Archive for the ‘physics’ Category

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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Formulas 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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Notes for technicians
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

The entry rate is thus the rate at which radon atoms enter the enclosed space without escaping.

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.

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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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.

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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.


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


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.


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;”


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.


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.


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


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


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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У меня имеется цифровая метеостанция с беспроводным датчиком, расположенным вне помещения. На фотографии: в верхнем правом квадранте отображается температура и относительная влажность вне помещения (6,2°C/94%) и в помещении (21,6°C/55%).

Я считаю, что эта разница (в помещении и вне) очень важна для определенных целей. Давайте посмотрим на цифры. Когда я смотрю на показания, то всегда задаюсь вопросом о том, различается ли количество водяного пара в воздухе внутри и вне помещения? Простой вопрос, а ответ потребует некоторых рассуждений. За основание возьмем уравнение идеального газа; для вычисления абсолютной влажности по температуре и относительной влажности необходим еще специальный алгоритм расчета давления насыщенного пара как функции от температуры. А это не очень простая вещь.

Формула для вычисления абсолютной влажности

В формуле ниже, температура (Т) измерена в градусах Цельсия, относительная влажность (rh) — в %, а е — это основание натурального логарифма 2,71828 [возведенное в степень, указанную в скобках]:

Абсолютная влажность (г/м3) =
6,112 x e^[(17,67 x T)/(T+243,5)] x rh x 18,02
(273,15+T) x 100 x 0,08314

что упрощается до

Абсолютная влажность (г/м3) =
6,112 x e^[(17,67 x T)/(T+243,5)] x rh x 2,1674

Точность этой формулы в пределах 0,1% на диапазоне температур от –30°C до +35°C

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формат gif


формат jpg


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Дополнительные примечания для студентов

Стратегия вычисления абсолютной влажности, определяемой как плотность водяного пара (г/м3) по температуре (Т) и относительной влажности (rh):

1. Водяной пар — это газ, поведение которого при обычной температуре атмосферы приближено к поведению идеального газа.

2. Применимо уравнение идеального газа PV = nRT. Газовая постоянная R и переменные T и V в этом случае известны (Т измерена, V = 1 m3). Для вычисления n необходимо рассчитать Р.

3. Чтобы получить значение Р можно применить следующий вариант формулы [см. eq.10] Магнуса-Тетенса, которая дает давление насыщенного пара Psat (гектопаскали) как функцию от температуры Т (в градусах Цельсия):

Psat = 6,112 x e^[(17,67 x T)/(T+243,5)]

4. Psat — это давление при относительной влажности 100%. Для вычисления давления P при любом значении относительной влажности, выраженном в %, мы умножаем выражение для Psat на коэффициент (rh/100):

P = 6,112 x e^[(17,67 x T)/(T+243,5)] x (rh/100)

5. Теперь мы знаем P, V, R, T и можем вычислить n, а это и есть количество водяного пара в молях. Значение затем умножается на 18,02 — это молекулярный вес воды. Ответ получается в граммах.

6. Обобщение:
Формула абсолютной влажности получена из уравнения идеального газа. Она выражает n всего через две переменные: температуру (Т) и относительную влажность (rh). Давление вычисляется как функция от обеих этих переменных; объем указан (1 m3), а газовая постоянная R известна.

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YouTube: Умный гараж, часть 3, Управление вытяжным вентилятором в подвале

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Игорь пользуется моей формулой, чтобы поддерживать ячейку погреба сухой.


Октябрь 2016: Я впечатлился системой управления влажностью основания здания, разработанной Игорем, и даже опубликовал отчет на форуме

Внутри короткой трубки установлен вентилятор с круговым уплотнением, распечатанным на 3D-принтере. Вентилятор замещает воздух, находящийся в основании, на воздух снаружи. Он включается, если абсолютная влажность в ячейке выше, чем на улице на 0,5 г/м3. Предполагается, что температура снаружи ниже. Это как раз и гарантирует, что вода в ячейке превращается в пар и вытягивается, а обратный процесс не может произойти.


Полное описание с набором отличных фотографий → здесь

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Формула позволяет измерять AH по данным от высокоточного датчика RH и T


Датчик SHT75 RH и T от SENSIRION

Апрель 2016: Проф. Антониетта Франи (Prof. Antonietta Frani) на основе моей формулы создала миниатюрный прибор для измерения абсолютной влажности. Миниатюрный микроконтроллер Arduino Uno оборудован датчиком SHT75 RH и T и подключается к компьютеру по кабелю USB. Системный интегратор Роберто Валголио (Roberto Valgolio) разработал интерфейс для передачи данных в листы Excel и отображения графиков.

