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UNITED STATES PARTMENT OF )MMERCE

IBLICATION

^»*'0'ca

*«»mO» ^*

lational Bureau of Standards

library, E-01 Admin. Bldg.

m 2 1973

NBS TECHNICAL NOTE 628

00

U.S.

ARTMENT

OF

)MMERCE

National

Bureau

of

standards

On the Utility of tlie ni-6-8

Potential Function
NATIONAL BUREAU OF STANDARDS

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On the Utility of tlie m-6-8

Potential Function

H.J.M. Hanley

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

Max Klein

Heat Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

NBS Technical notes are designed to supplement the

Bureau's regular publications program. They provide

a means for making available scientific data that are of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

<"»EAU Of

U.S. DEPARTMENT OF COMMERCE, Peter G. Peterson, Secretary

NATIONAL BUREAU OF STANDARDS, Lawrence M. Kushner, Acting Director

Issued November 1972
National Bureau of Standards Technical Note 628,

Nat. Bur. Stand.(U.S.), Tech. Note 628, 77 pages (November 1972)

CODEN: NBTNAE

For sale by tbe Superintendent of Documents, U.S. Government Printing Office, Wasbington, D. C. 20402

(Order by SD Catalog No. €13.46:628)
CONTENTS

Page

1. Model Potential Functions 1

2. The Four- Parameter , m-6-8. Potential 4

2, 1 Definition of the Reduced nn-6-8 Potential 5

3. Testing The m-6-8 Potential 6

3. 1 Fit to a Single Property: Satisfaction of Requirement (1),

Section 1 8 3.2 Simultaneous Fit of the Viscosity and Second Virial Coefficients:

Satisfaction of Requirem.ent (2), Section 1 10

3.3 Deviation Curves for Argon 12

4. Further Checks on the Use of the 11-6-8 (y = 3.0) Potential for

Argon 12

5. Tests on the Validity of the 11-6-8 (y = 3.0) Potential 14

5. 1 Repulsion: The Potential for r < a 15

5.2 Attraction: A Comparison of the Dispersion Coefficients 15

6. Application of the m-6-8 Potential to Krypton and Xenon 17

6. 1 Deviation Curves for Krypton and Xenon 18

7. Application to Polyatomiic Molecules 18

7. 1 Simultaneous Fit of the Second Virial Coefficient and Viscosity

Coefficient for Nitrogen 19

7.2 Fits to Single Properties: The Viscosity Coefficient and the

Second Virial Coefficient 19

7. 3 Fits for Methane, Oxygen and Carbon Dioxide 20

8. Kinetic Theory of Polyatomic Molecules 20

8. 1 Inelastic Collisions 20

8.2 Nonspherical Potentials 21

8. 3 Summary 21

9. Thermal Conductivity of Polyatonnic Molecules 21

10. Thernnal Diffusion of Polyatomic Molecules 23

11. Conclusion 23

12. References 58

APPENDIX 66
List of Figures

Figure

1. Plots of the m-6-8 potential computed from eq (10) to illustrate how the

attractive part varies with (a) an increase in m at fixed y, (b) an

increase in y at fixed m 26

2. Schematic plot of the solution of eq (13) for temperatures T, , Ty, T,,

and T^ given a potential function. The point at each temperature

corresponds to the b^ and e/k values which give a simultaneous fit of the viscosity coefficient and second virial coefficient at that

temperature 27

3. Solutions of eq (13) for argon at several exper innental temperatures for

the m-6 potential function. Note the "turn around," with respect to temperature, of the solutions at 129. 56 K and 143. 16 K, circled in the

figure 28

4. Solutions of eq (13) for argon at several experimental temperatures as

in figure 3. Note the very small area given by the 11-6-8 with

Y

= 3.0 29

5. Continuation of figure 4. Note the "turn around" for the 12-6-8,

Y

= 2.5 and the 13-6, y = 2.0, (circled) 30

6. Deviation plot of experimental second virial coefficients for argon [17]

compared to (corresponding) values determined from eq (12) using

the m-6-8 potential with m = 11, y = 3.0, a = 3.292 A, e/k = 153 K 31

7. Percentage deviation plot of experimental dilute gas viscosity

coefficients of argon compared to theoretical values determined from

eq (11) with the m-6-8 potential: m = 11, y = 3.0, a = 3.292 A, e/k = 153 K. Data [13], A [14], n[15], v[16], 0[19], #[20],

[21]; not distinguished [22,23] 32

8. Percentage deviation plot of experimental dilute gas thermial

conductivity coefficients for argon compared to eq (14) with the m-6-8

potential as for the viscosity and second virial coefficients. ^[24],

• [25],v[26], ©[27], [28],t[29], O[30],0[3l], n[32],

A [33], e [34], [35] 33

9. Percentage deviation plot of experimental self- diffusion coefficients

of argon compared to theoretical values determined from eq (15) with

the na-6-8 potential as for the viscosity and second virial

coefficient A [36], ^[37], 0[38]. Note: the data from [37] is relative to D at 298. 15 K. We used the calculated value for this

reference point 34

10. Experimental and theoretical [eq (16)] thermal diffusion factors for

argon. Data from [39]. We use the same m-6-8 potential as for the

other transport coefficients 35
Figure Page

111. The repulsive potential at small values of r. We compare the 11-6-8

(Y = 3.0) potential, found to be satisfactory for the dilute gas

properties of argon, with the potential estimated frona high energyscattering of beams of neutral argon atoms [41]. Also shown, as a

•' matter of interest, are the curves for the well-known lZ-6 and

18-6 potentials. [This latter represents the best three-parameter

potential fit for argon at low temperatures] 36

12. Deviation plots for the second virial and viscosity coefficients of argon based on the m-6-8 potential with the inverse sixth power

coefficient fixed at~60 x lO'*^ ergs/cm^: m=ll, y = 2.0,

a = 3.356 A, e/k = 137 K. Data not distinguished 37

13. The m-6-8 potential with m=ll, y = 2.0, a = 3.356A and

e/k = 137 K. This potential is the best potential that we have obtained

which can approximately fit both the viscosity and second virial

coefficient for argon and which gives approximately the correct

coefficient for the sixth power dispersion term (C^). Also shown

is the multiparanneter potential of Barker [43] for which C/ was

chosen to be correct 38

14. Deviation curves for the viscosity and second virial coefficients

of krypton. Data for viscosity, 0[14], A [15], v[l6], n [44]

.

