Taking a break from studying thermodynamics: 17th century soldiers playing at dice

In 17th century France, dice games were a popular and fashionable habit. All kinds of people played at dice – soldiers, sailors, socialites, aristocrats, celebrities … and professional gamblers. Every era in history has featured this latter group, often colourful characters with sharp minds living off their wits.

Such an individual was Antoine Gombaud (1607-1684), who had adopted the high-flown title of Chevalier de Méré and was known as a flambuoyant big spender in gambling circles. He also had good connections with Renaissance intellectuals, and was himself a notable Salon theorist.

Professional gamblers have one overriding aim in life, which is to win. This is easier said than done however as gambling necessarily involves games of chance, and chance is a fickle creature. So the professional gambler has to proceed with caution until he finds a wager whose odds are in his favor. Then he brings big money to the table, plays long and hard, and walks away a richer man.

Calculating success

This was how Gombaud operated. But unlike most other gamblers, he did not rely solely on experience to show him which bets were favorable. He had an analytical turn of mind, and had started to work out on the basis of mathematical principle whether a certain game had favorable odds.

He first applied his thinking to a popular dice game in which players wagered on a six appearing in four throws of a die. He correctly reasoned that since each number on a six-sided die was equally likely to occur, the chance of getting a six on a single throw must be 1/6. He then considered the chance of getting a six if a die were thrown four times instead of once. He reasoned that the chance of success would be four times greater since each throw represented a separate opportunity for a six to occur, and he calculated that chance as 4 x 1/6 = 2/3. In other words, the odds of winning were favorable (i.e. >1/2).

Long before the law of large numbers was formulated, Gombaud seems to have intuitively understood that these favorable odds meant that although the outcome of an individual game could not be predicted, success would be assured if enough games were played.

Gombaud made piles of money out of this game, thereby cementing belief in his method of mathematical analysis as a means of identifying a winning bet. Buoyed by this success he extended the same reasoning to another game where he calculated that the odds of winning were favorable. But an unpleasant surprise was in store.

Unexpected losses

Gombaud’s new focus of attention was a dice game in which players wagered on getting a double six in twenty four throws of two dice. He correctly reasoned that the chance of getting a double six in a single throw of two dice was 1/36. Then applying his formula he calculated the chance of success as 24 x 1/36 = 2/3, the same favorable odds as in the previous game.

Emboldened by this analysis, Gombaud brought a stack of money to the dice table to wager on getting double sixes. But his expectations did not materialise; in fact the more he played this game, the more his losses mounted. Gombaud simply could not understand it. Both games had exactly the same favorable odds. So why did he win at one and lose at the other?

Desperate for an explanation of his losses, he wrote in 1654 to one the foremost thinkers of his time, Blaise Pascal (1623-1662), who in turn shared the news of “De Méré’s paradox” with the profoundly talented amateur mathematician Pierre Fermat (1607-1665). The two began a legendary correspondence, out of which the theory of probability was born and the paradox was solved.

Fallacious reasoning

The work of Pascal and Fermat revealed Gombaud’s mistake in thinking that the chances of success with n throws could be calculated by multiplying the chance of success for a single throw by n.

Taking the first game as an example, Gombaud thought that after two throws of the die the chance of success doubled from 1/6 to 1/3. But a chart of all 36 possible outcomes shows that the total number of favorable outcomes (shown in gold) is not 12, but 11.

Pascal and Fermat no doubt saw that the ratio of favorable outcomes to total outcomes after two throws could be written as

and after three throws as

and so on. Since 5/6 was the chance of failure in a single throw, and the exponent was the number of throws, the formula for the chance of getting at least one success in n throws could be generalised as

where q is the chance of failure in a single throw. This was the correct formula for computing whether the odds were favorable or not. Contrast this with the formula Gombaud used

It is instructive to compare Gombaud’s incorrect formula with the correct one for the first game

and the second game

The correct figures in the final column make it clear why Gombaud won at the first game and lost at the second. They also show what a knife-edge situation it was. If the second game had been played with just one more throw (n=25, 1-q^n = 0.506) Gombaud would have won both games. There would have been no paradox to explain, and the genius minds of Pascal and Fermat might never have been applied to founding probability theory!

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Probability profiles

Game 1

Game 2

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P Mander November 2017

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This prototype displays temperature, relative humidity, dew point temperature and absolute humidity

As shown in previous posts on the CarnotCycle blog, it is possible to compute dew point temperature and absolute humidity (defined as water vapor density in g/m^3) from ambient temperature and relative humidity. This adds value to the output of RH&T sensors like the DHT22 pictured above, and extends the range of useful parameters that can be displayed or toggled on temperature-humidity gauges employing these sensors.

Meteorological opinion* suggests that dew point temperature is a more dependable parameter than relative humidity for assessing climate comfort especially during summer, while absolute humidity quantifies water vapor in terms of mass per unit volume. In effect this added parameter turns an ordinary temperature-humidity gauge into a gas analyzer.

