Müller I. A history of thermodynamics. The doctrine of energy and entropy (1185104), страница 42
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Wehave learned about this before, cf. Chap. 2. In an ideal gas the effect is nil,or very tiny indeed – to the extent that the gas is not really ideal. Thismeans that before throttling can be applied efficiently, the gas has toundergo Cailletet’s adiabatic expansion, which converts it into a vapour18The reader has surely noticed the author’s special liking for the science essays of IsaacAsimov. Actually the present treatment of gas liquefaction also makes use of two suchessays, namely I. Asimov: “Liquefying gases” and “Toward absolute zero” both in“Exploring the earth and the cosmos.” Penguin Books, London (1990). These essays,however, see Asimov wrong, because he confuses Cailletet’s adiabatic expansion and theadiabatic Joule-Thomson effect.
The former is an essentially reversible process at constantentropy, while the latter is an inherently irreversible process with an unchanged enthalpybetween beginning and end.19 Oxygen, nitrogen and hydrogen come in blue, green and red bottles, respectively, under apressure of 150 bar.1766 Third Law of Thermodynamicsclose to liquefaction. Linde used several steps of throttling andregeneration, i.e. he pre-cooled the incoming flow of vapour by making itexchange heat with the already throttled one. The Linde process is still usednow. And Linde’s firm – founded in 1879 – thrives on selling liquefiedgases, although it is mostly putting out the ubiquitous compressionrefrigerators, another invention of Linde’s.Fig.
6.4 . Carl Ritter von Linde (1842–1934). Schematic view of his air liquefying apparatusJohannes Diderik van der Waals (1837 -- 923)Van der Waals was the person who made sense out of the concept of thecritical point and who corroborated Andrew’s conjecture that all gasesshould have such a point.
He considered that the ideal gas law R 1v MP 6 isan idealization which ignores inter-atomic forces. Van der Waals reasonedthat the interaction force – now called van der Waals force – is mildlyattractive at large distances and strongly repulsive when the atoms are close.Thus the potential ij(r) of the force between two atoms in the distance r hasthe form shown qualitatively in Fig. 6.5.20 On the grounds of thisassumption van der Waals was able to derive a modified form of the idealgas law, namely, cf. Insert 6.1R206C 2v D vMP21.Van der Waals could not know the nature of the attractive force.
It is an electric dipoledipole interaction, and the dipoles are due to a mutually induced differential shift of theelectron shells and the nuclei of adjacent atoms.21 J.D. van der Waals: “Over de continuiteit van den gas- en vloeistoftoestand.” [On thecontinuity of the gaseous and the liquid state].
Dissertation, Leiden (1873).Johannes Diderik Van Der Waals (1837–1923)177Fig. 6.5. Schematic form of the interatomic interaction potential as a function of the distanceof two atoms. Also: van der Waals coefficientsFig. 6.6. Isotherms of a van der Waals gas.
Also: Maxwell constructionThis has become known as the van der Waals equation for a real gas.Obviously the modification lies in the positive coefficients a and b. Thecoefficient b represents the volume of an atom which clearly must detractfrom the total available volume. And the coefficient a represents the rangeand size of the attractive interaction which reduces the pressure exerted onthe wall.In a certain range of temperatures the van der Waals equation describesisotherms in a (p,v)-diagram that are non-monotone, as shown qualitativelyin Fig.
6.6. Thus there is the possibility to have two – actually three –specific volumes for one pressure and one temperature. Ignoring the middleone, van der Waals interpreted the two remaining volumes as those of theliquid and the vapour, and came up with the surprising conclusion that histheory, intended for real gases – as opposed to ideal gases –, could perhapsdescribe a liquid-vapour transition.
This is what the title of his worksuggests. Accordingly the temperature, whose isotherm develops ahorizontal point of inflection, has to be interpreted as the critical isothermand the inflection point itself as the critical point. By the van der Waalsequation that point has the coordinates1786 Third Law of Thermodynamicsv%3D ,R%1 C27 D 2M6%P8 C.27 DAlthough van der Waals’s work was presented as a doctoral thesis, –rather than in a scientific journal – it became quickly known. Boltzmannrecognized it as a masterpiece, and he was so enthusiastic about thederivation that he called van der Waals the Newton of real gases.22 AndMaxwell discovered a graphical method for the determination of thesaturated vapour pressure p(T) for the van der Waals gas, see.
