Posts Tagged ‘Puy de Dôme’

Torricelli using a lot more mercury than necessary to demonstrate the barometer principle

The scientific study of the atmosphere can be said to have begun in 1643 with the invention of the mercury barometer by Evangelista Torricelli (1608-1647). Although the phenomenon had been observed and discussed by others – including Galileo – in the preceding decade, it was Torricelli who provided the breakthrough in understanding.

The prevailing view at the time was that air was weightless and did not exert any pressure on the mercury in the bowl. Instead, it was thought that the vacuum above the liquid in the barometer tube exerted a force of attraction that held the liquid suspended in the tube.

Torricelli challenged this view by proposing the converse argument. He asserted that air did have weight, and that the atmosphere exerted pressure on the mercury in the bowl which balanced the pressure exerted by the column of mercury. The vacuum above the mercury in the closed tube, in Torricelli’s opinion, exerted no attractive force and had no role in supporting the column of mercury in the tube*.

The assertion that air had weight, Torricelli realized, could be tested. In elevated places like mountains the reduced weight of the overlying atmosphere would exert less pressure, so the corresponding height of the mercury column in the barometer tube should be lower. It seems that Torricelli did not have the opportunity in his short life to do this experiment, but in the year following his death the experiment was carried out in France at the behest of the scientific philosopher Blaise Pascal (1623-1662).

*CarnotCycle wonders if Torricelli tilted the barometer tube and observed the disappearance of the space above the mercury – see diagram below. This would have shown that something other than a vacuum held the liquid suspended in the tube.

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The Torricelli experiment

In 1644 the French salon theorist Marin Mersenne (1588-1648) travelled to Italy where he learned of Torricelli’s barometer experiment. He brought news of the experiment back with him to Paris, where the young Blaise Pascal was a regular attendant at Mersenne’s salon meetings.

Puy de Dôme in south-central France, close to Clermont-Ferrand

Pascal had moved to Paris from his childhood home of Clermont-Ferrand. The 1,465 meter high Puy de Dôme was a familiar feature in the landscape he knew as a youngster, and it provided an ideal means of testing Torricelli’s thesis. Pascal’s brother-in-law Florin Périer lived in Clermont-Ferrand, and after some friendly persuasion, Périer ascended Puy de Dôme with a Torricellian barometer, taking measurements as he climbed.

The Torricelli experiment was conducted by Florin Périer on Puy de Dôme on Saturday 19 September 1648

At the base of the mountain, Périer recorded a mercury column height of 26 inches and 3½ lines. He then asked a colleague to observe this barometer throughout the day to see if any change occurred, while he set off with another barometer to climb the mountain. At the summit he recorded a mercury column height of 23 inches and 2 lines, substantially less than the measurement taken 1,465 meters below, where the barometer had remained steady.

The Puy de Dôme experiment provided convincing evidence that it was the weight of air, and thus atmospheric pressure, that balanced the weight of the mercury column.

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Measuring pressure

A 14th century shopkeeper weighing out sugar cubes using hand-held scales

When Florin Périer conducted the Torricelli experiment on Puy de Dôme in 1648, the measurements he recorded were the heights of mercury columns in barometer tubes. From these measurements, Blaise Pascal inferred a comparison of atmospheric pressures at the top and bottom of the mountain.

This experiment took place, we should remind ourselves, when Isaac Newton was only 5 years old and had not yet formulated his famous laws which gave concepts like mass, weight, force and pressure a systematic, mathematical foundation. In the pre-Newtonian world of Torricelli and Pascal, their thinking was based on the balancing of weights in the familiar sense of a shopkeeper’s scales. The weight of the mercury column in the barometer tube, which acted on the mercury in the bowl, was balanced by the weight of the air acting on the mercury in the bowl. Since the height of the mercury column was directly proportional to its weight, it was valid to use a length scale marked on the barometer tube to compute the weight of the air acting on the mercury in the bowl.

It is instructive to compare the language of Robert Boyle (1627-1691) and Isaac Newton (1643-1727) when discussing the barometer in the decades which followed. In the second edition of Boyle’s New Experiments Physico-Mechanicall of 1662 – which contains the first statement of Boyle’s Law – the word pressure appears frequently and has a meaning synonymous with weight. In Isaac Newton’s Principia of 1687, pressure is regarded as a manifestation of force. Boyle and Newton are thus speaking in essentially the same terms since according to Newtonian principles, weight is a force.

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Newtonian principles applied

The crucial advance in atmospheric science that Newton supplied in his Principia was the second law, which gave mathematical expression to force, and thus to weight and pressure, through the famous formula

The weight of a mercury column of cross-sectional area A and height h is

where ρ is the mass density of mercury and g is the acceleration due to gravity. The pressure exerted by the mercury column, which balances the atmospheric pressure, is

Thus P is directly proportional to h.

