Archive for the ‘thermodynamics’ Category

Carbon dioxide – aka CO2 – has a lousy reputation in today’s world. Most of us regard it as a significant greenhouse gas contributor to atmospheric heating and all the bad climate stuff that comes with it: heat domes, wildfires, droughts, flash floods, mudslides, loss of land, loss of property, loss of life. The best available science warns us not to keep adding more CO2 to what’s already up in the air, so it’s understandable that our basic instinct is to capture it before emission, pump it into caverns underground, and leave it there.

Carbon Capture and Sequestration (CCS) is already being implemented in subterranean spaces such as depleted oilfields. The CO2 is captured from power stations and industrial facilities such as cement works, and either directly piped or shipped to the disposal point.

As a business model it is capital intensive and relies on the costs of sequestration being sufficiently lower than the costs of emission to make the undertaking economically viable. Also CCS intrinsically demonstrates a preference for permanent disposal over carbon re-utilization and implicit in that choice is a value assessment of the carbon contained in CO2.

Such an assessment is not easily made without familiarity with carbon’s capacity to form bonds with itself and other atoms, a knowledge of what that means in terms of carbon’s oxidation state range, and an understanding of how that range makes carbon a suitable vehicle for energy storage and release, as well as feedstock for a broad spectrum of industrially useful molecules.

It is asking a lot of those educated in political science rather than physical science to make reasoned judgements on the desirability of permanent disposal or carbon re-utilization. And it is rare indeed for heads of government to have skills in both disciplines, although it has been known.

Fortunately however, state-backed investments in CCS need not carry the risk of being the wrong choice long term because it has been shown possible to re-utilize sequestered CO2. Here’s how.

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The Sabatier-Senderens reaction

In 1897, two French chemists discovered that carbon dioxide could be reduced by hydrogen over a nickel catalyst at elevated temperature and pressure to form methane and water vapor. This is the same carbon transformation that occurs in the Carbon Cycle via photosynthesis and the gradual degradation of biomass to coal, oil and natural gas – a process that takes Nature millions of years to accomplish. Sabatier and Senderens found a way to do it in real time.

Now consider the conditions in the underground caverns where carbon dioxide gas is sequestered. The temperature is elevated and so is the pressure. And in the pores of the geological formations holding the gas there are microbes capable of catalyzing the Sabatier-Senderens reaction. All it needs is to add hydrogen, which can be produced electrolytically from water using solar or wind.

To get technical for a moment, it is instructive to apply Hess’s Law (G.H. Hess, 1840) to the electrolysis of water and the Sabatier-Senderens reaction

In effect the energy released by the combustion of hydrogen is being used to reverse the combustion of methane, the energy being stored in the stable C-H bonds of the methane molecule. Note that the oxygen is formed above ground during electrolysis and is either stored for commercial use or vented to the atmosphere.

The pressurized subterranean gas can be piped up and passed through a separator where methane is extracted and carbon dioxide and hydrogen are returned underground to continue the reaction. Now if the sequestered CO2 was captured before emission, it would defeat the object to use the methane in the energy supply infrastructure since combustion would simply release CO2 to the atmosphere. So what can be done with this methane? The next section supplies the answer.

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Tri-Reforming Methane


An LNG tanker

Methane (CH4) has certain characteristics which make it attractive as an energy carrier. It is not difficult to liquefy at 1 atmosphere pressure, it is energy dense and relatively efficient to transport as LNG (Liquid Natural Gas). So it is a practical proposition for methane gas produced in underground caverns to be transported to plants where CO2 capture is taking place.

Again to get technical for a moment, we notice that the oxidation states of carbon in the two molecules are at opposite ends of the scale. Methane has the most reduced form of carbon (-4) while carbon dioxide has the most oxidized form of carbon (+4). A redox reaction between the two looks possible and indeed is possible, albeit at elevated temperatures:

This catalyzed process, by which two greenhouse gases are converted into two non-greenhouse gases, is called dry reforming of methane (DRM) and was first introduced by Franz Fischer and Hans Tropsch in Germany in the 1920s. The 1:1 mixture of carbon monoxide and hydrogen is called syngas (synthesis gas), a key intermediate in the production of industrially useful molecules.

