present can be explained by the general laws of happenings, as studied in this paper.”

In summary then, it can be said that having resolved the issues that restricted the applicability of the malaria function, Ross and Hudson found that their generalized model – taking the happening to be a malaria outbreak – endorsed Ross’s original assertions, with the attendant implications for management and prevention.

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But all this lay ahead of Ronald Ross in February 1908 as he completed the groundwork for his first report. We leave him as he packs his bags to depart Mauritius, his mind full of island impressions, malaria statistics and mathematical ideas that he will contemplate at leisure on the month-long journey home.

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Appendix 1

What f.s.b.a.i.m.p means

(terms as defined in the 1908 report; note that Ross later revised some of these definitions)

p = the average population in the locality (units: people)
m = the proportion of p which are already infected with malaria in the start month (dimensionless)
i = the proportion of m which are infectious to mosquitoes (dimensionless)
a = the average number of mosquitoes in the locality (units: mosquitoes)
b = the proportion of a that feed on a single person (units: 1/people)

hence baimp = the average number of mosquitoes infected with malaria in the month

s = the proportion of mosquitoes that survive long enough to bite human beings (dimensionless)
f = the proportion of a which succeed in biting human beings (dimensionless)

hence fsbaimp = the average number of infected mosquitoes which succeed in biting human beings

If the constraint is applied that each of these mosquitoes infects a separate person and only one person, then fsbaimp will denote the average number of persons infected with malaria during the month. Since the constraint imposes a one-to-one correspondence, the units of fsbaimp may equally be taken as ‘infected mosquitoes’ and ‘infected people’.

Note also that, given p, either b or f is technically redundant since p = f/b

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Further reading

Ronald Ross, Report on the Prevention of Malaria in Mauritius (1908)
https://archive.org/details/b21352720

Paul Fine, Ross’s a priori Pathometry – a Perspective (1976)
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1864006/pdf/procrsmed00040-0021.pdf

Smith DL et al., Ross, Macdonald, and a Theory for the Dynamics and Control of Mosquito-Transmitted Pathogens (2012)
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3320609/pdf/ppat.1002588.pdf

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P Mander August 2016

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Wilhelmy’s birthplace – Stargard, Pomerania – in less happy times

The mid-point of the 19th century – 1850 – was a milestone year for the neophyte science of thermodynamics. In that year, Rudolf Clausius in Germany gave the first clear joint statement of the first and second laws, upon which Josiah Willard Gibbs in America would develop chemical thermodynamics. 1850 was also the year that the allied discipline of chemical kinetics was born, thanks to the pioneering work of Ludwig Wilhelmy.

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Ludwig Ferdinand Wilhelmy was born on Christmas Day 1812 in Stargard, Pomerania (now Poland). After completing his schooling, he studied pharmacy and subsequently bought an apothecary shop. In 1843, at the age of 31, he sold the shop in order to pursue research interests at university, where he made the acquaintance of Rudolf Clausius and Hermann von Helmholtz. In 1846, Wilhelmy received his doctorate from Heidelberg University, and it was here in 1850 that he conducted the first quantitative experiments in chemical kinetics, using a polarimeter to study the rate of inversion of sucrose by acid-mediated hydrolysis.

Wilhelmy’s work had a seminal quality to it because – apart from being a talented individual – he observed that great guiding principle when commencing an exploration of the unknown: he kept things simple.

He chose a monomolecular decomposition reaction, used a large volume of water to keep the acid concentration unchanged during the experiment, maintained constant temperature and followed the inversion process with a polarimeter, which did not physically disturb the conditions of the system under study. By rigorously limiting system variables, Wilhelmy discovered a simple truth: the rate of change of sucrose concentration at any moment is proportional to the sucrose concentration at that moment.

Now it just so happened that in Wilhelmy’s earlier doctoral studies, he had become familiar with utilizing differential equations. So it was a straightforward task for him to model his new discovery as an initial value problem, which he wrote as

where Z is the concentration of sucrose, T is time, S is the acid concentration (presumed unchanging throughout the reaction), and M is a constant today called the reaction velocity constant. Wilhelmy integrated this equation to

where C is the constant of integration. Recognising that when T = 0 the sucrose concentration is its initial value Z0, he wrote

or

He then proceded to show that this equation was consistent with his experimental results, and thus became the first to put chemical kinetics on a theoretical foundation.

