Archive for the ‘thermodynamics’ Category

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.

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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.


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:

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


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

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

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

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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

Über die vom Relativitätsprinzip geforderte Trägheit der Energie; von A. Einstein
Annalen der Physik 23 (1907) 371-384

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P Mander June 2017



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.

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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


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:




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


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


For N = 2


For N = 3


For N = 4


It appears that in general


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


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).

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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



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


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


where the terms in the denominator are as defined above.

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P Mander April 2016



The Arrhenius equation explains why chemical reactions generally go much faster when you heat them up. The equation was actually first given by the Dutch physical chemist JH van ‘t Hoff in 1884, but it was the Swedish physical chemist Svante Arrhenius (pictured above) who in 1889 interpreted the equation in terms of activation energy, thereby opening up an important new dimension to the study of reaction rates.

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Temperature and reaction rate

The systematic study of chemical kinetics can be said to have begun in 1850 with Ludwig Wilhelmy’s pioneering work on the kinetics of sucrose inversion. Right from the start, it was realized that reaction rates showed an appreciable dependence on temperature, but it took four decades before real progress was made towards quantitative understanding of the phenomenon.

In 1889, Arrhenius penned a classic paper in which he considered eight sets of published data on the effect of temperature on reaction rates. In each case he showed that the rate constant could be represented as an explicit function of the absolute temperature:


where both A and C are constants for the particular reaction taking place at temperature T. In his paper, Arrhenius listed the eight sets of published data together with the equations put forward by their respective authors to express the temperature dependence of the rate constant. In one case, the equation – stated in logarithmic form – was identical to that proposed by Arrhenius


where T is the absolute temperature and a and b are constants. This equation was published five years before Arrhenius’ paper in a book entitled Études de Dynamique Chimique. The author was J. H. van ‘t Hoff.

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Dynamic equilibrium

In the Études of 1884, van ‘t Hoff compiled a contemporary encyclopædia of chemical kinetics. It is an extraordinary work, containing all that was previously known as well as a great deal that was entirely new. At the start of the section on chemical equilibrium he states (without proof) the thermodynamic equation, sometimes called the van ‘t Hoff isochore, which quantifies the displacement of equilibrium with temperature. In modern notation it reads:


where Kc is the equilibrium constant expressed in terms of concentrations, ΔH is the heat of reaction and T is the absolute temperature. In a footnote to this famous and thermodynamically exact equation, van ‘t Hoff builds a bridge from thermodynamics to kinetics by advancing the idea that a chemical reaction can take place in both directions, and that the thermodynamic equilibrium constant Kc is in fact the quotient of the kinetic velocity constants for the forward (k1) and reverse (k-1) reactions


Substituting this quotient in the original equation leads immediately to


van ‘t Hoff then argues that the rate constants will be influenced by two different energy terms E1 and E-1, and splits the above into two equations


where the two energies are such that E1 – E-1 = ΔH

In the Études, van ‘t Hoff recognized that ΔH might or might not be temperature independent, and considered both possibilities. In the former case, he could integrate the equation to give the solution


From a starting point in thermodynamics, van ‘t Hoff engineered this kinetic equation through a characteristically self-assured thought process. And it was this equation that the equally self-assured Svante Arrhenius seized upon for his own purposes, expanding its application to explain the results of other researchers, and enriching it with his own idea for how the equation should be interpreted.

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Activation energy

It is a well-known result of the kinetic theory of gases that the average kinetic energy per mole of gas (EK) is given by


Since the only variable on the RHS is the absolute temperature T, we can conclude that doubling the temperature will double the average kinetic energy of the molecules. This set Arrhenius thinking, because the eight sets of published data in his 1889 paper showed that the effect of temperature on the rates of chemical processes was generally much too large to be explained on the basis of how temperature affects the average kinetic energy of the molecules.

The clue to solving this mystery was provided by James Clerk Maxwell, who in 1860 had worked out the distribution of molecular velocities from the laws of probability. Maxwell’s distribution law enables the fraction of molecules possessing a kinetic energy exceeding some arbitrary value E to be calculated.

It is convenient to consider the distribution of molecular velocities in two dimensions instead of three, since the distribution law so obtained gives very similar results and is much simpler to apply. At absolute temperature T, the proportion of molecules for which the kinetic energy exceeds E is given by


where n is the number of molecules with kinetic energy greater than E, and N is the total number of molecules. This is exactly the exponential expression which occurs in the velocity constant equation derived by van ‘t Hoff from thermodynamic principles, which Arrhenius showed could be fitted to temperature dependence data from several published sources.

Compared with the average kinetic energy calculation, this exponential expression yields very different results. At 1000K, the fraction of molecules having a greater energy than, say, 80 KJ is 0.0000662, while at 2000K the fraction is 0.00814. So the temperature change which doubles the number of molecules with the average energy will increase the number of molecules with E > 80 KJ by a factor of more than a hundred.

