Archive for the ‘physics’ Category

CarnotCycle is a thermodynamics blog but occasionally its enthusiasm spills over into other subjects, as is the case here.
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When one considers the great achievements in radioactivity research made at the start of the 20th century by Ernest Rutherford and his team at the Victoria University, Manchester it seems surprising how little progress they made in finding an answer to the question posed above.

They knew that radioactivity was unaffected by any agency applied to it (even temperatures as low as 20K), and since the radioactive decay law discovered in 1902 by Rutherford and Soddy was an exponential function associated with probabilistic behavior, it was reasonable to think that radioactivity might be a random process. Egon von Schweidler’s work pointed firmly in this direction, and the Geiger-Nuttall relation, formulated by Hans Geiger and John Nuttall at the Manchester laboratory in 1911 and reformulated in 1912 by Richard Swinne in Germany, laid a mathematical foundation on which to construct ideas. Yet despite these pointers, Rutherford wrote in 1912 that “it is difficult to offer any explanation of the causes operating which lead to the ultimate disintegration of the atom”.

The phrase “causes operating which lead to” indicates that Rutherford saw the solution in terms of cause and effect. Understandably so, since he came from an age where probability was regarded as a measure of uncertainty about exact cause, rather than something reflecting a naturally indeterministic process. C.P. Snow once said of Rutherford, “He thought of atoms as though they were tennis balls”. And therein lay the essence of his problem: he didn’t have the right kind of mind to answer this kind of question.

But someone else did, namely the pioneer who introduced the term radioactivity and gave it a quantifiable meaning – Maria Sklodowska, better known under her married name Marie Curie.

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Mme. Curie’s idea

The 2nd Solvay Conference (1913) La structure de la matière (The structure of matter)

When all the great men of science (and one woman) convened for the second Solvay Conference in 1913, the hot topic of the day was the structure of the atom. Hans Geiger and Ernest Marsden at Rutherford’s Manchester lab had recently conducted their famous particle scattering experiment, enabling Rutherford to construct a model of the atom with a central nucleus where its positive charge and most of its mass were concentrated. Rutherford and his student Thomas Royds had earlier conducted their celebrated experiment which identified the alpha particle as a helium nucleus, so the attention now focused on trying to explain the process of alpha decay.

It was Marie Curie who produced the most fruitful idea, foreshadowing the quantum mechanical interpretation developed in the 1920s. Curie suggested that alpha decay could be likened to a particle bouncing around inside a box with a small hole through which the particle could escape. This would constitute a random event; with a large number of boxes these events would follow the laws of probability, even though the model was conceptually based on simple kinetics.

Now it just so happened that a probability distribution based on exactly this kind of random event had already been described in an academic paper, published in 1837 and rather curiously entitled Recherches sur la probabilité des jugements en matière criminelle et matière civile (Research on the probability of judgments in criminal and civil cases). The author was the French mathematician Siméon Denis Poisson (1781-1840).

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The Poisson distribution

At the age of 57, just three years before his death, Poisson turned his attention to the subject of court judgements, and in particular to miscarriages of justice. In probabilistic terms, Poisson was considering a large number of trials (excuse the pun) involving just two outcomes – a correct or an incorrect judgement. And with many years of court history on the public record, Poisson had the means to compute a time-averaged figure for the thankfully rare judicial failures.

In his 1837 paper Poisson constructed a model which regarded an incorrect judgement as a random event which did not influence any other subsequent judgement – in other words it was an independent random event. He was thus dealing with a random variable in the context of a binomial experiment with a large number of trials (n) and a small probability (p), whose product (pn) he asserted was finite and equal to µ, the mean number of events occurring in a given number of dimensional units (in this case, time).

In summary, Poisson started with the binomial probability distribution

where p is the probability of success and q is the probability of failure, in which successive terms of the binomial expansion give the probability of the event occurring exactly r times in n trials

Asserting µ = pn, he evaluated P(r) as n goes to infinity and found that

This is the general representation of each term in the Poisson probability distribution

which can be seen from

As indicated above, the mean µ is the product of the mean per unit dimension and the number of dimensional units. In the case of radioactivity, µ = λt where λ is the decay constant and t is the number of time units

If we set t equal to the half-life t½ the mean µ will be λt½ = ln 2. Mapping probabilities for the first few terms of the distribution yields

Unlike the binomial distribution, the Poisson distribution is not symmetric; the maximum does not correspond to the mean. In the case of µ = ln2 the probability of no decays (r = 0) is exactly a half, as can be seen from

At this point we turn to another concept introduced by Poisson in his paper which was taken further by the Russian mathematician P.L. Chebyshev – namely the law of large numbers. In essence, this law says that if the probability of an event is p, the average number of occurrences of the event approaches p as the number of independent trials increases.

In the case of radioactive decay, the number of independent trials (atoms) is extremely large: a µg sample of Cesium 137 for example will contain around 10^15 nuclei. In the case of µ = ln2 the law of large numbers means that the average number of atoms remaining intact after the half-life period will be half the number of atoms originally present in the sample.

The Poisson distribution correctly accounts for half-life behavior, and has been successfully applied to counting rate experiments and particle scattering. There is thus a body of evidence to support the notion that radioactive decay is a random event to which the law of large numbers applies, and is therefore not a phenomenon that requires explanation in terms of cause and effect.