For a column of mercury 1 mm in height in a standard gravitational field (g = 9.80665 ms-2) at 273K, P is equal to 133.322 pascals. This is a unit of pressure called the torr. Pascal and Torricelli are thus both commemorated in units of pressure.

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A question of balance

Torricelli, Pascal and Boyle were in agreement with the proposition that air has weight. According to Newton’s interpretation the atmosphere possesses mass which is subject to gravitational acceleration, resulting in a downward force. This raises the question – Why doesn’t the sky fall down?

Since the sky is observed to remain aloft, there must exist a counteracting upward force. The vital clue as to the nature of this force was obtained on Pascal’s behalf by Florin Périer on Puy de Dôme in 1648 – namely that pressure decreases with height in the atmosphere.

A difference in pressure produces a force. In this way a parcel of air in a vertical column of cross-sectional area A exerts a force in the opposite direction to the gravitational force, as shown in the diagram.

At equilibrium, the forces are equal. Thus

where ρ is the density of the air.

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The decrease of temperature with altitude

A snow-capped Puy de Dôme

The appearance of snow above a certain height in elevated places provides plain evidence that temperature decreases with altitude, at least in that part of the atmosphere into which our earthly landscape protrudes. No doubt Torricelli, Pascal and other scientific philosophers of their time noticed this phenomenon and pondered upon it. But the explanation had to wait for another two centuries until the industrial revolution began, ushering in the age of steam and the associated science of thermodynamics.

The air in the troposphere, the lowest layer of the atmosphere where almost all weather phenomena occur, exhibits convection currents which continually transport air from lower regions to higher ones, and from higher regions to lower ones. When air rises it expands as the pressure decreases and so does work on the air around it. Thermodynamic principles dictate that this work requires the expenditure of heat, which has to come from within since air is a poor conductor and very little heat is transferred from the surroundings. As a result, rising air cools.

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Atmospheric convection processes fall within the province of the first law of thermodynamics, which can be expressed mathematically (see Appendix I) as

This equation states an energy conservation principle that applies to processes involving heat, work and internal energy. The atmospheric convection process is adiabatic meaning that no heat flows into or out of the system i.e. dQ = 0. Applying this constraint and using the combined gas equation to eliminate pressure p the above equation becomes

Integration yields

Converting from logarithms to numbers gives

Since by Mayer’s relation R = CP – CV

where γ = CP / CV. Using the combined gas equation to substitute V, the above equation can be rendered (with the help of γ√) as

Applying logarithmic differentiation gives

Assuming hydrostatic equilibrium, dp can be substituted giving

Since ρ = m/(RT/p) the above becomes

This adiabatic convection equation gives the rate at which the temperature of dry air falls with increasing altitude. Taking the following values: γ = 1.4 (dimensionless) ; R = 8.314 kgm2s-2 K-1 mol-1 ; m = 0.0288 kg mol-1 ; g = 9.80665 ms-2 gives

At the top of Puy de Dôme (1465 meters), dry air will be 14°C cooler than at the base of the mountain. This explains why snow can appear on the summit while the grass is still growing on the lower slopes.

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Appendix I

Benoît Émile Clapeyron (1799-1864)

In 1834, more than a century after Newton’s death, the French physicist and engineer Émile Clapeyron wrote a monograph entitled Mémoire sur la Puissance Motrice de la Chaleur (Memoir on the Motive Power of Heat). It contains the first appearance in print of the ideal gas equation, which combines the gas law of Boyle-Mariotte (PV)T = k with that of Gay-Lussac (V/T)P = k. Clapeyron wrote it in the form

where R is a constant and the sum of the terms in parentheses can be regarded as the thermodynamic temperature.

Sixteen years later in 1850, the German physicist Rudolf Clausius wrote a monograph on the same subject entitled Ueber die bewegende Kraft der Wärme und die Gesetze, welche sich daraus fuer die Wärmelehre selbst ableiten lassen (On the Motive Power of Heat, and on the Laws which can be deduced from it for the Theory of Heat). Seeking an analytical expression of the principle that a certain amount of work necessitates the expenditure of a proportional quantity of heat, he arrived at the following differential equation in the case of an ideal gas

where Q is the heat expended, U is an arbitrary function of temperature and volume, and A is the mechanical equivalent of heat. Earlier in his paper Clausius had represented Clapeyron’s combined statement of the laws of Boyle-Mariotte and Gay-Lussac as pv = R(a + t) so he recognized the right-hand term as corresponding to pdv, the external work done during the change

We know this equation today as an expression of the first law of thermodynamics, where U is the internal energy of the system under consideration.