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Формула позволила создать калькулятор RH←→AH


Март 2016: Немецкий веб-сайт использует мою формулу для своего онлайн-калькулятора RH/AH

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Формулу процитировали в академической исследовательской публикации


Январь 2016: Исследовательская публикация в Landscape Ecology (октябрь 2015) посвящена микроклиматическим образцам в городской среде США. Там для вычисления абсолютной влажности по температуре и относительной влажности использована моя формула.

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Формула нашла применение и в блоках управления влажностью

Август 2015: ПО с открытым исходным кодом (проект Arduino) также использует в микроконтроллере управления влажностью основания здания мою формулу для расчета абсолютной влажности:


«Вся идея состоит в том чтобы измерить температуру и относительную влажность в подвале и на улице, на основании температуры и относительной влажности рассчитать абсолютную влажность и принять решение о включении вытяжного вентилятора в подвале. Теория для расчета изложена здесь –»

Дополнительные фотографии по ссылке

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Процедура вычисления AH применена в калибровке спутника погоды Американского Национального Космического Агентства (NASA)

Июнь 2015: Моя общая процедура расчета AH по RH и T применена для абсолютной калибровки Глобальной Спутниковой Системы Навигации Циклона (CYGNSS), причем именно в отношении данных RH, предоставленных Системой Непрерывного Анализа и Прогноза Климата (CFSR). Единственное изменение в моей формуле Psat состоит в том, что используется выражение Августа-Роше-Мангуса, а не Болтона.

Система CYGNSS имеет сеть из восьми спутников. Она предназначена для улучшенного прогноза силы ураганов. Запущена 15 декабря 2016 г.

Ссылка (Технический отчет “An Antenna Temperature Model for CYGNSS”, июнь 2015)

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Sir Joseph Banks, President of the Royal Society in London, sat in the splendour of his office in Somerset House. It was an April morning in the year 1800. The clatter of horse-drawn carriages in the Strand rose to his window, but he did not notice; his attention was elsewhere. Staring into space, he clutched a letter that had been delivered to him that very morning. Dated March 20th and written in French, it had been sent from Como in Lombardy by an Italian professor of experimental physics named Alessandro Volta.

Professor Volta’s letter was clearly attended by some haste, since he had dispatched the first four pages in advance of the remainder, which was to follow. The subject matter was experiments on electricity, and the first pages of Volta’s letter to Banks described the invention of an apparatus “which will no doubt astonish you”.


Alessandro Volta (1745-1827) and his amazing invention

On that April morning in London, Banks read the letter and was duly astonished. Volta’s apparatus, consisting of a series of discs of two different metals in contact separated by brine-soaked pasteboard, was capable of generating a continuous current of electricity. This was a world apart from the static electricity of the celebrated Leyden jar and indeed a most astonishing discovery; no wonder Volta was so anxious to communicate it without delay to Banks and thereby to the Royal Society – of which Volta was also a fellow.

Still clutching the letter, Joseph Banks regained his composure and collected his thoughts. He must of course arrange for the letter to be read to the Society, after which it would duly appear in print in the Society’s Philosophical Transactions.

In the meantime, Banks was naturally obliged to keep Volta’s discovery confidential. But then again, with a such an astonishing discovery as this, it was sorely tempting to show Signor Volta’s letter – in the strictest confidence of course – to certain individuals in his large circle of scientific acquaintances, who would surely be fascinated by its contents. What could be the harm in that?

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Yielding to temptation

The state of mind into which Sir Joseph Banks was propelled by Professor Volta’s letter was pertinently observed (albeit almost a century later) by Oscar Wilde in his play Lady Windermere’s Fan, in which Lord Darlington famously quips “I can resist everything but temptation.”


32 Soho Square (right), the London home of Sir Joseph Banks

The London home of Joseph Banks, in Soho Square, was the centre of bustling scientific activity and attracted all the leading members of the scientific establishment. Within a month of receiving Volta’s letter, Banks had yielded to temptation and shown it to a number of acquaintances.

Among them was Anthony Carlisle, a fashionable London surgeon who was shortly to display remarkable abilities in the realm of physical chemistry. Having perused the letter, Carlisle immediately arranged for his friend the chemist William Nicholson to look over the pages with him, after which Carlisle set about constructing the apparatus according to Volta’s instructions – the fabled instrument we now call the Voltaic Pile.


Sir Anthony Carlisle (1768-1840), painted by Henry Bone in 1827

So within a month of Volta’s hastened communication to Banks, the details of the construction of the Voltaic Pile had been leaked to, among others, Carlisle and Nicholson, enabling the latter to begin experiments with Volta’s apparatus that would lead to their privileged discovery of electrolysis, before Volta’s letter had even been read to – let alone published by – the Royal Society.