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Geiger and Nuttall

Despite Ernest Rutherford’s protestations that atomic disintegration defied explanation, it was in fact Rutherford who took the first step along the path that would eventually lead to a quantum mechanical explanation of α-decay. In 1911 and again in 1912, Rutherford communicated papers by two of his Manchester co-workers, physics lecturer Hans Geiger (of Geiger counter fame) and John Nuttall, a graduate student.

Rutherford’s team at the Physical Laboratories was well advanced with identifying radioactive decay products, several of which were α-emitters. It had been noticed that α-emitters with more rapid decay rates had greater α-particle ranges. Geiger and Nuttall investigated this phenomenon, and when they plotted the logarithms of the decay constants (they called them transformation constants) against the logarithms of the corresponding α-particle ranges for decay products in the uranium and actinium series they got this result (taken from their 1911 paper):

This implies the existence of a relationship log λ = A + B log R, where A has a characteristic value for each series and B has the same value for both series. Curiously, Geiger and Nuttall did not express the straight lines in mathematical terms in either of their papers; they were more interested in using the lines to calculate the immeasurably short periods of long-range α-emitters. But they made reference in their 1912 paper to somebody who had “recently shown that the relation between range and transformation constant can be expressed in another form”.

That somebody was the German physicist Richard Swinne (1885-1939) who sent a paper entitled Über einige zwischen den radioaktiven Elementen bestehene Beziehungen (On some relationships between the radioactive elements) to Physikalische Zeitschrift, which the journal received on Tuesday 5th December 1911 and published in volume XIII, 1912.

The other form that Swinne had found, which he claimed to represent the experimental data at least as well as the (unstated) formula of Geiger and Nuttall, was log λ = a + bv^n, where a and b are constants and v is the particle velocity.

When it came to n, Swinne was rangefinding: he tried various values of n and found that “n kann am besten gleich 1 gesetzt werden”; he was thus edging towards what we now call the Geiger-Nuttall law, namely that the logarithm of the α-emitter’s half-life is inversely proportional to the square root of the α-particle’s kinetic energy

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Gurney and Condon, and Gamow

The potential well diagram in Gurney and Condon’s article

In 1924, the British mathematician Harold Jeffreys developed a general method of approximating solutions to linear, second-order differential equations. This method, rediscovered as the WKB approximation in 1926, was applied to the Schrödinger equation first published in that year and resulted in the discovery of the phenomenon known as quantum tunneling.

It was this strange effect, by which a particle with insufficient energy to surmount a potential barrier can effectively tunnel through it (the dotted line DB) that was seized upon in 1928 by Ronald Gurney and Edward Condon at Princeton – and independently by George Gamow at Göttingen – as a way of explaining alpha decay. Gurney and Condon’s explanation of alpha emission was published in Nature in an article entitled Wave Mechanics and Radioactive Disintegration, while Gamow’s considerably more academic (and mathematical) paper Zur Quantentheorie des Atomkernes was published in Zeitschrift für Physik.

In the quantum mechanical treatment, the overall rate of emission (i.e. the decay constant λ) is the product of a frequency factor f – the rate at which an alpha particle appears at the inside wall of the nucleus – multiplied by a transmission coefficient T, which is the (independent) probability that the alpha particle tunnels through the barrier. Thus

At this point it is instructive to recall Marie Curie’s particle-in-a-box idea, a concept which involves the product of two quantities: a large number of escape attempts and a small probability of escape.

The frequency factor f – or escape attempt rate – is estimated as the particle velocity v divided by the distance across the nucleus (2R) where R is the radius

Here, V0 is the potential well depth, Qα is the alpha particle kinetic energy and µ is the reduced mass. The escape attempt rate is quite large, usually of the order of 1021 per second. By contrast the probability of alpha particle escape is extremely small. In calculating a value for T, Gamow introduced the Gamow factor 2G where

Typically the Gamow factor is very large (2G = 60-120) which makes T very small (T = 10-55-10-27).

Combining the equations

or

which is the Geiger-Nuttall law.

The work of Gurney, Condon and Gamow provided a convincing theoretical explanation of the Geiger-Nuttall law on the basis of quantum mechanics and Marie Curie’s hunch, and put an end to the classical notions of Rutherford’s generation that radioactive decay required explanation in terms of cause and effect.

So to return to the question posed at the head of this post – What determines the moment at which a radioactive atom decays? – the answer is chance. And the law of large numbers.

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An important consequence

George Gamow and John Cockroft

The successful application of quantum tunneling to alpha particle emission had an important consequence, since it suggested to Gamow that the same idea could be applied in reverse i.e. that projectile particles with lower energy might be able to penetrate the nucleus through quantum tunneling. This led Gamow to suggest to John Cockroft, who was conducting atom-smashing experiments with protons, that protons with more moderate speeds could be used. Gamow’s suggestion proved correct, and the success of these trials ushered in a new era of intensive development in nuclear physics.

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Links to original papers mentioned in this post

G. Gamow (1928) Zur Quantentheorie des Atomkernes, Zeitschrift für Physik; 51: 204-212
https://link.springer.com/article/10.1007/BF01343196

H. Geiger and J.M. Nuttall (1911) The ranges of the α particles from various radioactive substances and a relation between range and period of transformation, Phil Mag; 22: 613-621
https://archive.org/stream/londonedinburg6221911lond#page/612/mode/2up

H. Geiger and J.M. Nuttall (1912) The ranges of α particles from uranium, Phil Mag; 23: 439-445
https://archive.org/stream/londonedinburg6231912lond#page/438/mode/2up

R.W. Gurney and E.U. Condon (1928) Wave Mechanics and Radioactive disintegration, Nature; 122 (Sept. 22): 439
http://www.nature.com/physics/looking-back/gurney/index.html

R. Swinne (1912) Über einige zwischen den radioaktiven Elementen bestehene Beziehungen, Physikalische Zeitschrift; XIII: 14-21
https://babel.hathitrust.org/cgi/pt?id=mdp.39015023176806;view=1up;seq=52;size=125

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© P Mander August 2017

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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 dewpoint 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 dewpoint temperature TD from measured RH and T.

