Torricelli e la fondazione della chimica fisica

La presente cronologia evidenzia come l’invenzione del barometro da parte di Torricelli e la relativa rivoluzionaria intuizione abbiano portato alla scoperta della Legge di Boyle, formula fondamentale della chimica fisica

1643

In Italia, Evangelista Torricelli inventa il barometro che, etimologicamente, significa “misura del peso”. È il primo ad affermare che l’atmosfera ha un peso e che tale peso controbilancia esattamente il peso della colonna di mercurio nel tubo barometrico. In una lettera a un collega, scrive la famosa frase:

“Noi viviamo sommersi nel fondo d’un pelago d’aria”.

Basandosi su tale affermazione, prevede che l’altezza del mercurio all’interno del tubo sarebbe risultata inferiore a un’altitudine superiore, poiché il peso dell’aria che agisce su di esso sarebbe stato inferiore.

1644

La notizia dell’invenzione di Torricelli, e del pensiero rivoluzionario che l’ha accompagnata, arriva in Francia. Blaise Pascal, talentuoso matematico e fisico allora solo 21enne, esegue una verifica della previsione di Torricelli misurando l’altezza del barometro ai piedi e in vetta al vulcano Puy de Dôme, alto 1465 metri, situato nella zona centro-meridionale della Francia.

La misura di 23 unità in vetta e di 26 unità ai piedi del vulcano confermano la previsione di Torricelli e forniscono convincenti prove a favore della teoria che l’atmosfera abbia un peso, esercitando una pressione che controbilancia il peso del mercurio nel tubo barometrico.

1647

Torricelli muore a Firenze, probabilmente di tifo, alla giovane età di 39 anni. Viene sepolto nella Basilica di San Lorenzo nel cui chiostro si trova una lapide commemorativa che lo ricorda.

1653

Il barometro di Torricelli viene discusso in Inghilterra nei centri scientifici di Londra, Oxford e Cambridge. Henry Power, uno studente di Cambridge, verifica l’osservazione di Pascal e inizia a studiare le caratteristiche di espansione e compressione dell’aria.

1661

Photo acknowledgement: Lee Pilkington

Nell’aprile del 1661, Henry Power e l’aristocratico inglese Richard Towneley conducono un esperimento a Pendle Hill, nel nord dell’Inghilterra. Dopo aver introdotto una quantità d’aria sopra al mercurio in un tubo torricelliano ne misurano il volume; utilizzando quindi il tubo come barometro, misurano la pressione dell’aria. Salgono poi sulla collina, alta 557 metri, e sulla vetta, ripetono le misurazioni di volume e pressione. Come previsto, si verifica una diminuzione della pressione, accompagnata da un corrispondente aumento del volume dell’aria nel tubo.

Tale esperimento permette loro di dedurre una semplice relazione numerica tra la pressione e il volume dell’aria nel tubo

dove p₁ e v₁ rappresentano le misurazioni della pressione e del volume dell’aria nel tubo ai piedi della collina e p₂ e v₂ le corrispondenti misurazioni in vetta. Tale formula, caposaldo della chimica fisica, viene comunicata e pubblicata in Inghilterra nel 1662 dal filosofo naturalista anglo-irlandese Robert Boyle e diventa nota come la Legge di Boyle.

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P Mander maggio 2022

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The Ozone Experiment

I have always had a fondness for classical experiments that revealed fundamental things about the particulate nature of our world. Examples that spring to mind include JJ Thomson’s cathode ray tube experiment (1897), Robert Millikan’s oil drop experiment (1909), and CTR Wilson’s cloud chamber (1912). The particles of interest in these cases were subatomic, but during this era of discovery there was another pioneering experiment that focused on molecules and their chemical reactivity. The insight this experiment provided was important, but the curious fact is that relatively few people have ever heard of it.

So to resurrect this largely forgotten piece of scientific history, CarnotCycle here tells the story of the Ozone Experiment conducted by the Hon. Robert John Strutt FRS at Imperial College of Science, South Kensington, London in 1912.

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The experimenter

RJ Strutt (1875-1947) photographed in 1923

The Honorable Robert John Strutt, 4th Baron Rayleigh, might be an unfamiliar name to some of you. But you will undoubtedly have heard of his father, Lord Rayleigh of Rayleigh scattering fame. Where his father led, Robert John followed: first as a research student at the Cavendish Laboratory in Cambridge where his father had been Cavendish professor, and then at Imperial College of Science in South Kensington, London where he followed up his father’s eponymous work on light scattering.