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A page from Wilhelmy’s pioneering work “Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet”, published in Annalen der Physik und Chemie 81 (1850), 413–433, 499–526

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Inversion of sucrose

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Sucrose has a dextrorotatory effect on polarized light, but on acid hydrolysis the resulting mixture of glucose and fructose is levorotatory, because the levorotatory fructose has a greater molar rotation than the dextrorotatory glucose. As the sucrose is used up and the glucose-fructose mixture is formed, the angle of rotation to the right (as the observer looks into the polarimeter tube) becomes less and less. It can be demonstrated that the angle of rotation is directly proportional to the sucrose concentration at any moment during the inversion process.

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A Laurent polarimeter from around 1900.

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A neglected pioneer

Ludwig Wilhelmy’s groundbreaking research into the kinetics of sucrose inversion was published in Annalen der Physik und Chemie in 1850, but it failed to garner attention and in 1854 he left Heidelberg, retiring to private life in Berlin at the age of 42. He died ten years later in 1864, his pioneering work still unrecognized.

It was not until 1884, twenty years after Wilhelmy’s death and thirty four years after his great work, that Wilhelm Ostwald – one of the founding fathers of physical chemisty – called attention to Wilhelmy’s paper. Among those who took notice was the talented Dutch theoretian JH van ‘t Hoff, who in 1884 was engaged on kinetic studies of his own, soon to be published in the milestone monograph Études de dynamique chimique (Studies in Chemical Dynamics). In this book, van ‘t Hoff extended and generalized the mathematical analysis that had originally been given by Wilhelmy.

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A digression on half-life

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In his study of sucrose inversion, Ludwig Wilhelmy showed that the instantaneous reaction rate was proportional to the sucrose concentration at that moment, a result he expressed mathematically as

In his published paper, Wilhelmy did not mention the fact that the fraction of sucrose consumed in a given time is independent of the initial amount. But he might well have noticed that by expressing Z as a fraction of Z0, the left hand side of the equation simply becomes the logarithm of a dimensionless number.

For example the half-life time (T0.5) i.e. the time at which half of the substance present at T0 has been consumed

JH van ‘t Hoff was well aware of this fact – he derived a half-life expression on page 3 of the Études. And in all likelihood he was also aware that this kinetic truth produced a conflict with the thermodynamic necessity for a chemical reaction to reach equilibrium.

Reconciling kinetics and thermodynamics

The starting concentration of sucrose in Wilhelmy’s inversion experiment is Z0. So after the half life period T0.5 has elapsed, the sucrose concentration will be Z0/2. After further successive half life periods the concentrations will be Z0/4, Z0/8, Z0/16 and so on. The fraction of sucrose consumed after n half lives is

This is a convergent series whose sum is Z0 – corresponding to the total consumption of the sucrose and the end of the reaction. The problem with this formula is that it implies that Wilhelmy’s sucrose inversion reaction – or any first order reaction – will take an infinitely long time to complete. This is not consistent with the fact that chemical reactions are observed to attain thermodynamic equilibrium in finite timescales.

In the Études, van ‘t Hoff successfully reconciled kinetic truth and thermodynamic necessity by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant Kc is in fact the quotient of the kinetic velocity constants for the forward (k1) and reverse (k-1) reactions

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Wilhelmy’s legacy

Wilhelmy’s pioneering work may not have been recognised in his lifetime, but the science of chemical kinetics which began with him developed into a major branch of physical chemistry, involving many famous names along the way.

In the 1880s, van ‘t Hoff and the Swedish physical chemist Svante Arrhenius – both winners of the Nobel Prize – made important theoretical advances regarding the temperature dependence of reaction rates, which proved a difficult problem to crack.

In 1899, DL Chapman proposed his theory of detonation. The chemical kinetics of explosive reactions was then taken forward by Jens Anton Christiansen, whose idea of chain reactions was further developed by Nikolaj Semyonov and Cyril Norman Hinshelwood, both of whom won the Nobel Prize for their development of the concept of branching chain reactions, and the factors that influence initiation and termination.