*https://www.weather.gov/arx/why_dewpoint_vs_humidity

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Hardware

I used an Arduino Uno microprocessor and a wired DHT22 sensor with data output to a 16×2 liquid crystal display. Circuit components are uncomplicated: a 10 kΩ potentiometer, 220 Ω resistor and a few jumper and breadboard wires are all that is needed, power supplied by a 9V battery* after programming via USB.

*http://www.instructables.com/id/Powering-Arduino-with-a-Battery/

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Circuitry

I wired the LCD as per guidance on the Arduino website. The pot controls contrast on the LCD. The DHT22 was wired to take 5V from the breadboard power rail with sensor data routed to digital pin 7. The sensor version that I used (Adafruit AM2302) has a built-in 5.1 kΩ pull-up resistor.

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Code

The DHT22 has a sampling rate of 0.5 Hz which some regard as a weakness, but in the context of a temperature-humidity gauge the criticism is rather academic since it would serve no purpose to output data to the LCD at such a rapid rate. I set the display refresh to 30 seconds. Note the built-in option to display ambient temperature and dew point temperature in Celsius or Fahrenheit.

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Experiment

I used the unit to investigate the change in temperature and humidity parameters in a bathroom (enclosed volume 11.6 m^3) before and after operating the shower at a temperature of 40°C for about 5 minutes. The sensor was placed 60 cm above floor level at the midpoint of the room.

Here is the data display before the shower

and after the shower

The displayed data shows that bathroom temperature stayed constant during the experiment while the relative humidity increased markedly. This result could have been obtained with an ordinary temperature-humidity gauge, but the smart gauge gives additional information.

In contrast to the steady ambient temperature, the dewpoint temperature shows a sharp rise from a comfortable 11.8°C (53°F) to a humid 18.3°C (65°F). The absolute humidity data shows an even greater increase – a 50% hike in water vapor concentration from 10 to 15 grams per m^3 in a matter of minutes.

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© P Mander, June 2018

A couple of blocks down from the Metro station Jussieu in Paris’s 5th arrondisement lies Rue Cuvier, which runs along the north-western edge of the botanic gardens which houses the Natural History Museum. The other side of the road is bordered by various institutes of the Sorbonne, notably UPMC (formerly Pierre and Marie Curie University).

The Curies have historical associations with a number of streets in the Latin Quarter, and Rue Cuvier in particular. Pierre Curie was born at No.16 and it was in a science faculty building in this street that the Curies conducted their fundamental research on radium between 1903 and 1914. The building still exists, shielded from public curiosity by a set of prison-style metal gates, and it was in this laboratory that the first pioneering research into what would later be recognized as nuclear energy was conducted in 1903.

Yet it was not the renowned husband-and-wife team which carried out this experiment. It was in fact Pierre Curie and his young graduate assistent Albert Laborde who did the work and reported it in Comptes Rendus in a note entitled Sur la chaleur dégagée spontanément par les sels de radium (On the spontaneous production of heat by radium salts). The note, which barely covers two pages, was published in March 1903.

The laboratory in Rue Cuvier where the Curies and Laborde worked was at No.12. Just across the street is No.57, which once housed the Appled Physics laboratory of the Natural History Museum. It was here in 1896 that Henri Becquerel serendipitously discovered the strange phenomenon of radioactivity.

Between that moment of discovery on one side of Rue Cuvier and Curie and Laborde’s remarkable experiment on the other, lay the years of backbreaking work in a shed in nearby Rue Vauquelin where the Curies, together with chemist Gustave Bémont, processed tons of waste from an Austrian uranium mine in order to extract a fraction of a gram* of the mysterious new element radium.

*the maximum amount of radium coexisting with uranium is in the ratio of their half-lives. This means that uranium ores can contain no more than 1 atom of radium for every 2.8 million atoms of uranium.

– – – –

The Curie – Laborde experiment

Albert Laborde (left) and Pierre Curie, in 1901 and 1903 respectively

Pierre Curie and Albert Laborde were the first to make an experimental determination of the heat produced by radium because they were the first to have enough radium-enriched material to make the experiment practicable. It was a close-run thing though. Ernest Rutherford and Frederick Soddy had been busy working on radioactivity at McGill University in Canada since 1900, but they were hampered by lack of access to radium and were using much weaker thorium preparations. This situation would quickly change however when concentrated radium samples became available from Friedrich Giesel in Germany. By the summer of 1903, Soddy (now at University College London) and Rutherford would have their hands on Giesel’s supply. But Curie and Laborde had a head start, and they turned their narrow time advantage to good account.

Methodology

To determine the heat produced by their radium preparation, they used two different approaches – a thermoelectric method, and an ice calorimeter method.

