ventus001

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
(273,15+T)

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

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формат gif

ah3a

формат jpg

ah1a

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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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ОБНОВЛЕНИЯ

– – – –

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

– – – –

Игорь пользуется моей формулой, чтобы поддерживать ячейку погреба сухой.

igor01

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

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.

igor02

Полное описание с набором отличных фотографий → здесь

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

sht75

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.

– – – –

Формула позволила создать калькулятор RH←→AH

reckoner

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

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Формулу процитировали в академической исследовательской публикации

ahcitat

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:

arduino

«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 – carnotcycle.wordpress.com/2012/08/04/how-to-convert-relative-humidity-to-absolute-humidity/.»

Дополнительные фотографии по ссылке http://arduino.ru/forum/proekty/kontrol-vlazhnosti-podvala-arduino-pro-mini

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vol05

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”.

vol02

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?

– – – –

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.”

vol03

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.

vol04

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.

– – – –

The chronology of the case

1800

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.

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

May
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.

July
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.

September
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.

– – – –

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)

– – – –

P Mander August 2015

pan01

The Kinetics of Chemical Change in Gaseous Systems by Cyril Norman Hinshelwood, published by Oxford University Press and printed in Great Britain. This well-preserved copy is a second edition from 1929, the first edition having been published in 1926.

hinsh

CN Hinshelwood 1897-1967

The key difference between the first and second edition is the addition of a chapter on chain reactions. Hinshelwood’s theoretical and experimental work in this area, especially in the study of explosions, led to him receiving the Nobel Prize for Chemistry in 1956, which he shared with Nikolay Semyonov.

This particular copy of Hinshelwood’s masterwork has further historical interest since it once belonged to another famous chemist, who inscribed his name on the title page:

pan02

The signature is that of Friedrich Paneth, an Austrian-born chemist of Jewish parentage who took refuge in Britain in 1933 and became a naturalized British citizen in 1939. After a distinguished scientific career in Britain, Paneth was invited to become director of the Max Planck Institute for Chemistry in Mainz, a position he held from 1953 until his death in 1958.

pan03

Friedrich Paneth 1887-1958

The inscription XI.29. indicates that Paneth acquired this copy in November 1929, shortly after its publication. In the chapter on heterogeneous reactions, Hinshelwood refers to work published by Paneth and includes him in the index of authors. So it is possible that this book was a courtesy copy sent by Hinshelwood to Paneth, who at the time was heading the chemical institute at Königsberg University.

– – – –

Postscript

Friedrich Paneth was a multitalented individual; besides his extensive achievements as a physical chemist, he was also an accomplished photographer and early user of autochrome photography, as can be seen on this link:

http://mashable.com/2015/11/28/traveling-with-the-paneths/#UTaWotXqEkql

This autochrome photograph of his children Eva and Heinz, on the shore of Lake Luzern, was probably taken in 1927.

pan04

– – – –

lc01

8 octobre 1850 – 17 septembre 1936

History

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:

lc02

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:

lc05

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.

– – – –

The Principle behind the Principle

lc06

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:

lc07

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

lc08

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.

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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

lc10

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:

lc11

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

lc12

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

lc13

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

lc14

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

lc15

Setting dP=0 yields the result

lc16

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

lc17

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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P Mander February 2016

ae01

In my previous post JH van ‘t Hoff and the Gaseous Theory of Solutions, I related how van ‘t Hoff deduced a thermodynamically exact relation between osmotic pressure and the vapor pressures of pure solvent and solvent in solution, and then abandoned it in favor of an erroneous idea which seemed to possess greater aesthetic appeal, on account of a chance encounter with a colleague in an Amsterdam street.

Rendered in modern notation, the thermodynamically exact equation van ‘t Hoff deduced in his Studies in Chemical Dynamics (1884), was

ts06

Following his flawed moment of inspiration upon learning the results of osmotic experiments conducted by Wilhelm Pfeffer, he leaped to the conclusion that the law of dilute solutions was formally identical with the ideal gas law

ts04

It would seem van ‘t Hoff was so enamored with the idea of solutions and gases obeying the same fundamental law, that he failed to notice that the latter equation is actually a special case of the former. Viewed from this perspective, the latter’s resemblance to the gas law is entirely coincidental; it arises solely from a sequence of approximations applied to the original equation.

As a footnote to history, CarnotCycle lays out the path by which the latter equation can be reached from the former, and shows how accuracy reduces commensurately with simplification.

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We begin with van ‘t Hoff’s thermodynamically exact equation from the Studies in Chemical Dynamics

ts06 (1)

where Π is the osmotic pressure, V1 is the partial molal volume of the solvent in the solution, p0 is the vapor pressure of the pure solvent and p is the vapor pressure of the solvent in the solution.

Assuming an ideal solution, in the sense that Raoult’s law is obeyed, then

ae02

where x1 and x2 are the mole fractions of solvent and solute respectively. So for an ideal solution, equation 1 becomes

ae03

If the ideal solution is also dilute, the mole fraction of the solute is small and hence

ae04

so that

ae05

For a dilute solution x2 approximates to n2/n1, where n2 and n1 are the moles of solute and solvent, respectively, in the solution. The above equation may therefore be written

ae06

In dilute solution, the partial molal volume of the solvent V1 is generally identical with the ordinary molar volume of the solvent. The product V1n1 is then the total volume of solvent in the solution, and V1n1/n2 is the volume of solvent per mole of solute. Representing this quantity by V’, the above equation becomes

ae07 (2)

which is identical with the empirical equation proposed by HN Morse in 1905. For an extremely dilute solution the volume V’ may be replaced by the volume V of the solution containing 1 mole of solute; under these conditions we have

ts04 (3)

which is the van ‘t Hoff equation.

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It is instructive to compare the osmotic pressures calculated from the numbered equations shown above and those obtained by experiment. It is seen that Eq.1, which involves measured vapor pressures, is in good agreement with experiment at all concentrations. Eq.3 fails in all but the most dilute solutions, while Eq.2 represents only a modest improvement.

ae08

These figures give a measure of van ‘t Hoff’s talent as a theoretician in deducing Eq.1, and the error into which he fell when abandoning it in favor of Eq.3.

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Mouse-over links to works referred to in this post

Jacobus Henricus van ‘t Hoff Studies in Chemical Dynamics

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P Mander June 2015