Several other Nobel Prize winners have their names associated with chemical kinetics, including Walther Nernst, Irving Langmuir, George Porter and JC Polanyi. The work of all these illustrious men has enriched this important subject.

But for now, we must return to Ludwig Wilhelmy in Heidelberg. It is 1854, and having failed to garner any interest in his seminal studies, he has packed his bags at the university, handed in his keys at the porter’s lodge, and is ready to begin the long journey home to Berlin.

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Heidelberg, Germany in the 1850s

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Mouse-over links to works referred to in this post

Jacobus Henricus van ‘t Hoff Studies in Chemical Dynamics

Ludwig Wilhelmy “Ueber das Gesetz, nach welchem die Einwirkung der Säuren auf den Rohrzucker stattfindet” , published in Annalen der Physik und Chemie 81 (1850), 413–433, 499–526

P Mander April 2016

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Tottenham Court Road, London WC2 in 1880

In the study of chemical reactions, thermodynamics enables us to calculate changes in state functions such as enthalpy, entropy and free energy, and determine the direction in which a reaction is spontaneous. But it tells us nothing about the speed of reaction; that is the province of chemical kinetics. Thermodynamics and chemical kinetics can be viewed as complementary disciplines, which together provide the means by which the course of a reaction can be elucidated.

A classic case which exemplifies the dual application of thermodynamics and chemical kinetics is the Tottenham Court Road gas explosion which occurred in July 1880.

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The incident

It was a time of great expansion of the network for gas pipeline transport in London. Gas lighting of streets and buildings was well-established, but now the gas stove was about to become a commercial success, and new gas mains were being laid to supply the anticipated demand.

The Gas Light and Coke Company, which supplied coal gas from a number of gasworks in London, had laid a new 1.2 kilometer (0.75 mile) section of main from Bedford Square to Fitzroy Square, the pipeline crossing Tottenham Court Road at the junction with Bayley Street and running along Percy Street before turning north along the entire length of Charlotte Street.

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On the evening of Monday 5th July 1880, workmen were preparing to connect the new main to the existing network at Bayley Street. Unknown to them however, a faulty valve at the other end of the new main was leaking coal gas, which had mingled with the air in the pipe to form an explosive mixture. In a presumed act of carelessness by one of the workmen at Bayley Street, a flame or other ignition source came in close proximity to the pipe.

The gas mixture detonated and the explosion ripped through the entire length of the new 1.2 kilometer main. A number of people were killed and injured in the blast, and 400 houses were damaged by flying debris. The entire incident lasted about 12 seconds.

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The investigation

A singularly worrying feature of the Tottenham Court Road gas explosion was that it had ripped through over a kilometer of pipeline in a matter of seconds. How could this happen? And how easily could this happen again? For the safety of millions of Londoners, answers had to be found.

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Augustus Vernon Harcourt (1834-1919)

The authorities turned to one the country’s leading chemists, Augustus Vernon Harcourt, who was conducting a program of research in chemical kinetics at Oxford University. Together with his student Harold Baily Dixon (1852-1930), Harcourt began to investigate the rates of propagation of gaseous explosions.

In what sounds like a rather risky experiment, they set up long metal pipes under the Dining Hall of Balliol College Oxford to measure the speed with which explosion waves travel when a mixture of air and coal gas detonates.

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The Dining Hall of Balliol College, Oxford

Twenty three years earlier, the German chemist Robert Bunson (of Bunsen burner fame) had investigated the rate of propagation for the ignition of coal gas and oxygen and concluded that the flame front velocity was less than 1 meter per second. From the experiments at Balliol however, Harcourt and Dixon arrived at a very different answer. In a report to the Board of Trade on the Tottenham Court Road blast, Harcourt concluded that the velocity of a coal gas/air explosion wave exceeded 100 yards per second (91 meters per second).

From the safety point of view, Harcourt and Dixon had shown how absolutely essential it was to prevent air becoming mixed with coal gas in the gas pipeline network. But it would take decades before sufficient theoretical progress was made to allow a detailed understanding of what exactly happened in the great gas explosion of 1880.