This diagram of their thermoelectric device, taken from Mme Curie’s Traité de Radioactivité (1910), Tome II, p272, unfortunately lacks an explanation of the key, but the set-up essentially comprises a test ampoule containing the chloride salt of radium-enriched barium and a control ampoule of pure barium chloride. These are marked A and A’. The ampoules are placed in the cavities of brass blocks enclosed in inverted Dewar flasks D, D’ with some unstated packing material to keep the ampoules from falling down. The flasks are enclosed in containers immersed in a further medium-filled container E supported in a space enclosed by a medium F, all of which was presumably designed to ensure a constant temperature environment. The key feature is C and C’ which are iron-constantan thermocouples, embedded in the brass cavities, with their associated circuitry.

The current produced by the Seebeck effect resulting from the temperature difference between C and C’ was measured by a galvanometer. The radium ampoule was then replaced by an ampoule containing a platinum filament through which was passed a current whose heating effect was sufficient to obtain the same temperature difference. The equivalent rate of heat production by the radium ampoule could then be calulated using Joule’s law.

The second method used was a Bunsen calorimeter, which was known to be capable of very exact measurements using only a small quantity of the test substance. For details of the operational principleof this calorimeter, the reader is referred to this link:

http://thewaythetruthandthelife.net/index/2_background/2-1_cosmological/physics/j9.htm

The above diagram of the Bunsen calorimeter is taken from Mme Curie’s Traité de Radioactivité (1910), Tome II, p273.

Results

For most of their experiments, Curie and Laborde used 1 gram of a radium-enriched barium chloride preparation, which liberated approximately 14 calories (59 joules) of heat per hour. It was estimated from radioactivity measurements – no doubt using the quartz electrometer instrumentation invented by Curie – that the gram of test substance contained about one sixth of a gram of radium.

Measurements were also made on a 0.08 gram sample of pure radium chloride. These yielded results of the same order of magnitude without being absolutely in agreement. Curie and Laborde made it clear in their note that these were pathfinding experiments and that their aim was solely to demonstrate the fact of continuous, spontaneous emission of heat by radium and to give an approximate magnitude for the phenomenon. They stated:

» 1 g of radium emits a quantity of heat of approximately 100 calories (420 joules) per hour.

In other words, a gram of radium emitted enough heat in an hour to raise the temperature of an equal weight of water from freezing point to boiling point. And it was continuous emission, hour after hour for year after year, without any detectable change in the source material.

Curie and Laborde had quantified the capacity of radium to generate heat on a scale which was far beyond that known for any chemical reaction. And this heat was continuously produced at a constant rate, unaffected by temperature, pressure, light, magnetism, electricity or any other agency under human control.

The scientific world was astonished. This phenomenon seemed to defy the laws of thermodynamics and the question was immediately raised: Where was all this energy coming from?

Speculation and insight

In 1903, little was known about the radiation emitted by radioactive substances and even less about the atoms emitting them. The air-ionizing emissions had been grouped into three categories according to their penetrating abililities and deflection by a magnetic field, but the nature of the atom – with its nucleus and orbiting electrons – was a mystery yet to be unveiled.

Illustration from Marie Curie’s 1903 doctoral thesis of the deflection of rays by a magnetic field. Note the variable velocities shown for the β particle, whose charge-mass ratio Becquerel had demonstrated to be identical to that of the electron.

Radioactivity had been discovered by Henri Becquerel as an accidental by-product of his main area of interest, optical luminescence – which is the emission of light of certain wavelengths following the absorption of light of other wavelengths. By association luminescence was seen as a possible explanation of radioactivity, that radioactive substances might be absorbing invisible cosmic energy and re-emitting it as ionizing radiation. But no progress was made on identifying a cosmic source.

Meanwhile, from her detailed analytical work that she began in 1898, Marie Curie had discovered that uranium’s radioactivity was independent of its physical state or its chemical combinations. She reasoned that radioactivity must be an atomic property. This was a crucial insight, which directed thinking towards the idea of conversion of mass into energy as an explanation of the continuous and prodigious production of heat by radium that Pierre Curie and Albert Laborde had observed.

One of the major theories in physics at this time was electromagnetic theory. Maxwell’s equations predicted that mass and energy should be mathematically related to each other, and it was by following this line of thought that Frederick Soddy, previously Ernest Rutherford’s collaborator in Canada, came to the conclusion that radium’s energy was obtained at the expense of its mass.

Writing in the very first Annual Report on the Progress of Chemistry, published by the Royal Society of Chemistry in 1904, Soddy said this:

” … the products of the disintegration of radium must possess a total mass less than that originally possessed by the radium, and a part of the energy evolved must be considered as being derived from the change of a part of the mass into energy.”

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A different starting point

While Pierre Curie and Albert Laborde were conducting their radium experiment in Rue Cuvier, Paris, Albert Einstein – a naturalized Swiss citizen who had recently completed his technical high school studies in Zurich – was working as a clerk at the Patent Office in Bern. Much of his work related to questions about signal transmission and time synchronization, and this may have influenced his own thoughts, since both of these issues feature prominently in the conceptual thinking that led Einstein to his theory of special relativity submitted in a paper entitled Zur Elektrodynamik bewegter Körper (On the electrodynamics of moving bodies) to Annalen der Physik on Friday 30th June 1905.

