Posts Tagged ‘kinetic theory’

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I have always had a fondness for classical experiments that revealed fundamental things about the particulate nature of our world. Examples that spring to mind include JJ Thomson’s cathode ray tube experiment (1897), Robert Millikan’s oil drop experiment (1909), and CTR Wilson’s cloud chamber (1912). The particles of interest in these cases were subatomic, but during this era of discovery there was another pioneering experiment that focused on molecules and their chemical reactivity. The insight this experiment provided was important, but the curious fact is that relatively few people have ever heard of it.

So to resurrect this largely forgotten piece of scientific history, CarnotCycle here tells the story of the Ozone Experiment conducted by the Hon. Robert John Strutt FRS at Imperial College of Science, South Kensington, London in 1912.

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The experimenter

NPG x122578; Lord Robert John Rayleigh, 4th Baron Rayleigh by Bassano

RJ Strutt (1875-1947) photographed in 1923

The Honorable Robert John Strutt, 4th Baron Rayleigh, might be an unfamiliar name to some of you. But you will undoubtedly have heard of his father, Lord Rayleigh of Rayleigh scattering fame. Where his father led, Robert John followed: first as a research student at the Cavendish Laboratory in Cambridge where his father had been Cavendish professor, and then at Imperial College of Science in South Kensington, London where he followed up his father’s eponymous work on light scattering.

But Robert John did some interesting work of his own. For one thing, he was the first to prove the existence of ozone in the upper atmosphere, and for another he studied the effect of electrical discharges in gases. Interestingly it was a combination of these two things – ozone produced in an electrical discharge tube – that formed the basis of Strutt’s groundbreaking 1912 experiment.

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The experiment

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Here is the apparatus that Strutt employed in his experiment. As the arrows indicate, air enters from the right via stopcock a, where the pressure is significantly reduced by the action of the air pump at left. Low-pressure air then passes through the discharge tube b, where ozone is formed from oxygen according to the reaction

The air, containing ozone at a few percent, enters chamber c where it encounters a silver gauze partition d, mounted between two mica discs e in each of which there is a hole 2 millimeters in diameter. A sealed-in glass funnel g supports the mica discs as shown. As the air passes the gauze, ozone reacts with the silver in what is thought to be an alternating cycle of oxidation and reduction which destroys the ozone while constantly refreshing the silver

The chambers on either side of the gauze partition are connected by tubes f, either of which could be put into communication with a McLeod pressure gauge. The rate of air intake was measured by drawing in air at atmospheric pressure from a graduated vessel standing over water. From this data, combined with the McLeod pressure gauge measurements, the volume v of the low-pressure air stream passing the gauze per second could be calculated.

So to recap, in Strutt’s steady-state experiment, air passes through the apparatus at a constant rate as ozone is generated in the discharge tube and destroyed by the silver gauze. The question then arises – What proportion of the ozone is destroyed as it passes the gauze?

This brings us to the luminous aspect of the ozone experiment, which enabled Strutt to provide an answer.

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The green glow

The conversion of oxygen into its allotrope ozone was not the only reaction taking place in the discharge tube of Strutt’s apparatus. There was also a reaction between nitrogen and oxygen – known to occur in lightning strikes – which produces nitrogen(II) oxide

Now it just so happens that nitrogen(II) oxide and ozone react in the gas phase to produce activated nitrogen(IV) dioxide, which exhibits chemiluminescence in the form of a green glow as it returns to its ground state

This was a crucial factor in Strutt’s experiment. The air flowing into the chamber c was glowing green due to the above reactions taking place in the gas phase. But as the flow passed the silver gauze, ozone molecules were destroyed with the result that the green glow was weaker in the left-hand chamber compared with the right-hand chamber.

By adjusting the rate of air flow through the apparatus, Strutt could engineer a steady state in which the green glow was just extinguished by the silver gauze – in other words he could find the flow rate at which all of the ozone molecules were destroyed by the silver/silver oxide of the gauze partition.

[To allay doubts, Strutt introduced ozone gas downstream of the gauze where the green glow had been extinguished. The glow was restored.]