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The chronology of the case


March 20th
Volta sends a letter (in French) from Como, Lombardy, to Sir Joseph Banks at the Royal Society in London, announcing his invention of the Voltaic pile.

Banks leaks the contents of Volta’s letter to several acquaintances, including Anthony Carlisle, who arranges for William Nicholson to view the letter.

Carlisle and Nicholson construct a Voltaic Pile according to Volta’s instructions. With this apparatus they discover the electrolysis of water into hydrogen and oxygen.

June 26th
Volta’s letter is read to the Royal Society.

William Nicholson publishes a paper in The Journal of Natural Philosophy, Chemistry & the Arts, announcing the discovery of electrolysis by Anthony Carlisle and himself, using the Voltaic Pile.

Volta’s letter announcing his invention of the Voltaic pile is published in French in the Philosophical Transactions of the Royal Society, and in English in The Philosophical Magazine.

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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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8 octobre 1850 – 17 septembre 1936


Le Châtelier’s principle is unusual in that it was conceived as a generalization of a principle first stated by someone else.

In 1884, the Dutch theoretician JH van ‘t Hoff published a work entitled Etudes de Dynamique Chimique [Studies in Chemical Dynamics]. In it, he stated a principle drawn from observations of different forms of equilibrium:

“Lowering the temperature displaces the equilibrium between two different conditions of matter (systems) towards the system whose formation produces heat.”

The converse statement was also implied, leading van ‘t Hoff to the realization that application of the principle made it possible “to predict the direction in which any given chemical equilibrium will be displaced at higher or lower temperatures.”

A few months after the publication of the Etudes, the following note appeared on page 786 of volume 99 of Comptes-rendus de l’Academie des Sciences:


The note covers two pages, but the crucial paragraph is the one shown immediately above, in which Le Châtelier extends van ‘t Hoff’s recently published principle to include pressure and (in modern terms) chemical potential. Rendered in English, the paragraph reads

“Any system in stable chemical equilibrium, subjected to the influence of an external cause which tends to change either its temperature or its condensation (pressure, concentration, number of molecules in unit volume), either as a whole or in some of its parts, can only undergo such internal modifications as would, if produced alone, bring about a change of temperature or of condensation of opposite sign to that resulting from the external cause.”

Just as van ‘t Hoff used inductive reasoning to relate temperature change to displacement of equilibrium, so Le Châtelier adopts the same technique to extend the principle to changes of pressure and potential.

Having arrived at a generalized principle – that systems in stable equilibrium tend to counteract changes imposed on them – Le Châtelier then sought to deduce this result mathematically from equations describing systems in equilibrium. During this quest, he discovered that the American physicist Josiah Willard Gibbs had done a good part of the groundwork in his milestone monograph On The Equilibrium of Heterogeneous Substances (1876-1878). In 1899, Le Châtelier translated this hugely difficult treatise into French, thereby helping many scientists in France and beyond to access Gibbs’ powerful ideas.

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Early misunderstandings

Le Châtelier’s principle, first stated in 1884 and extended as the Le Châtelier-Braun principle in 1887, has stood the test of time. Today we view it as a very useful law, but that was not how it was viewed by some of the academic establishment in the early 20th century. Critics including the illustrious Paul Ehrenfest and Lord Rayleigh regarded the principle as vaguely worded and impossible to apply without ambiguity. As late as 1937, Paul Epstein in his Textbook of Thermodynamics wrote that this criticism “has been generally accepted since”.

This was news to me; when I was taught Le Châtelier’s principle at school, the wording was the same as in Epstein’s day but we had no issues with vagueness or ambiguity. I wondered what this criticism was all about, so I delved into the online archive of ancient journals. And came up with this:


From J Chem Soc, 1917; vol 111. CarnotCycle hopes that the misspelling of Braun in the title was a genuine typo, and not the deliberate use of irony to mock the authors of the principle.

It is clear from the first paragraph that the charge of ambiguity by Ehrenfest and Rayleigh arose from a failure to distinguish between cause and effect. Perturbations of systems in stable chemical equilibrium are caused by changes in generalized forces which, as Le Châtelier documents, are intensive variables. The ‘response of the system’, or generalized displacements, are the extensive conjugates. This answers Rayleigh’s question as to why we are to choose the one (pressure) rather than the other (volume) as the independent variable.

What surprised me was that this misunderstanding persisted for three decades. It just goes to show that in thermodynamics, even the most perspicacious individuals can have enduring blind spots.

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The Principle behind the Principle


In the Etudes of 1884, van ‘t Hoff stated his principle on the basis of different observations of equilibrium displacement with temperature. But while reaching his conclusion inductively, he still managed to give a precise mathematical expression of the principle. In modern notation it reads:


This famous equation, sometimes called the van ‘t Hoff isochore, was stated without proof in the 1884 edition, but in the second edition of 1896 a proof was provided which is based – as with many proofs of that era – on a reversible cycle of operations involving heat and work.