Fig. 6.6. Hewrote the phase-equilibrium condition of Insert 3.7 for the free energyF = U-TS in the formÈ F ØFƎ – Fƍ = – p(T)(VƎ – Vƍ)or withp ÉÊ V ÙÚ TV cc³ p(V , T )dVp(T )(V cc V c) ,Vcwhere the integration must be taken along the isotherm. Thus p(T) is theisobar that makes the two shaded areas in Fig. 6.6 equal in size, Thisgraphical method to determine p(T), and vƍ(T), vƎ(T) has become known asthe construction of the Maxwell line.An interesting corollary of the van der Waals equation emerges when oneintroduces dimensionless variablesSR,R%Qvv%W6,6%because in that case the equation becomes universal, i.e independent ofparameters relating to the particular fluidS8W3 2 .3Q 1 QVan der Waals called this relation the law of corresponding states: Stateswith equal non-dimensional variables correspond (sic) to each otherirrespective of the material properties.
This implies that the liquid-vapourproperties of all substances are alike:x convex, monotonically increasing vapour pressure curves,x similar wet steam regions and, of coursex critical points.The underlying reason for such conformity is the fairly plain (ij,r)-relation,cf. Fig. 6.5, which is common – qualitatively – to all gases.22..In: Encyclopadie der mathematischen Wissenschaften, Bd.
V.1. p. 550.Johannes Diderik Van Der Waals (1837–1923)179From a practical point of view, and with regard to liquefying gases, themost important conclusion from the van der Waals equation concerns theJoule-Thomson effect in a throttling experiment. It turned out that throttlingdid not necessarily lead to cooling. One thing was well-known, however:The energy flux before and behind an adiabatic throttle must be equal;therefore the first law requires that the specific enthalpy h is unchanged,provided that the kinetic energy of the flow can be neglected. Thatcondition could be used for the calculation of the temperature change ǻT fora given pressure drop ǻp, cf.
Insert 6.2. One obtains the criteria1 È v Ø1v Ê T Ú pTÉÙ ! cooling0 for no change .heatingRather obviously the equality holds for ideal gases, so that ideal gases donot change their temperature upon adiabatic throttling. And for a van derWaals gas the criteria imply that the initial state must lie below the graphswhich define the inversion curve in the (Ȟ,IJ)- , the (ʌ,IJ)-, or the (ʌ,Ȟ)diagram, viz.Q1,3 2 13 WS24 3W 12W 27,S9Q218Q.Obviously we have used here the dimensionless variables of the law ofcorresponding states. If a state lies on the inversion curves, it does notchange temperature upon throttling; if it lies above the curves, the gas heatsup.Figure 6.7 shows the inversion curves in the (ʌ,IJ)-diagram and in the(ʌ,Ȟ)-diagram along with – for better orientation – the critical isochor andthe critical isotherm, respectively.
Inspection of the (ʌ,IJ)-diagram – and ofthe mini-table in Fig. 6.7 with critical data for oxygen and hydrogen –shows that hydrogen of 1atm heats up, if throttled above T = 140 K.Therefore the Linde process for the liquefaction of hydrogen must start at alower temperature. For oxygen, on the other hand, the process may start atroom temperature. To be sure, it is not very efficient there; the coolingeffect at room temperature was barely big enough to have been noticed byJoule and Kelvin.The van der Waals equation with its two parameters a and b isquantitatively not good for any actual gas no matter how a and b arechosen.
It does, however, have great heuristic value, because it is based onmolecular considerations, cf. Insert 6.1, and it represents a fairly simpleanalytic thermal equation of state. It is therefore revisited over and overagain. Fairly recently I have come across an instructive article entitled1806 Third Law of ThermodynamicsFig.
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