A complete iterative inversion procedure for second virial coefficient data

Smith, E. B.; Tindell, Andrew R.; Wells, Bryan H.; Crawford, F. W.

doi:10.1080/00268978100100721

“…However, for monatomic gases other than helium, where the repulsive potential energy between two helium atoms was obtained by an inverse Laplace transform of B(T) data in the range 300 K-1500 K, 178 rather sophisticated, less formal, inversion techniques have been devised that allow the extraction of the complete potential-energy function from experimental second virial coefficients. 86,[179][180][181] For molecular gases it is not possible to write Equation (1.11) as a Laplace transform, that is a formal inversion procedure does not exist in this case. The majority of virial coefficient data has been obtained by PVT (or PrT) measurements, [182][183][184][185][186][187][188][189][190][191][192][193][194][195][196][197][198] of which the Burnett technique 182-188 is perhaps the most commonly used.…”

Section: @S @Tmentioning

confidence: 94%

CHAPTER 1. Volumetric Properties: Introduction, Concepts and Selected Applications

Wilhelm¹

2014

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“…In practice, with judicious selection of ϕ 0 , convergence for the CIIM was found to be so rapid that only three iterations sufficed (see [3], p. 707; [18], p. 939; [19]). Their claim for the CIIM was that the final result ϕ 3 will closely approximate the potential ϕ related to the given second virial coefficient by (1.2).…”

Section: Iterationmentioning

confidence: 97%

“…In light of the negative mathematical answer to the general inversion question for second virial coefficient data, the claim [3,12,18] of a group of scientists led by Maitland and Smith to possess a complete iterative inversion method, i.e. one which obtains the potential in both the repulsive and attractive regions, was met with skepticism.…”

Section: Introductionmentioning

confidence: 98%

Mathematical analysis of the complete iterative inversion method — I

Grow

¹

,

Insall

²

2009

Differ Equ Dyn Syst

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A gas composed of identical isotropic molecules has a potential energy of interaction between pairs of particles that depends only on their separation distance. The pair potential is encoded in the virial coefficients of the virial equation of state for a gas.The complete iterative inversion method is a technique employed in an attempt to recover the pair potential from the second virial coefficient. Implicit in the complete iterative inversion method is the requirement that various mathematical expressions are meaningful: improper integrals converge, derivatives exist, etc. We provide a mathematical framework in which all these implicit assumptions are valid. We show that the complete iterative inversion method cannot recover the pair potential even if the target potential and the initial estimate are infinitely differentiable.Keywords Second virial coefficient · Spherical intermolecular potential · Equations of state for a gas · Integral equation · Iterative inversion. Mathematics Subject Classification (2000)45G (Nonlinear integral equations), 47J06 (Nonlinear ill-posed problems), 82B40 (Kinetic theory of gases).

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“…Frisch and Helfand [3] showed that if ϕ has a short-range repulsive branch and a long-range attractive branch, then knowledge of B(T ) for all T > 0 is insufficient to permit reconstruction of the potential in the attractive region. Despite this negative answer to the general inversion question, a group of scientists led by Maitland and Smith (see [2,6,10]) claimed to possess a complete iterative inversion method (CIIM), i.e. one which obtains the potential in both the repulsive and attractive regions.…”

Section: Introductionmentioning

confidence: 98%

Mathematical Analysis of the Complete Iterative Inversion Method II

Grow

¹

,

Insall

²

2010

Differ Equ Dyn Syst

2

0

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A gas composed of identical isotropic molecules has a potential energy of interaction between pairs of particles that depends only on their separation distance. The pair potential is encoded in the virial coefficients of the virial equation of state for a gas. The complete iterative inversion method (CIIM) is an algorithm employed in an attempt to recover the pair potential from the second virial coefficient through successive approximations. In an earlier investigation we identified a very general class of "admissible" pair potentials for which the implicit assumptions of the CIIM are valid: improper integrals converge, derivatives exist, etc. Furthermore, we showed that the CIIM cannot recover the pair potential even if the target potential and the initial estimate are infinitely differentiable. For analytic pair potentials, it is known that the second virial coefficient uniquely determines the potential. The present work represents significant progress in the development of the mathematical framework suitable for confirming this uniqueness result for admissible analytic potentials within the universe of discourse of the CIIM. In particular, we formulate the CIIM convergence question as a classical fixed point problem in the local metric space of admissible analytic potentials. A further result exhibits a set of simple, natural conditions sufficient to guarantee that the CIIM operator is a self-map on a subspace of "normal" analytic potentials.Keywords Second virial coefficient · Spherical inter molecular potential · Equations of state for a gas · Integral equation · Iterative inversion · Pseudometric · Local metric space Mathematics Subject Classification (2000) 45G · 47J06 · 82B40 · 54E35

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A complete iterative inversion procedure for second virial coefficient data

Cited by 22 publications

References 4 publications

CHAPTER 1. Volumetric Properties: Introduction, Concepts and Selected Applications

CHAPTER 1. Volumetric Properties: Introduction, Concepts and Selected Applications

Mathematical analysis of the complete iterative inversion method — I

Mathematical Analysis of the Complete Iterative Inversion Method II

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