Formula for computing dewpoint temperature TD

In this formula (P Mander 2017) the measured temperature T and the computed dewpoint 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 dewpoint 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 saturated 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

Rearranging

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 = .)

=243.5*(LN(RH/100)+((17.67*T)/(243.5+T)))/(17.67-LN(RH/100)-((17.67*T)/(243.5+T)))

Formula for TD (decimal separator = ,)

=243,5*(LN(RH/100)+((17,67*T)/(243,5+T)))/(17,67-LN(RH/100)-((17,67*T)/(243,5+T)))

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 dewpoint 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

From the perspective of classical thermodynamics, osmosis has a rather unclassical history. Part of the reason for this, I suspect, is that osmosis was originally categorised under the heading of biology. I can remember witnessing the first practical demonstration of osmosis in a biology class, the phenomenon being explained in terms of pores (think invisible holes) in the membrane that were big enough to let water molecules through, but not big enough to let sucrose molecules through. It was just like a kitchen sieve, we were told. It lets the fine flour pass through but not clumps. This was very much the method of biology in my day, explaining things in terms of imagined mechanism and analogy.

And it wasn’t just in my day. In 1883, JH van ‘t Hoff, an able theoretician and one of the founders of the new discipline of physical chemistry, became suddenly convinced that solutions and gases obeyed the same fundamental law, pv = RT. Imagined mechanism swiftly followed. In van ‘t Hoff’s interpretation, osmotic pressure depended on the impact of solute molecules against the semipermeable membrane because solvent molecules, being present on both sides of the membrane through which they could freely pass, did not enter into consideration.

It all seemed very plausible, especially when van ‘t Hoff used the osmotic pressure measurements of the German botanist Wilhelm Pfeffer to compute the value of R in what became known as the van ‘t Hoff equation

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where Π is the osmotic pressure, and found that the calculated value for R was almost identical with the familiar gas constant. There really did seem to be a parallelism between the properties of solutions and gases.

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JH van ‘t Hoff (1852-1911)

The first sign that there was anything amiss with the so-called gaseous theory of solutions came in 1891 when van ‘t Hoff’s close colleague Wilhelm Ostwald produced unassailable proof that osmotic pressure is independent of the nature of the membrane. This meant that hypothetical arguments as to the cause of osmotic pressure, such as van ‘t Hoff had used as the basis of his theory, were inadmissible.

A year later, in 1892, van ‘t Hoff changed his stance by declaring that the mechanism of osmosis was unimportant. But this did not affect the validity of his osmotic pressure equation ΠV = RT. After all, it had been shown to be in close agreement with experimental data for very dilute solutions.

It would be decades – the 1930s in fact – before the van ‘t Hoff equation’s formal identity with the ideal gas equation was shown to be coincidental, and that the proper thermodynamic explanation of osmotic pressure lay elsewhere.

But long before the 1930s, even before Wilhelm Pfeffer began his osmotic pressure experiments upon which van ‘t Hoff subsequently based his ideas, someone had already published a thermodynamically exact rationale for osmosis that did not rely on any hypothesis as to cause.

That someone was the American physicist Josiah Willard Gibbs. The year was 1875.

gibbs

J. Willard Gibbs (1839-1903)

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Osmosis without mechanism

It is a remarkable feature of Gibbs’ On the Equilibrium of Heterogeneous Substances that having introduced the concept of chemical potential, he first considers osmotic forces before moving on to the fundamental equations for which the work is chiefly known. The reason is Gibbs’ insistence on logical order of presentation. The discussion of chemical potential immediately involves equations of condition, among whose different causes are what Gibbs calls a diaphragm, i.e. a semipermeable membrane. Hence the early appearance of the following section

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In equation 77, Gibbs presents a new way of understanding osmotic pressure. He makes no hypotheses about how a semipermeable membrane might work, but simply states the equations of condition which follow from the presence of such a membrane in the kind of system he describes.

This frees osmosis from considerations of mechanism, and explains it solely in terms of differences in chemical potential in components which can pass the diaphragm while other components cannot.

In order to achieve equilibrium between say a solution and its solvent, where only the solvent can pass the diaphragm, the chemical potential of the solvent in the fluid on both sides of the membrane must be the same. This necessitates applying additional pressure to the solution to increase the chemical potential of the solvent in the solution so it equals that of the pure solvent, temperature remaining constant. At equilibrium, the resulting difference in pressure across the membrane is the osmotic pressure.

Note that increasing the pressure always increases the chemical potential since

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is always positive (V1 is the partial molar volume of the solvent in the solution).

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Europe fails to notice (almost)

Gibbs published On the Equilibrium of Heterogeneous Substances in Transactions of the Connecticut Academy. Choosing such an obscure journal (seen from a European perspective) clearly would not attract much attention across the pond, but Gibbs had a secret weapon. He had a mailing list of the world’s greatest scientists to which he sent reprints of his papers.