Theoretical values deternnined for eqs (11) and (12) with the

m-6-8 potential, parameters as given in table 1 39

15. Deviation curves for the viscosity and second virial coefficients

of xenon. Data for viscosity, 0[14], A [15], o[45], V [46,47].

See table 1 40

16. Deviation curve for the thermal conductivity of krypton. Theoretical

values calculated in eq (14) with parameters given in table 1. Data: [31], • [35], [48], A [49], V [50], D [51] 41

17. Deviation curve for the thermal conductivity of xenon. Theoretical

values calculated in eq (14) with parameters given in table 1. Data: "[31], [48], A [49], V [50], O [51], D [52] 42

18. The isotopic thermial diffusion factor for krypton calculated from

eq (16), using parameters given in table 1, conapared to experiment [53] 43

19. The isotopic thermal diffusion factor for xenon calculated from

eq (16), using parameter given in table 1, compared to experiment [54] 44

20. Deviation curve for the self-diffusion coefficient of krypton from

eq (15) with the parameters given in table 1, experiment from [55] .... 45
Figure Page

21. Deviation curves for the viscosity and second virial coefficients of nitrogen based on a simultaneous fit. Parameters: m = 10,

Y=1.0, a =3.694 A and e/k=87.0K. Data (not distinguished):

viscosity [13-16], second virial coefficient [17] 46

22. Fits of the viscosity and second virial coefficients of nitrogen based

on a fit of the viscosity data. Parameters m = 12, y = 2. 0, a = 3.54 A, e/k=118.0K. Viscosity data, V [13] , A [14] , n[15],

[16], [22], 0[56], A [57], • [58] 47

23. Deviation curves for the viscosity and second virial coefficients of

nitrogen based on a fit of the second virial coefficient alone.

Parameters, m=12, y=2.0, a=3.60A, e/k=118.0K.

Note (a) the unusual deviation pattern for the viscosity, (b) that

the potentials used here and in figure 22 differ only by a snnall

anaount in a 48

24. Deviation curve for the viscosity of methane based on the fit of viscosity data alone. Parameters , m=ll, y = 3.0, a=3.68A,

e/k = 168 K. Note that we have fitted both high and low temperature data with one set of parameters. Data: [56], A [60],

[61], D [63] 49

25. Deviation curve for the viscosity of oxygen based on the fit of

viscosity alone. Parameters, m = 10, y = l-O, a - 3.437 A, e/k = 113 K. IMPORTANT NOTE: Some of the data have been

adjusted. See the Appendix. Data: v [22], [47], A [56 adjusted].

Not shown [65,66] 50

26. Deviation curve for the viscosity of carbon dioxide based on a fit of viscosity data alone. Parameters , m=14, y=1'0> a=3.68A,

e/k = 282 K. Data: v [16], D [62], • [56] 51

27. Deviation curve for the thermal conductivity of nitrogen based on

the parameters found for viscosity. Calculated via eq (29).

RDtential parameters: Table 2. Data, [27], [28], [32], 9 [35],

A [70], D [71], V [72], 0[73], © [74], • [75], A [76], A [77] 52

28. Deviation curve for the thernnal conductivity of oxygen based on the

viscosity paranneters, table 2. Calculated via eq (29). Data: [27],

A [78-80], <i[82], D [81], t[83], A [74], #[85], ^[84], O [75] 53

29. Deviation curve for the thernnal conductivity of methane calculated

via eq (29). Potential paranneters from viscosity, table 2. Data:

[27], O [32], n [81], v [87], [88]. Not shown [86], [34] 54

30. Isotopic thernnal diffusion factor for nitrogen calculated from eq (16)

with the potential paranneters used for viscosity. Data fronn

reference [92] 55
Figure Page

o 31. Isotopic thermal diffusion factor for oxygen calculated from eq (16)

with the viscosity potential parameters. Data from

reference [93] 56

32. Isotopic thermal diffusion factor for carbon dioxide calculated frona

eq (16) with the viscosity potential parameters. Data from

reference [94] 57

Al. Deviation plot between modern data for argon (open circles) and

older data for argon (closed circles). Similar deviation curves could

be constructed for most other conamon gases. We demonstrate this

in the figure by showing representative deviations for helium,

neon, krypton, xenon, air, nnethane and nitrogen (triangles) ........ 6i

List of Tables

Table Page

1. Values of the parameters of the m-6-8 potential function which

fit both transport and equilibrium properties 25

2. Values of the parameters of the m-6-8 potential function which

' fit transport data 25

Al. Adjusted experimental viscosities for oxygen 67
ON THE UTILITY OF THE m-6-8 POTENTIAL FUNCTION

t tt H. J. M. Hanley and Max Klein

The calculation of transport and equilibrium properties of simple gases with the

m-6-8 potential function is discussed. Properties referred to specifically are the

viscosity, diffusion, thermal conductivity and second virial coefficients, and the thermal

diffusion factor , Gases referred to are argon, krypton, xenon, nitrogen, oxygen, carbon

dioxide and methane. We also discuss in detail the genera] behavior of a model potential

with respect to the fitting of data and the selection of potential parameters. The m-6-8

appears to be the simplest model potential which can satisfactorily be used to fit data over

a wide temperature range. The relationship of the potential to independent theory is

briefly examined.

