Archive for the ‘physics’ Category

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

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

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The experimenter

NPG x122578; Lord Robert John Rayleigh, 4th Baron Rayleigh by Bassano

RJ Strutt (1875-1947) photographed in 1923

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

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

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The experiment

str03

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

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

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

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

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

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The green glow

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

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

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

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

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

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

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The analysis

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

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

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

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

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

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

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

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

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The ratio

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

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

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

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

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

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

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

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

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

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Further reading

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

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

P Mander August 2016

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:

oder

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

or

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)
https://archive.org/details/20Avogadro/mode/2up

English translation
https://web.lemoyne.edu/~giunta/avogadro.html

Bust of Avogadro in Vercelli, Italy.

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P Mander April 2021

The absolute humidity formula posted in 2012 on this blog has a range of -30°C to 35°C. To expand this range I have developed a new formula to compute absolute humidity from relative humidity and temperature based on a simple but little known polynomial expression (Richards, 1971) for the saturation vapor pressure of water, valid to ±0.1% over the temperature range -50°C to 140°C.

Formula for calculating absolute humidity

In the formula below, temperature (T) is expressed in degrees Celsius, relative humidity (rh) is expressed in %, and e is the Euler number 2.71828 [raised to the power of the contents of the square brackets]:

Absolute Humidity = 1013.25 × e^[13.3185t – 1.9760t^2 – 0.6445t^3 – 0.1299t^4] × rh × 18.01528
(grams/m^3)                                                   100 × 0.083145 × (273.15 + T)

where the parameter t = 1 – 373.15/(273.15 + T)

The above formula simplifies to

Absolute Humidity = 1013.25 × e^[13.3185t – 1.9760t^2 – 0.6445t^3 – 0.1299t^4] × rh × 2.1667
(grams/m^3)                                                                   273.15 + T

To cite this formula please quote: P Mander (2020), carnotcycle.wordpress.com/2020/08/01/compute-absolute-humidity-from-relative-humidity-and-temperature-50c-to-140c

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Study notes

Strategy for computing absolute humidity, defined as water vapor density in grams/m3, from temperature (T) and relative humidity (rh):

1. Water vapor is a gas whose behavior in air approximates that of an ideal gas due to its very low partial pressure.

2. We can apply the ideal gas equation PV = nRT. The gas constant R and the variables T and V are known in this case (T is measured, V = 1 m3), but we need to calculate P before we can solve for n.

3. To obtain a value for P, we can use the polynomial expression of Richards (ref) which generates saturation vapor pressure Psat (hectopascals) as a function of temperature T (Celsius) in terms of a parameter t

Psat = 1013.25 × e^[13.3185t – 1.9760t2 – 0.6445t3 – 0.1299t4]
where t = 1 – 373.15/(273.15 + T)

4. Psat is the vapor pressure when the relative humidity is 100%. To compute the pressure P for any value of relative humidity expressed in %, the expression for Psat is multiplied by the factor rh/100:

P = 1013.25 × e^[13.3185t – 1.9760t2 – 0.6445t3 – 0.1299t4] × rh/100

5. We now know P, V, R, T and can solve for n, which is the amount of water vapor in moles. This value is then multiplied by the molecular weight of water to give the answer in grams.

Absolute humidity (grams/m3) = Psat  ×  rh  ×  mol wt
                                                          100 × R × (273.15 + T)

Saturation vapor pressure Psat is expressed in hectopascals hPa
Relative humidity rh is expressed in %
Molecular weight of water mol wt = 18.01528 g mol-1
Gas constant R = 0.083145 m3 hPa K-1 mol-1
Temperature T is expressed in degrees Celsius

6. Summary:
The formula for absolute humidity is derived from the ideal gas equation. It gives a statement of n solely in terms of the variables temperature (T) and relative humidity (rh). Pressure is computed as a function of both these variables; the volume is specified (1 m3) and the gas constant R is known.

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Formula jpgs

decimal separator = .

decimal separator = ,

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P Mander, July 2020

Ventus W636 Weather Station with outdoor sensor

If your weather station displays barometric pressure, temperature and relative humidity like the one pictured above, you can calculate the amount of water vapor in the air expressed either as grams of water vapor per kilogram of dry air (known as Mixing Ratio) or as grams of water vapor per kilogram of vapor-containing air (known as Specific Humidity). The two measures are very similar for cooler air; differences only become apparent for warmer air.

Formulas for calculating Mixing Ratio and Specific Humidity

In the formulas below, barometric pressure P is expressed in hectopascals (hPa), temperature T is expressed in degrees Celsius, relative humidity rh is expressed in %, and e is the Euler number 2.71828 [raised to the power of the contents of the square brackets]:


The decimal separator is shown as a full point (.) In developing these formulas, the following textbook was consulted: Atmospheric Thermodynamics by Grant W. Petty, Sundog Publishing, Madison Wisconsin. ISBN-10: 0-9729033-2-1

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P Mander, June 2020

Taking a break from thermodynamics, a subject which deeply fascinated Einstein and to which he made significant contributions.

Just to be clear, the headline does not refer to any of Einstein’s academic papers in which he presented the theory of relativity. The error occurs in the popular account Einstein later wrote under the title Über die spezielle und die allgemeine Relativitätstheorie; it then reappears in curiously modified form in the authorised English translation Relativity – the special and the general theory.

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The German original

Part 1 of Einstein’s book deals with Special Relativity. Having discussed coordinate systems, the Principle of Relativity (in the restricted sense), the Lorentz Transformation and the behavior of measuring rods and clocks in motion, Einstein presents in §15 the general results of the theory.

It is at this point that he gives the classical expression for kinetic energy, then the new expression for it according to relativity theory, and then employs a binomial expansion to extend this new expression into a series

From the 24th edition, 2009

And this is where the error occurs.

The binomial expansion of the first term in the parentheses is

Einstein seems to have neglected the “–1” term in the parentheses which cancels out the first term in the expansion. The actual result is

This is what you would expect the relativistic kinetic energy expression to look like, since when v<<c all terms other than the first can be neglected and it reduces to the classical expression ½mv2.

Einstein’s result, on the other hand, reduces to mc2 + ½mv2 (*). Since this includes the non-zero rest-mass energy it cannot be a purely kinetic energy expression which he states it to be.

(*) the mass m in these expressions indicates the rest mass m0

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The English translation

In 1920 an authorised English translation of Einstein’s popular book was published. The curious feature of this translation is that the parenthetic “–1” term in the relativistic kinetic energy expression has been removed, as can be seen below

From the 15th edition, reprinted 1979

This does not rectify the original error, but simply transfers it from one expression to the other. For while the series expression is a correct extension, the relativistic kinetic energy expression from which it is obtained is incorrect as it now contains the non-zero rest-mass energy.

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