But Robert John did some interesting work of his own. For one thing, he was the first to prove the existence of ozone in the upper atmosphere, and for another he studied the effect of electrical discharges in gases. Interestingly it was a combination of these two things – ozone produced in an electrical discharge tube – that formed the basis of Strutt’s groundbreaking 1912 experiment.

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The experiment

Here is the apparatus that Strutt employed in his experiment. As the arrows indicate, air enters from the right via stopcock a, where the pressure is significantly reduced by the action of the air pump at left. Low-pressure air then passes through the discharge tube b, where ozone is formed from oxygen according to the reaction

The air, containing ozone at a few percent, enters chamber c where it encounters a silver gauze partition d, mounted between two mica discs e in each of which there is a hole 2 millimeters in diameter. A sealed-in glass funnel g supports the mica discs as shown. As the air passes the gauze, ozone reacts with the silver in what is thought to be an alternating cycle of oxidation and reduction which destroys the ozone while constantly refreshing the silver

The chambers on either side of the gauze partition are connected by tubes f, either of which could be put into communication with a McLeod pressure gauge. The rate of air intake was measured by drawing in air at atmospheric pressure from a graduated vessel standing over water. From this data, combined with the McLeod pressure gauge measurements, the volume v of the low-pressure air stream passing the gauze per second could be calculated.

So to recap, in Strutt’s steady-state experiment, air passes through the apparatus at a constant rate as ozone is generated in the discharge tube and destroyed by the silver gauze. The question then arises – What proportion of the ozone is destroyed as it passes the gauze?

This brings us to the luminous aspect of the ozone experiment, which enabled Strutt to provide an answer.

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The green glow

The conversion of oxygen into its allotrope ozone was not the only reaction taking place in the discharge tube of Strutt’s apparatus. There was also a reaction between nitrogen and oxygen – known to occur in lightning strikes – which produces nitrogen(II) oxide

Now it just so happens that nitrogen(II) oxide and ozone react in the gas phase to produce activated nitrogen(IV) dioxide, which exhibits chemiluminescence in the form of a green glow as it returns to its ground state

This was a crucial factor in Strutt’s experiment. The air flowing into the chamber c was glowing green due to the above reactions taking place in the gas phase. But as the flow passed the silver gauze, ozone molecules were destroyed with the result that the green glow was weaker in the left-hand chamber compared with the right-hand chamber.

By adjusting the rate of air flow through the apparatus, Strutt could engineer a steady state in which the green glow was just extinguished by the silver gauze – in other words he could find the flow rate at which all of the ozone molecules were destroyed by the silver/silver oxide of the gauze partition.

[To allay doubts, Strutt introduced ozone gas downstream of the gauze where the green glow had been extinguished. The glow was restored.]

Strutt was now in a position to interpret the experiment from a new and pioneering perspective – his 1912 paper was one of the very first to consider a chemical reaction in the context of molecular statistics.

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The analysis

In terms of chemical process, Strutt’s steady-state experiment was unremarkable. Air flowed through the apparatus and the ozone generated in the discharge tube was destroyed by the silver gauze. The novel feature was in the analysis, where Strutt applied both classical physics and the kinetic theory of gases to calculate the ratio of the mass of ozone impinging upon the silver gauze in a second to the mass passing the gauze in a second.

As mentioned above, Strutt could compute the volume v of the stream passing through the apparatus in a second, so the mass of ozone passing the gauze in a second was simply ρv, where ρ is the density of ozone in the stream as it arrives at the gauze.

In his paper, Strutt states a formula for calculating the mass of ozone impinging upon the silver surface in a second

without showing the steps by which he reached it. These steps are salient to the analysis, so I include the following elucidation due to CN Hinshelwood* in which u_rms is the root mean square velocity (i.e. the average velocity, with units taken to be cm/s) of the gas molecules:

Suppose we have a solid surface of unit area exposed to the bombardment of gas molecules. Approximately one-sixth of the total number of molecules may be regarded as moving in the direction of the surface with the average velocity. In one second all those within distance u_rms could reach and strike the surface, unless turned back by a collision with another molecule, but for every one so turned back, another, originally leaving the surface, is sent back to it. Thus the number of molecules striking the surface in a second is equal to one-sixth of the number contained in a prism of unit base and height u_rms. This number is 1/6.n’.u_rms,, n’ being the number of molecules in 1 cm^3. Thus the mass of gas impinging upon the surface per second is