One of the names on that list was James Clerk Maxwell, who instantly appreciated Gibbs’ work and began to promote it in Europe. On Wednesday 24 May 1876, the year that ‘Equilibrium’ was first published, Maxwell gave an address at the South Kensington Conferences in London on the subject of Gibbs’ development of the doctrine of available energy on the basis of his new concept of the chemical potentials of the constituent substances. But the audience did not share Maxwell’s enthusiasm, or in all likelihood share his grasp of Gibbs’ ideas. When Maxwell tragically died three years later, Gibbs’ powerful ideas lost their only real champion in Europe.

It was not until 1891 that interest in Gibbs masterwork would resurface through the agency of Wilhelm Ostwald, who together with van ‘t Hoff and Arrhenius were the founders of the modern school of physical chemistry.

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Wilhelm Ostwald (1853-1932) He not only translated Gibbs’ masterwork into German, but also produced a profound proof – worthy of Sadi Carnot himself – that osmotic pressure must be independent of the nature of the semipermeable membrane.

Although perhaps overshadowed by his colleagues, Ostwald had a talent for sensing the direction that the future would take and was also a shrewd judge of intellect – he instinctively felt that there were hidden treasures in Gibbs’ magnum opus. After spending an entire year translating ‘Equilibrium’ into German, Ostwald wrote to Gibbs:

“The translation of your main work is nearly complete and I cannot resist repeating here my amazement. If you had published this work over a longer period of time in separate essays in an accessible journal, you would now be regarded as by far the greatest thermodynamicist since Clausius – not only in the small circle of those conversant with your work, but universally—and as one who frequently goes far beyond him in the certainty and scope of your physical judgment. The German translation, hopefully, will more secure for it the general recognition it deserves.”

The following year – 1892 – another respected scientist sent a letter to Gibbs regarding ‘Equilibrium’. This time it was the British physicist, Lord Rayleigh, who asked Gibbs:

“Have you ever thought of bringing out a new edition of, or a treatise founded upon, your “Equilibrium of Het. Substances.” The original version though now attracting the attention it deserves, is too condensed and too difficult for most, I might say all, readers. The result is that as has happened to myself, the idea is not grasped until the subject has come up in one’s own mind more or less independently.”

Rayleigh was probably just being diplomatic when he remarked that Gibbs’ treatise was ‘now attracting the attention it deserves’. The plain fact is that nobody gave it any attention at all. Gibbs and his explanation of osmosis in terms of chemical potential was passed over, while European and especially British theoretical work centered on the more familiar and more easily understood concept of vapor pressure.

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Gibbs tries again

Although van ‘t Hoff’s osmotic pressure equation ΠV = RT soon gained the status of a law, the gaseous theory that lay behind it remained clouded in controversy. In particular, van ‘t Hoff’s deduction of the proportionality between osmotic pressure and concentration was an analogy rather than a proof, since it made use of hypothetical considerations as to the cause of osmotic pressure. Following Ostwald’s proof that these were inadmissible, the gaseous theory began to look hollow. A better theory was needed.

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Lord Kelvin (1824-1907) and Lord Rayleigh (1842-1919)

This was provided in 1896 by the British physicist, Lord Rayleigh, whose proof was free of hypothesis but did make use of Avogadro’s law, thereby continuing to assert a parallelism between the properties of solutions and gases. Heavyweight opposition to this soon materialized from the redoubtable Lord Kelvin. In a letter to Nature (21 January 1897) he charged that the application of Avogadro’s law to solutions had “manifestly no theoretical foundation at present” and further contended that

“No molecular theory can, for sugar or common salt or alcohol, dissolved in water, tell us what is the true osmotic pressure against a membrane permeable to water only, without taking into account laws quite unknown to us at present regarding the three sets of mutual attractions or repulsions: (1) between the molecules of the dissolved substance; (2) between the molecules of water; (3) between the molecules of the dissolved substance and the molecules of water.”

Lord Kelvin’s letter in Nature elicited a prompt response from none other than Josiah Willard Gibbs in America. Twenty-one years had now passed since James Clerk Maxwell first tried to interest Europe in the concept of chemical potentials. In Kelvin’s letter, with its feisty attack on the gaseous theory, Gibbs saw the opportunity to try again.

In his letter to Nature (18 March 1897), Gibbs opined that “Lord Kelvin’s very interesting problem concerning molecules which differ only in their power of passing a diaphragm, seems only to require for its solution the relation between density and pressure”, and highlighted the advantage of using his potentials to express van ‘t Hoff’s law:

“It will be convenient to use certain quantities which may be called the potentials of the solvent and of the solutum, the term being thus defined: – In any sensibly homogeneous mass, the potential of any independently variable component substance is the differential coefficient of the thermodynamic energy of the mass taken with respect to that component, the entropy and volume of the mass and the quantities of its other components remaining constant. The advantage of using such potentials in the theory of semi-permeable diaphragms consists partly in the convenient form of the condition of equilibrium, the potential for any substance to which a diaphragm is freely permeable having the same value on both sides of the diaphragm, and partly in our ability to express van’t Hoff law as a relation between the quantities characterizing the state of the solution, without reference to any experimental arrangement.”

But once again, Gibbs and his chemical potentials failed to garner interest in Europe. His timing was also unfortunate, since British experimental research into osmosis was soon to be stimulated by the aristocrat-turned-scientist Lord Berkeley, and this in turn would stimulate a new band of British theoreticians, including AW Porter and HL Callendar, who would base their theoretical efforts firmly on vapor pressure.

