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10
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84950899131
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A notable exception is A. D. Buckingham, The Laws and Applications of Thermodynamics (Pergamon, New York, 1964), pp. 81, 189. Buckingham gives a purely thermodynamic proof that a particular example of a violation of the 180° rule cannot occur.
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The proof proceeds from the detailed features of the example rather than dealing with the general case.
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11
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84950756458
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A different type of proof is sometimes given which relies upon one of the phases being an ideal gas and upon the latene heat of melting being positive
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(See, for example, Glasstone, Treatise on Physical Chemistry/.).
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13
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84950894607
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This is the behavior found in all known natural systems. The usual thermodynamic proofs of the equality of fields in coexisting phases are circular in that they assume the continuity of the fields as functions of the densities within each individual phase. Fisher
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has constructed systems with long-ranged infinitely many body forces which violate this assumpiton. Nevertheless, for systems satisfying reasonable restrictions on the potential energy, one expects continuity of the fields in the densities.
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(1972)
Commun. Math. Phys.
, vol.26
, pp. 6
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20
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33749432190
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Condensed Phase Diagram of the System Argon-Nitrogen
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While this is a condensed phase diagram in which the pressure is equal to the equilibrium vapor pressure rather than constant, the pressure is low (<100 mm Hg) and slowly varying near the condensed phase triple point, so that the 180° rule should hold.
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(1963)
Thermodynamics Chem. Eng. Prog. Symp. Ser.
, vol.59
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Long, H.M.1
Di Paolo, F.S.2
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21
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84950911254
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See also, Ref. 3, p. 176; Ref. 4, p. 104; Ref. 5
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24
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84950884754
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Part 2, by E. H. Buchner
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25
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84950919655
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Part 3, by A. H. W. Aten
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28
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84950754904
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Reference 4, p. 244; Ref. 5, pp. 133, 282.
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35
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84950577151
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Reference 4.
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37
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84950803366
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A function is concave if and only if its negative is convex/. A function f/(x/), where [formula omitted] is convex if, for any two points [formula omitted] and [formula omitted] in its domain, [formula omitted] for every choice: ([formula omitted] [formula omitted] [formula omitted]). For a discussion of the properties of convex functions see
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Convexity Cambridge Tracts in Mathematics and Mathematical Physics No. 47 (Cambridge U.P., London), Chap3.
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(1966)
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Eggleston, H.G.1
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48
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84950534815
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Reference 6
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49
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84950795314
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Reference 34
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50
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84950824558
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Reference 36
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51
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84950577151
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Reference 4
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52
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84950519042
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Reference 5
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53
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84950595546
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Reference 22.
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55
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84950752526
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This may be seen by considering the required phase diagram in the [formula omitted] plane at fixed [formula omitted] below the quadruple point.
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Corresponding to the triple points, [formula omitted] [formula omitted] [formula omitted] in the [formula omitted] plane, there will be 3 three-phase triangles in the [formula omitted] plane-each with one corner labeled δ. As [formula omitted] is raised through the quadrupole point, these triangles must coallesce to give a single triangle, [formula omitted] above the quadruple point. This can happen only if δ falls inside the triangle [formula omitted] at the quadruple point. (See, for example, Ref. 4, pp. 217–219, Figs. 10–23.) A rigorous demonstration can be given from the Clausius-Calpeyron equations (7.4), which constitute an orthogonality relationship between the triple lines in field space and the corresponding faces of the four-phase tetrahedron in density space.
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56
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84950931529
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The question might be raised as to whether cancellation could occur among the terms in the first square bracket in Eq. (7.11) so that the entire quadratic form remains finite.
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We have argued elsewhere (see Ref. 12) that this is to be expected only if [formula omitted] is replaced by [formula omitted]
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