Johannes Diderik The Nobel Prize in Physics 1910

biography

The simple rules for gases about pressure, volume and temperature that you learnt in Scuba for beginners are only an approximation. Good enough for some poor lad or lass who has to worry about doing a mask clear without a total sinus washout but now you're grown up and want to breath fancy stuff it just won't do anymore. You haven't forgotten but we'll recap anyway. Do you remember PV=kT? The ideal gas law? Oh yes. The one that summarised all the others ones that we can't remember the names of. Well let us get rid of that nasty k for an arbitrary constant and turn it into the real one PV=nRT that scientists use. This is much better because we can calculate things with it rather than just do ratios. Let's name the parts in useful units:

P Pressure in bar

V Volume in Litres

n Quantity of gas in mols

R The universal gas constant 0.0831451 if we want to do bar and litres

T The temperature in Kelvins (virtually Centigrade+273)

mols? Well it's a chemists trick to have a measure of something rather than work in boring units like grams. A mol is a gram molecule. It is the amount of a substance that weighs X grams where X is the molecular weight of the molecule in question. So Oxygen has an atomic weight of 16 (well 15.994 if you want to get all isotopic about it) so O2 has a molecular weight of 32 so 32 grams of O2 is one mol and 64 grams is 2 mols etc. Plutonium Oxide has a molecular weight of 536 so it take over half a kilo of that stuff to make a mol but chemists don't care. The neat trick is that 1 mol contains 6.02x1023 molecules. A mol is not so much a quantity as a head count and is great when you are working out how things react. Great. Real numbers. Let's do an example. The 100% oxygen deco bottle is 3L, it is January at Stoney Cove so it is 4C, that is 277K and we want the whole 300 bar. So n = PV/RT so n = (fumbles with calculator) n = 39 mols so at 32 grams to the mol we have 1.248 Kilograms of Oxygen.

Then along comes Van der Waals in spoil sport mode and says "well not really". The trouble is that this thing is the Ideal gas law. That is it only really would work for an ideal gas and all we have are real ones. An ideal gas is about as common as an ideal husband/boy friend or an ideal wife/girl friend. Yup. They don't exist. An ideal gas would be composed of infinitely small molecules that did not attract one another but real gases have real sized molecules taking up the space and they tend to attract and repel one another. Van der Waals was the man who set about solving the problem by working out how to allow for real gases. What he came up with was not the exact answer but a much better gas equation.

If you look at it and remember PV=nRT you can see the old ideal gas equation in here but with two extra terms, one applying a fiddle factor to the pressure to allow for the attraction between molecules drawing them inwards and reducing the pressure on the outside world and the other fiddle factor is on the volume where it is effectively reduced to allow for the fact that all these molecules are taking up space. We get two new constants a and b which depend on the gas we are considering. Back to the example for our tank of O2 and use the Oxygen a value of 1.382 and the b value of 0.03186 and we get a different situation. Putting our 1.248Kgs ie. our 39 mols of Oxygen into our 3L tank gives 277bar.

OK lets graph that with mols along the bottom and pressure in bar up the left. What's this mean? The nice straight purple line is gas pressure against mols of gas for the Ideal law. The blue line sweeping upwards is Van der Waals' calculation. Down at 0 to 20 bar, where you did you school physics and your diving it is very good. Up at 100 bar it takes 15 mols of Oxygen to get 100 bar and for 200 bar it takes 30 mols but for 300 bar you do not have the 45 mols you expected but just 41. This is why divers get edgy about mixing Nitrox to 300 bar final pressure.

It gets worse.

We assume for ideal gases that we can work out the partial pressures independently and just add them up (Dalton's Law) and moreover we assume that the ratios don't change with pressure. Now for nitrox it happens that the a and b values for Nitrogen are similar to Oxygen but if (simplistically) we put 100 bar of Oxygen in a tank and then topped it off to 300 bar with pure Nitrogen we do not have 33% Nitrox. It is a bit higher. More like 37%. We can calculate it but Van der Waals' equation as stated above only applies to the simple monatomic gases and as the molecules of one gas see the others nothing is simple. However we can produce modified a and b constants for the mixed gas formed using the values for each gas combined using:

What on Earth? Yes. I did university physics and I winced a bit at that one. What it is saying that for a mixed gas made of n gases (1 to n) whose fractions (ratio of mols) are x1, x2, x3... xn and whose a and b values are a1, b1, a2, b2 etc. then you get the global a and b values by taking the formula to the right of the two sigma signs and adding up all the bits. If you have three gases (Oxygen, Nitrogen and Helium for example) then a = ?(a1*a1)*x1*x1 + ?(a1*a2)*x1*x2 + ?(a1*a3)*x1*x3 + ?(a2*a1)*x2*x1 + ?(a2*a2)*x2*x2 + ?(a2*a3)*x2*x3 + ?(a3*a1)*x3*x1 + ?(a3*a2)*x3*x2 + ?(a3*a3)*x3*x3 and naturally b looks much the same. This was probably grief to poor old Van der Waals but we have spread-sheets on our home computers... Once you have done this you can work out the total pressure but don't think in partial pressures, Dalton style, any more because they don't exist. Definitely don't try to work back from pressure to ratio. It's horrible

Johannes Diderik van der Waals was born on November 23, 1837 in Leyden, The Netherlands, the son of Jacobus van der Waals and Elisabeth van den Burg. After having finished elementary education at his birthplace he became a schoolteacher. Although he had no knowledge of classical languages, and thus was not allowed to take academic examinations, he continued studying at Leyden University in his spare time during 1862-65. In this way he also obtained teaching certificates in mathematics and physics. In 1864 he was appointed teacher at a secondary school at Deventer; in 1866 he moved to The Hague, first as teacher and later as Director of one of the secondary schools in that town.