Key words: Dispersion coefficients; m-6-8 potential; parameter selection; second

virial coefficients; transport properties.

This work was carried out at the National Bureau of Standards and supported by the

Office of Standard Reference Data.

Cryogenics Division, Institute for Basic Standards, National Bureau of Standards,

Boulder, Colorado 80302.

tt Heat Division, Institute for Basic Standards, National Bureau of Standards,

Washington, D, C. 20234.
1. Model Potential Functions

In this paper we discuss a new model potential function, the m-6-8, and demonstrate how it can be used to correlate and predict experimental data. Before going into

any details, however, we would like to review very briefly the general development of model functions and their relationship to experiment and statistical mechanics.

Roughly speaking, the evolution of model potential functions parallels theoretical

developments in molecular structure and inter molecular forces. For example, the

simplest concept of molecular structure is that of the hard spherical molecule and the

corresponding potential accounts only for repulsion between molecules on contact. The

potential has one parameter, the effective diameter of the spheres. While this concept is obviously too simple --it cannot, for example, explain the stability of solids and liquids, or give a temperature dependence to the second virial coefficient -- it was nevertheless

very useful, and remains so today. The hard sphere model can be modified to incorporate a simple inverse power repulsive term consistent with the picture of a "soft"

repulsive spherical molecule whose diameter decreases as the energy of the molecular

system increases.

Realistic potentials have to consider both the repulsion and attraction between

molecules. This could be taken into account by using inverse power potentials of the type

$(r) = A/r - C/r (1)

generally associated with the name of Lennard- Jones. Here $ (r) is the potential, a

function of inter molecular separation, r, while m and n, (m.>n) are fixed at arbitrary

values, and A and C are parameters to be found from experiment. One method for

finding such parameters is described below.

Several authors have attempted to be more specific by predicting values for m

and n. For example, details of the attraction between m.olecules were developed fron:i quantum mechanical perturbation theory, i.e. , attraction was seen to be equivalent to the

concept of induced molecular polarizability. Specifically, quantum mechanics shows that

the leading term of the induction effect varies as the inverse sixth power of r thus

suggesting that the attractive index, n of eq (1), should be taken as 6. It is not possible,

however, to come up with a satisfactory a priori value for the repulsive index m from

quantum mechanics, so m is often taken to be 12 for mathematical convenience. With

this choice of m, we have the 12-6 or, as it is usually known, the Lennard- Jones

potential:

I (r) = A/r - C, /r (la)
where A and C, -- the coefficients of repulsion and inverse sixth attraction,

respectively -- are two parameters to be found fronn experiment as before. This potential

has been the standard for many years and still serves as the model potential for many

molecular theories.

In the last fifteen years or so, several three-parameter potential families have

been proposed which try to add flexibility to the 12-6. Most often, the inverse sixth term

is retained for the attraction but an extra parameter is introduced for the repulsion. The

simplest example is the m-6 family of potentials with the form of eq (la) but m replacing

the index 12 as a parameter; m has then to be found from experinnent along with A and

C, • Other examples of such three parameter potentials are the Kihara, the exp:6, and

6 the Morse potentials [l] . Unfortunately, the application of the three- parameter fannilies

to experiment often gave misleading or even erroneous results: thus, there seemed to be

no definite criteria to judge how effectively a potential represented a given property --in

naany cases, even, it was not clear if the introduction of the extra parameter led to a fit between theory and experiment which was an improvemient over the corresponding fit with

the 12-6. Consequently, it also was difficult to say which of the new three-paranneter

potential families was the most generally useful.

To try to clear up some of the confusion, we undertook a study [1-3] to investigate

the relationship between certain experimental properties and the potential function as required by statistical mechanics. The nnain results of the study are of interest to the

present work so they are discussed briefly here.

First of all, we found that temperature ranges exist for different properties in which these properties are only sufficiently sensitive to the details of the function to produce information about two parameters: in other words, in these temperature ranges,

the properties are insensitive to any extra parameters which might be introduced. For

equilibrium properties this range is, in reduced form, approxinnately

2.t;^_^.10 (2a)

and for transport properties the range is approximately

2 . T*^_^ . 5 (2b)

Numbers in brackets refer to references given at the end of the paper,

2
where T , - T / {e ,/k) with T the temperature in kelvin, k Boltzmann's constant

and e^^ / the maximum potential energy of attraction for the Lennard- Jones, 12-6

potential [4]

.

The definition of the insensitive temperature range enabled us to give a quantitative

significance to the terms "high" and "low" temperatures -- terms which are important in

discussions of the limiting behavior of potential functions. Thus, a high temperature is a temperature above T ?- 10 for equilibrium properties (or 5 for transport properties)

while a low temperature is a temperature below T , r^ 2.

Note that the insensitive temperature range covers the normal experimental range

for most simple substances (e.g. , the second virial coefficients of argon, oxygen, or nitrogen are insensitive to more than two parameters in the potential function in the temperature range 200 < T(K) < 1000). In our opinion, many authors were not familiar with

the insensitive temperature ranges and this lack of awareness led to most of the confusion

in dealing with three-parameter functions. For example, use of data in the insensitive

range alone often forces a decision in favor of one potential function based, in the final

analysis, on the particular methods used for treating the experimental errors. The

selection of the potential function can therefore be made on a totally irrelevant basis.