On the basis of electromagnetic theory, supplemented by the principle of relativity (in the restricted sense) and the principle of the constancy of the velocity of light contained in Maxwell’s equations, Einstein proves Doppler’s principle by demonstrating the following:

Ist ein Beobachter relativ zu einer unendlich fernen Lichtquelle von der Frequenz ν mit der Geschwindigkeit v derart bewegt, daß die Verbindungslinie “Lichtquelle-Beobachter” mit der auf ein relativ zur Lichtquelle ruhendes Koordinatensystem bezogenen Geschwindigkeit des Beobachters den Winkel φ bildet, so ist die von dem Beobachter wahrgenommene Frequenz ν’ des Lichtes durch die Gleichung gegeben:

If an observer is moving with velocity v relatively to an infinitely distant light source of frequency ν, in such a way that the connecting “source-observer” line makes the angle φ with the velocity of the observer referred to a system of co-ordinates which is at rest relatively to the source of light, the frequency ν’ of the light perceived by the observer is given by:

where Einstein uses V (not c) to represent the velocity of light. He then finds that both the frequency and energy (E) of a light packet (cf. E=hν) vary with the velocity of the observer in accordance with the same law:

It was to this equation Einstein returned in a paper entitled Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (Does the inertia of a Body depend on its Energy Content?) submitted to Annalen der Physik on Wednesday 27th September 1905.

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Mass-energy equivalence

Marie Curie and Albert Einstein, Geneva, Switzerland, 1925

Einstein’s paper of September 1905 – the last of the famous set published in Annalen der Physik in that memorable year – is less than three pages long and constitutes little more than a footnote to the preceding 30-page relativity paper. Yet despite its brevity, it is a difficult and troublesome work over which Einstein brooded for some years.

The paper describes a thought experiment in which a body sends out a light packet in one direction, and simultaneously another light packet of equal energy in the opposite direction. The energy of the body before and after the light emission is determined in relation to two systems of co-ordinates, one at rest relative to the body (where the before-and-after energies are E0 and E1) and one in uniform parallel translation at velocity v (where the before-and-after energies are H0 and H1).

Einstein applies the law of conservation of energy, the principle of relativity and the above-mentioned energy equation to arrive at the following result for the rest frame and the frame in motion relative to the body, the light energy being represented by a capital L:

At this point, things start getting a little tricky. Einstein subtracts the rest frame energies from the moving frame energies for both the before-emission and after-emission cases, and then subtracts these differences:

These differences represent the before-emission kinetic energy (K0) and after-emission kinetic energy (K1) with respect to the moving frame

Since the right hand side is a positive quantity, the kinetic energy of the body diminishes as a result of the emission of light, even though its velocity v remains constant. To elucidate, Einstein performs a binomial expansion on the first term in the braces, although he makes no mention of the procedure; nor does he show the math. So this next bit is my own contribution:

Let (v/V)2 = x

The appropriate form of the binomial expansion is

Setting x = v2/V2 and n = ½

The contents of the braces in the kinetic energy expression thus become

Now back to Einstein. At this point he introduces a new condition into the scheme of things, namely that the velocity v of the system of co-ordinates moving with respect to the body is much less than the velocity of light V. We are in the classical world of v<<V, and so Einstein allows himself to neglect magnitudes of fourth and higher orders in the above expansion. Hence he arrives at

This equation gives the amount of kinetic energy lost by the body after emitting a quantity L of light energy. In the classical world of v<<V the kinetic energy of the body is also given by ½mv2, and since the velocity v is the same before and after the light emission, Einstein is led to identify the loss of kinetic energy in his thought experiment with a loss of mass:

Gibt ein Körper die Energie L in Form von Strahlung ab, so verkleinert sich seine Masse um L/V2. Hierbei ist es offenbar unwesentlich, daß die dem Körper entzogene Energie gerade in Energie der Strahlung übergeht, so daß wir zu der allgemeineren Folgerung geführt werden: Die Masse eines Körpers ist ein Maß für dessen Energie-inhalt.

If a body gives off the energy L in the form of radiation, its mass diminishes by L/V2. The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that: The mass of a body is a measure of its energy content.

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Testing the theory

The pavilion where Curie and Laborde did their famous work

When Einstein wrote Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? in 1905, he was certainly aware of the phenomenon of continuous heat emission by radium salts as measured by Curie and Laborde, and confirmed by several others in 1903 and 1904. In fact he saw in this a possible means of putting relativity theory to the test:

Es ist nicht ausgeschlossen, daß bie Körpern, deren Energieinhalt in hohem Maße veränderlich ist (z. B. bei den Radiumsaltzen) eine Prüfung der Theorie gelingen wird.

It is not impossible that with bodies whose energy content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.