A more precise investigation allowing for the unequal speeds of different molecules shows that the factor 1/6 should really be

We therefore arrive at the result that the mass of gas striking an area A in one second is

*CN Hinshelwood, The Kinetics of Chemical Change in Gaseous Systems, 2nd Ed. (1929), Clarendon Press

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The ratio

Strutt takes the above formula for the mass of ozone impinging on the gauze per second and divides it by the formula for the mass of ozone passing the gauze per second, ρv. This operation cancels out the unknown value of ρ, giving

The values of v (200 cm³s^-1) and A (0.0369 cm²) were obtained by Strutt using direct measurements, while u_rms for ozone molecules is simply stated without mentioning that it is necessarily computed from the fundamental kinetic equation

If n is Avogadro’s number, v is the molar volume and pv = RT, whence

where M is the molar mass. The u_rms figure Strutt gives for ozone is 3.75 × 10⁴; typically for the time he neglects to state the units which are presumed to be cm/s. This velocity seems a little low, implying a temperature of 270.6K for the air flow in his apparatus. But then again, the pressure dropped significantly at the stopcock so in all likelihood there would have been some Joule-Thomson cooling.

Inserting the values for A, v and u_rms in the ratio expression gives

Since we can interpret mass in terms of the number of ozone molecules, the ratio expresses the number of collisions to the number of molecules passing, or the average number of times each ozone molecule must strike the silver surface before it passes.

As the experiment is arranged so that no ozone molecules pass the silver gauze, the ratio must represent the average number of collisions that an ozone molecule makes with the silver surface before it is destroyed.

The 1.6 ratio reveals the astonishing fact that practically every ozone molecule which strikes the silver (oxide) surface is destroyed. To a chemical engineer that is a hugely important piece of information, which amply illustrates the value of applying kinetic theory to chemical reactivity.

The application of analogous calculations to the passage of gas streams over solid catalysts in industrial processes is obvious. All of which makes it even more curious that Robert John Strutt’s apparatus, and the pioneering work he did with it, is not better known.

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Further reading

Hon. RJ Strutt, The Molecular Statistics of some Chemical Actions (1912)
The principal source for this blog post.
http://rspa.royalsocietypublishing.org/content/royprsa/87/596/302.full.pdf

CTR Wilson, On an expansion apparatus for making visible the tracks of ionizing particles in gases and some results obtained by its use (1912)
The Cloud Chamber – a truly historic piece of apparatus and one of my favorites. This paper was published in September 1912, just a month before Strutt’s paper.
http://rspa.royalsocietypublishing.org/content/royprsa/87/595/277.full.pdf

P Mander August 2016

Thomas Graham and the path to uranium enrichment

The statue of Thomas Graham, sculpted by William Brodie in 1872

On the south-eastern corner of Glasgow’s George Square is a fine statue of Thomas Graham (1805-1869). Born and raised in the city, he became a chemistry student at the University of Glasgow and graduated there in 1826. At some point in his studies he happened to read about an observation made by the German chemist Johann Döbereiner (1780-1849) that hydrogen gas leaked out from a crack in a glass bottle faster than air leaked in. It was this simple fact that set Thomas Graham on the path to scientific fame. But before we continue, a few words about Johann Döbereiner.

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Döbereiner’s lamp – the first lighter

Johann Döbereiner, professor of chemistry at the University of Jena, invented this amazing piece of apparatus in 1823, while Thomas Graham was still an undergraduate student in Scotland. It consists of a glass container (a) filled with dilute sulfuric acid and inside it an inverted cup (b) in which is suspended a lump of zinc metal (c,d). When the tap (e) is opened, the acid enters the cup and reacts with the zinc, producing hydrogen gas which flows out of a tube (f) and onto a piece of platinum gauze (g). Now here is the interesting part. The gauze catalyzes the reaction of hydrogen with atmospheric oxygen, producing a lot of heat in the process. The platinum gauze gets red hot and ignites the hydrogen flowing out of the tube, producing a handy flame for lighting candles, cigars, etc. In the days before matches, this gadget was a godsend and became a commercial hit with thousands being mass produced in a wonderful range of styles. A YouTube demonstration of Döbereiner’s Lamp can be seen here.