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Branching chains

The development of chemical kinetics involved many different contributors in the decades after Harcourt and Dixon’s pioneering work at Oxford. Theories were advanced on several different aspects of the subject, but one piece of theoretical work had particular relevance to the study of explosions.

In 1921, a Danish physical chemist by the name of Jens Anton Christiansen (1888-1969) completed his PhD studies in reaction kinetics at Copenhagen University. In his thesis he incorporated an idea first suggested by Bodenstein in 1913 and introduced the term “kædereaktion”. This term, and the conceptual idea behind it, attracted considerable attention and the equivalent English expression “chain reaction” came into use. Two years later, Christiansen and the Dutch physicist Hendrick Anthony Kramers (1894-1952) published a paper in which they suggested the possibility of branching chains. Their idea was that a chain reaction could involve steps in which one chain carrier (an atom or radical) might not only regenerate itself but also produce an additional chain carrier. If such chain branching occurred, the number of chain carriers could increase extremely rapidly and result in an explosion.

The idea proved to be well-founded, and was further developed by Nikolai Semyonov (1896-1986) and Cyril Norman Hinshelwood (1897-1967). Their work also showed that chain carriers were removed at the walls of the reaction vessel. If the rate of removal of the chain carriers was fast enough to counteract the effect of chain branching, a steady reaction ensued. But if the removal rate could not keep pace with the chain branching rate, an explosion would result.

On the basis of their thinking, the reaction rate expression assumed the form

where F is a function of the concentrations characteristic of the chain branching step, fa is a function determining the removal of chain carriers, and fb is a function expressing the branching nature of the chain reaction.

In steady reaction conditions, fa is sufficiently greater than fb. But if conditions change so that fa and fb converge, a point will be reached where the difference between them becomes vanishingly small. The reaction rate will soar towards infinity however small F may be, and the evolution of heat in the system will be so great as to cause an explosion.

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Semyonov and Hinshelwood were awarded the Nobel Prize in 1956 for their work on reaction rates

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Piecing the facts together

From the information contained in newspaper reports, and the application of kinetic theory and thermodynamics, it is possible to arrive at a likely explanation of why the great gas explosion of 1880 happened in the way it did.

It is known that coal gas leaked into the newly laid main at its northern end, and that detonation occurred at the other end in Bayley Street. From this it can be inferred that the entire pipeline between these two points contained coal gas admixed with the air that the pipe originally contained. On the assumption that the leaking valve was introducing coal gas at a modest and steady rate, it is likely that the partial pressures of the gases in the pipe were being brought into equilibrium as the coal gas seeped along the pipe.

Newspaper reports stated that the new main between Bayley Street and Fitzroy Square was a metal pipe of fixed (3 ft/0.91 m) diameter. The ratio of the surface area to the enclosed volume or, which is the same thing, the ratio of the circumference to the cross-sectional area

was therefore constant along its length*.

*assuming the geometry of the bend had no effect on fa. This point is examined later.

At the moment of detonation at Bayley Street, it is a reasonable hypothesis that the function F in the Semyonov-Hinshelwood rate expression was not subject to large variations along the length of the new main. The same can be said of fb, and since the ratio of the circumference to the cross-sectional area of the pipeline was constant, the function fa determining the removal of chain carriers at the walls of the pipe was also constant. In short, the reaction rate expression applying at the end of the pipe – where detonation is known to have occurred – applied at every other point along its length.

At this juncture, it is convenient to recall the combustion reactions of the principal components of coal gas, namely hydrogen, methane and carbon monoxide:

We observe that from a stoichiometric perspective, none of the reactions involves an increase in volume; in fact two of them result in a decrease. The overall entropy of reaction is negative, and this tells us that the conversion of reactants into products, however rapidly it took place, could not in itself have resulted in any pressure increase under the constant volume conditions of the pipe.