Although thermodynamically exact, the equation provides little insight into why a system in stable equilibrium tends to resist actions which alter that state. Not that this would have bothered van ‘t Hoff, who was much more interested in practicality than philosophical pondering.

But in the early 1900s, physical chemists began to look for an explanation. In A Textbook of Thermodynamics with special reference to Chemistry (1913), J.R. Partington remarked that Le Châtelier’s principle is an expression of “a very general theorem … called the Principle of Least Action. We can state it in the form that, if the system is in stable equilibrium, and if anything is done so as to alter this state, then something occurs in the system itself which tends to resist the change, by partially annulling the action imposed on the system.”

Partington was hinting at a more general notion underlying Le Châtelier’s original description. That notion was more concisely expressed in another volume entitled A Textbook of Thermodynamics, written by Frank Ernest Hoare in 1931, in which he stated “every system in equilibrium is conservative”.

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Interlude : Mapping chemical reactions


It is one the conditions of stable equilibrium in thermodynamic systems that for a given temperature and pressure, the Gibbs free energy is a minimum. In the context of a chemical reaction, it means that the Gibbs free energy of the reaction mixture will decrease in the manner shown above, where the difference between P (pure products) and R (pure reactants) is the standard free energy of reaction and E is the equilibrium point at the minimum point of the curve.

If the reactants are initially present in stoichiometric proportions, the x-axis represents the mole fraction of products in the reaction mixture. In 1920, a Belgian mathematician and physicist called Théophile de Donder proposed another name for this dimensionless extensive variable. He called it “the degree of advancement of a chemical reaction”, and represented it by the Greek letter ξ (xi).

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Defining conservative behavior

In 1937, Professor Mark Zemansky – at the time an associate professor of physics at what was then called the College of the City of New York – published a textbook entitled Heat and Thermodynamics.

In the last section of the last chapter of the book, Zemansky turns his attention to Le Châtelier’s principle. He considers a heterogeneous chemical reaction which is in phase equilibrium but not chemical equilibrium; under these circumstances the Gibbs free energy G is a function of temperature T, pressure P and degree of advancement ξ.


When the chemical reaction reaches stable equilibrium at temperature T and pressure P, it follows that ∂G/∂ξ = 0. Zemansky then considers a neighboring equilibrium state at temperature T+dT and pressure P+dP. The new degree of reaction will be ξ+dξ, but the change in the slope of the curve during this process is zero. Therefore


Zemansky thus arrives at a mathematical definition of conservative behavior for a thermodynamic system consisting of a reaction mixture in stable equilibrium with respect to the reaction to which ξ refers.

The next task is to use the operations of calculus to find expressions for the derivatives ∂ξ/∂T and ∂ξ/∂P in terms of ΔS (=ΔH/T) and ΔV respectively. The first step is to write out fully the condition on dT, dP and dξ required to maintain conservative behavior:


Zemansky then employs a neat device to introduce S and V into the calculation. The order of differentiation of a state function is immaterial, so he reverses the order of differentiation in the first two terms


Since (∂G/∂T)P,ξ = –S and (∂G/∂P)T,ξ = V,


For the sake of brevity, I will introduce at this point a shortcut that Zemansky did not use, but which does not in any way alter the results of his reasoning.

For any extensive property X which varies according to the degree of advancement of a chemical reaction ξ at constant temperature and pressure, the slope of the curve has the following property


Applying this fact to the above equation, we find that in order to maintain the equilibrium condition ∂G/∂ξ=0, dT, dP and dξ must be such that


Setting dP=0 yields the result


When ΔG=0, the denominator is positive. At equilibrium therefore, (∂ξ/∂T)P and ΔH have the same sign. So for an endothermic reaction (positive ΔH) the degree of reaction advancement at equilibrium increases as the temperature increases. This accords with Le Châtelier’s principle.

Setting dT=0 yields the result


When ΔG=0, the denominator is positive. At equilibrium therefore, (∂ξ/∂P)T and ΔV have opposite signs. For a reaction resulting in a reduction of volume, the degree of reaction advancement at equilibrium increases as the pressure increases. This accords with Le Châtelier’s principle.

Zemansky thus demonstrates that deductions from a mathematical definition of conservative behavior for a thermodynamic system consisting of a reaction mixture in stable equilibrium result in equations which “express in a rigorous form that part of Le Châtelier’s principle which concerns chemical reaction in heterogeneous systems”.

Le Châtelier never got to see this deduction of his principle. He died in 1936, just a year before Zemansky’s book was published.

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