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Assemblies of Imperfect Gases by the method of Partition Functions

Published online by Cambridge University Press:  24 October 2008

R. H. Fowler
R. H. Fowler
Affiliation:
Trinity College
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Extract

In a recent series of papers analytical methods have been introduced which allow of a mathematically simple treatment of the theorems of statistical mechanics for the usual assemblies of isolated or effectively isolated systems. By this we mean that the individual component systems may be treated for energy content as if they were never interfered with. It is only then that energy can be assigned to systems rather than to the assembly as a whole, and it is on this partition of energy among the systems that the analysis is based. When this independence, for example between separate atoms, breaks down as in a molecule, and still more in a crystal, we can take the whole complex to be a system. The analysis will still apply, and if we can formulate the dynamical motions of the complex system, we can still make progress. The essential step for any system is to construct its partition function. Examples of such constructions for molecules and crystals will be found in the papers quoted, and are of course otherwise well known.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 22 , Issue 6 , November 1925 , pp. 861 - 885
Copyright
Copyright © Cambridge Philosophical Society 1925

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References

* Darwin, I. and Fowler, , Phil. Mag. vol. XLIV, p. 450 (1922); II. Phil. Mag. p. 823;CrossRefGoogle ScholarIII. Proc. Camb. Phil. Soc. vol. xxi, p. 262 (1922);Google ScholarIV. Fowler, , Phil. Mag. vol. XLV, p. 1 (1923); V. Phil. Mag.. p. 497;CrossRefGoogle ScholarVI. Darwin, and Fowler, , Proc. Camb. Phil. Soc. vol. xxi, p. 392 (1923); VII. Proc. Camb. Phil. Soc.. p. 730.Google Scholar

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* Jeans, , Dynamical theory of gases, ed. 3, p. 91. The discussion of dissociation and aggregation there given by Jeans on a classical basis by means of the above device is of course inadmissible in general. The phenomena are essentially phenomena of the quantum theory. The device however is admirably adapted to the discussion of such classical systems and is so applied here to our classical compound “systems.” It is essentially equivalent to Boltzmann's discussion of dissociation, Vorlesungen üiber Gastheorie, II, Abschnitt VI.Google Scholar

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* The internal quantized energies of the molecules are accounted for by separate partition functions, which need not be further referred to here.Google Scholar

* Equation (27) is more usually arrived at by saying that the coefficient of N α divided by B() is the probability or frequency with which a selected α-molecule lies in d ωα and that therefore the expectation of all α-molecules in d ωα, or their average number, is given by (27). The arrangement of the proof in the text follows more closely the normal line of development adopted in these papers.Google Scholar

* Defined in this way W αβ is the free energy of β in the field of an a (Einstein, , Ann. de Phys. vol. XVII, p. 549 (1905)). This connection between the free energy and the partition functions is expressed in its most general form in the structure of Ψ.CrossRefGoogle Scholar

* Special cases of equations equivalent to (37) and (38) were first given in connection with high temperature atmospheres by Urey, , Astrophys. J. vol. LIX, p. 1 (1924),CrossRefGoogle Scholar and independently by Fermi, , Zeit. fär Phys. vol. xxvi, p. 54 (1924).CrossRefGoogle Scholar

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* At exceptional points and the fluctuation of course does not vanish.Google Scholar

* In order to discuss such an assembly completely it is necessary to suppose the problem somewhat idealized so that the mass of gas is contained in a reflecting enclosure so large that molecular impacts on the walls do not effectively alter the position of the centre of gravity of the mass of gas, or consequently its momentum, which must be fixed by the conditions of the problem. [This is not strictly realizable.] Such conditions can be formally accounted for by additional selector variables both for momenta and positional coordinates.Google Scholar

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* Rosseland, , M.N.R.A.S. vol. LXXXIV, p. 720 (1924), independently arrived at the same conclusion (p. 728).CrossRefGoogle Scholar

It must be necessary ultimately to fall back on direct calculations from B (), in the manner attempted by Milner, , Phil. Mag. vol. XXIII, p. 551 (1912); vol. xxv, p. 742 (1913). It should be placed to Milner's credit that he gave the first correct explanation of the anomaly of strong electrolytes.CrossRefGoogle Scholar

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† Milner, , loc. cil. 2nd paper, p. 748.Google Scholar

* This similarity argument is well known for 8 = 2, when it plays a great part in the theory of radiative equilibrium of gaseous stars (Eddington). It can be verified by direct calculation in any case for which calculations can be carried through.Google Scholar