Germany’s dynamic duo: Franz Fischer and Hans Tropsch

Because of the high process temperature, DRM also results in thermal decomposition of both methane and carbon dioxide and the deactivating deposition of carbon on the catalyst. This problem can however be mitigated in a very neat way by combining DRM with another methane reforming process, namely steam reforming (SRM). This not only re-utilizes the deposited carbon but also adds another syngas product with a 3:1 ratio:

All these reactions are endothermic (requiring heat). This heat can be supplied by adding oxygen to the reactant stream, which allows partial oxidation of methane (POM) and catalytic combustion of methane (CCM) to take place, both of which are exothermic reactions (producing heat):

Putting three reforming agents – carbon dioxide, water and oxygen – together in the reactant stream with methane feedstock creates a sufficiently energy-efficient overall process known as Tri-reforming.

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Bringing it all together

We have seen that the decision to avoid CO2 emissions to the atmosphere by employing capital intensive Carbon Capture and Sequestration (CCS) in subterranean locations such as depleted oilfields does not preclude the subsequent or concurrent addition of hydrogen to facilitate a gas phase redox reaction in which carbon dioxide is converted into energy-rich methane.

Methane feedstock can efficiently be transported to plants where CO2 capture is taking place and fed into a tri-reforming reactor together with carbon dioxide, oxygen and steam to create commercially valuable syngas and obtain a return on CCS investment through the production of industrially useful carbon-containing molecules that do not pose a greenhouse gas risk.

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Suggested further reading

A mini-review on CO2 reforming of methane
Published in June 2018 this is a useful and easily readable grounder covering the thermodynamic, kinetic, catalysis and commercial aspects of the subject.

Tri-reforming: a new process for reducing CO2 emissions
A bedrock paper (January 2001) from the legendary Chunshan Song at Penn State.

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P Mander February 2022

As a follow-up to my 2018 Smart Temperature and Humidity Gauge here is a new and improved wireless version. A smartphone screen replaces the previous LCD display, allowing the data to be read remotely from the device location. The four displayed parameters are the same, but the previous Absolute Humidity formula with its -30°C to +35°C range has been replaced with a new formula (Mander 2020) with an extended temperature range.

The WiFi-enabled microcontroller can be programmed to operate in Station mode, Access Point mode or both modes simultaneously so you can use the device at home, at work, on the beach, wherever you want to know the values of the parameters which determine your comfort.

The original Smart Gauge prototype with attached LCD display

The wireless function also enables you to read temperatures inside a closed compartment such as a fridge. By following the readings on your smartphone (I set mine to refresh every 20 secs) you can see what the upper and lower thermostat settings are. For example the air in the 4°C compartment of our fridge cycles between 2.9°C and 7.5°C. I was surprised by this to begin with, but then realized that things stored in fridges generally have significantly larger heat capacities than air so their temperatures will fluctuate over narrower ranges.

A sensor mounted on jumper leads is ideal for testing condenser coil ventilation on a fridge, with the sensor placed directly in the airflow of the lower inlet and upper outlet.

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The CarnotCycleAIR Smart Gauge is built around an Arduino IDE compatible Sparkfun ESP8266 Thing Dev microcontroller featuring an integrated FTDI USB-to-Serial chip for easy programming. A 2-pin JST connector has been soldered to the footprint alongside the micro-USB port and the board is powered by a rechargeable 3.7V lithium polymer flat pouch battery which fits neatly underneath the standard 400 tie-point breadboard. The system is designed to work with DHT sensors such as the DHT22 with a temperature range of -40°C to +80°C and relative humidity range of 0-100%, or the DHT11 with a temperature range of 0°C to +50°C and relative humidity range of 20-90%. These are available mounted on 3-pin breakout boards which feature built-in pull-up resistors. Both sensors are designed to function on the 3.3V supplied by the board, enabling the signal output to be connected directly to one of the board’s I/O pins without the need for a logic level converter [the ESP8266’s I/O pins do not easily tolerate voltages higher than 3.3V. Using 5V will blow it up].

