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This article was downloaded by: [Columbia University] On: 08 December 2014, At: 06:46 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Chemical Engineering Communications Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gcec20 TRANSPORT PROPERTIES AND SECOND VIRIAL COEFFICIENTS FOR NEON MARTIN J. SLAMAN a & RONALD A. AZIZ a a Department of Physics , University of Waterloo , Waterloo, Ontario, N2L 3G1, Canada Published online: 29 Oct 2007. To cite this article: MARTIN J. SLAMAN & RONALD A. AZIZ (1991) TRANSPORT PROPERTIES AND SECOND VIRIAL COEFFICIENTS FOR NEON, Chemical Engineering Communications, 104:1-3, 139-152, DOI: 10.1080/00986449108910880 To link to this article: http://dx.doi.org/10.1080/00986449108910880 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

TRANSPORT PROPERTIES AND SECOND VIRIAL COEFFICIENTS FOR NEON

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This article was downloaded by: [Columbia University]On: 08 December 2014, At: 06:46Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Chemical Engineering CommunicationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/gcec20

TRANSPORT PROPERTIES AND SECOND VIRIALCOEFFICIENTS FOR NEONMARTIN J. SLAMAN a & RONALD A. AZIZ aa Department of Physics , University of Waterloo , Waterloo, Ontario, N2L 3G1, CanadaPublished online: 29 Oct 2007.

To cite this article: MARTIN J. SLAMAN & RONALD A. AZIZ (1991) TRANSPORT PROPERTIES AND SECOND VIRIAL COEFFICIENTSFOR NEON, Chemical Engineering Communications, 104:1-3, 139-152, DOI: 10.1080/00986449108910880

To link to this article: http://dx.doi.org/10.1080/00986449108910880

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: TRANSPORT PROPERTIES AND SECOND VIRIAL COEFFICIENTS FOR NEON

Chem. Eng. Comm. 1991, Vol. 104, pp. 139-152 Reprints available directly from the publisher. Photocopying permitted by license only. 0 1991 Gordon and Breach Science Publishers S.A. Printed in the United States of America

TRANSPORT PROPERTIES AND SECOND VIRIAL COEFFICIENTS FOR NEON

MARTIN J. SLAMAN and RONALD A. AZIZl

Department of Physics University of Waterloo

Waterloo, Ontario, N2L 3G1, Canada

(Received October 8, 1990; in final form March 5, 1990)

Recently, an accurate interatomic potential for neon was constructed (Aziz and Slaman, 19R9) by a multiproperty fit to several sets of experimental data. Along with the data to which it was fitted, the potential accurately reproduces a wide range of other bulk and microscopic data, and appears to be the best available characterization of the neon interaction. Based on this potential, second virial coefficients and transport properties of dilute neon gas are tabulated from 50 to 10,000 K, and empirical functions are presented for interpolating between the tabulated values. KEYWORDS Virials Transport properties Neon Potential.

Knowledge of interatomic forces for spherical systems has advanced to the stage where transport properties and virial coefficients for the dilute gas can be calculated to a degree of precision which is equivalent or superior to experimen- tally obtained results. Since the behaviour of an atomic system is ultimately determined by the interatomic potential associated with these forces, the various properties of the system can be calculated over a wide temperature range with high accuracy once an accurate potential has been constructed. This method of calculating the properties of the system has advantages over the experimental approach e.g., it can provide results for temperatures which are difficult or impossible to determine experimentally.

An important consideration in the construction of an accurate potential is the choice of functional form for the potential energy curve, V ( r ) . The form should contain just enough adjustable parameters to make it flexible enough to reproduce all the reliable data. Furthermore, it should be realistic at short and long range, and lend itself readily to the inclusion of theoretical values if available. (For neon, only the long range is well known theoretically because accurate dispersion coefficients have been determined). Potentials of the Lennard-Jones, exp-6, Morse, Morse-6, Maitland-Smith, or m-6-8 type are not suitable candidates because they do not satisfy all these requirements.

