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 t_{1} for the first gas and t_{2} for the second, the rates of effusion are in the ratio t_{2}/t_{1}.

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^{-1}T^{-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**

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 ^{235}U. 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 UF_{6}. Since fluorine has a single naturally occurring isotope, the difference in weights of ^{235}UF_{6} and ^{238}UF_{6} 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 ^{235}U 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