Strutt was now in a position to interpret the experiment from a new and pioneering perspective – his 1912 paper was one of the very first to consider a chemical reaction in the context of molecular statistics.

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The analysis

In terms of chemical process, Strutt’s steady-state experiment was unremarkable. Air flowed through the apparatus and the ozone generated in the discharge tube was destroyed by the silver gauze. The novel feature was in the analysis, where Strutt applied both classical physics and the kinetic theory of gases to calculate the ratio of the mass of ozone impinging upon the silver gauze in a second to the mass passing the gauze in a second.

As mentioned above, Strutt could compute the volume v of the stream passing through the apparatus in a second, so the mass of ozone passing the gauze in a second was simply ρv, where ρ is the density of ozone in the stream as it arrives at the gauze.

In his paper, Strutt states a formula for calculating the mass of ozone impinging upon the silver surface in a second

without showing the steps by which he reached it. These steps are salient to the analysis, so I include the following elucidation due to CN Hinshelwood* in which urms is the root mean square velocity (i.e. the average velocity, with units taken to be cm/s) of the gas molecules:

Suppose we have a solid surface of unit area exposed to the bombardment of gas molecules. Approximately one-sixth of the total number of molecules may be regarded as moving in the direction of the surface with the average velocity. In one second all those within distance urms could reach and strike the surface, unless turned back by a collision with another molecule, but for every one so turned back, another, originally leaving the surface, is sent back to it. Thus the number of molecules striking the surface in a second is equal to one-sixth of the number contained in a prism of unit base and height urms. This number is 1/6.n’.urms,, n’ being the number of molecules in 1 cm^3. Thus the mass of gas impinging upon the surface per second is

A more precise investigation allowing for the unequal speeds of different molecules shows that the factor 1/6 should really be

We therefore arrive at the result that the mass of gas striking an area A in one second is

*CN Hinshelwood, The Kinetics of Chemical Change in Gaseous Systems, 2nd Ed. (1929), Clarendon Press

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The ratio

Strutt takes the above formula for the mass of ozone impinging on the gauze per second and divides it by the formula for the mass of ozone passing the gauze per second, ρv. This operation cancels out the unknown value of ρ, giving

The values of v (200 cm3s-1) and A (0.0369 cm2) were obtained by Strutt using direct measurements, while urms for ozone molecules is simply stated without mentioning that it is necessarily computed from the fundamental kinetic equation

If n is Avogadro’s number, v is the molar volume and pv = RT, whence

where M is the molar mass. The urms figure Strutt gives for ozone is 3.75 × 104; typically for the time he neglects to state the units which are presumed to be cm/s. This velocity seems a little low, implying a temperature of 270.6K for the air flow in his apparatus. But then again, the pressure dropped significantly at the stopcock so in all likelihood there would have been some Joule-Thomson cooling.

Inserting the values for A, v and urms in the ratio expression gives

Since we can interpret mass in terms of the number of ozone molecules, the ratio expresses the number of collisions to the number of molecules passing, or the average number of times each ozone molecule must strike the silver surface before it passes.

As the experiment is arranged so that no ozone molecules pass the silver gauze, the ratio must represent the average number of collisions that an ozone molecule makes with the silver surface before it is destroyed.

The 1.6 ratio reveals the astonishing fact that practically every ozone molecule which strikes the silver (oxide) surface is destroyed. To a chemical engineer that is a hugely important piece of information, which amply illustrates the value of applying kinetic theory to chemical reactivity.

The application of analogous calculations to the passage of gas streams over solid catalysts in industrial processes is obvious. All of which makes it even more curious that Robert John Strutt’s apparatus, and the pioneering work he did with it, is not better known.