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Things Come Full Circle

As the new century dawned, van ‘t Hoff cemented his reputation with the award of the very first Nobel Prize for Chemistry “in recognition of the extraordinary services he has rendered by the discovery of the laws of chemical dynamics and osmotic pressure in solutions”.

The osmotic pressure law was held in high esteem, and despite Lord Kelvin’s protestations, Britain was well disposed towards the Gaseous Theory of Solutions. The idea circulating at the time was that the refinements of the ideal gas law that had been shown to apply to real gases, could equally well be applied to more concentrated solutions. As Lord Berkeley put it in the introduction to a paper communicated to the Royal Society in London in May 1904:

“The following work was undertaken with a view to obtaining data for the tentative application of van der Waals’ equation to concentrated solutions. It is evidently probable that if the ordinary gas equation be applicable to dilute solutions, then that of van der Waals, or one of analogous form, should apply to concentrated solutions – that is, to solutions having large osmotic pressures.”

Lord Berkeley’s landmark experimental studies on the osmotic pressure of concentrated solutions called renewed attention to the subject among theorists, who now had some fresh and very accurate data to work with. Alfred Porter at University College London attempted to make a more complete theory by considering the compressibility of a solution to which osmotic pressure was applied, while Hugh Callendar at Imperial College London combined the vapor pressure interpretation of osmosis with the hypothesis that osmosis could be described as vapor passing through a large number of fine capillaries in the semipermeable membrane. This was in 1908.

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H L Callendar (1863-1930)

So seventeen years after Wilhelm Ostwald conclusively proved that hypothetical arguments as to the cause of osmotic pressure were inadmissible, things came full circle with hypothetical arguments once more being advanced as to the cause of osmotic pressure.

And as for Gibbs, his ideas were as far away as ever from British and European Science. The osmosis papers of both Porter (1907) and Callendar (1908) are substantial in referenced content, but nowhere do either of them make any mention of Gibbs or his explanation of osmosis on the basis of chemical potentials.

There is a special irony in this, since in Callendar’s case at least, the scientific papers of J Willard Gibbs were presumably close at hand. Perhaps even on his office bookshelf. Because that copy of Gibbs’ works shown in the header photo of this post – it’s a 1906 first edition – was Hugh Callendar’s personal copy, which he signed on the front endpaper.

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Hugh Callendar’s signature on the endpaper of his personal copy of Gibbs’ Scientific Papers, Volume 1, Thermodynamics.

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Epilogue

Throughout this post, I have made repeated references to that inspired piece of thinking by Wilhelm Ostwald which conclusively demonstrated that osmotic pressure must be independent of the nature of the membrane.

Ostwald’s reasoning is so lucid and compelling, that one wonders why it didn’t put an end to speculation on osmotic mechanisms. But it didn’t, and hasn’t, and probably won’t.

Here is how Ostwald presented the argument in his own Lehrbuch der allgemeinen Chemie (1891). Enjoy.

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“… it may be stated with certainty that the amount of pressure is independent of the nature of the membrane, provided that the membrane is not permeable by the dissolved substance. To understand this, let it be supposed that two separating partitions, A and B, formed of different membranes, are placed in a cylinder (fig. 17). Let the space between the membranes contain a solution and let there be pure water in the space at the ends of the cylinder. Let the membrane A show a higher pressure, P, and the membrane B show a smaller pressure, p. At the outset, water will pass through both membranes into the inner space until the pressure p is attained, when the passage of water through B will cease, but the passage through A will continue. As soon as the pressure in the inner space has been thus increased above p, water will be pressed out through B. The pressure can never reach the value P; water must enter continuously through A, while a finite difference of pressures is maintained. If this were realised we should have a machine capable of performing infinite work, which is impossible. A similar demonstration holds good if p>P ; it is, therefore, necessary that P=p; in other words, it follows necessarily that osmotic pressure is independent of the nature of the membrane.”

(English translation by Matthew Pattison Muir)

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P Mander July 2015

Radon gas levels in indoor spaces are known to fluctuate considerably, so continuous monitoring is necessary to compute long-term averages. This particular radon detector, which uses continuous air sampling coupled to algorithm-based alpha spectrometry, is designed to do this job and has gained good reviews on Amazon. It is made in Norway by Corentium AS.

My unit has been in continuous operation since October 2015. Although the short-term average figure goes up and down from day to day, and to a lesser extent from week to week (the display shows alternating 1-day and 7-day figures), the long-term average figure is really quite steady.

I was thinking about this the other day, and it occurred to me that since the long-term figure varies so little in a month’s turning, I could use it to estimate the entry rate of radon gas into the enclosed space where the device is located.

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Formula for computing entry rate

1. Units in becquerels per cubic meter (Bq/m3)

If the device is showing a steady long-term average figure (n) and the enclosed space has a volume of v cubic meters, the entry rate of radon gas is computed as follows:
Entry rate = 8.78nv attomoles per month
For example, if n = 79 and v = 5
Entry rate = 8.78 × 79 × 5 = 3468 attomoles per month
(1 attomole = 10-18 moles)

2. Units in picocuries per litre (pCi/L)

If the device is showing a steady long-term average figure (n) and the enclosed space has a volume of v cubic meters, the entry rate of radon gas is computed as follows:
Entry rate = 324.74nv attomoles per month
For example, if n = 4.64 and v = 5
Entry rate = 324.74 × 4.64 × 5 = 7534 attomoles per month
(1 attomole = 10-18 moles)

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Explanatory notes
First I should clarify what I mean by entry rate. Radon is a gas that seeps into enclosed spaces through conduits, joints and cracks; it is also exhaled by diffusion through surfaces. Having infiltrated the space, some of the radon will escape, either through back-diffusion or infiltration into adjacent spaces. Without knowing the rates of ingress and escape, one can conclude that a steady long-term average figure on the detector, which implies a steady concentration of radon in the enclosed space, indicates equilibrium between the rate of radon ingress on the one hand, and the rate of radon escape and decay on the other. In other words at equilibrium

Defining entry rate as the difference between ingress rate and escape rate, we have

Given the premise that the concentration of radon gas in the enclosed space is steady, the decay rate can be taken as constant since it is determined solely by the concentration – i.e. the number of radon atoms present in a given volume. So a steady long-term average on the detector means that the entry rate, as here defined, is also constant.