As a second result of our study, we found that all three-parameter function

families are essentially equivalent as far as fitting low density data is concerned. This

particular result can be used to advantage since one can choose any convenient threeparameter family, say the m- 6 , to be typical of all three-parameter functions. We have

often done this in our previous work [3].

A third, and more negative, conclusion drawn was the clear recognition that the

three-parameter functions, while an improvement over the two- parameter 12-6, were

still not flexible enough to fit data as one would like. In particular, they failed to satisfy

two important requirements:

(1) The ability to correlate data for a given substance and property

at high and at low temperatures with the same set of parameters.

and

(2) The ability to correlate, for a given substance, two different

properties using the same set of parameters in a sensitive

temperature range.
Other multiparameter functions, where the parameters are related to theory as far as possible, have been proposed by several authors. One recent attennpt is that of Barker and Pompe [5]. Essentially, their potential is based on a series of exponential

ternns for the repulsion, which are determined empirically, along with a series of inverse

powers for the attraction, which are estimated from quantum mechanical dispersion theory.

An alternative technique for developing a multiparameter function is to start with a large drawing of a reasonable potential and then, using tabular values read off the drawing,

fit different properties simultaneously. By modifying the potential graphically in some

consistent manner until the best overall agreement between theory and experiment is obtained [6,7], a function might possibly be selected out whose shape closely approaches

that of the "actual" potential function. See in particular, the work of Dymond and

Alder [7].

This covers a brief outline of the potential models used up to the time this work

was undertaken. Very recently, potential functions have been determined from low energy

scattering experiments and from spectroscopy. See, for example, reference [99].

2. The Four- Parameter , m-6-8. Potential

Although generally successful, the complex potentials, such as the multiparameter

semi- theoretical potential of Barker and Pompe, or the numerical potential of Dymond

and Alder, are very specialized. These potentials are derived for a particular substance

and are not always easy to generalize to other substances. It is also not easy to take into

account possible changes either in the assunnptions or in the data on which these functions are based. We felt that one had to connpromise between these sonnewhat inflexible complex

potentials and procedures on the one hand, and the more flexible but over-simplified

three-parameter functions on the other hand. Our approach is straightforward: we consider the addition of only one parameter to the three-parameter models used previously

and, furthermore, select a paranneter with a reasonable physical basis.

Extension of quantum mechanical perturbation theory indicates that, for separated

molecules, the attractive part of the inter molecular potential function should be described

by a series of inverse powers of the intermolecular distance, the first term of which

depends on the inverse sixth power of the intermolecular separation, as already mentioned. The next term of the series depends on the inverse eighth power and is, therefore,

a logical choice for an additional term for a model potential. Theoretical estimates have

been made of the relative magnitudes of the coefficients of the inverse eighth and inverse

sixth power terms for a nunnber of substances [107], which enables one to introduce an

inverse eighth power dependence in terms of the coefficient of the inverse sixth power

4
term without additional parameterization. However, since the validity of these theoretical

estinnates has not been clearly established, this cannot be done with certainty. It seems

reasonable, therefore, to add an inverse eighth power term but to take the corresponding

coefficient as an additional potential parameter, thereby defining a four-paranaeter

function fanaily .

The particular choice of a three-parameter potential function on which to add an

inverse eighth power attraction is arbitrary because, as we have pointed out, all of the

three-paranrieter functions are equivalent from the point of view of correlating data. The

equivalence shows up especially in the arbitrary choice of the repulsive terms. One could

argue, however, based on our knowledge of high energy scattering data, that a shielded

coulonnb form for this repulsion would be most suitable. Nevertheless, we decided to continue to use the single term inverse power repulsion contained in the m-6 function.

This choice was made because (a) such a form is sinaple, and (b) adding an inverse eighth

power to an m-6 potential is a natural extension of the work we already have done with the

nn.-6

.

2. 1 Definition of the Reduced m-6-8 Potential

The definition of the m-6-8 potential follows fronn eq (1):

l(r) = A/r""- C^/r^ - Cg/r^ (3)

where m is the repulsive index, and A, C , , and C are the coefficients for repulsion

D o and for the sixth and eighth power attraction terms respectively. We introduce the more

convenient reduced form of potential (3) as follows.

If r denotes the intermolecular separation at the maximum attractive energy,

m

then we define $ (r ) = -e. Using that at r , d$/dr = 0, one can show that

m m

(4) c c„ me 8 / 1 ^

\ 8 / 1 m-6 8 \ va-6/)y - 8 [T'

r r m m

where

= ^ r^r - (^f (5) m \ r

y - z(^r + (-8)(-Lf (6)
with r' - r/r . We now define a parameter, y, characteristic of the inverse eighth

term,

Y

= -Cg/er^ (7)

thus

The potential is given in eq (8) in terms of r', i.e. , scaled by r , but it is frequentlymore convenient to work in terms of the effective hard sphere diameter, a, defined by

$ (r) - 0. Introduction of the definitions

d = r /a and r* = r/o (9) m

then converts eq (8) into

,, (,.,) __ Ii_L|ll (A.y _ l^-yi^-^)] (^f _ (Af (10) m-6 \r*/ m-6 Vrv/ \r*/

where we have defined f^l^ (r=;=) as l(r*)/e. Potentials (8) and (10) are, of course,

equivalent. Care must be taken, however, when converting from one form to the other to ensure that the factor d is used properly. We will work with potential (10).