Once a suitable form has been chosen, the next stage in constructing an accurate potential is the use of a multiproperty fit to the most reliable theoretical and experimental results available. It is essential that a multiproperty fit be

t Author to whom correspondence should be addressed

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140 MARTIN J . SLAMAN AND RONALD A. AZIZ

employed because various properties sense the potential (and hence constrain it) in different regions. The particular region of the potential which is probed by each of the properties has been adequately described elsewhere (Aziz, 1984, 1987). If one attempts to determine the potential by fitting to only one or two properties, no unique solution can be found. By increasing the range of measurement, one increases the range of the potential sensed by the property. Consequently, one must perform a fit of the potential to a judiciously selected set of properties. In this way the interaction will be accurately characterized. Hence, properties can be accurately predicted over a temperature range which includes higher and lower reduced temperatures than that range known to be insensitive to the potential energy function (2 < T* < 5 for transport properties and 2 < T* < 10 for virials, Hanley and Klein, 1967).

Using a multiproperty fit, Aziz and Slaman (1989) recently presented a reliable potential for neon which appears to be a better characterization of that system than any potential previously found in the literature. The potential form they chose to represent the neon interaction was the HFD-B form (Aziz and Chen, 1977), which is given by:

V(r) = EV*(X), (1)

where

V*(x) =A* exp(-cy*~ + /?*x2) - F(X) [C~/X~ + c8/x8 + c ~ ~ / x ~ ~ ] (2) with

F(x) = exp[-(Dlx - I)'], x < D = 1, x z D (3)

where

x=r/r,,, and c,=C,/€r;

TABLE I

Parameters for HFD-B neon potential

The values for C,, C,, C,,, 0, elk, r, and D are used to define the potential and are not truncated. Not all figures displayed are significant. Some are displayed only to avoid round off errors.

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TRANSPORT AND VIRIAL COEFFICIENTS FOR NEON 141

The parameters for the new neon potential are listed in Table I. For a detailed explanation of the fitting procedure used to determine the potential, the reader is referred to Aziz and Slaman, 1989. We discuss next a sketch of that fitting procedure, but only insofar as it pertains to the suitability of the potential for providing correlation functions for the properties presently of interest. In this discussion, we also show how the predictions of the potential for these properties compare with the primary data and earlier correlations.

The long range part of the potential was determined by inserting known theoretical values for the dispersion coefficients into the realistic van der Waals form for the tail of the potential. This part of the potential is dominated by the C, dispersion coefficient which is known to within 1% (Kumar and Meath, 1985). Similarly, the values for Cg, and C,, were held to within the bounds assigned by Standard and Certain (1985). Once these three dispersion coefficients are fixed, the long range part has been determined.

TABLE I1

Deviations of some hulk properties, calculated on the basis of the HFD-B potential, from macroscopic data

Temperature Error Rms Maximum Reference range [K] barse deviation" deviation"

Second uiriak (Units of ml mol-')

Michels era/. (1960) Nicholson/Schneider (1955) Levelt-Sengers eral. (1971) Levelt-Sengers er a/. (1971) Najafi eral. (1983)

Viscosity

Vogel (1984) Vogel (1984) Kestin er al. (1980) Kestin er al. (1972) ClarkeISmith (1969)b GuevaraIStensland (1971)b Guevara/Stensland (1971)~ Najafi er 01. (1983) Najafi er a/. (1983) Bich n a/. (1990)

Thermal conducriuity

Assael et a/. (1981) Kestin er a/. (1980) SpringerlWingeier (1973, 1982)

"Error bars and maximum deviations are given in percent, except for virials where they are in units of ml mol-'. Similarly, rms deviations are listed for virials (in ml mol-'), whereas rms percentage deviations are quoted in the case of the other properties.

bThe relative viscosity measurements of these authors have been re-normalized using the result for the HFD-B potential near room temperature as a reference value. The potential was fit to the accurate measurement (f 0.1% error bar) of Vogel (1984) at 298 K.

