mau01

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.

– – – –

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.

– – – –

Genesis

mau02

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.

– – – –

Statistics

mau03

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.

– – – –

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)

– – – –

Ratios

mau10

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.

– – – –

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.

– – – –

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?

mau12

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.

—-

mau13

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.

– – – –

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

– – – –

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

– – – –

P Mander August 2016

Peter Mander, author and editor of CarnotCycle

“Thank you to everyone who has visited this blog since its inception in August 2012. CarnotCycle’s country statistics show that thermodynamics interests many, many people. They come to this blog from all over the world, and they keep coming.

It’s wonderful to see all this activity, but perhaps not so surprising. After all, thermodynamics is the branch of science that can help us to use energy wisely and to understand and manage the effects of global warming on our atmosphere, our ice caps and our oceans.

Today’s students of physics and physical chemistry, for whom thermodynamics is a core subject, will be at the forefront of future efforts to keep this planet habitable. They must be knowledgeable and competent, and CarnotCycle is one of the many free resources to help them achieve that aim.”

Exploring the mutual solubility of phenol and water at the Faculty of Pharmacy, National University of Malaysia.

The phenol-water system is a well-studied example of what physical chemists call partially miscible liquids. The extent of miscibility is determined by temperature, as can be seen from the graph below. The inverted U-shaped curve can be regarded as made up of two halves, the one to the left being the solubility curve of phenol in water and the other the solubility curve of water in phenol. The curves meet at the temperature (66°C) where the saturated solutions of water in phenol, and phenol in water, have the same composition.

The thermodynamic forces driving the behavior of the phenol-water system are first visible in the upwardly convex mutual solubility curve, showing that the enthalpy of solution (ΔHs) in the saturated solution is positive i.e. that the system absorbs heat and so solubility increases with temperature in accordance with Le Châtelier’s Principle.

More rigorously, one can ascertain whether solubility of the minor component increases or decreases with temperature by computing

where s is the solubility expressed in mole fraction units, f2 is the corresponding activity coefficient and Ts is the temperature at which the solution is saturated.

The line of the umbrella curve charts the variation in composition of saturated solutions – phenol in water and water in phenol – with temperature. The area to the left of the curve represents unsaturated solutions of phenol in water and the area to the right represents unsaturated solutions of water in phenol, while the area above the curve represents solutions of phenol and water that are fully miscible i.e. miscible in all proportions.

But what about the area inside the curve, which is beyond the saturation limits of water in phenol and phenol in water? In this region, the system exhibits its most striking characteristic – it divides into two coexistent phases, the upper phase being a saturated solution of phenol in water and the lower phase a saturated solution of water in phenol. The curious feature of these phases is that for a given temperature their composition is fixed even though the total amounts of phenol and water composing them may vary.

To analyze how this comes about, consider the dotted line on the diagram below, which represents the composition of the phenol-water system at 50°C.

Starting with a system which consists of water only we gradually dissolve phenol in it, maintaining the temperature at 50°C, until we reach the point Y on the curve at which the phenol-in-water solution becomes saturated.

Now imagine adding to the saturated solution a small additional amount of phenol. It cannot dissolve in the solution and therefore creates a separate coexistent phase. Since this newly-formed phenol phase contains no water, the chemical potential of water in the solution provides the driving force for water to pass from the aqueous solution into the phenol phase. This cannot happen on its own however since water passing out of a phenol-saturated solution would cause the solution to become supersaturated. This would constitute change from a stable state to an unstable state which cannot occur spontaneously.

What can be postulated to occur is that the movement of water from the solution into the phenol phase simultaneously lowers the chemical potential of phenol in that coexistent phase, allowing phenol to move with the water in such proportion that the phenol-in-water phase remains saturated – as it must do since the temperature remains constant. In other words, saturated solution passes spontaneously from the aqueous phase into the phenol phase, diminishing the amount of the former and increasing the amount of the latter. Because water is the major component of the phenol-in-water phase, this bulk movement will continuously increase the proportion of water in the coexistent water-in-phenol phase until it reaches the saturation point whose composition is given by point Z on the mutual solubility curve.

In terms of chemical potential in the two-phase system, equilibrium at a given temperature will be reached when:

upper phase = sat. soln. of phenol in water
lower phase = sat. soln. of water in phenol

– – – –

Applying the Phase Rule

F = C – P + 2

First derived by the American mathematical physicist J. Willard Gibbs (1839-1903), the phase rule computes the number of system variables F which can be independently varied for a system of C components and P phases in a state of thermodynamic equilibrium.

Applying the rule to the 1 Phase region of the phenol-water system, F = 2 – 1 + 2 = 3 where the system variables are temperature, pressure and composition. So for a chosen temperature and pressure, e.g. atmospheric pressure, the composition of the phase can also be varied.

In the 2 Phase region of the phenol-water system, F = 2 – 2 + 2 = 2. So for a chosen temperature and pressure, e.g. atmospheric pressure, the compositions of the two phases are invariant.