In hindsight, it was unlikely that Einstein could have made this test work and he soon abandoned the idea. Not only would the mass difference have been extremely small, but also the process of nuclear decay was conceptually different to Einstein’s thought experiment. In Curie and Laborde’s calorimeter, the energy emitted by the body (radium nucleus) was not initially in the form of radiant energy; it was in the form of kinetic energy carried by an ejected alpha particle (helium nucleus) and a recoiling radon nucleus.

But Einstein had a knack of getting ahead of himself and ending up in the right place. The mass-energy equivalence relation he obtained from his imagined light-emitting body turned out to be valid also in relation to the kinetic energy of radioactive decay particles.

To see this in relation to Curie and Laborde’s experiment, consider the nuclear reaction equation

Here Q is the mass difference in atomic mass units (u) required to balance the equation:
Mass of Ra = 226.02536 u
Mass of Rn (222.01753) + He (4.00260) = 226.02013 u
Mass difference = Q = 0.00523 u
The kinetic energy equivalent of 1 u is 931.5 MeV
So Q = 4.87 MeV

The kinetic energy is shared by the ejected alpha particle and recoiling radon nucleus. Since the velocities are non-relativistic, this can be calculated on the basis of the momentum conservation law and the classical expression for kinetic energy. Given the masses of the Rn and He nuclei, their respective velocities must be in the ratio 4.00260 to 222.01753. Writing the kinetic energy expression as ½mv.v and recognizing that ½mv has the same magnitude for both nuclei, the kinetic energies of the Rn and He nuclei must also be in the ratio 4.00260 to 222.01753. The kinetic energy carried by the alpha particle is therefore

4.87 x 222.01753/226.02013 = 4.78 MeV

This result has been confirmed by experiment.

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Links to original papers mentioned in this post

Sur la chaleur dégagée spontanément par les sels de radium ; par MM. P. Curie et A. Laborde
Comptes Rendus, Tome 136, janvier – juin 1903

http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-3091&I=673&M=tdm

Zur Elektrodynamik bewegter Körper; von A. Einstein
Annalen der Physik 17 (1905) 891-921

https://archive.org/stream/annalenderphysi108unkngoog#page/n1020/mode/2up

Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? von A. Einstein
Annalen der Physik 18 (1905) 639-641

https://archive.org/stream/annalenderphysi143unkngoog#page/n707/mode/2up

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Postscript

In Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? Einstein arrived at a general statement on the dependence of inertia on energy (Δm = ΔE/V2, in today’s language E = mc2) from the consideration of a special case. He was deeply uncertain about this result, and returned to it in two further papers in 1906 and 1907, concluding that a general solution was not possible at that time. He had to wait a few years to discover he was right. I include links to these papers for the sake of completeness.

Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie; von A. Einstein
Annalen der Physik 20 (1906) 627-633
http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1906_20_627-633.pdf

Über die vom Relativitätsprinzip geforderte Trägheit der Energie; von A. Einstein
Annalen der Physik 23 (1907) 371-384
http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1907_23_371-384.pdf

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P Mander June 2017

mir01

Taking a break from studying thermodynamics: Mme. Curie measuring picoamp leakage currents

Having read biographies of the Curies, Marie Curie’s doctoral thesis, and a number of scholarly articles about the radium phenomenon, I have come to the conclusion that the Marie Curie legend in popular culture tends to sideline her scientific achievements, focusing more on her imagined saintliness and perceived role as bringer of medical marvels than on her pioneering work as a physical chemist, in which her husband Pierre (above left) played an important facilitating role.

To my mind, the popular press of her day was largely responsible for the misconstruction of the Marie Curie legend, filling the public’s mind with the discovery of a miracle cure for cancer brought about by what it portrayed as an angelic young foreign-born mother slaving away in a dark shed in Paris for no wages.

The media frenzy around radium had consequences. Once radium production was established on a commercial scale, ignorant and unscrupulous marketers quickly morphed the Curies’ discovery of an element that glowed in the dark into revitalizing radium baths, radium drinking water, radium chocolate, radium toothpaste, radium cigarettes and even radium suppositories for restoring male potency while eradicating hemorrhoids:

Also splendid for piles and rectal sores. Try them and see what good results you get!

The dreadful damage these products must have caused doesn’t bear thinking about. Sadly, the Curies themselves seemed carried along similarly radium-dazzled tracks. They failed to connect Pierre’s rapidly deteriorating health with exposure to radioactive emissions, while stoically accepting the painful damage to Marie’s hands as a price worth paying for the greater good they somehow imagined radium to represent.

Irène Curie points to her mother’s radiation-damaged hands

The sensationalist aspects of the Curie legend, while an education in themselves, are however not the subject of this post. Physics and chemistry are the subjects here. When you look at Marie Curie as a physical chemist, and examine her contributions to the science of natural radioactivity, it is clear how crucial a role was played by the miracle machine designed and developed by Pierre Curie.