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142 MARTIN J . SLAMAN AND RONALD A. AZIZ

The repulsive wall of the potential was determined in two ways. First, the lower part of the repulsive branch was adjusted until the accurate oscillating disk viscosity measurements of Vogel (1984) were predicted. This locates the wall in the lower repulsive region. Secondly, the highly repulsive part of the wall was fitted to the high energy total cross section (TCS) values of Rol and co-workers (Aziz, 1984). When these two regions are fitted with a realistic potential model such as the HFD-B, the intermediate portion of the wall is reliably determined as well. For instance, a perusal of Table I1 shows that the high temperature transport properties, which are sensitive to this intermediate portion of the wall, have been satisfactorily predicted by this procedure. This table also shows how the predictions of the HFD-B compare favourably with the Vogel data to which it was fitted, as well as other intermediate and low temperature measurements of bulk properties. Also, Table I11 shows the agreement between the HFD-B and the high energy TCS data, which have error bars of f 10% (Rol, 1988).

Deviations of the sets of primary experimental virial data (Michels et al. (1960) and Nicholson and Schneider (1955)) from values calculated on the basis of the HFD-B potential are summarized in Figure la . While not fit to these data, the HFD-B predicts them very well. A similar plot is given in Figure l b for the correlation of the virials of Levelt-Sengers et al. (1971) and the improved corresponding states principle correlation of Najafi et al. (1983). Above 1000 K, the latter correlation agrees extremely well with the predictions of the potential but is somewhat inconsistent with the potential and the primary data below 1000 K.

In Figure 2a various sets of primary and correlated viscosity data are plotted. Deviations exceed the experimental error of 1.0% above 1800 K for the data of Guevara and Stensland (1971). But, according to Millat (1990), these data at high temperatures were not corrected for slip effects. The correlation of Bich et al. (1990), shown in Figure 2b, gives better agreement with the potential. The correlation of Najafi et al. also gives excellent agreement above 1000 K, but deviates somewhat (about 0.6%) from the prediction of the potential at about 280-300 K where measurements are expected to be most accurate. For thermal

TABLE I11

Percentage deviations of neon HFD-B potential from the values of Rol and co-workers (Aziz, 1984) for the highly repulsive region

r [A] R 1 V(~)I,,.B [Kl Percent deviation of V,,,.. from V,,,

1 .25 136621.7 135583.9 -0.76 1.30 107868.8 107924.9 0.05 1.40 67243.2 68208.0 1.43 1 S O 41918.0 42935.2 2.43 1.60 26130.8 26903.0 2.96 1.70 16289.4 16769.6 2.95 1.80 10154.5 10389.1 2.31 1.90 6330.1 6387.9 0.91 1.98 4337.2 4299.4 -0.87

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TRANSPORT AND VIRIAL COEFFICIENTS FOR NEON 143

FIGURE 1 Deviations for various experimental second virial data from the predictions of the HFD-B potential.

Primary Data (Figure la top) Correlations (Figure l b bottom) I3 Michels er al. (1960) O Levelt-Sengers et al. (1971) x NicholsonlSchneider (1955) + Najafi et a/. (1983)

5

3

1

-1

- 3 .

conductivity, (Figure 3) all the primary data are in good agreement with the potential.

The properties which are most sensitive to the bowl of the potential include second virial coefficients, spectroscopic data, and low energy differential collision cross sections (DCCS). While not fitted to second virials, the HFD-B predicts

I 1 I 1 I I I

Ne-Ne HFD-B: Reference 1 ine

-

+ - 0

+ m * * + + * + + + + + + + + + + s s s * s s

a D + D Y + + + + - . + + -

I I I I I I I

0 1000 2000 3000 4000

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144 MARTIN J . SLAMAN AND RONALD A. AZIZ

HFD-B: Reference 1 ine

FIGURE 2 Deviations for various experimental viscosity data from the predictions of the HFD-B Potential.