Radon is a very dense gas, almost 8 times as dense as air, and this tempts many to think that radon accumulates at the bottom of an enclosed space. This is not what happens. Like any gas, radon exhibits the phenomenon of diffusion – which is the tendency of a substance to spread uniformly throughout the space available to it. What the density of a gas does affect is the rate at which the gas diffuses. But given sufficient time to reach a state of equilibrium, it can be assumed that the concentration of radon gas will be uniform throughout the enclosed space.

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Assigning a unit of time
So far I have said nothing concerning the unit of time to be applied in relation to the foregoing rate equation. Now we can address this issue, which constitutes the novelty of the computation scheme.

— Let the unit of time by which rate is measured be set equal to the half-life of the isotope (Rn-222) of which radon gas is largely composed.

Theorem
Let the entry rate of radon gas into a previously radon-free bounded space be x atoms per unit of time corresponding to the half-life of Rn-222. At the end of the first half-life period, x/2 atoms will have decayed (via α emission) while x/2 atoms remain. At the end of the second half-life period, the first x atoms will have decayed to x/4 while the second x atoms will have decayed to x/2. At the end of the third half-life period, the first x atoms will have decayed to x/8 and the second x atoms to x/4, while the third x atoms will have decayed to x/2 … and so on according to the following scheme

This process forms an absolutely convergent geometric series in which the number of radon atoms remaining in the space after n half-life periods will be

The conclusion is reached that if the entry rate of radon gas into a previously radon-free bounded space is x radon atoms in the unit of time corresponding to the half-life of Rn-222, the number of radon atoms in this space will over successive half-lives approach a steady-state value of x.

Assuming diffusion throughout the space, a steady state value of x should be realized in little more than a month since when n = 8 (equivalent to 30.6 days) the series sum is 99.6% of x.

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Computing entry rate
Given a steady long-term average figure on the detector, which implies a steady concentration of radon gas throughout the bounded space, the number of radon atoms in this space can be estimated as follows.

For decay rates measured in becquerels per cubic meter (Bq/m3)
Let the long-term average figure on the detector, measured in decays per second per cubic meter be n, and let the bounded space be v cubic meters.
In the half-life of Rn-222 (3.8235 days) the number of decays in volume v will be n × v × 330,350
This equals x/2 where x is the steady state population of radon atoms in volume v
Therefore x = n × v × 660,701 radon atoms
By the theorem, x is also the number of radon atoms entering the bounded space in a unit of time equal to the half-life of the isotope (Rn-222) of which radon gas is largely composed – 3.8235 days. The magnitude 8x is therefore the number of radon atoms entering the space in a month (8 x 3.8235 = 30.6). Dividing 8x by the Avogadro number converts the number of radon atoms into moles of radon gas:
Entry rate (moles of radon gas per month) = 8 × n × v × 660,701 / (6.022 x 10^23)

The entry rate of radon gas ≅ 8.78nv attomoles per month
(1 attomole = 10-18 moles)

For decay rates measured in picocuries per litre (pCi/L)
Let the long-term average figure on the detector, measured in picocuries per litre be n, and let the bounded space be v cubic meters.

The entry rate of radon gas ≅ 324.74nv attomoles per month
(1 attomole = 10-18 moles)

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Disclaimer
Please note that the theorem on which the above calculations are based is untested. Until the theorem has been tested and the accuracy of results obtained with it has been determined, the calculation of entry rate as herein defined can only be regarded as a theoretical prediction and should be viewed accordingly.

P Mander May 2017

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Anthony Carlisle (left) and William Nicholson, London, May 1800

Anthony Carlisle (left) and William Nicholson, London, May 1800

The rise of physical chemistry in the 19th century has at its root two closely connected events which took place in the final year of the 18th century. In 1800, Alessandro Volta in Lombardy invented an early form of battery, known as the Voltaic pile, which Messrs. Carlisle and Nicholson in England promptly employed to discover electrolysis.

Carlisle and Nicholson’s discovery that electricity can decompose water into hydrogen and oxygen caused as big a stir as any scientific discovery ever made. It demonstrated the existence of a relationship between electricity and the chemical elements, to which Michael Faraday would give quantitative expression in his two laws of electrolysis in 1834. Faraday also introduced the term ‘ion’, a little word for a big idea that Arrhenius, Ostwald and van ‘t Hoff would later use to create the foundations of modern physical chemistry in the 1880s.

About this post

The story of Carlisle and Nicholson’s discovery properly begins with a letter that Volta wrote on March 20th, 1800 to the President of the Royal Society in London, Sir Joseph Banks. The leaking of that letter (which contained confidential details of the construction of the Voltaic pile) to among others Anthony Carlisle, forms the narrative of my previous post “The curious case of Volta’s leaked letter”.