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Further reading

Hon. RJ Strutt, The Molecular Statistics of some Chemical Actions (1912)
The principal source for this blog post.
http://rspa.royalsocietypublishing.org/content/royprsa/87/596/302.full.pdf

CTR Wilson, On an expansion apparatus for making visible the tracks of ionizing particles in gases and some results obtained by its use (1912)
The Cloud Chamber – a truly historic piece of apparatus and one of my favorites. This paper was published in September 1912, just a month before Strutt’s paper.
http://rspa.royalsocietypublishing.org/content/royprsa/87/595/277.full.pdf

P Mander August 2016

The statue of Thomas Graham, sculpted by William Brodie in 1872

On the south-eastern corner of Glasgow’s George Square is a fine statue of Thomas Graham (1805-1869). Born and raised in the city, he became a chemistry student at the University of Glasgow and graduated there in 1826. At some point in his studies he happened to read about an observation made by the German chemist Johann Döbereiner (1780-1849) that hydrogen gas leaked out from a crack in a glass bottle faster than air leaked in. It was this simple fact that set Thomas Graham on the path to scientific fame. But before we continue, a few words about Johann Döbereiner.

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Döbereiner’s lamp – the first lighter

Johann Döbereiner, professor of chemistry at the University of Jena, invented this amazing piece of apparatus in 1823, while Thomas Graham was still an undergraduate student in Scotland. It consists of a glass container (a) filled with dilute sulfuric acid and inside it an inverted cup (b) in which is suspended a lump of zinc metal (c,d). When the tap (e) is opened, the acid enters the cup and reacts with the zinc, producing hydrogen gas which flows out of a tube (f) and onto a piece of platinum gauze (g). Now here is the interesting part. The gauze catalyzes the reaction of hydrogen with atmospheric oxygen, producing a lot of heat in the process. The platinum gauze gets red hot and ignites the hydrogen flowing out of the tube, producing a handy flame for lighting candles, cigars, etc. In the days before matches, this gadget was a godsend and became a commercial hit with thousands being mass produced in a wonderful range of styles. A YouTube demonstration of Döbereiner’s Lamp can be seen here.

In a paper published in 1823, Döbereiner recorded the observation that hydrogen stored in a glass jar over water leaked out from a crack in the glass much faster than the surrounding air leaked in, causing the level of the water to rise significantly. This was the trigger for Graham’s research into the phenomenon of diffusion, during which he discovered not only an important quantitative relation between diffusion and gas density but also a means by which the separation of mixed gases could be achieved.

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Graham’s experiments on effusion

Taking his cue from Döbereiner’s leaking glass jar, Graham developed apparatus by which he could study the rate of escape of a contained gas through a small hole in a piece of platinum foil. This particular kind of diffusion, where the flux is restricted to a tiny orifice between one gaseous environment and another, is called effusion.

The rates of effusion of two gases can be compared using the apparatus illustrated. The first gas is introduced through the three-way tap C to fill the entire tube B. The tap is closed and the gas is then allowed to effuse through the hole in the platinum foil A. The time taken for the liquid level to rise from X to Y is recorded as the gas escapes into the atmosphere. The experiment is then repeated with the second gas. If the recorded times are t1 for the first gas and t2 for the second, the rates of effusion are in the ratio t2/t1.

Using this method Graham discovered that the rate of escape of a gas was inversely related to its density: for example hydrogen escaped 4 times faster than oxygen. Given that the density of oxygen is 16 times that of hydrogen, the nature of the inverse relation suggested itself and was confirmed by comparisons with other gases.

In 1829, Graham submitted an internal research paper in which he recorded his experimentally determined relation between the effusion rates of gases and their densities

Graham also experimented with binary gases, and noted that the greater rate of escape of the lighter gas made it possible to achieve a measure of mechanical separation by this means.

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Graham’s Law

By 1831 Graham had recognized that the comparative rates of effusion of two gases into the atmosphere could equally be applied to the diffusion of two gases in contact.

On Monday 19th December 1831, Graham read a paper before the Royal Society of Edinburgh in which he stated his eponymous square root law. This paper was published in the Philosophical Magazine in 1833 while he was professor of chemistry at Anderson’s College in Glasgow. Four years later he moved to London to became professor of chemistry at University College, where in 1848 he embraced Avogadro’s hypothesis by stating that the rate of diffusion of a gas is inversely proportional to the square root of its molecular weight.