Primary Data (Figure 2a top) Carrelatons (Figure 2b bottom) 0 Vogel (1984) Najafi el 01. (1983)

Kestin el al. (1980) -- Bich el al. (1990) ClarkeISmith (1969)

'A GuevaralStensland (1971)

5 I I I I I I I

Ne-Ne HFD-B: Reference 1 ine

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TRANSPORT AND VIRIAL COEFFICIENTS FOR NEON 145

FIGURE 3 Deviations for various experimental thermal conductivity data from the predictions of the HFD-B potential.

0 Assael el al. (1981) Kestin er al. (1980)

iXI SpringerlWingeier (1973,1982)

4

them well (Table 11). Further, it predicts the values for the spectroscopic parameters given by LeRoy er al. (1974) and Tanaka and Yoshino (1972). as well as the DCCS of Farrar el al. (1973) very well (Aziz and Slaman, 1989).

In summary, we have briefly discussed the predictive abilities of the HFD-B in the regions of the potential bowl, the long range van der Waals tail, and the repulsive wall. Clearly, the neon interaction is well characterized by the HFD-B potential at all separations. Since this is the case, we present recommended second virial coefficients and transport properties based on this potential in Table IV.

I I I

N e - N e - - HFD-B: Reference 1 i ne

In the table, the first two quantum corrections have been used to calculate the virials, and Chapman-Cowling expressions have been used to evaluate the transport properties. Viscosity coefficients, thermal conductivity coefficients, and thermal diffusion factors have been evaluated in third order and diffusion coefficients in second order. Some details about how these properties are calculated from the interatomic potential now follow.

For central potentials, the classical statistical-mechanical expression for the second virial coefficient of a dilute gas is B,,.,(T) = boB:,(T*), where bo = 2nNArL/3, and B:,,, the reduced classical second virial coefficient is given by

Bt,(T*) = 3 [1 - exp(-V*(x)l~*)Jr' dx 0

(4)

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146 MARTIN J . SLAMAN AND RONALD A. AZIZ

TABLE IV

Properties of neon as a function of temperature

D (1.013 bar) T K or "C B ml mol-' q pPa s A m~ m-' K-' lo-' m2 s-' no

(Maitland et al., 1981). Here, T is the temperature (in Kelvin), k represents the Boltzmann constant, N,, is Avogadro's number, T'= kTIe is the reduced temperature, and r, is the location of the potential minimum. For low temperatures, a semi-classical approach is needed to achieve sufficient accuracy. The semi-classical virial, B(T) = b,B*(T*), is expressed in terms of a series in which Br,,,, and the reduced first the second quantum corrections, Bf, and Bf2,

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TRANSPORT AND VIRIAL COEFFICIENTS FOR NEON 147

appear as follows;

Here, A* = h / r , , , ( m ~ ) ' ~ is the de Boer parameter, and m the mass of the molecule. A full quantum procedure is also available (Hirschfelder et al . , 1954), but the semi-classical approach is quite sufficient for neon even at temperatures as low as 50 K (Maitland et al., 1981), the lowest temperature for which we present our values (Table IV). At this temperature the HFD-B potential predicts B(T) = -37.666 ml mol-l, with B,,,,(T) contributing -39.717 ml mol-', the B,, term contributing +2.150 ml mol-', and the B,, term contributing just -0.099 ml mol-I. At temperatures of 100 K and higher, the B,, correction term is smaller than -0.01 ml mol-'. The expressions for the reduced quantum correc- tions are,

where V*' = dV*/dr, and V*" = d2V*/dr2 (Kihara et al., 1955). The dilute gas transport properties are expressed in terms of the reduced

collision integrals, R".")', which are functions of reduced temperature T*. The first order expressions for the viscosity, thermal conductivity, and self diffusion coefficients of a pure gas are, respectively, (Maitland et al . , 1981),

In Eq. (lo), n represents the number density. The isotopic thermal diffusion factor can be written in first order (Chapman-Cowling) approximation as (Maitland et al., 1981),

Higher order corrections involving more collision integrals are available for Eqs. (8)-(ll), and these corrections must be included for accurate work. Although the higher order terms are too lengthy to reproduce here, their evaluation is straightforward and the transport properties are determined fairly readily once the reduced collision integrals are known.