The m-6-8 potential is sketched in figure 1. Figure la shows how the attractive

part is affected by an increase in m with y = O-O- Figure lb shows how the attractive

part is affected by increasing y with m constant. Note that increasing y at constant

m has roughly the same effect as increasing m at constant y. [We have not shown the

associated small changes in the repulsive part of the potential, such changes being very

small when compared to the changes in the attractive part.]

3. Testing The m-6-8 Potential

We next outline our approach for testing the potential. We clarify what we propose

to do: Any potential function obtained fromi fitting data alone is only the "best" function

associated with a particular functional form, a particular set of experimental data, and a particular criterion of best fit. A successful potential function, from a correlation point

of view, is therefore not necessarily a valid one from a theoretical viewpoint, although

the reverse must always be true, i. e. , a valid theoretical potential must, of necessity,

correlate good data. When testing with experimental data, we therefore refer to testing
the utility of a function rather than testing its validity. Of course, if the potential

parameters are then found to be compatible with independent theoretical estimates, the

potential takes on a significance beyond its ability to fit the data.

Substances : Since the potential function assumied is spherical, the necessary first

test has to be tnade with nnonatomic molecules because the potential function can, in principle, describe the interactions between such molecules without further approximation.

[Non- sphericity not only involves the potential function, but also a description of the

interactions between the internal degrees of freedom of the molecules.] We exclude

quantuni fluids for now. Hence, initial tests should be made with argon, krypton and

xenon. Of these, argon is the most convenient because argon data for several properties are available outside the reduced insensitive temperature ranges defined by eq (2). Since

this is not always the case for the heavier rare gases, much of this paper will refer to comparisons between predicted and experimental values for argon.

Properties : There are a number of fundamental theoretical expressions which

give macroscopically measurable properties in terms of potential functions and which can,

in principle, be used to test the utility of such potentials. Only a few of these are

practical, however. The reason is simply that the corresponding experimental data must

be available and, furthermore, must be precise enough to allow for a choice to be made

annong potential functions [1-3,9]. In practice, the usable expressions are those for the

viscosity coefficient, Tj, and the second virial coefficient, B. Of the two, the viscosity

is usually to be preferred since the insensitive range, eq (2), covers a smaller range of temperature than does the corresponding range for the second virial. Thus, data for a given substance covering a given experimental temperature range are nnore likely to overlap both high and low temperature regions for the viscosity coefficient than for the

second virial coefficient. This must, however, be balanced against the fact that, of the

two properties, the second virial coefficient is the more sensitive to the potential at low

temiperatures.

The expressions connecting these quantities with the potential function are [lO]:

Viscosity (to the first Chapman- Enskog approximation [ll])

'I 16 2 (2,2)*

rra Q (T-'-)
Second Virial Coefficient

B = b B* (T*)

o

where (IZ)

2 3 b = — TT Na

o 3

(2 2)*

In eqs (11) and (12), the intermolecular potential function enters via Q ' (T*) -- the

collision integral accounting for the dynamics of a binary collision in the viscosity

process, and B* (T*) -- the reduced second virial coefficient. For a given potential

(2,2)- function, §(r), both Q (T*) and B* (T*) can be calculated as functions of T*,

where T* = T/(e/k) with e the maximum well-depth for the given potential. Also, in these equations, m is the mass of a naolecule, N is Avagadros' number and a has

been defined previously.

Data : The argon data selected for test purposes [12] were extracted from the

following sources: viscosity [13-16]; second virial [17]. We have carefully evaluated

these data and they appear to be the best available at this time. There are problems,

nevertheless: values of both the viscosity and second virial coefficients at the lower tenn- peratures (T < 100 K) remain uncertain. Also, the reliability of the published viscosity

coefficients at high temperatures has been questioned recently. [This latter problem,

however, now seems resolved [12].]

Collision Integrals and Reduced Second Virial Coefficients: Collision integrals

and reduced second virial coefficients have been computed for the m-6-8 potential with

9 s m < 18 for values of y such that < [m - y('^ - 8)]. The calculation procedure and

tables are available in another publication [8].

3. 1 Fit to a Single Property: Satisfaction of Requirement (1), Section 1 We began our investigation by studying how the m-6-8 potential can be used to fit

a single property, namely the viscosity. The dilute gas viscosity coefficients were

examined separately at low temperatures, at high temperatures, and at all temperatures.

Several techniques are available for optimizing potential function parameters.

For example, one can select the parameters which give the best fit of theory to experime

according to least squares. We use a graphical technique, however, which we have four

to be both successful and instructive. To illustrate this approach, let us first consider

argon viscosity data in a given temperature range.

8
In our fitting procedure, a potential with particular values of m and y is first

selected, i. e. , only e and a are to be varied initially, a is then also fixed initially at some sensible preliminary value. This leaves only e to be determined. Let us consider

a particular experimental temperature T. Given exper innental data for T\ and T and the

initial choice of a, eq (11) can be used to obtain an experimental reduced collision inte-

(2 Z)* gral Q ' at that value of T. Our purpose i.s to make a correspondence between such

"experimental collision integrals" obtained at the exper innental tennperature s and calculated from (11), and those calculated theoretically (as in [8]) for the particular choice of m and y at theoretical temperatures T*. The correspondence results in a value of e /k

for the given experimental temperature T from the relation between T and T*. This is

accomplished by interpolating in the calculated set listed as a function of T* for the mi and Y values chosen for the value of the "experimental collision integral" found using (11).