From an enthalpy of reaction perspective however, the situation is very different. The above reactions are all significantly exothermic processes – the calorific value of coal gas is typically around 20 megajoules per cubic meter. In the circumstances of detonation, the virtually instantaneous release of a large amount of heat would result in a similarly rapid rise in temperature, causing sudden compression of the adjacent volume element in the pipe and heating it to the point of detonation. This sequence would be repeated from one volume element to the next, with a wave of adiabatic compression intensifying the pressure as it traversed the pipe. A continuously propagating explosion would then follow the pressure wave along the course of the main as the pipe ruptured.

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The bend in the pipe

The junction of Percy Street with Charlotte Street was the only point along the entire length of the new main which deviated from a straight line. Here the pipeline executed a 90 degree turn, and it raises the question of how a detonation wave can go round corners. The exact construction of the bend is not recorded, but it is likely that an elbow joint was used.

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Geometrically, the bend itself is a quadrant of a torus, whose geometry is such that regardless of whether the elbow has a long or short major radius R, the ratio of the surface area to the enclosed volume is constant

This is the same ratio as that of the straight pipe. The bend at the junction of Percy Street with Charlotte Street introduced no changes to the fa term in the Semyonov-Hinshelwood rate expression, and thus the conditions for detonation were met at every point of the bend.

So the 90 degree elbow made no difference to the detonation wave. It simply turned sharp right and carried on up to Fitzroy Square, at a velocity of almost 100 meters per second.

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Estimating the power of the explosion

It is known from the analysis of coal gas that one volume of coal gas requires approximately 10 volumes of air for its complete combustion. This means that an explosive mixture with air cannot be formed at coal gas concentrations much above 9%, since there would be insufficient oxygen to support the necessary rate of reaction. Below 7% coal gas concentration, the mixture is also non-explosive, for other reasons.

An average coal gas concentration of 8% throughout the pipeline is therefore a fair estimate, and seems plausible given that the new main contained air when laid and that coal gas was introduced at a modest rate from a leaking valve. We know that the new 1.2 kilometer main had a radius of 0,455 meters, giving a total volume of 780 cubic meters. At the moment of detonation, coal gas is estimated to have filled 8% of this volume i.e. 62 cubic meters. The calorific value of coal gas is typically 20 megajoules per cubic meter, so we can conclude that the Tottenham Court Road gas explosion released around 1,240 MJ in the 12 seconds it took to traverse the pipeline. The power of the explosion was therefore 1240/12 = 103 MW.

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The 3×2 flagstones used on London sidewalks weigh around 70 kg each. The energy released by the Great Gas Explosion of 1880 was sufficient to blast 59,000 flagstones to a height of 30 meters.

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Contemporary accounts

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Charlotte Street after the blast

Newspaper accounts remarked on the rapid progression of the explosion, with one commenting:

“[The main pipe at Bayley Street] burst with a terrific report, and sheets of flame issued suddenly from the earth. Instantly the report seemed to run along Percy Street, which was torn up for sixty or seventy yards (ca. 60 meters), the paving stones flying on each side against the houses.”

“At the corner of Charlotte Street the basements of two houses were shattered. The paving stones were here also sent into the air, falling on and through the roofs of the houses opposite. Further on, the pipe burst again, near the corner of Bennett Street, where there is a large gap in the roadway. Another burst-up occurred near the corner of Howland Street, and at the corner of London Street (now Maple Street) still further on…”

One eye-witness was in Percy Street when the explosion occurred. He experienced the effect of not only the pressure wave from the bursting pipe, but also the decompression wave which followed in its wake:

“I was walking down Percy Street, when I felt the ground shaking under my feet. I immediately saw the centre of the street rising in the air. A tremendous report followed, and then there was a shower of bricks and stones. I felt myself lifted from the ground, and the next moment I was lying among the debris at the bottom of a deep hole in the roadway.”

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P Mander December 2015

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I can’t think of a better introduction to this post than Ludwig Boltzmann gave in his Vorlesungen über Gastheorie (Lectures on Gas Theory, 1896):

“General thermodynamics proceeds from the fact that, as far as we can tell from our experiences up to now, all natural processes are irreversible. Hence according to the principles of phenomenology, the general thermodynamics of the second law is formulated in such a way that the unconditional irreversibility of all natural processes is asserted as a so-called axiom … [However] general thermodynamics (without prejudice to its unshakable importance) also requires the cultivation of mechanical models representing it, in order to deepen our knowledge of nature—not in spite of, but rather precisely because these models do not always cover the same ground as general thermodynamics, but instead offer a glimpse of a new viewpoint.”