Note: Comparing temperature readings in ambient conditions with an accurate glass thermometer has shown that when the sensors are plugged directly into the breadboard they record a temperature approx. 1°C higher than the true temperature, which in turn affects the accuracy of the computed absolute humidity and dewpoint. The cause appears to be Joule heating in the breadboard circuitry. Using jumper leads to distance the sensor from the board solves the problem.

The DHT22 sensor works happily down to –40°C, as does the new absolute humidity formula.

When used in Station mode the range is determined largely by the router. In Access Point mode where the device and smartphone communicate directly with each other, the default PCB trace antenna was found to work really well with an obstacle-free range of at least 35 meters (115 feet).

A Sparkfun LiPo Charger Basic, an incredibly tiny device just 3 cm long with a JST connector at one end and a micro-USB connector at the other, was used to recharge the battery. Charging time was about 4 hours.

The Sparkfun LiPo Charger Basic

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The circuitry for CarnotCycleAIR Smart Gauge couldn’t be simpler.

The ESP8266 (with header pins) is placed along the center of the breadboard and 3.3V is supplied to the power bus from the left header (3V3, GND). The DHT sensor is powered from this bus and the signal is routed to a suitable pin. I used pin 12 on the right header. That’s all there is to it.

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CarnotCycleAIR Smart Gauge is a further development of an ESP8266 project – “ESP8266 NodeMCU Access Point (AP) for Web Server” – published online by Random Nerd Tutorials which displays temperature and relative humidity on a smartphone with the ESP8266 set up as an Access Point. Code for the Station-only version using Arduino IDE can be copied from this link:

Note: The coding for the Absolute Humidity and the Dewpoint Temperature formulas must be written on one line

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© P Mander December 2021

Every day this blog gets visits from all over the world, and in numbers which show that thermodynamics interests many, many people. They come from lands big and small, rich and poor, happy and less-than-happy. And they are all united in their desire for knowledge.

Knowledge is power. And in the case of thermodynamics, that knowledge is especially powerful.

There are now 100 posts and pages on this blog, covering a sizeable range of topics in thermodynamics and allied disciplines. They are written for enquiring minds, and it is truly gratifying to see so much of the CarnotCycle resource being accessed by so many.

Thank you for visiting.

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


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


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P Mander, February 2021

For a physicist, Albert Einstein (1879-1955) took a remarkable interest in physical chemistry. His doctoral thesis, submitted in 1905, was concerned with determining the dimensions of molecules. And his famous paper from the same year on Brownian motion has at its core the molecular-kinetic theory, a cornerstone of physical chemistry. In both these works, and incidentally in his equally famous paper on the photoelectric effect, Einstein is noticeably occupied with the determination of Elementarquanta (fundamental atomic constants), principal among them being the Avogadro number N. Indeed, following the publication of his epoch-making paper on Special Relativity, he went back twice to his thesis, in 1906 and 1911, to revise his estimate of this number. Not only that, but in between these revisits, he published a paper in 1907 describing yet another method of determining N which will constitute the principal content of this blogpost. But first, a few words about Amedeo Avogadro (1776-1856) and the number that is named for him.

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Avogadro’s hypothesis

In 1811 Amedeo Avogadro, an Italian of noble birth who studied ecclesiastical law before teaching science at a school in Vercelli in northern Italy, sent a paper to the Journal de Physique, de Chemie et d’Histoire naturelle. In it, he said this:

It was the first statement of what became known as Avogadro’s hypothesis, that equal volumes of all gases [at the same temperature and pressure] contain the same number of molecules. In one sentence, Avogadro took science a crucial step forward. He reconciled the atomic theory (1803-1806) of John Dalton and the gas volume studies (1808) of Joseph Gay-Lussac through the inspired idea that the elemental substances in Gay-Lussac’s experiments existed as divisible polyatomic units i.e. molecules and not as single atoms upon which Dalton insisted.