U is a function of T and V so we may write the partial differential equation

Since U for an ideal gas is independent of volume and dU/dT is the heat capacity at constant volume CV, the first law for an ideal gas takes on the form

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P Mander January 2018

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The Honorable Robert Boyle FRS (1627-1691)

The fourteenth child of the immensely wealthy Richard Boyle, 1st Earl of Cork, Robert Boyle inherited land and property in England and Ireland which yielded a substantial income. He never had to work for a living, and following three years of travel and study as a teenager in Europe, Boyle decided at the age of 17 to devote his life to scientific research and the cultivation of what was called the “new philosophy”.

In Britain, Boyle was the leading figure in a move away from the Aristotelian view that knowledge was best obtained by the use of reason and logic. Boyle rejected this argument, and insisted that the path to knowledge was through empiricism and experiment. He won over many to his view, notably Isaac Newton, and in 1660 Boyle was a founding member of a society which believed that knowledge should be based first on experiment; we know it today as the Royal Society.

Boyle carried out a wealth of experiments in many areas of physics and chemistry, yet he seems to have been content with obtaining experimental results and generally stopped short of formulating theories to explain them.

Leibniz expressed astonishment that Boyle “who has so many fine experiments, had not come to some theory of chemistry after meditating so long on them”.

But what about Boyle’s law? you ask.

Well, it may surprise you to know that Robert Boyle did not originate the pressure-volume law commonly called Boyle’s law. A description of the reciprocal relation between the volume of air and its pressure does first appear in a book written by Boyle, but he refers to it as “Mr Towneley’s hypothesis”, for reasons we shall see.

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Torricelli leads the way

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Torricelli using a lot more mercury than necessary to demonstrate the barometer.

It was Evangelista Torricelli (1608-1647) in Italy who started it all in the summer of 1644 with the invention of the mercury barometer. It was an impressive device, made all the more impressive by the insight that came with it. For one thing, Torricelli had no problem accepting the space above the mercury as a vacuum, in contrast to the vacuum-denying views of Aristotle and Descartes. For another, he was the first to appreciate the fact that air had weight, and to understand that the column of mercury was supported by the pressure of the atmosphere. As he put it: “We live submerged at the bottom of an ocean of air, which by unquestioned experiments is known to have weight.”

This led Torricelli to surmise that atmospheric pressure should be less in elevated places like mountains, an idea which was put to the test in France by Blaise Pascal, or more precisely by Pascal’s brother-in-law Florin Périer, who happened to live in Clermont-Ferrand, which has Puy de Dôme nearby.

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Puy de Dôme in south-central France, close to Clermont-Ferrand.

On Saturday, September 19, 1648, Florin Périer and some friends performed the Torricelli experiment on the top of Puy de Dôme in south-central France. The height of the mercury column was substantially less – 85 mm less – than the control instrument stationed at the base of the mountain 1,400 metres below.

The Puy de Dôme experiment provided Pascal with convincing evidence that it was the weight of air, and thus atmospheric pressure, that balanced the weight of the mercury column. Torricelli’s instrument provided a convenient means of measuring this pressure. It was a barometer. The news quickly spread to England.

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Reaching new heights

The Torricellian experiment was demonstrated and discussed at the scientific centres of learning in London, Oxford and Cambridge from 1648 onwards. At Cambridge, there was an enthusiastic experimental scientist by the name of Henry Power, who was studying for a degree in medicine at Christ’s College. Power’s home was at Halifax in Yorkshire, and this gave him the opportunity to verify the Puy de Dôme experiment because unlike London, Oxford and Cambridge, the land around Halifax rises to significant heights.

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The hills around Halifax in northern England

On Tuesday, May 6, 1653, Henry Power carried the Torricellian experiment to the summit of Halifax Hill, to the east of the town, where he was able to verify Pascal’s observation. In further experiments, he began to investigate the elasticity of air – i.e. its expansion and compression characteristics. And it was this change of focus that was to characterise England’s contribution to the scientific study of air.

The pioneering work in Italy and France had been concerned with the physical properties of the atmosphere. In England, attention was turning to the physical properties of air itself.

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The Spring of the Air

In 1654, the year after Henry Power’s excursion on Halifax Hill, Robert Boyle arrived in Oxford where he rented rooms to pursue his scientific studies. With the assistance of Robert Hooke, he constructed an air pump in 1659 and began a series of experiments on the properties of air. An account of this work was published in 1660 under the title “New Experiments Physico-Mechanicall, Touching The Spring of the Air, and its Effects”.