Today, the work of Ludwig Boltzmann (1844-1906) is considered among the finest in physics. But in his own lifetime he faced considerable hostility from those of his contemporaries who did not believe in the atomic hypothesis. As late as 1900, the kinetic-molecular theory of heat developed by Maxwell and Boltzmann was being vigorously attacked by a school of scientists including Wilhelm Ostwald, who argued that since mechanical processes are reversible and heat conduction is not, thermal phenomena cannot be explained in terms of hidden, internal mechanical variables.

Boltzmann refuted this argument. Mechanical processes, he pointed out, are irreversible if the number of particles is sufficiently large. The spontaneous mixing of two gases is a case in point; it is known from experience that the process cannot spontaneously reverse – mixed gases don’t unmix. Today we regard this as self-evident, but in Boltzmann’s time his opponents did not believe in atoms or molecules; they considered matter to be continuous. So the attacks on Boltzmann’s theories continued.

Fortunately, this did not deter Boltzmann from pursuing his ideas, at least not to begin with. He saw that spontaneous processes could be explained in terms of probability, and that a system of many particles undergoing spontaneous change would assume – other things being equal – the most probable state, namely the one with the maximum number of arrangements. And this gave him a new way of viewing the equilibrium state.

One can see Boltzmann’s mind at work, thinking about particle systems in terms of permutations, in this quote from his Lectures on Gas Theory:

“From an urn, in which many black and an equal number of white but otherwise identical spheres are placed, let 20 purely random drawings be made. The case that only black spheres are drawn is not a hair less probable than the case that on the first draw one gets a black sphere, on the second a white, on the third a black, etc. The fact that one is more likely to get 10 black spheres and 10 white spheres in 20 drawings than one is to get 20 black spheres is due to the fact that the former event can come about in many more ways than the latter. The relative probability of the former event as compared to the latter is the number 20!/10!10!, which indicates how many permutations one can make of the terms in the series of 10 white and 10 black spheres, treating the different white spheres as identical, and the different black spheres as identical. Each one of these permutations represents an event that has the same probability as the event of all black spheres.”

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By analyzing the ways in which systems of particles distribute themselves, and the various constraints to which particle assemblies are subject, important links came to be established between the statistical properties of assemblies and their bulk thermodynamic properties.

Boltzmann’s contribution in this regard is famously commemorated in the formula inscribed on his tombstone: S = k log W. There is powerful new thinking in this equation. While the classical thermodynamic definition of entropy by Rankine and Clausius was expressed in terms of temperature and heat exchange, Boltzmann gave entropy – and its tendency to increase in natural processes – a new explanation in terms of probability. If a particle system is not in its most probable state then it will change until it is, and an equilibrium state is reached.

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P Mander April 2016

Torricelli using a lot more mercury than necessary to demonstrate the barometer principle

The scientific study of the atmosphere can be said to have begun in 1643 with the invention of the mercury barometer by Evangelista Torricelli (1608-1647). Although the phenomenon had been observed and discussed by others – including Galileo – in the preceding decade, it was Torricelli who provided the breakthrough in understanding.

The prevailing view at the time was that air was weightless and did not exert any pressure on the mercury in the bowl. Instead, it was thought that the vacuum above the liquid in the barometer tube exerted a force of attraction that held the liquid suspended in the tube.

Torricelli challenged this view by proposing the converse argument. He asserted that air did have weight, and that the atmosphere exerted pressure on the mercury in the bowl which balanced the pressure exerted by the column of mercury. The vacuum above the mercury in the closed tube, in Torricelli’s opinion, exerted no attractive force and had no role in supporting the column of mercury in the tube*.

The assertion that air had weight, Torricelli realized, could be tested. In elevated places like mountains the reduced weight of the overlying atmosphere would exert less pressure, so the corresponding height of the mercury column in the barometer tube should be lower. It seems that Torricelli did not have the opportunity in his short life to do this experiment, but in the year following his death the experiment was carried out in France at the behest of the scientific philosopher Blaise Pascal (1623-1662).