In the diagram below, the compositions of the upper and lower phases remain invariant along the line joining Y and Z, the pressure being atmospheric and the temperature being maintained at 50°C. As we have seen, the upper layer will be a saturated solution of phenol in water where the point Y determines the % weight of phenol (= 11%). Correspondingly, the lower layer will be a saturated solution of water in phenol where the point Z determines the % weight of phenol (= 63%).

– – – –

Relationship between X, Y and Z

If a mixture of phenol and water is prepared containing X% by weight of phenol where X is between the points Y and Z as indicated on the above diagram, the mixture will form two phases whose phenol content at equilibrium is Y% by weight in the upper phase and Z% by weight in the lower phase.

Let the mass of the upper phase be M1 and that of the lower phase be M2. The mass of phenol in these two phases is therefore Y% of M1 + Z% of M2. Conservation of mass dictates that this must also equal X% of M1 + M2. Therefore

The relative masses of the upper and lower phases change according to the position of X along the line Y-Z. As X approaches Y the upper phase increases as the lower phase diminishes, becoming one phase of saturated phenol-in-water at point Y. Conversely as X approaches Z the lower phase increases as the upper phase diminishes, becoming one phase of saturated water-in-phenol at point Z.

– – – –

Further reading

Logan S.R. Journal of Chemical Education 1998

This paper uses the well-known thermodynamic equation ΔG = ΔH + TΔS as a theoretical basis for determining the circumstances under which spontaneous mixing occurs when two partially miscible liquids are brought together at constant temperature and pressure.

The approach involves the construction of equations for estimating both the enthalpy and entropy of mixing in terms of the mole fraction x of one component, and graphing the change in Gibbs free energy ΔG against x to determine the position of any minimum/minima. The paper goes on to examine the criteria for the existence of two phases on the basis of determining the circumstances under which a system of two phases will have a combined ΔG value that is lower than the corresponding ΔG for a single phase.

The conclusion is reached that the assessment of ΔG on mixing two liquids can provide a qualitative explanation of some of the phenomena observed in relation to the miscibility of two liquids.

The paper is available from the link below (pay to view)
https://pubs.acs.org/doi/abs/10.1021/ed075p339?src=recsys&#

– – – –

P Mander November 2018

wil01

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.

– – – –

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.

wil06

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

– – – –

Inversion of sucrose

wil07

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.

wil08

A Laurent polarimeter from around 1900.

– – – –

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.

– – – –

A digression on half-life

wil09

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

– – – –

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.

wil15

Heidelberg, Germany in the 1850s

– – – –

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

CarnotCycle is a thermodynamics blog but occasionally it takes five for recreational purposes

Poker dice is played with five dice each with playing card images – A K Q J 10 9 – on the six faces. There are a total of 6 × 6 × 6 × 6 × 6 = 7776 outcomes from throwing these dice, of which 94% are scoring hands and only 6% are busts.

Here are the number of outcomes for each hand and their approximate probabilities

And here is the data presented as a pie chart

poker03

The percentage share of 5 of a kind (0.08%) is omitted due to its small size

A noticeable feature of the data is that the number of outcomes for 1 pair is exactly twice that for 2 pairs, and likewise the number of outcomes for Full house is exactly twice that for 4 of a kind. But when outcomes are calculated in the conventional way, it is not obvious why this is so.

Taking the first case, the conventional calculation runs as follows:

1 pair – There are 6C1 ways to choose which number will be a pair and 5C2 ways to choose which of five dice will be a pair, then there are 5C3 × 3! ways to choose the remaining three dice

6C1 × 5C2 × 5C3 × 3! = 3600 outcomes

2 pairs – There are 6C2 ways to choose which two numbers will be pairs, 5C2 ways to choose which of five dice will be the first pair and 3C2 ways to choose which of three dice will be the second pair. Then there are 4C1 ways to choose the last die

6C2 × 5C2 × 3C2 × 4C1 = 1800 outcomes

Conventional calculation gives no obvious indication of why there are twice as many outcomes for a 1-pair hand than a 2-pair hand.

But there is an alternative method of calculation which does make the difference clear.

– – – –

A different approach

Instead of starting with component parts, consider the hand as a whole and count the number (n) of different faces on view. The number of ways to choose n from six faces is computed by calculating 6Cn. Now multiply this by the number of distinguishable ways of grouping the faces, which is given by n!/s! where s is the number of face groups sharing the same size. The number of combinations for the hand is 6Cn × n!/s!

Since the dice are independent variables, each combination is subject to permutation taking into account the number of indistinguishable dice in each of the n groups according to the general formula n!/n1! n2! . . nr!

The total number of outcomes for any poker dice hand is therefore

It is easy to see from the table why there are twice as many outcomes for a 1-pair hand than a 2-pair hand. The number of combinations (6Cn × n!/s!) is the same in both cases but there are twice as many permutations for 1 pair. Similarly with Full house and 4 of a kind, the number of combinations is the same but there are twice as many permutations for Full house.

– – – –

Note that

is reducible to

– – – –

P Mander April 2018