This post is concerned with the construction details themselves, which have their own story to tell, and the experimental activities of Messrs. Carlisle and Nicholson after they had seen the letter, which were reported in July 1800 by Nicholson in The Journal of Natural Philosophy, Chemistry & the Arts – a publication that Nicholson himself owned.

The Voltaic pile

“The apparatus to which I allude, and which will no doubt astonish you, is only the assemblage of a number of good conductors of different kinds arranged in a certain manner.”
Alessandro Volta’s letter to Joseph Banks, introducing the Voltaic pile

Volta’s arrangement comprised a pair of different metals in contact (Z = Zinc, A = Silver), followed by a piece of cloth or other material soaked in a conducting liquid; this ‘module’ could be repeated an arbitrary number of times to build a pile in the manner illustrated below.

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The Voltaic Pile: Volta’s own illustration enclosed with the letter to Banks

Volta believed the electrical current was excited by the mere contact of two different metals, and that the liquid-soaked material simply conducted the electricity from one metal pair to the next. This explains why Volta’s illustration shows the metals always in pairs – note the silver disc below the zinc at the bottom of the pile and a zinc disc above the silver at the top.

It was later shown that these terminal discs are unnecessary: the actual electromotive unit is zinc-electrolyte-silver. Volta’s arrangement can therefore be seen as a happy accident, in that his mistaken belief regarding the generation of electromotive force led him to the correct arrangement of repeated electrochemical cells, in which the terminal discs act merely as connectors for the external circuit wires.

Volta’s pile thus contained one less generating unit than he thought; it also caused the association of the two metals with the positive and negative poles of the battery to be reversed.

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Enter Mr. Carlisle

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London’s Soho Square in the early 19th century. Animals were often driven to market through the square.

The president of the Royal Society, Sir Joseph Banks, lived in a house at No.32 Soho Square. Here he entertained all the leading members of the scientific establishment, and it was here in April 1800 that he yielded to temptation and disclosed the contents of Signor Volta’s confidential letter to certain chosen acquaintances. Among them was another resident of Soho Square, the fashionable surgeon Anthony Carlisle, who had just moved in at No.12.

Volta’s announcement of his invention made an instant impression on Carlisle, who immediately arranged for his friend the chemist William Nicholson to look over the letter with him, after which Carlisle set about constructing the apparatus according to the instructions in Volta’s letter.

Nicholson records in his paper that by 30th April 1800, Carlisle had completed the construction of a pile “consisting of 17 half crowns, with a like number of pieces of zinc, and of pasteboard, soaked in salt water”. Using coinage for the silver discs was smart thinking by Carlisle – with a diameter of 1.3 inches (3.3 cm), the half crown was an ideal size for the purpose, and was made of solid silver.

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Silver half crown, diameter 1.3 inches

From Nicholson’s account, it seems likely that Carlisle obtained a pound (approx. ½ kilo) of zinc from a metal dealer called John Tappenden who traded from premises just opposite the church of Saint Vedast Foster Lane, off Cheapside in the City of London. A pound of zinc was enough to make 20 discs of the diameter of a half crown.

Having constructed the pile exactly according to Volta’s illustration above, Carlisle and Nicholson were ready to begin their experiments. But before describing their work, it is pertinent to draw attention to the way in which they approached their program of research, which was quite unlike that of Volta.

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Differences in approach

Alessandro Volta’s letter to Joseph Banks, apart from briefly detailing the construction of the pile, comprises a lengthy account of electric shocks administered to various parts of the human anatomy and the nature of the resulting sensations.

Volta does first prove with a charging condenser that the pile generates electricity, but having ascertained this fact, he makes no further observations on the pile, other than asserting that the device has “an inexhaustible charge, a perpetual action” and later commenting: “This endless circulation of the electric fluid (this perpetual motion) may appear paradoxical and even inexplicable, but it is no less true and real;”

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One of Volta’s arrangements, using electrodes dipped in bowls of water for delivering electric shocks to the hands. If Volta had just put both electrodes in one bowl, he would have discovered electrolysis.

Volta appears not to have observed that the zinc discs quickly oxidise during operation; perhaps it was because he enclosed the pile in wax to prevent it from drying out. But nonetheless it seems strange that Volta did not discover during the course of his many experiments that the zinc discs do not have an unlimited lifetime.

William Nicholson also found it strange, commenting in his paper, “I cannot here look back without some surprise and observe that … the rapid oxidation of the zinc should constitute no part of his [Volta’s] numerous observations.”

Reading Volta’s communication to Banks, one is struck by the brevity of the text pertaining to his fabulous invention, and contrarily, the abundant descriptions of the shocks he administered with it. Volta is demonstrably more occupied with how humans experience the shocks that the pile delivers, than with the pile itself.

With Carlisle and Nicholson, the situation is very much the reverse. Having given themselves an obligatory shock with their newly-built machine, the attention immediately shifts to the pile itself. Their experiments and attendant reasoning show an approach that is more analytical in character.

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The path to discovery

On May 1st, 1800, Carlisle and Nicholson set up their pile – most likely in Carlisle’s house at 12 Soho Square – and began by forming a circuit with a wire and passing a current through it. To assist contact with the wire, a drop of river water was placed on the uppermost disc. As soon as this was done, Nicholson records

“Mr. Carlisle observed a disengagement of gas round the touching wire. This gas, though very minute in quantity, evidently seemed to me to have the smell afforded by hydrogen”

It is amazing that Nicholson was able to identify hydrogen from such a minute sample. But even more amazing was the thought that occurred to him next

“This [release of hydrogen gas], with some other facts, led me to propose to break the circuit by the substitution of a tube of water between two wires.”