In a paper published in 1823, Döbereiner recorded the observation that hydrogen stored in a glass jar over water leaked out from a crack in the glass much faster than the surrounding air leaked in, causing the level of the water to rise significantly. This was the trigger for Graham’s research into the phenomenon of diffusion, during which he discovered not only an important quantitative relation between diffusion and gas density but also a means by which the separation of mixed gases could be achieved.

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Graham’s experiments on effusion

Taking his cue from Döbereiner’s leaking glass jar, Graham developed apparatus by which he could study the rate of escape of a contained gas through a small hole in a piece of platinum foil. This particular kind of diffusion, where the flux is restricted to a tiny orifice between one gaseous environment and another, is called effusion.

The rates of effusion of two gases can be compared using the apparatus illustrated. The first gas is introduced through the three-way tap C to fill the entire tube B. The tap is closed and the gas is then allowed to effuse through the hole in the platinum foil A. The time taken for the liquid level to rise from X to Y is recorded as the gas escapes into the atmosphere. The experiment is then repeated with the second gas. If the recorded times are t₁ for the first gas and t₂ for the second, the rates of effusion are in the ratio t₂/t₁.

Using this method Graham discovered that the rate of escape of a gas was inversely related to its density: for example hydrogen escaped 4 times faster than oxygen. Given that the density of oxygen is 16 times that of hydrogen, the nature of the inverse relation suggested itself and was confirmed by comparisons with other gases.

In 1829, Graham submitted an internal research paper in which he recorded his experimentally determined relation between the effusion rates of gases and their densities

Graham also experimented with binary gases, and noted that the greater rate of escape of the lighter gas made it possible to achieve a measure of mechanical separation by this means.

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Graham’s Law

By 1831 Graham had recognized that the comparative rates of effusion of two gases into the atmosphere could equally be applied to the diffusion of two gases in contact.

On Monday 19th December 1831, Graham read a paper before the Royal Society of Edinburgh in which he stated his eponymous square root law. This paper was published in the Philosophical Magazine in 1833 while he was professor of chemistry at Anderson’s College in Glasgow. Four years later he moved to London to became professor of chemistry at University College, where in 1848 he embraced Avogadro’s hypothesis by stating that the rate of diffusion of a gas is inversely proportional to the square root of its molecular weight.

Hence if the rates of diffusion of two gases are known and the molar mass of one is known, the molar mass of the other can be calculated from the relation

In 1910 the French chemist André-Louis Debièrne, a close associate of Pierre and Marie Curie, used this relation to calculate the molecular weight of Radon gas. (Trivial Fact: Debièrne was one of those fortunate Frenchmen to be born on France’s national day, Le Quatorze Juillet – 14 July. So every year his birthday was a national holiday :-)

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Thermodynamics and Kinetics

In the first half of the 19th century, the understanding of gases rested on the gas laws which Sadi Carnot’s compatriot Émile Clapeyron synthesized into the ideal gas law pv = RT in 1834. Meanwhile Avogadro’s hypothesis of 1811 laid the foundations of molecular theory from which developed the idea that gases consisted of large numbers of very small perfectly elastic particles moving in all directions through largely empty space.

These two strands of thought came together in the notion that gas pressure could be attributed to the random impacts of molecules on the walls of the containing vessel. In Germany Rudolf Clausius produced a paper in 1857 in which he derived a formula connecting pressure p and volume v in a system of n gas molecules of mass m moving with individual velocity c

In this equation we see the meeting of thermodynamics on the left with kinetic theory on the right. And it points up a feature of thermodynamic expressions that commonly escapes notice. We are taught that in classical thermodynamics, the time dimension is absent as a unit of measure although entropy is sometimes cast in this role as “the arrow of time”. But the fact is that time is very much present when you apply dimensional analysis.

Pressure is force per unit area and has dimensions ML^-1T^-2. And there is the time dimension T, in the definition of the thermodynamic intensive variable pressure. This is what enabled Clausius to equate a time-dependent expression on the right with a seemingly time-independent one on the left.

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Kinetic Theory and Graham’s Law

Since density ρ is mass per unit volume, the above equation can be written

If the rate of effusion/diffusion of a gas is taken to be proportional to the root mean square velocity of the gas molecules, then at constant pressure

which is the first statement of Graham’s law.

For 1 mole of gas, the aforementioned Clausius equation can be written as

where V is the molar volume, R is the gas constant, T is the temperature and N is the Avogadro number. Since the product of the Avogadro number N and the molecular mass m is the molar mass M, it follows that

Again, if the rate of effusion/diffusion of a gas is taken to be proportional to the root mean square velocity of the gas molecules, then at constant temperature

which is the second statement of Graham’s law.