New legislation whereby university students in science were exempted from the conditions concerning prior classical education enabled Van der Waals to sit for university examinations. In 1873 he obtained his doctor's degree for a thesis entitled Over de Continu?teit van den Gas - en Vloeistoftoestand (On the continuity of the gas and liquid state), which put him at once in the foremost rank of physicists. In this thesis he put forward an "Equation of State" embracing both the gaseous and the liquid state; he could demonstrate that these two states of aggregation not only merge into each other in a continuous manner, but that they are in fact of the same nature. The importance of this conclusion from Van der Waals' very first paper can be judged from the remarks made by James Clerk Maxwell in Nature, "that there can be no doubt that the name of Van der Waals will soon be among the foremost in molecular science" and "It has certainly directed the attention of more than one inquirer to the study of the Low-Dutch language in which it is written" (Maxwell probably meant to say "Low-German", which would also be incorrect, since Dutch is a language in its own right). Subsequently, numerous papers on this and related subjects were published in the Proceedings of the Royal Netherlands Academy of Sciences and in the Archives N?erlandaises, and they were also translated into other languages.

When, in 1876, the new Law on Higher Education was established which promoted the old Athenaeum Illustre of Amsterdam to university status, Van der Waals was appointed the first Professor of Physics. Together with Van't Hoff and Hugo de Vries, the geneticist, he contributed to the fame of the University, and remained faithful to it until his retirement, in spite of enticing invitations from elsewhere. The immediate cause of Van der Waals' interest in the subject of his thesis was R. Clausius' treatise considering heat as a phenomenon of motion, which led him to look for an explanation for T. Andrews' experiments (1869) revealing the existence of "critical temperatures " in gases. It was Van der Waals' genius that made him see the necessity of taking into account the volumes of molecules and the intermolecular forces ("Van der Waals forces", as they are now generally called) in establishing the relationship between the pressure, volume and temperature of gases and liquids.

A second great discovery - arrived at after much arduous work - was published in 1880, when he enunciated the Law of Corresponding States. This showed that if pressure is expressed as a simple function of the critical pressure, volume as one of the critical volume, and temperature as one of the critical temperature, a general form of the equation of state is obtained which is applicable to all substances, since the three constants a, b, and R in the equation, which can be expressed in the critical quantities of a particular substance, will disappear. It was this law which served as a guide during experiments which ultimately led to the liquefaction of hydrogen by J. Dewar in 1898 and of helium by H. Kamerlingh Onnes in 1908. The latter, who in 1913 received the Nobel Prize for his low-temperature studies and his production of liquid helium, wrote in 1910 "that Van der Waals' studies have always been considered as a magic wand for carrying out experiments and that the Cryogenic Laboratory at Leyden has developed under the influence of his theories ". Ten years later, in 1890, the first treatise on the "Theory of Binary Solutions" appeared in the Archives N?erlandaises - another great achievement of Van der Waals. By relating his equation of state with the Second Law of Thermodynamics, in the form first proposed by W. Gibbs in his treatises on the equilibrium of heterogeneous substances, he was able to arrive at a graphical representation of his mathematical formulations in the form of a surface which he called "Psi-surface" in honour of Gibbs, who had chosen the Greek letter Psi as a symbol for the free energy, which he realised was significant for the equilibrium. The theory of binary mixtures gave rise to numerous series of experiments, one of the first being carried out by J. P. Kuenen, who found characteristics of critical phenomena fully predictable by the theory. Lectures on this subject were subsequently assembled in the Lehrbuch der Thermodynamik (Textbook of thermodynamics) by Van der Waals and Ph. Kohnstamm. Mention should also be made of Van der Waals' thermodynamic theory of capillarity, which in its basic form first appeared in 1893. In this, he accepted the existence of a gradual, though very rapid, change of density at the boundary layer between liquid and vapour - a view which differed from that of Gibbs, who assumed a sudden transition of the density of the fluid into that of the vapour. In contrast to Laplace, who had earlier formed a theory on these phenomena, Van der Waals also held the view that the molecules are in permanent, rapid motion. Experiments with regard to phenomena in the vicinity of the critical temperature decided in favour of Van der Waals' concepts.

Van der Waals was the recipient of numerous honours and distinctions, of which the following should be particularly mentioned. He received an honorary doctorate of the University of Cambridge; was made honorary member of the Imperial Society of Naturalists of Moscow, the Royal Irish Academy and the American Philosophical Society; corresponding member of the Institut de France and the Royal Academy of Sciences of Berlin; associate member of the Royal Academy of Sciences of Belgium; and foreign member of the Chemical Society of London, the National Academy of Sciences of the U.S.A., and of the Accademia dei Lincei of Rome.

In 1864, Van der Waals married Anna Magdalena Smit, who died early. He never married again. They had three daughters and one son. The daughters were Anne Madeleine who, after her mother's early death, ran the house and looked after her father; Jacqueline Elisabeth, who was a teacher of history and a well-known poetess; and Johanna Diderica, who was a teacher of English. The son, Johannes Diderik Jr., was Professor of Physics at Groningen University 1903-08, and subsequently succeeded his father in the Physics Chair of the University of Amsterdam. Van der Waals' main recreations were walking, particularly in the country, and reading. He died in Amsterdam on March 8, 1923.

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