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The Curie quartz electrometer apparatus

The above diagram is taken from Méthodes de Mesure employées en Radioactivité published in Paris in 1911 by Albert Laborde, a graduate engineer who became Pierre Curie’s assistant in 1902. It shows the quartz electrometer apparatus developed by Pierre Curie and his brother Jacques for the precise measurement of very weak currents (of the order of tenths of picoamperes) following their discovery of piezoelectricity in 1880. It was this discovery that prompted the brothers Curie to build a calibrated electrostatic charge generator using a thin quartz lamella (center) to compensate and thus measure the leakage current from a charged capacitor (left) using a quadrant electrometer (right).

This apparatus was later adapted by Pierre Curie to allow accurate quantification of the tiny leakage currents produced in an ionization chamber by samples of radioactive material.

This is the experimental set-up that Marie Curie can be seen using in the header photograph, which dates from 1898.

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Amazing coincidences

When you consider the train of coincidences that led to Marie Curie’s choice of subject for her doctorate (Recherches sur les substances radioactives) it is nothing short of amazing. At the time she was looking around for suitable topic, five years after having journeyed from Poland to Paris to enroll as a Sorbonne student, Henri Becquerel had just accidentally discovered mysterious rays emanating from uranium which had the property of weakly ionizing air. This was in 1896. Just a year previously, Marie had married Pierre Curie who happened to possess the one instrument capable of accurately measuring small ionization currents, following his discovery of piezoelectricity sixteen years earlier.

Because uranic rays were a new phenomenon, Marie was saved the task of first researching the topic which otherwise would have entailed reading a lot of academic papers in unfamiliar French. This saving of time and effort attracted her to choose to study Becquerel’s uranic rays, something she admitted in later life. Furthermore she had no competition since Becquerel had shown little interest in pursuing his original finding – the big news at the time was X rays, discovered by Wilhelm Röntgen in 1895. No fewer than 1044 papers on X rays were published in 1896, when Becquerel first announced his discovery. Not surprisingly, nobody took any notice. Marie Curie had the field to herself.

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The weight, the watch and the light spot

This magnified portion of the header image shows Marie Curie as she sits at the quartz electrometer apparatus. Her right hand can be seen holding an analytical balance weight in a controlled manner while in her left hand the edge of a stopwatch can be seen. Her eyes are looking fixedly at a horizontal measuring scale above a light source (square hole) mounted on a wooden pedestal.

The light source is shining a beam onto the mirror of a quadrant electrometer out of view to the left. The light spot is reflected onto the horizontal scale (cf. diagram above) and Marie is endeavoring to keep the light spot stationary. She does this by gradually releasing the weight which is attached to the quartz lamella, thereby generating charge to compensate the ionization current produced by the radioactive sample in the ionization chamber also out of view to the left. The entire process of weight release is timed by a stopwatch. Once the weight is fully released the watch is stopped. The weight generates a specific amount of charge Q* on the quartz lamella during the measured time T. Hence Q/T is equal to the ionization current, which is directly proportional to the intensity of the ionizing radiation emitted by the sample, or to use the term Marie Curie coined, its radioactivity.

*The amount of charge Q is calculated from Q = W × K × L/B where W is the applied weight, K is the quartz specific constant, L is the lamella length and B is the lamella thickness.

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Thesis

On Thursday 25th June 1903, at La Faculté des Sciences de Paris, Marie Curie presented her doctoral thesis to the examination committee, two of whose members were later to become Nobel laureates. The committee was impressed; in fact it expressed the view that her findings represented the greatest scientific contribution ever made in a doctoral thesis.

At the outset, Curie coined a new term – radioactivity – to describe the ionizing radiation emitted by the uranium compounds studied by Henri Becquerel. She announced her discovery that the element thorium also displays radioactivity. And she presented a method, using the quartz electrometer apparatus developed by Jacques and Pierre Curie, by which the intensity of radioactive emissions could be precisely quantified and expressed as ionization currents. This was a game-changing advance on the essentially qualitative methods that had been used hitherto e.g. electroscopes and photographic plates.

As one would expect, Curie began her experimental work with a systematic study of uranium and its compounds, measuring and tabulating their ionization currents. There was a considerable range from the largest to the smallest currents, and within the limits of experimental error it was evident that the ionization currents were proportional to the amounts of uranium present in the sample. The same was true for thorium.

From the chemist’s perspective this was a puzzling result. The properties of chemical compounds of the same element generally depend on what it is compounded with and the arrangement of atoms in the molecule. Yet here was a very different finding – the radioactivity Curie measured was independent of compounding or molecular structure.

Curie drew the conclusion that radioactivity was a property of the atom – une propriété atomique she called it. She wasn’t referring to the uranium atom or the thorium atom, but to the atom as a generalized material unit with an implied interior from which radioactive emissions issued. That is a profound conception, with which Marie Curie made a significant contribution to the advancement of physics.

And at this point in her thesis she hadn’t even mentioned radium.