Using the value of T* corresponding to this value of the theoretical collision integrals,

the experimental temperature T and the relation e/k = T/T*, a value of s/k is obtained

for T. Proceeding in this way for all of the experimiental temperatures, we then obtain

e/k as a function of T. If, now, the value taken for a is changed, a second set of values

of e/k is obtained as a function of T. Proceeding in this way a set of curves is obtained, one for each value of a. These curves represent the variation of e/k with T and a

for the given choices of m and y. Now, if the potential function were valid and the

"correct" value of a used, e/k would not vary with temperature. Hence, a reasonable

criterion for selecting the best a value is to choose that one which is associated with the

least variation of e/k with T. The appropriate value of e/k to take is one within its range of variation with T for that a. Thus, this procedure leads to a best pair, e/k

and a, for the particular m and y. By repeating the process for different vn, y pairs, one can select the a, e/k, m, y combination which gives the smallest overall

variation of e/k with temperature.

Results

Both at high and at low temperatures, fitted separately , we were able to find

several sets of m, y and a which gave a constant e/k (constant within experimental

error). This is just the kind of result found with the three- parameter functions. We

were, however, able to reduce the possibilities to a single set of m, y> ^ ^Y requiring

that the same values of a and e/k be applicable to both high and low temperatures.

Values thus obtained for argon were m=ll, y = 3'0' a=3.29A, e/k=153K

-10

(1 A = 10 m).

This result was encouraging since it meant that the m-6-8 potential can satisfy

requirement (1) of Section 1 -- the simplest function found thus far to do so. It is

9
important to remark, however, that, in order to do this, data are required for both the

high and low temperature ranges. In other words, a fit of data at low temperatures only

is not necessarily applicable to data at high temperatures, and vice versa.

3.2 Simultaneous Fit of the Viscosity and Second Virial Coefficients:

Satisfaction of Requirement (2), Section 1

In a previous paper [2] we mentioned another graphical and interpolation scheme

whereby a simultaneous fit of the viscosity and second virial coefficient for a given gas

could be used to give potential parameters and which would, at the same time, demonstrate how the fit was modified as the parameters change. This discussion is repeated

briefly here.

From eqs (11) and (12), one can obtain a single relation which needs to be satisfied

for a simultaneous fit of the experimental second virial and viscosity coefficients at the

experimental temperature, T. This relation is

3/2 3/4

T1(T) B(T) = K M B-t (T-K) ^3/4 n^^'^^* ^^*'

where K is a constant: K=137.93xl0 c.g.s., units, and M is the molecular

weight. The left-hand- side of eq (13) is a function of the experimental temperature, T,

whereas the right-hand- side is a function of the reduced temperature, T*. The equation

is used as follows: We tabulate the right-hand- side as a function of T* for a chosen

potential. At a given experimental temperature, T , the left-hand- side is evaluated

from experiment. By interpolation in the values tabulated for the right-hand- side, a value

of T*, say T *, is obtained for which the right-hand- side of (13) equals the left-handside. Using e/k = T /T * we thus find e/k at T and also the corresponding value

of a (or b ) from either eq (11) or (12). In a graphical sense, the pair of values a

(or b ) and e/k can be represented by a point in a graph having a (or b ) and e/k

as axes. This point r .presents the simultaneous solutLon of eqs (11) and (12) and

necessarily corresponds to a simultaneous fit of the viscosity and second virial coefficients at the experimental temperature T . One can determine points for each of several

experimental temperatures, T , T , T , T . If the data are exact and the potential

perfect, a and e/k values obtained at all temperatures are identical and there results

a single point in a plot of e/k versus a as a function of T. In an actual calculation

process, however, an area is traced out since, obviously, the data are not exact and the

potential is not perfect. Thus, both e/k and g are tennperature dependent. See

figure 2. If, however, the potential is reasonable, i.e., if one could use it to fit both

10
I

second virial and viscosity coefficients simultaneously within experimental error with the

same paranaeters at all temperatures, then the scatter in figure Z would be small and of a size consistent with the scatter in the experimental data.

Application to Argon

h We illustrate how this method works for argon. [We are restricted here to the low

and insensitive temperature ranges since the high temperature range for the second virial

coefficient, as given by eq (2a), is inaccessible if one requires data of the necessary

precision [l l] . ] The m- 6 Potential : The method was first applied to the thr ee-paranneter potentials

of the m-6 family (corresponding to the value y = in the four-parameter nn-6-8 function

family). Figure 3 shows the curves for m = 9, 12, 17 and 24. As already described, and

as sketched in figure 2, each of the points on these curves corresponds to a solution of

eq (13) at a given experimental temperature. Notice, (a) no curve is reduced to a comnaon

solution point independent of temperature; or, more realistically, there is no small area whose size could be described as a point consistent with the experimental error; (b) there

is, however, a "turn around" with respect to temperature as a function of m. For

example, examine the relative positions of the solutions for the exper innental tennperatures

129.56 K and 143. 16 K for the 9-6 and the 24-6 functions [18]. From this one can surmise

that, if a common solution point were to exist, it would have to be found for m less than

24.

The m-6-8 Potential : We indicated in Section 2. 1 and figure 1 that increasing the

Y paraineter (which represents the strength of the inverse eight attraction) effectively

tends to increase nn. For instance, an 11-6 potential with y = 2.0 is similar to a 12-6

potential with y = I'O- Thus, based on the curves for the m-6 potential shown in figure

3, a commion solution point cannot occur with the m-6-8 potential for m > 24 (and,

possibly, based on an exannination of figure 3, cannot occur for m > 17) since rerjipving

the restriction y = is equivalent, in this sense, to increasing m. This initial crude

selection process is helpful since it results in a reduction in the number of m-6-8

functions which need to be tested for argon. In keeping with this, we concentrated our

attention on potentials with m values between 9 and 18 and show the particular results

obtained for m = 10, 11, 12 and 13 for various values of y in figures 4 and 5. We found

that a "turn around" is observed for the 12-6-8 and 13-6-8 potentials at high values of y. This turn around indicates an upper limit for y associated with each m, i. e. , for the

12-6-8, y 2 2.0 and for the 13-6-8 y < 1.5. We also found that the common solution area becanae progressively larger as m increased beyond m = 12 no matter what value

11
of Y was selected. Based on this latter behavior, therefore, we can further restrict the

parameter range and say that, for the m-6-8 potential, m must be less than 12 for argon.