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Graham’s Law and uranium enrichment

Plaque marking the site of the K-25 plant at Oak Ridge Tennessee.

Back in 1829 Thomas Graham noted from his effusion experiments on binary gases that a measure of mechanical separation could be achieved by this means. Over a century later, that observation was of crucial importance to the scientists engaged in the Manhattan Project which produced the first nuclear weapons during WW2.

To produce an atomic bomb required a considerable quantity of the fissile uranium isotope ²³⁵U. The problem was that this isotope makes up only about 0.7% of naturally occurring uranium. Substantial enrichment was necessary, and this was achieved in part by employing gaseous effusion of uranium hexafluoride UF₆. Since fluorine has a single naturally occurring isotope, the difference in weights of ²³⁵UF₆ and ²³⁸UF₆ is due solely to the difference in weights of the uranium isotopes and so a degree of separation can be achieved.

The optimal effusion rate quotient (√ 352/349) is only 1.0043 so it was clear to the Manhattan Project engineers that a large number of separation steps would be necessary to obtain sufficient enrichment, and this was done at Oak Ridge Tennessee with the construction of the K-25 plant which ultimately consisted of 2,892 stages.

In more recent times, the development of the Zippe-type centrifuge made the gas diffusion method of ²³⁵U isotope separation redundant and led to the closure of the K-25 plant in 2013. The Zippe-type centrifuge is considerably more energy-efficient than gaseous diffusion, has less gaseous material in circulation during separation, and takes up less space.

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Further reading

Thomas Graham biography at encyclopedia.com

link: https://www.encyclopedia.com/people/science-and-technology/chemistry-biographies/thomas-graham

Thomas Graham Contributions to diffusion of gases and liquids, colloids, dialysis and osmosis Jaime Wisniak, 2013 (contains comprehensive references to Graham’s published work)

link: https://www.sciencedirect.com/science/article/pii/S0187893X13725217#bib0005

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P Mander, February 2021

Konovalov’s Thought Experiment

Dmitiri Konovalov (1856-1929) was a Russian chemist who made important contributions to the theory of solutions. He studied the vapor pressure of solutions of liquids in liquids and in 1884 published a book on the subject which gave a scientific foundation to the distillation of solutions and led to the development of industrial distillation processes.

On the subject of partially miscible liquids forming conjugate solutions, Konovalov in 1881 established the following fact: “If two liquid solutions are in equilibrium with each other, their vapor pressures, and the partial pressures of the components in the vapor, are equal.”

J. Willard Gibbs in America had already developed the concept of chemical potential to explain the behavior of coexistent phases in his monumental treatise On the Equilibrium of Heterogeneous Substances (1875-1878). Konovalov was unaware of this work, and independently found a proof on the basis of this astutely reasoned thought experiment:

«Consider Figure 77 shown above. Two liquid layers α and β in coexistent equilibrium are contained in a ring-shaped tube, and above them is vapor. If the pressure of either component in the vapor were greater over α than over β, diffusion of vapor would cause that part lying over β to have a higher partial pressure of the given component than is compatible with equilibrium. Condensation occurs and β is enriched in the specified component. By reason of the changed composition of β however, the equilibrium across the interface of the liquid layers is disturbed and the component deposited by the vapor will pass into the liquid α. The whole process now commences anew and the result is a never-ending circulation of matter round the tube i.e. a perpetual motion, which is impossible. Hence the partial pressures of both components are equal over α and β and therefore also their sum i.e. the total vapor pressure.»

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Equivalence of vapor pressure and chemical potential

Konovalov showed that the condition of equilibrium in coexistent phases was equality of vapor pressure p for each component. This is consistent with the concept of ‘generalized forces’, a set of intensive variables which drive a thermodynamic system spontaneously from one state to another in the direction of equilibrium. Vapor pressure is one such variable, and chemical potential is another. Hence Gibbs showed that chemical potential μ is a driver of compositional change between coexistent phases and that equilibrium is reached when the chemical potential of each component in each phase is equal. In shorthand the equilibrium position for partially miscible liquids containing components 1 and 2 in coexistent phases α, β and vapor can be stated as:

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P Mander, July 2020

Mauritius, Malaria and Mathematics

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

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

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

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?

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