The classical reduced collision integrals are determined from the reduced

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148 MARTIN J . SLAMAN AND RONALD A. AZIZ

potential V*(x) according to the expressions,

[ 1''-I)']-'[ Q(')'(E*) = 2 1 - 2(1+ 1)

(1 - cos' ~ ) b * db *

In these expressions, b* is the reduced impact parameter, E* is the reduced relative energy between colliding molecules, x is the angle of deflection casued by the collision, and Q")' is the reduced cross section of order 1. For the full details regarding the physical significance of these quantities, see Hirschfelder et al. (1954). The Qc'.")' were calculated classically for reduced temperatures T* 2 60 with the ACQN FORTRAN program of O'Hara and Smith (1971). Using this program, which was modified to run on IBM machines (Neufeld and Aziz, 1972). we evaluated the Q"."" integrals to the specified relative accuracy of 0.01%. O'Hara and Smith (1971) give a detailed account of the numerical techniques employed. For the values we present in Table IV, higher order correction factors (Hirschfelder et al., 1954) to the first order transport coefficients were applied.

For reduced temperatures T* 5 60, a full quantum calculation of the collision integrals was performed. This means that the reduced cross sections Q")', were determined from formulae which are analogous to those presented above (Hirschfelder el al., 1954).

For purposes of interpolation between the values of Table IV, we have empirical functions for the five properties. These functions are given by

the following expressions, in which T represents the temperature in degrees

TABLE V

Coefficients of the interpolating functions for B(T)"

50-150 K

A -0.36122813 X lo4 B 0.26944503 x lo4 C -0.77182828 X 10' D 0.10044793 X I d E -0.49863814 X 10' 4 16 Std. dev 0.021 Max. dev 0.037

Note: For each temperature range we have listed the number of points, N,, used to generate the interpolating function from a least squared fit to the values calculated directly from the potential. The standard deviation and maximum deviation of the Np empirical values from the values calculated directly from the potential have also been listed.

"Units of ml mol-'.

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TRANSPORT AND VIRIAL COEFFICIENTS FOR NEON 149

TABLE VI

Coefficients of the interpolating functions for transport properties

T [Kl 50-200 K A 0.25694357 B -0,16693409 C 0.11359046 D -0.19689812 X lo-' E 0.1 1624387 X lo-*

21 $d. % dev 0.049 Max. % dev 0.091

Viscosity, q ( T ) , pPa s 200-2000 K 2000-10000K

$d. % dev Max. % dev

Thermal Conductivity, A(T), mW m-' 50-200 K 200-2000 K

0.75715963 -0.548f34496 X 10' -0.16877059 X 10' 0.38721414 X 10' 0.11314931 X 10' -0.68238885

-0.19472574 0.63804563 x lo-' 0.11428656 X lo-' -0.22060041 x

Self diffusion, D(T)P, m2 s-' 50-200 K 200-2000 K 2000-10000K

A -0.10967206 X I d -0.14943363 X I d -0.11618863 x 1 6 B 0.75654867 0.43400455 x 10' 0.23645392 x 10' C 0.61972403 -0.55802812 -0.13020678 D -0.11724802 0.51441178 X lo-' 0.10117013 X lo-' E 0.71811452 X -0.17491799 X lo-' -0.26044092 x lo-"