Closer examination showed, with very little doubt, that the graph of e/k versus b had

o the smallest common solution area for the 11-6-8 with y = 3.0, i.e., this is the

potential for which the best simultaneous fits are obtained for the viscosity and second

virial coefficients for argon over the widest temperature range.

From within the common solution area obtained in an e/k versus b plot for the

3 ° 11-6-8 potential with y = 3.0, we then chose the parameters b =45 cm /mol

o [a - 3.29Z A], and e/k = 153 K, table 1. We have therefore been able to satisfy requirement (2) for argon and with parameters which fit the viscosity data of argon

simultaneously at high and at low temperatures. This ls a very strong argument in favor

of using this potential for argon.

One might argue that, because we now have four parameters, a unique solution to

the problem of producing a simultaneous fit to the viscosity and second virial coefficients

is not possible. In other words, although we find the 11-6-8, y - "i.O potential to be the

best potential, we might equally well have chosen another as best, e.g. , the 10-6-8,

Y

= 4. 5 (say) or the 9-6-8, y = ^' Collision integrals were not available to check on this

but, in any case, if multiple solutions occur, they most probably indicate that the data are

not sufficiently precise to make a truly definitive choice of the potential possible. The data

are, however, sufficiently precise to reduce the number of possibilities quite drastically.

For instance, we have already ruled out any solutions for m > 13, regardless of the value

of Y'

3.3 Deviation Curves for Argon _

Given the potential m = 1 1 , y = 3. 0, with a = 3 . 29 A (1 A = 1 0~ m) and

e/k = 153 K, we constructed deviation curves for the dilute gas viscosity coefficients

[13-16, 19-23] and the second virial coefficient [17], figures 6 and 7. We regard them as

satisfactory.

4. Further Checks on the Use of the 11-6-8 (y = 3.0) Potential for Argon

Thermal Conductivity: The statistical mechanical expression for the thermal

conductivity coefficient of a dilute n:ionatomic gas, \' involves the same collision

integrals as does the viscosity coefficient:

, 25 V^ mkT %

32 2 (2, 2)>.- m

n o ii (T='=)

12
^ / 15 k ^ 4 m '

3 where c '/m = — k/m is the translational specific heat per molecule. A check on thermal V Z conductivity thus serves as a cross check on the viscosity and thermal conductivity data

and not necessarily as a check on the potential function. Figure 8 gives the deviation curve for the thermial conductivity. Data are taken from references [Z4-35].

Self- Diffusion Coefficient D : This coefficient is of more importance than the

thermal conductivity as a check on the potential because a collision integral other than

(2 2)* Q ' (T*) enters into the theoretical expression. This expression is:

_ 5 TT mkT ,, r. P° = 8 2 (1,1)* ^

^''^

T^ o Q ' (T*)

where p is the mass density of the gas and Q ' (T*) is the collision integral for

diffusion. Figure 9 gives a plot of deviations of self- diffusion data, references [36-38],

and values calculated from eq (15) using the 11-6-8 {y = 3.0) potential.

Isotopic Thermal Diffusion Factor, a ^: Since combinations of collision integrals are required by the kinetic theory expression for this property, it is especially sensitive

to the potential function. According to Kihara [3],

a = a ' a + 6) (16) o o

where

with 6 given by

, _ 15(6C» - 5) (2A* + 5)

^'o "

2A>1= (16A* - 12B* + 55)

(17)

2A* r 1 /7(5 - 6C*) + A*l* \ / (35/8) + 28A* - 6F*

(35/4) + 7A* + 4F*7: [-4 L 2V 5 + 2A* / \ 21A*

(18)

5 r 7 /5 - 6C* \ 31*

[„*.

7 L 5 \5 -f 2A* / 10

13
where

I* = 8 - 8E*

and

(35/4) - 3B=:= - bC^<

«* = TT-G^^ (19)

The potential function enters into the equation for q' via the collision integrals which

here are expressed as ratios, A*, B-!', C-~l% E* and F* given by

A* . q(2'^)*/q(1'1)%

B* = (5q(1'2)>._^^(1,3)*^/^(1,1)*^

C* = n^ ' ' /n* ' '

; (20)

Figure 10 shows a plot of a determined from eq (l6) compared to the data of Paul

at al. [39]. Since thermal diffusion data are difficult to obtain with precision, we regard

the fit indicated by this figure as satisfactory.