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Boyle’s book was a landmark work, in which were reported the first controlled experiments on the effects of reduced air pressure. The experiments are divided into seven groups, the first of which concern “the spring of the air” i.e. the pressure exerted by the air when its volume is changed. It is clear that Boyle had an interest not only in demonstrating the elastic nature of air, but also in finding a quantitative expression of its elasticity. Due to inefficiencies in the air pump and the inherent difficulties of the experiment, Boyle failed in his first attempt. But it was not for want of trying.

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Enter Mr Towneley

Richard Towneley, whose home was near Burnley in Lancashire, northern England, was curiously similar to Robert Boyle in that the Towneley family estate also generated an income which meant that Richard did not have to work for a living. And just like Boyle, Towneley devoted his time to scientific studies.

Richard Towneley’s home, Towneley Hall in northern England, painted by JMW Turner in 1799

Now it just so happened that the previously-mentioned Henry Power of Halifax was the Towneley family’s physician. This gave Richard and Henry regular opportunities to share their enthusiasm for scientific experiments and discuss the latest scientific news.

In 1660 they both read Robert Boyle’s New Experiments Physico-Mechanicall, and this prompted further interest in the study of air pressure which Power had conducted on Halifax Hill seven years earlier. Not far to the north of Towneley Hall is Pendle Hill, whose summit is 1,827 feet (557 meters) above sea level, and it was here that Power and Towneley conducted an experiment that made history.

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Pendle Hill, photographed by Lee Pilkington

On Wednesday 27th April 1661, they introduced a quantity of air above the mercury in a Torricellian tube. They measured its volume, and then using the tube as a barometer, they measured the air pressure. They then ascended Pendle Hill and at the summit repeated the measurements of volume and air pressure. As expected, there was an increase in volume and a decrease in pressure.

Although their measurements were only roughly accurate and doubtless affected by temperature differences between the base and summit of the hill, the numerical data was sufficient to give them the intuitive realization of a reciprocal relationship between the pressure and volume of air

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On a high hill in northern England, nature revealed one of its secrets to Henry Power and Richard Towneley. Now they needed to communicate their finding.

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Boyle reports the news

It was not until two years after the Pendle Hill experiment that Henry Power eventually got around to publishing the results in Experimental Philosophy (1663). This delay explains why history has never associated the names of Power and Towneley with the gas law they discovered, because the news from Pendle Hill was first given to, and first reported by, Robert Boyle.

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The second edition of 1662

In the appendix to the second edition of his New Experiments Physico-Mechanical published in 1662, Boyle reports Power and Towneley’s experimental activities, which in error he ascribes solely to Richard Towneley. It was on the basis of this second edition of Boyle’s book that the reciprocal relationship between the volume of air and its pressure became known as “Mr. Towneley’s hypothesis” by contemporary authors such as Robert Hooke and Isaac Newton.

A complication was added by the fact that Power and Towneley’s experiment led Boyle to the idea of compressing air in a J-shaped tube by pouring mercury into the long arm and measuring the volume and applied pressure.

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From this experiment, conducted in September 1661, Boyle discovered that the volume of air was halved when the pressure was doubled. Curiously, he translated this finding into the hypothesis that there was a direct proportionality between the density of air and the applied pressure.

It didn’t occur to Boyle that the direct relation between density and pressure was the same thing as the reciprocal relation between volume and pressure discovered earlier by Power and Towneley. Boyle seems to have thought that compression was somehow different to expansion, and that his experiment broke new ground.

For this reason Boyle believed he had discovered something new, and since his name was far better known in scientific circles than that of Henry Power and Richard Towneley, it was inevitable that Boyle’s name would be associated with the newly-published hypothesis, which in time became Boyle’s law.

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Postscript : La loi de Mariotte

The pressure-volume law discovered by Power and Towneley, and confirmed by Boyle, is known as Boyle’s law only in Britain, America, Australia, the West Indies and other parts of what was once called the British Empire. Elsewhere it is called Mariotte’s law, after the French physicist Edme Mariotte.

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Mariotte stated the pressure-volume relationship in his book De la nature de l’air, published in 1679, some 17 years after the second edition of Boyle’s New Experiments Physico-Mechanical. Mariotte made no claim of originality, nor did he make any reference to Boyle. But he was an effective publicist for the law, with the result that his name became widely associated with it.

Some might argue that the attribution is not quite deserved. However, Edme Mariotte stated something immensely important that Robert Boyle neglected to mention. He pointed out that temperature must be held constant for the pressure-volume relationship to hold:

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In the view of CarnotCycle, the pressure-volume law so stated can with justification be called Mariotte’s law.

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P Mander January 2016