– – – –

New elements

For the next part of her thesis, Marie Curie turned her attention to the study of uranium-containing minerals, one of which was the mineral pictured above. Today we call it uraninite but in Curie’s day it was called pitchblende. The sample she obtained was from a uranium mine near the town of Joachimsthal in Austria, now Jáchymov in the Czech Republic. She measured its ionization current and found it to be considerably higher than its uranium content warranted. If her radioactivity hypothesis was correct, there was only one explanation: pitchblende contained atoms that emitted much more intense radiation than uranium atoms, which meant that another radioactive substance must be present in the ore. Curie now had the task of finding it, and was joined in this quest by her husband Pierre and the chemist Gustave Bémont.

The quartz electrometer demonstrated its value yet again, since the various fractions derived from the pitchblende sample during chemical analysis could be tested for radioactivity. In this way, the radioactivity was followed to two fractions: one containing the post-transition metal bismuth* and another containing the alkaline earth metal element barium. The Curies announced their findings in July 1898, stating their belief that these fractions contained two previously unknown metal elements, and suggesting that if the existence of these metals were confirmed, the bismuth-like element should be called polonium and the barium-like element radium.

*unknown to the Curies, the uranium decay series actually produces two radioisotopes of bismuth along with the isotopes of polonium, so the presence of radioactivity in this fraction did not solely indicate the presence of a new element.

– – – –

The shed

The fabled shed where Marie Curie labored for four years, chemically processing tons of pitchblende to produce a tiny spoonful of radium.

The heroic work which made Marie Curie a legend took place in a shed at the back of a Grande École* in Rue Vauquelin. In order to produce sufficient quantities to isolate the new elements and determine their atomic weights, tons of pitchblende were needed. This was because the maximum amounts of radium and polonium that can coexist in secular equilibrium with uranium are in the ratios of their respective half lives. This fact and the limited human resources available rendered any attempt to isolate polonium impossible, and the situation with radium was not much better. At best, a quantity of uranium ore containing 3 metric tons of elemental uranium is needed to extract 1 gram of radium at a yield of 100%. In the primitive conditions of the shed, obtaining a gram of radium meant processing 8 or 9 tons of uranium ore. One can only wonder at how Marie Curie found the physical and mental strength for such an arduous task.

*At the time, it was called École supérieure de physique et de chimie industrielles de la ville de Paris. Today it is called ESCPI Paris.

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Did she deserve two Nobel Prizes?

Marie Curie was awarded a quarter of the Nobel Prize for Physics in 1903 for her work on radioactivity. In 1911 she was the sole recipient of the Nobel Prize for Chemistry, awarded for the discovery of radium and polonium.

There can be no doubt about her credentials for the 1903 award, but some biographers have questioned whether the 1911 Prize was deserved, claiming that the discoveries of radium and polonium were part of the reason for the first prize.

As described in this post, the experimental evidence which Marie Curie set forth to reason that radioactivity is an atomic property was based solely on her experiments with uranium and thorium. Neither radium nor polonium had anything to do with it. On these grounds the claims of those biographers can be rejected.

Which leaves the question of under what circumstances the discovery of a new element qualifies for a Nobel Prize in chemistry. Clearly the discovery of a naturally radioactive element is not sufficient, otherwise Marguerite Perey – who worked as Marie Curie’s lab assistant and discovered francium in 1939 – would have qualified. Other aspects of the discovery need to be taken into account, and in 1911 there were many such aspects to Marie Curie’s discovery of radium and polonium, and the isolation of radium.

Reading the award citation, what comes across to me – albeit between the lines – is a recognition of the monumental personal effort and dedication involved in the discovery and characterization of these remarkable elements that led to the modern science of nuclear physics.

The miracle machine

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P Mander June 2017

tp01

A thermodynamic system doesn’t have to be big. Although thermodynamics was originally concerned with very large objects like steam engines for pumping out coal mines, thermodynamic thinking can equally well be applied to very small systems consisting of say, just a few atoms.

Of course, we know that very small systems play by different rules – namely quantum rules – but that’s ok. The rules are known and can be applied. So let’s imagine that our thermodynamic system is an idealized solid consisting of three atoms, each distinguishable from the others by its unique position in space, and each able to perform simple harmonic oscillations independently of the others. At the absolute zero of temperature, the system will have no thermal energy, one microstate and zero entropy, with each atom in its vibrational ground state.

Harmonic motion is quantized, such that if the energy of the ground state is taken as zero and the energy of the first excited state as ε, then 2ε is the energy of the second excited state, 3ε is the energy of the third excited state, and so on. Suppose that from its thermal surroundings our 3-atom system absorbs one unit of energy ε, sufficient to set one of the atoms oscillating. Clearly, one unit of energy can be distributed among three atoms in three different ways – 100, 010, 001 – or in more compact notation [100|3].