Hence if the rates of diffusion of two gases are known and the molar mass of one is known, the molar mass of the other can be calculated from the relation

In 1910 the French chemist André-Louis Debièrne, a close associate of Pierre and Marie Curie, used this relation to calculate the molecular weight of Radon gas. (Trivial Fact: Debièrne was one of those fortunate Frenchmen to be born on France’s national day, Le Quatorze Juillet – 14 July. So every year his birthday was a national holiday :-)

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Thermodynamics and Kinetics

In the first half of the 19th century, the understanding of gases rested on the gas laws which Sadi Carnot’s compatriot Émile Clapeyron synthesized into the ideal gas law pv = RT in 1834. Meanwhile Avogadro’s hypothesis of 1811 laid the foundations of molecular theory from which developed the idea that gases consisted of large numbers of very small perfectly elastic particles moving in all directions through largely empty space.

These two strands of thought came together in the notion that gas pressure could be attributed to the random impacts of molecules on the walls of the containing vessel. In Germany Rudolf Clausius produced a paper in 1857 in which he derived a formula connecting pressure p and volume v in a system of n gas molecules of mass m moving with individual velocity c

In this equation we see the meeting of thermodynamics on the left with kinetic theory on the right. And it points up a feature of thermodynamic expressions that commonly escapes notice. We are taught that in classical thermodynamics, the time dimension is absent as a unit of measure although entropy is sometimes cast in this role as “the arrow of time”. But the fact is that time is very much present when you apply dimensional analysis.

Pressure is force per unit area and has dimensions ML-1T-2. And there is the time dimension T, in the definition of the thermodynamic intensive variable pressure. This is what enabled Clausius to equate a time-dependent expression on the right with a seemingly time-independent one on the left.

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Kinetic Theory and Graham’s Law

Since density ρ is mass per unit volume, the above equation can be written

If the rate of effusion/diffusion of a gas is taken to be proportional to the root mean square velocity of the gas molecules, then at constant pressure

which is the first statement of Graham’s law.

For 1 mole of gas, the aforementioned Clausius equation can be written as

where V is the molar volume, R is the gas constant, T is the temperature and N is the Avogadro number. Since the product of the Avogadro number N and the molecular mass m is the molar mass M, it follows that

Again, if the rate of effusion/diffusion of a gas is taken to be proportional to the root mean square velocity of the gas molecules, then at constant temperature

which is the second statement of Graham’s law.

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Graham’s Law and uranium enrichment

Plaque marking the site of the K-25 plant at Oak Ridge Tennessee.

Back in 1829 Thomas Graham noted from his effusion experiments on binary gases that a measure of mechanical separation could be achieved by this means. Over a century later, that observation was of crucial importance to the scientists engaged in the Manhattan Project which produced the first nuclear weapons during WW2.

To produce an atomic bomb required a considerable quantity of the fissile uranium isotope 235U. The problem was that this isotope makes up only about 0.7% of naturally occurring uranium. Substantial enrichment was necessary, and this was achieved in part by employing gaseous effusion of uranium hexafluoride UF6. Since fluorine has a single naturally occurring isotope, the difference in weights of 235UF6 and 238UF6 is due solely to the difference in weights of the uranium isotopes and so a degree of separation can be achieved.

The optimal effusion rate quotient (√ 352/349) is only 1.0043 so it was clear to the Manhattan Project engineers that a large number of separation steps would be necessary to obtain sufficient enrichment, and this was done at Oak Ridge Tennessee with the construction of the K-25 plant which ultimately consisted of 2,892 stages.

In more recent times, the development of the Zippe-type centrifuge made the gas diffusion method of 235U isotope separation redundant and led to the closure of the K-25 plant in 2013. The Zippe-type centrifuge is considerably more energy-efficient than gaseous diffusion, has less gaseous material in circulation during separation, and takes up less space.

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Further reading

Thomas Graham biography at encyclopedia.com

link: https://www.encyclopedia.com/people/science-and-technology/chemistry-biographies/thomas-graham

Thomas Graham Contributions to diffusion of gases and liquids, colloids, dialysis and osmosis Jaime Wisniak, 2013 (contains comprehensive references to Graham’s published work)

link: https://www.sciencedirect.com/science/article/pii/S0187893X13725217#bib0005

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P Mander, February 2021