*CarnotCycle wonders if Torricelli tilted the barometer tube and observed the disappearance of the space above the mercury – see diagram below. This would have shown that something other than a vacuum held the liquid suspended in the tube.

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The Torricelli experiment

In 1644 the French salon theorist Marin Mersenne (1588-1648) travelled to Italy where he learned of Torricelli’s barometer experiment. He brought news of the experiment back with him to Paris, where the young Blaise Pascal was a regular attendant at Mersenne’s salon meetings.

Puy de Dôme in south-central France, close to Clermont-Ferrand

Pascal had moved to Paris from his childhood home of Clermont-Ferrand. The 1,465 meter high Puy de Dôme was a familiar feature in the landscape he knew as a youngster, and it provided an ideal means of testing Torricelli’s thesis. Pascal’s brother-in-law Florin Périer lived in Clermont-Ferrand, and after some friendly persuasion, Périer ascended Puy de Dôme with a Torricellian barometer, taking measurements as he climbed.

The Torricelli experiment was conducted by Florin Périer on Puy de Dôme on Saturday 19 September 1648

At the base of the mountain, Périer recorded a mercury column height of 26 inches and 3½ lines. He then asked a colleague to observe this barometer throughout the day to see if any change occurred, while he set off with another barometer to climb the mountain. At the summit he recorded a mercury column height of 23 inches and 2 lines, substantially less than the measurement taken 1,465 meters below, where the barometer had remained steady.

The Puy de Dôme experiment provided convincing evidence that it was the weight of air, and thus atmospheric pressure, that balanced the weight of the mercury column.

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Measuring pressure

A 14th century shopkeeper weighing out sugar cubes using hand-held scales

When Florin Périer conducted the Torricelli experiment on Puy de Dôme in 1648, the measurements he recorded were the heights of mercury columns in barometer tubes. From these measurements, Blaise Pascal inferred a comparison of atmospheric pressures at the top and bottom of the mountain.

This experiment took place, we should remind ourselves, when Isaac Newton was only 5 years old and had not yet formulated his famous laws which gave concepts like mass, weight, force and pressure a systematic, mathematical foundation. In the pre-Newtonian world of Torricelli and Pascal, their thinking was based on the balancing of weights in the familiar sense of a shopkeeper’s scales. The weight of the mercury column in the barometer tube, which acted on the mercury in the bowl, was balanced by the weight of the air acting on the mercury in the bowl. Since the height of the mercury column was directly proportional to its weight, it was valid to use a length scale marked on the barometer tube to compute the weight of the air acting on the mercury in the bowl.

It is instructive to compare the language of Robert Boyle (1627-1691) and Isaac Newton (1643-1727) when discussing the barometer in the decades which followed. In the second edition of Boyle’s New Experiments Physico-Mechanicall of 1662 – which contains the first statement of Boyle’s Law – the word pressure appears frequently and has a meaning synonymous with weight. In Isaac Newton’s Principia of 1687, pressure is regarded as a manifestation of force. Boyle and Newton are thus speaking in essentially the same terms since according to Newtonian principles, weight is a force.

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Newtonian principles applied

The crucial advance in atmospheric science that Newton supplied in his Principia was the second law, which gave mathematical expression to force, and thus to weight and pressure, through the famous formula

The weight of a mercury column of cross-sectional area A and height h is

where ρ is the mass density of mercury and g is the acceleration due to gravity. The pressure exerted by the mercury column, which balances the atmospheric pressure, is

Thus P is directly proportional to h.

For a column of mercury 1 mm in height in a standard gravitational field (g = 9.80665 ms-2) at 273K, P is equal to 133.322 pascals. This is a unit of pressure called the torr. Pascal and Torricelli are thus both commemorated in units of pressure.

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A question of balance

Torricelli, Pascal and Boyle were in agreement with the proposition that air has weight. According to Newton’s interpretation the atmosphere possesses mass which is subject to gravitational acceleration, resulting in a downward force. This raises the question – Why doesn’t the sky fall down?