Different gases have different densities, and by relating these densities to the lightest known gas, hydrogen, the concept of molecular weight and the gram-molecule or mole was developed. A mole of any gas has the same volume (22.4 liters at 1 atmosphere pressure and a temperature of 273K) and therefore contains the same number of molecules. So what is this number? The answer is 6.022 x 1023, a fundamental constant fittingly known as the Avogadro number.

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Einstein’s forgotten idea

Einstein’s preoccupation with determining this number reflected his belief in the atomic view of matter and the validity of molecular-kinetic theory, which was not universally recognized at the turn of the 20th century. By presenting methods for the determination of the Avogadro number through experiments on observable phenomena, Einstein built a case for the real existence of molecules which came to fruition in 1908 with Jean Perrin’s famous work based on Einstein’s Brownian motion paper of 1905.

Perhaps this explains why Einstein’s interim paper of 1907, suggesting yet another route to determining the Avogadro number, seems to have been passed over – the reality of molecules had been demonstrated and there was no need for further proof.

Even so, it would be interesting to know whether any determination of the Avogadro number was ever conducted on the basis of the 1907 paper. And if not, whether anyone might be interested in putting Einstein’s forgotten idea to the test. Below I have posted both the original text of the paper and an English translation.

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Über die Gültigkeitsgrenze des Satzes vom thermodynamischen Gleichgewicht und über die Möglichkeit einer neuen Bestimmung der Elementarquanta; von A. Einstein

(Annalen der Physik 22 (1907): 569-572)

Der Zustand eines physikalischen Systems sei im Sinne der Thermodynamik bestimmt durch die Parameter λ, µ etc. (z. B. Anzeige eines Thermometers, Lange oder Volumen eines Körpers, Substanzmenge einer gewissen Art in einer Phase. Ist das System mit anderen Systemen nicht in Wechselwirkung, was wir annehmen, so wird nach der Thermodynamik Gleichgewicht bei bestimmten Werten λ0, µ0 etc. der Parameter statthaben, fur welche Werte die Entropie S des Systems ein Maximum ist. Nach der molekularen Theorie der Wärme jedoch ist dies nicht genau, sondern nur angenähert richtig; nach dieser Theorie besitzt der Parameter λ auch bei Temperaturgleichgewicht keinen konstanten Wert, sondern einen unregelmäßig schwankenden, der sich von λ0 allerdings nur äußerst selten beträchtlich entfernt.

Die theoretische Untersuchung des statistischen Gesetzes, welchem diese Schwankungen unterworfen sind, scheint auf den ersten Blick bestimmte Festsetzungen in betreff des anzuwendenden molekularen Bildes zu erfordern. Dies ist jedoch nicht der Fall. Es genügt vielmehr im wesentlichen, die bekannte Boltzmannsche Beziehung anzuwenden, welche die Entropie S mit der statistischen Wahrscheinlichkeit eines Zustandes verbindet. Diese Beziehung lautet bekanntlich

wobei R die Konstante der Gasgleichung und N die Anzahl der Moleküle in einem Grammäquivalent bedeutet.

Wir fassen einen Zustand des Systems ins Auge, in welchem der Parameter λ den von λ0 sehr wenig abweichenden Wert λ0 + ɛ besitzt. Um den Parameter λ auf umkehrbarem Wege vom Werte λ0 zum Werte λ bei konstanter Energie E zu bringen, wird man eine Arbeit A dem System zufuhren und die entsprechende Wärmemenge dem System entziehen müssen. Nach thermodynamischen Beziehungen ist:

oder, da die betrachtete Änderung unendlich klein und ʃ dE = 0 ist:

Andererseits ist aber nach dem Zusammenhang zwischen Entropie und Zustandswahrscheinlichkeit:

Aus den beiden letzten Gleichungen folgt:


Dies Resultat insolviert eine gewisse Ungenauigkeit, indem man ja eigentlich nicht von der Wahrscheinlichkeit eines Zustandes, sondern nur von der Wahrscheinlichkeit eines Zustands-gebietes reden kann. Schreiben wir statt der gefundenen Gleichung

so ist das letztere Gesetz ein exaktes. Die Willkur, welche darin liegt, daß wir das Differential von λ und nicht das Differential irgendeiner Funktion von λ in die Gleichung eingesetzt haben, wird auf unser Resultat nicht von Einfluß sein.