Nicholson does not say what those other facts are, but he does record that on the first appearance of hydrogen gas, both he and Carlisle suspected that the gas stemmed from the decomposition of water by the electric current. Following that wonderfully intuitive piece of reasoning, Nicholson’s suggestion can be seen as a natural next step in their investigation.

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William Nicholson (1753-1815)

On 2nd May, Carlisle and Nicholson began their experiment using brass wires in a tube filled with river water. A fine stream of bubbles, identifiable as hydrogen, immediately arose from the wire attached to the zinc disc, while the wire attached to the silver disc became tarnished and blackened by oxidation.

This was an unexpected result. Why was the oxygen, presumably formed at the same place as the hydrogen, not evolved at the same wire? Why and how does the oxygen apparently burrow through the water to the other wire where it produces oxidation of the metal? This finding, which according to Nicholson “seems to point at some general law of the agency of electricity in chemical operations” was to occupy physical chemists for the next 100 years…

Meanwhile, Carlisle and Nicholson responded to their new experimental finding with another wonderfully intuitive piece of reasoning. What would be the effect, they asked, of using electrodes made from a metal that resisted oxidation, such as platinum?

Immediately they set about finding the answer. With electrodes fashioned from platinum wire they observed a plentiful stream of bubbles from the wire attached to the zinc disc and a less plentiful stream from the wire attached to the silver disc. No tarnishing of the latter wire was seen. Nicholson wrote

“It was natural to conjecture, that the larger stream was hydrogen, and the smaller oxygen.”

The conjecture was correct. On a table top in Soho Square, Carlisle and Nicholson had successfully decomposed water into its constituent gases by the use of the Voltaic pile, and had thereby discovered electrolysis – a technique which was to prove of immeasurable importance to industry.

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Anthony Carlisle (1768-1840)

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Quantitative analysis

Carlisle and Nicholson realised that the decomposition of water using platinum wires “offered a means of obtaining the gases separate from each other”. This not only provided a new way of producing these gases, but also opened up a new avenue of analysis. By measuring the relative volumes of hydrogen and oxygen evolved from the wires, they could compare their result with known data for water. [It should be noted that Carlisle and Nicholson did not have the benefit of Avogadro’s law, which was not formulated until 1811].

Carlisle and Nicholson subjected water to electrolysis for 13 hours, after which they determined the weight of water displaced by each gas in the respective tubes. The weights were in the proportion 142:72 in respect of hydrogen and oxygen; this is very close to the whole number ratio of 2:1 which was known to be the proportions in which these gases combine to produce water. Here then was quantitative evidence that the hydrogen and oxygen observed in Carlisle and Nicholson’s electrolytic cell originated from the decomposition of water.

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The experimental observations – explained

It was that drop of water placed on the uppermost disc to assist contact with the metal wire that opened the path to discovery. The fact that gas was formed “round the touching wire” indicates that the contact was intermittent: when the wire was in contact with the water drop but not the uppermost disc, a miniature electrolytic cell was formed and hydrogen gas was evolved.

Illustrating this graphically requires some qualifying explanation, since as already mentioned the terminal discs of the Voltaic pile assembled according to Volta’s instructions were unnecessary, and acted merely as conductors. Electrochemically, the uppermost disc of Carlisle and Nicholson’s Voltaic pile was a silver cathode, connected to the water drop via a zinc disc; the lowest disc in the pile was a zinc anode, which via an interposed silver disc was connected to the water drop via an unspecified metal wire. The electrochemical processes can be illustrated as follows

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Carlisle and Nicholson’s first experiment, May 1st, 1800

The drop of water shown in blue acted as an electrolytic cell supplied by a zinc anode (the uppermost disc) and an unspecified metal cathode (the wire). When current was passed through this cell at moments when the wire lost contact with the zinc disc, reduction of hydrogen ions produced bubbles of hydrogen at the cathode, i.e. around the wire, as Carlisle observed. At the anode, the oxygen formed would have immediately oxidised the zinc with no visible evolution of gas.

The evolution of hydrogen gas between each pair of discs in the Voltaic pile, i.e. on the side in communication with the electrolyte, was also noted in Nicholson’s paper, as was the erosion of the zinc anode.

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And so to the experimental set-up with which Carlisle and Nicholson successfully decomposed water into its constituent gases by the use of the Voltaic pile, and thereby discovered electrolysis. Electrochemically, the uppermost disc in the pile was a silver cathode, which via an interposed zinc disc was connected to the water in the tube via a platinum electrode; the lowest disc in the pile was a zinc anode, which via an interposed silver disc was connected to the water in the tube via a platinum electrode. The electrochemical processes can be illustrated as follows

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Carlisle and Nicholson’s electrolysis of water, May 1800

The tube of water shown in blue acted as an electrolytic cell supplied by a platinum anode and cathode. When current was passed through this cell, reduction of hydrogen ions produced bubbles of hydrogen at the cathode, while the oxidation of water produced hydrogen ions and bubbles of oxygen at the anode.

The evolution of hydrogen gas between each pair of discs in the Voltaic pile, i.e. on the side in communication with the electrolyte, was also noted in Nicholson’s paper, as was the erosion of the zinc anode.

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Mouse-over links to original papers mentioned in this post

Volta’s letter to Banks (begins on page 289)

Nicholson’s paper (begins on page 179)

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© P Mander September 2015