Since the sky is observed to remain aloft, there must exist a counteracting upward force. The vital clue as to the nature of this force was obtained on Pascal’s behalf by Florin Périer on Puy de Dôme in 1648 – namely that pressure decreases with height in the atmosphere.

A difference in pressure produces a force. In this way a parcel of air in a vertical column of cross-sectional area A exerts a force in the opposite direction to the gravitational force, as shown in the diagram.

At equilibrium, the forces are equal. Thus

where ρ is the density of the air.

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The decrease of temperature with altitude

A snow-capped Puy de Dôme

The appearance of snow above a certain height in elevated places provides plain evidence that temperature decreases with altitude, at least in that part of the atmosphere into which our earthly landscape protrudes. No doubt Torricelli, Pascal and other scientific philosophers of their time noticed this phenomenon and pondered upon it. But the explanation had to wait for another two centuries until the industrial revolution began, ushering in the age of steam and the associated science of thermodynamics.

The air in the troposphere, the lowest layer of the atmosphere where almost all weather phenomena occur, exhibits convection currents which continually transport air from lower regions to higher ones, and from higher regions to lower ones. When air rises it expands as the pressure decreases and so does work on the air around it. Thermodynamic principles dictate that this work requires the expenditure of heat, which has to come from within since air is a poor conductor and very little heat is transferred from the surroundings. As a result, rising air cools.

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Atmospheric convection processes fall within the province of the first law of thermodynamics, which can be expressed mathematically (see Appendix I) as

This equation states an energy conservation principle that applies to processes involving heat, work and internal energy. The atmospheric convection process is adiabatic meaning that no heat flows into or out of the system i.e. dQ = 0. Applying this constraint and using the combined gas equation to eliminate pressure p the above equation becomes

Integration yields

Converting from logarithms to numbers gives

Since by Mayer’s relation R = CP – CV

where γ = CP / CV. Using the combined gas equation to substitute V, the above equation can be rendered (with the help of γ√) as

Applying logarithmic differentiation gives

Assuming hydrostatic equilibrium, dp can be substituted giving

Since ρ = m/(RT/p) the above becomes

This adiabatic convection equation gives the rate at which the temperature of dry air falls with increasing altitude. Taking the following values: γ = 1.4 (dimensionless) ; R = 8.314 kgm2s-2 K-1 mol-1 ; m = 0.0288 kg mol-1 ; g = 9.80665 ms-2 gives

At the top of Puy de Dôme (1465 meters), dry air will be 14°C cooler than at the base of the mountain. This explains why snow can appear on the summit while the grass is still growing on the lower slopes.

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Appendix I

Benoît Émile Clapeyron (1799-1864)

In 1834, more than a century after Newton’s death, the French physicist and engineer Émile Clapeyron wrote a monograph entitled Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the Motive Power of Heat). It contains the first appearance in print of the ideal gas equation, which combines the gas law of Boyle-Mariotte (PV)T = k with that of Gay-Lussac (V/T)P = k. Clapeyron wrote it in the form

where R is a constant and the sum of the terms in parentheses can be regarded as the thermodynamic temperature.

Sixteen years later in 1850, the German physicist Rudolf Clausius wrote a monograph on the same subject entitled Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus fuer die Wärmelehre selbst ableiten lassen (On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat). Seeking an analytical expression of the principle that a certain amount of work necessitates the expenditure of a proportional quantity of heat, he arrived at the following differential equation in the case of an ideal gas

where Q is the heat expended, U is an arbitrary function of temperature and volume, and A is the mechanical equivalent of heat. Earlier in his paper Clausius had represented Clapeyron’s combined statement of the laws of Boyle-Mariotte and Gay-Lussac as pv = R(a + t) so he recognized the right-hand term as corresponding to pdv, the external work done during the change

We know this equation today as an expression of the first law of thermodynamics, where U is the internal energy of the system under consideration.

U is a function of T and V so we may write the partial differential equation

Since U for an ideal gas is independent of volume and dU/dT is the heat capacity at constant volume CV, the first law for an ideal gas takes on the form

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P Mander January 2018