Wir setzen nun λ = λ0 + ɛ und beschränken uns auf den Fall, daß A nach positiven Potenzen von ɛ entwickelbar ist, und daß nur das erste nicht verschwindende Glied dieser Entwickelung zum Werte des Exponenten merklich beiträgt bei solchen Werten von ɛ, fur welche die Exponentialfunktion noch merklich von Null verschieden ist. Wir setzen also A = αɛ2 und erhalten:

Es gilt also in diesem Falle fur die Abweichungen ɛ das Gesetz der zufälligen Fehler. Für den Mittelwert der Arbeit A erhält man den Wert:

Das Quadrat der Schwankung ɛ eines Parameters λ ist also im Mittel so groß, daß die äußere Arbeit A, welche man bei strenger Gültigkeit der Thermodynamik anwenden müßte, um den Parameter λ bei konstanter Energie des Systems von λ0 auf zu verändern, gleich ½RT/N ist (also gleich dem dritten Teil der mittleren kinetischen Energie eines Atoms).

Führt man fur R und N die Zahlenwerte ein, so erhalt man angenähert:

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Wir wollen nun das gefundene Resultat auf einen kurz geschlossenen Kondensator von der (elektrostatisch gemessenen) Kapazität c anwenden. Ist die  Spannung (elektrostatisch), welche der Kondensator im Mittel infolge der molekularen Unordnung annimmt, so ist

Wir nehmen an, der Kondensator sei ein Luftkondensator und er bestehe aus zwei ineinandergeschobenen Plattensystemen von je 30 Platten. Jede Platte habe von den benachbarten des anderen Systems im Mittel den Abstand 1 mm. Die Größe der Platten sei 100 cm2. Die Kapazität c ist dann ca. 5000. Für gewöhnliche Temperatur erhält man dann

In Volt gemessen erhält man

Denkt man sich die beiden Plattensysteme relativ zueinander beweglich, so daß sie vollständig auseinander geschoben werden können, so kann man erzielen, daß die Kapazität nach dem Auseinanderschieben von der Größenordnung 10 ist.

Nennt man π die Potentialdifferenz, welche durch das Auseinanderschieben aus p entsteht, so hat man

Schließt man also den Kondensator bei zusammengeschobenen Plattensystemen kurz, und schiebt man dann, nachdem die Verbindung unterbrochen ist, die Plattensysteme auseinander, so erhält man zwischen den Plattensystemen Spannungsdifferenzen von der Größenordnung eines halben Millivolt.

Es scheint mir nicht ausgeschlossen zu sein, daß diese Spannungsdifferenzen der Messung zuganglich sind. Falls man nämlich Metallteile elektrisch verbinden und trennen kann, ohne daß hierbei noch andere unregelmäßige Potentialdifferenzen von gleicher Größenordnung wie die soeben berechneten auftreten, so muß man durch Kombination des obigen Plattenkondensators mit einem Multiplikator zum Ziele gelangen können. Es wäre dann ein der Brownschen Bewegung verwandtes Phänomen auf dem Gebiete der Elektrizität gegeben, daß zur Ermittelung der Größe N benutzt werden könnte.

Bern, Dezember 1906.
(Eingegangen 12. Dezember 1906)

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On the Limit of Validity of the Law of Thermodynamic Equilibrium and on the Possibility of a New Determination of the Elementary Quanta; by A. Einstein

(Annalen der Physik 22 (1907): 569-572)
Dated: Bern, December 1906
Received: 12 December 1906
Published: 5 March 1907

Let the state of a physical system be determined in the thermodynamic sense by parameters λ, µ etc. (e.g., readings of a thermometer, length or volume of a body, amount of a substance of a certain kind in one phase). If, as we assume, the system is not interacting with other systems, then, according to the laws of thermodynamics, equilibrium will occur at particular values λ0, µ0 etc. of the parameters, for which the system’s entropy S is a maximum. However, according to the molecular theory of heat, this is not exactly but only approximately correct; according to this theory, the value of the parameter λ is not constant even at temperature equilibrium, but shows irregular fluctuations, though it is very rarely much different from λ0.