Here was the clue Arrhenius was seeking to explain why increased temperature had such a marked effect on reaction rate. He reasoned it was because molecules needed sufficiently more energy than the average – the activation energy E – to undergo reaction, and that the fraction of these molecules in the reaction mix was an exponential function of temperature.

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The meaning of A

But back to the Arrhenius equation


I have always thought that calling the constant A the ‘pre-exponential factor’ is a singularly pointless label. One could equally write the equation as


and call it the ‘post-exponential factor’. The position of A in relation to the exponential factor has no relevance.

A clue to the proper meaning of A is to note that e^(–E/RT) is dimensionless. The units of A are therefore the same as the units of k. But what are the units of k?

The answer depends on whether one’s interest area is kinetics or thermodynamics. In kinetics, the concentration of chemical species present at equilibrium is generally expressed as molar concentration, giving rise to a range of possibilities for the units of the velocity constant k.

In thermodynamics however, the dimensions of k are uniform. This is because the chemical potential of reactants and products in any arbitrarily chosen state is expressed in terms of activity a, which is defined as a ratio in relation to a standard state and is therefore dimensionless.

When the arbitrarily chosen conditions represent those for equilibrium, the equilibrium constant K is expressed in terms of reactant (aA + bB + …) and product (mM + nN + …) activities


where the subscript e indicates that the activities are those for the system at equilibrium.

As students we often substitute molar concentrations for activities, since in many situations the activity of a chemical species is approximately proportional to its concentration. But if an equation is arrived at from consideration of the thermodynamic equilibrium constant K – as the Arrhenius equation was – it is important to remember that the associated concentration terms are strictly dimensionless and so the reaction rate, and therefore the velocity constant k, and therefore A, has the units of frequency (t^-1).

OK, so back again to the Arrhenius equation


We have determined the dimensions of A; now let us turn our attention to the role of the dimensionless exponential factor. The values this term may take range between 0 and 1, and specifically when E = 0, e^(–E/RT) = 1. This allows us to assign a physical meaning to A since when E = 0, A = k. We can think of A as the velocity constant when the activation energy is zero – in other words when each collision between reactant molecules results in a reaction taking place.

Since there are zillions of molecular collisions taking place every second just at room temperature, any reaction in these circumstances would be uber-explosive. So the exponential term can be seen as a modifier of A whose value reflects the range of reaction velocity from extremely slow at one end of the scale (high E/low T) to extremely fast at the other (low E/high T).

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P Mander September 2016

CarnotCycle is honored to announce that Rutgers, The State University of New Jersey, has included this thermodynamics blog on the reading list for students of physical chemistry. Founded in 1766, Rutgers is the eighth oldest college in the United States and is the largest institution for higher education in New Jersey.

CarnotCycle is committed to making topics in this area of science accessible to students worldwide. Thermodynamics has played – and continues to play – a major role in shaping our world. It can be a difficult subject, but time spent learning about thermodynamics is never wasted. It enriches knowledge and empowers the mind.


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Relative humidity (RH) and temperature (T) data from an RH&T sensor like the DHT22 can be used to compute not only absolute humidity AH but also dew point temperature TD

There has been a fair amount of interest in my formula which computes AH from measured RH and T, since it adds value to the output of RH&T sensors. To further extend this value, I have developed another formula which computes dew point temperature TD from measured RH and T.

Formula for computing dew point temperature TD

In this formula (P Mander 2017) the measured temperature T and the computed dew point temperature TD are expressed in degrees Celsius, and the measured relative humidity RH is expressed in %

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Strategy for computing TD from RH and T

1. The dew point temperature TD is defined in the following relation where RH is expressed in %

2. To obtain values for Psat, we can use the Bolton formula[REF, eq.10] which generates saturation vapor pressure Psat (hectopascals) as a function of temperature T (Celsius)

These formulas are stated to be accurate to within 0.1% over the temperature range –30°C to +35°C

3. Substituting in the first equation yields

Taking logarithms


Separating TD terms on one side yields

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Spreadsheet formula for computing TD from RH and T

1) Set up data entry cells for RH in % and T in degrees Celsius.

2) Depending on whether your spreadsheet uses a full point (.) or comma (,) for the decimal separator, copy the appropriate formula below and paste it into the computation cell for TD.

Formula for TD (decimal separator = .)


Formula for TD (decimal separator = ,)


3) Replace T and RH in the formula with the respective cell references. (see comment)

Your spreadsheet is now complete. Enter values for RH and T, and the TD computation cell will return the dew point temperature. If an object whose temperature is at or below this temperature is present in the local space, the thermodynamic conditions are satisfied for water vapor to condense (or freeze if TD is below 0°C) on the surface of the object.

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P Mander August 2017