N~ 2 1 51 71 Std. % dev 0.114 0.018 0.001 Max. % dev 0.268 0.086 0.004

Isotopic Thermal Diffusion, CY,,(T)~ T [KI 50-250 K 250-1000 K 1000-10000K

A 0.18648771 x I d -0.66435099 X 10' -0.25108703 X 10' B -0.16558614 x I d 0.32699526 X 10' 0.14855754 X 10' C 0.53120929 X 10' -0.50768143 -0.26453765 D -0.72919659 0.28824040 x lo-' 0.20571782 x lo-' E 0.36664130 X lo-' -0.30910299 x lo-' -0.61159969 x lo-"

38 29 91 2 % dev 0.064 0.070 0.012 Max. % dev 0.246 0.191 0.033

Note: For each temperature range we have listed the number of points, Np, used to generate the interpolating function from a least squared fit to the third order values (second order for Diffusion) calculated directly from the potential. The standard percentage deviation and maximum percentage deviation of the Np empirical values from the values calculated directly from the potential have also been listed.

" A t a pressure of 1.013 bar. Dimensionless.

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150 MARTIN J. SLAMAN AND RONALD A. AZIZ

Kelvin:

Second virials: B(T) = G(ln T) (15)

Viscosity: W ) = exp[G(ln 7-11 (16)

Thermal conductivity: I(T) = exp[G(ln T)] (17)

Diffusion: D(T) = exp[G(ln T)] (18)

Thermal Diffusion: %(T) = G(ln T) (19)

where

G(ln T) = A + B In T + C(ln T)' + D(ln T)3 + E(ln T)4 (20)

Coefficients for these functions are given in Tables V and VI. Also presented in these tables are the standard deviations (or standard percentage deviations) of the empirical values from the values calculated directly from the potential as described above.

In summary, the potential reproduces numerous bulk and microscopic pro- perties which include second virials, viscosity, thermal conductivity, spectroscopy, DCCS, as the high energy TCS of Rol and co-workers (Aziz, 1984). Also, the long range has been properly fixed because it contains the accurate new C, dispersion coefficient and improved values for C, and Clo. Based on such a potential, recommended second virials and transport properties have been presented over a large temperature range both in tabular form and in terms of empirical functions for the purpose of interpolating between the tabulated values.

ACKNOWLEDGEMENT

This research is supported in part by the Natural Sciences and Engineering Research Council of Canada.

NOMENCLATURE

Reduced potential parameter in Eq. (2). Ratios of collision integrals in Eq. (11).

Reduced impact parameter. Eq. (12). Second virial coefficient, ml mol-I.

Reduced second virial coefficient. Dimensionless.

Classical second virial coefficient, ml mol-'. Reduced classical second virial coefficient.

Reduced quantum corrections for B*(T*). Dipole-dipole dispersion energy coefficient, in atomic units.

Reduced dipole-dipole dispersion energy coefficient.

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E* DCCS F G

rm

T T* TCS v v*

TRANSPORT AND VIRIAL COEFFICIENTS FOR NEON 151

Higher multipole dispersion energy coefficients, atomic units.

Reduced higher multipole dispersion energy coefficients. Self diffusion coefficients at 1.013 bar, mZs-'. Also used to represent reduced potential parameter in Eq. (3).

REFERENCES

Reduced relative energy between colliding molecules. Differential collision cross sections. Reduced damping function in Eqs. (2) and (3).

Function of Eq. (20). Used in interpolation functions of Eqs. (15) to (19). Boltzman constant.

Atomic mass.

Avogadro's number. Number of points used in least square fit.

Reduced cross section of order I . Separation between atoms, A. Separation between atoms at potential minimum, A. Temperature, in K or "C.

Reduced temperature. Dimensionless. Total cross sections.

Potential energy of pair of isolated atoms, K. Reduced potential energy.

Reduced separation between atoms. Isotopic thermal diffusion factor, dimensionless.

Reduced potential parameter in Eq. (2).

Potential parameter in Table I. Reduced potential parameter in Eq. (2).

Well depth of potential. Well depth as given in Table I, in K. Viscosity coefficients, pPa s. Thermal conductivity coefficient, mW m-' K-l. Separation between atoms when V = 0.

Reduced collision integral.

Angle of deflection.

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