At first glance the theoretical examination of the statistical law that governs these fluctuations would seem to require that certain stipulations regarding the molecular model must be applied. However, this is not the case. Rather, essentially it is sufficient to apply the well-known Boltzmann relation connecting the entropy S with the statistical probability of a state. As we know, this relation is

where R is the constant of the gas equation and N is the number of molecules in one gram-equivalent. We consider a state of the system in which the parameter λ has a value λ0 + ɛ differing very little from λ0. To bring the parameter λ from the value λ0 to the value λ along a reversible path at constant energy E, one will have to supply some work A to the system and to withdraw the corresponding amount of heat. According to thermodynamic relations, we have

or, since the change in question is infinitesimally small and ʃ dE = 0

On the other hand, however, according to the connection between entropy and probability of state, we have

From the last two equations it follows that


The result involves a certain degree of inaccuracy, because in fact one cannot talk about the probability of a state, but only about the probability of a state range. If instead of the equation found we write

then the latter law is exact. The arbitrariness due to our having inserted the differential of λ rather than the differential of some function of λ into the equation will not affect our result.

We now put λ = λ0 + ɛ and restrict ourselves to the case that A can be developed in positive powers of ɛ, and that only the first non-vanishing term of this series contributes noticeably to the value of the exponent at such values of ɛ for which the exponential function is still noticeably different from zero. Thus, we put A = αɛ2 and obtain

Thus, in this case there applies the law of chance errors to the deviations ɛ. For the mean value of the work A one obtains

Hence, the mean value of the square of the fluctuation ɛ of a parameter λ is such that, in order to change the parameter λ from λ0 to at constant energy of the system, the external work A that one would have to apply, if thermodynamics were strictly valid, equals ½ RT/N (i.e., one-third of the mean kinetic energy of one atom).

If one inserts the numerical values for R and N, one obtains approximately

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We shall now apply the result obtained to a short-circuited condenser of (electrostatically measured) capacitance c. If is the mean (electrostatic) potential difference that the condenser assumes as a result of molecular disorder, then

We assume that the condenser is an air condenser consisting of two interlocking plate systems containing 30 plates each. The average distance between each plate and the adjacent plate of the other system shall be 1 mm. The size of the plates shall be 100 cm2. The capacitance c is then about 5,000. At normal temperature one then obtains

Measured in volts, one obtains

If one imagines that the two plate systems can move relative to one other, so that they can be completely separated, one can get the capacitance to be of order of magnitude 10 after the plates have been moved apart. If π denotes the potential difference resulting from p due to the separation, one obtains

Thus, if the condenser is short-circuited when the plate systems are pushed together, and the plates are pulled apart after the connection has been broken, potential differences of the order of magnitude of one-half millivolt will result between the plate systems.

It does not seem to me out of the question that these potential differences may be accessible to measurement. For if metal parts can be electrically connected and separated without the occurrence of other irregular potential differences of the same order of magnitude as those calculated above, then it must be possible to achieve the goal by combining the above plate condenser with a multiplier. We would then have a phenomenon akin to Brownian motion in the domain of electricity that could be used for the determination of the quantity N.

Bern, December 1906. (Received on 12 December 1906)

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References to Avogadro’s paper of 1811

D’une manière de déterminer les masses relatives des molécules élémentaires des corps, et les proportions selon lesquelles elles entrent dans ces combinaisons
Journal de Physique, de Chemie et d’Histoire naturelle, 73, 58-76 (1811)

English translation

Bust of Avogadro in Vercelli, Italy.

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P Mander April 2021