Now let’s consider 2ε of absorbed energy. Our system can do this in two ways, either by promoting one oscillator to its second excited state, or two oscillators to their first excited state. Each of these energy distributions can be achieved in three ways, which we can write [200|3], [110|3]. For 3ε of absorbed energy, there are three distributions: [300|3], [210|6], [111|1].

Summarizing the above information

Energy E (in units of ε) Total microstates W Ratio of successive W’s
0 1
1 3 3
2 6 2
3 10 1⅔

 

The summary shows that as E increases, so does W. This is to be expected, since as W increases, the entropy S (= k log W) increases. In other words E and S increase or decrease together; the ratio ∂E/∂S is always positive. Since ∂E/∂S = T, the finding that E and S increase or decrease together is equivalent to saying that the absolute temperature of the system is always positive.

– – – –

Adding an extra particle

It is instructive to compare the distribution of energy among three oscillators (N =3)*

E = 0: [000|1]
E = 1: [100|3]
E = 2: [200|3], [110|3]
E = 3: [300|3], [210|6], [111|1]

with the distribution among four oscillators (N = 4)*

E = 0: [0000|1]
E = 1: [1000|4]
E = 2: [2000|4], [1100|6]
E = 3: [3000|4], [2100|12], [1110|4]

*For any single distribution among N oscillators where n0, n1,n2 … represent the number of oscillators in the ground state, first excited state, second excited state etc, the number of microstates is given by

tp02

It is understood that 0! = 1. Derivation of the formula is given in Appendix I.

For both the 3-oscillator and 4-oscillator systems, the first excited state is never less populated than the second, and the second excited state is never less populated than the third. Population is graded downward and the ratios n1/n0 > n2/n1 > n3/n2 are less than unity.

Example calculations for N = 4, E = 3:

tp03

tp04

tp05

Comparisons can also be made of a single ratio across distributions and between systems. For example the values of n1/n0 for E = 0, 1, 2, 3 are

(N = 4) : 0, ⅓, ½, ⅗
(N = 3) : 0, ½, ⅔, ¾

Since for a macroscopic system

tp06

this implies that for a given value of E the 4-oscillator system is colder than the 3-oscillator system. The same conclusion can be reached by looking at the ratio of successive W’s for the 4-oscillator system sharing 0 to 3 units of thermal energy

Energy E (in units of ε) Total microstates W Ratio of successive W’s
0 1
1 4 4
2 10
3 20 2

 

For the 4-oscillator system the ratios of successive W’s are larger than the corresponding ratios for the 3-oscillator system. The logarithms of these ratios are inversely proportional to the absolute temperature, so the larger the ratio the lower the temperature.

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Finite differences

The differences between successive W’s for a 4-oscillator system are the values for a 3-oscillator system

W for (N =4) : 1, 4, 10, 20
Differences : 3, 6, 10

Likewise the differences between successive W’s for a 3-oscillator system and a 2-oscillator system

W for (N =3) : 1, 3, 6, 10
Differences : 2, 3, 4

Likewise for the differences between successive W’s for a 2-oscillator system and a 1-oscillator system

W for (N =2) : 1, 2, 3, 4
Differences : 1, 1, 1

This implies that W for the 4-particle system can be expressed as a cubic in n, and that W for the 3-particle system can be expressed as a quadratic in n etc. Evaluation of coefficients leads to the following formula progression

For N = 1

tp07

For N = 2

tp08

For N = 3

tp09

For N = 4

tp10

It appears that in general

tp11

Since n = E/ε and ε = hν, the above equation can be written

tp12

For a system of oscillators this formula describes the functional dependence of W microstates on the size of the particle ensemble (N), its energy (E), the mechanical frequency of its oscillators (ν) and Planck’s constant (h).

– – – –

Appendix I

Formula to be derived

For any single distribution among N oscillators where n0, n1,n2 … represent the number of oscillators in the ground state, first excited state, second excited state etc, the number of microstates is given by

tp02

Derivation

In combinatorial analysis, the above comes into the category of permutations of sets with the possible occurrence of indistinguishable elements.

Consider the distribution of 3 units of energy across 4 oscillators such that one oscillator has two units, another has the remaining one unit, and the other two oscillators are in the ground state: {2100}

If each of the four numbers was distinct, there would be 4! possible ways to arrange them. But the two zeros are indistinguishable, so the number of ways is reduced by a factor of 2! The number of ways to arrange {2100} is therefore 4!/2! = 12.

1 and 2 occur only once in the above set, and the occurrence of 3 is zero. This does not result in a reduction in the number of possible ways to arrange {2100} since 1! = 1 and 0! = 1. Their presence in the denominator will have no effect, but for completeness we can write

4!/2!1!1!0!

to compute the number of microstates for the single distribution E = 3, N = 4, {2100} where n0 = 2, n1 = 1, n2 = 1 and n3 = 0.

In the general case, the formula for the number of microstates for a single energy distribution of E among N oscillators is

tp02

where the terms in the denominator are as defined above.

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P Mander April 2016