5. Tests on the Validity of the 11-6-8 (y = 3. 0) Potential

Our results, as illustrated by the deviation curves, give us confidence that we

have indeed reached an important goal, that is, we now have a useful, relatively simple,

analytical model potential function for argon. As we shall show later, this same model

potential is also satisfactory for the other heavy rare gases and very reasonable for

simple polyatomic molecules. However, it is interesting to check on how valid our argon

potential is; i, e. , to see how close it connes to fulfilling requirements other than those

associated with the fitting of equilibrium and transport data [99]. For example, the

repulsive part of the model potential can be compared with results from molecular beam

scattering experiments and the coefficients of the attractive part, i.e. , C, and C^,

D O with independent semitheoretical quantum mechanical calculations [101]. Independent

checks on the values of e/k and a are also possible. For exannple, the value of o can

be compared to that obtained from the crystal lattice [5] and the value of e/k can be

compared with a value found appropriate in a theory of the liquid state [40]. In both these

latter cases, however, one has to introduce the concept of non-additivity of the pair

potentials which introduces an extra uncertainty [43],

14
5. 1 Repulsion: The Potential for r < a

The repulsive part of the pair potential for argon at small separations has been

estimated from molecular beam scattering data [41] and we plot a composite curve

obtained from these data as the heavy solid line in figure 11. The curve for the 11-6-8

(Y = 3.0), was calculated and is shown as a lighter solid curve. The two curves begin to separate at r ~ 2. 8 A or § (r) ~ 0. 1 eV. [Also shown as a matter of interest are the curves for the 12-6 function, long used as a good first approximation potential for the

description of high temperature argon data, and for the 18-6 which is typical of all three

parameter potentials which can be used to describe low temperature argon data [3,17].]

Taken literally this comparison indicates a need for a potential at small r which is much

softer than the 11-6-8. This conclusion must be taken with caution, however, because

repulsive contributions do not dominate the interaction potential in this region. For

example, our potential for argon has a magnitude of approxinnately 1 eV at r* = 0.75. For

this value of r*, the attractive inverse eighth term has a nnagnitude which is 38% of that

of the inverse eleventh and the attractive inverse sixth ternn has a magnitude which is 2%

of the inverse eleventh. Thus, if our potential for argon is at all valid, it follows that the

analysis of scattering data in terms of pure repulsion can produce misleading results.

An idea of the error introduced by neglecting attraction can be estimated as

follows. At each value of r*, let us, in the style of the scattering experiments, represent

our potential function by a pure inverse power repulsion. Thus

/ |(r*) = A/r*"^

If this were a true representation of the function, an effective repulsive exponent could

. r -I- d f then be obtained simply by calculating m = —-

-—- . But, for example, at r* = 0. 75 ' $ dr*

this leads to an effective value for our potential of m' = 13: different fronn the value

m ::= 11 associated with the repulsive branch of the complete potential.

5.2 Attraction: A Comparison of the Dispersion Coefficients

The expressions for the dispersion coefficients, C, and C , for the n:i-6-8

potential function are, frotn eq (10):

C/ = --^ [m - Y(m-8)] a d (21) 6 (m-6)

C_ = e y o d (22)

15
For argon we have found that m=ll, y = 3,0, e = 153Kergsand, ct = 3.Z9x10 cm.

Also, d = 1 . 14462. Using these values in eqs (21 ) and (22) one obtains C , s 21 x 10

ergs/cm and C =2000 x 10 ergs/cna . Senaitheoretical quantum nnechanical calculations, however, indicate that C, = 60 x 10 ergs/cm and C = 120 x 10 ergs/cnn

6 8 [42]. The coefficient C, is believed to be known quite accurately (within 5%), C. is 2

^ known somewhat less well, but is about twice C r , 6 Obviously, our 11-6-8 potential for argon is not giving the correct value for the

inverse sixth power coefficient C, . Let us then see what happens to the data correlation

if the C, of our potential is forced to have a value approximately equal to 60 x 10

6 ergs/cnn . In so doing, we effectively fix one of the four parameters. This we choose

to do by fixing y for a given m. From the simultaneous fits shown in figure 5 we select

m = 11, and y = 2 which gives C, = 55 x 10 ergs/cm -- sufficiently close to theory.

Corresponding best values of a and e/k are found from the graph to be 3.356 A and

137 K, respectively. Deviation curves for the second virial and viscosity using this

11-6-8 (y = 2.0) potential are presented in figure 12. One sees that these curves are not unreasonable, that is, the fit is within three percent, except for the second virial

coefficient at very low temperatures. They do, however, show systematic deviations and are definitely inferior to the curves given in figures 6 and 7.

These results can be sumniarized in three remarks.

1. The numerical values of C, and C„ for the m-6-8 potential are very

6 o sensitive to the choice of m and y because of the particular functional forms used,

i.e. , eq (21). The quantity a, which is also part of the expressions, is found from

experiment. Since a is raised to the sixth and eighth power respectively, uncertainty in

it obviously leads to increased uncertainty in C, and C . 6 8

2. A convenient argument says that our potential is not sufficiently flexible to

fit data while simultaneously giving a good value for C, (and to a lesser extent C ). 6 8 That our potential is not flexible enough is obviously true -- it is only meant to approximate the true potential (whatever that is). This argument, though necessary, is not

sufficient, however, to recommend the need for additional parameters. We can show this

by referring to a recent potential function due to Barker [43]. Because it has additional

parameters, Barker's potential is very flexible when compared with ours but, according

to figure 13, the attractive part of the 11-6-8 (y = 2.0) is close to Barker's potential even

though his C is fixed at the correct value. Consequently, it can be assumed that simply

6 adding parameters to the m-6-8 for greater flexibility might not result in a C, which

approaches the theoretical value. In fact, it is possible that the requirement of fitting

16
data as well as we have done in figures 6 and 7 might be incompatible with the requirement

for a value of C, in reasonable agreement with theory [99].

3. Since we regard the data we have fitted as reliable, the m-6-8 potential thus

fails in that it cannot give the dispersion coefficients correctly. We must emphasize,

however, that this failure must not be taken out of context. The m-6-8 potential, being

a model function, can only be approximate. Its advantage is in its utility and not

necessarily in its ability to give correct theoretical predictions for the form of the potential itself -- although it would obviously be a very powerful property of the potential if this were so.

6. Application of the m-6-8 Potential to Krypton and Xenon

We have tested the m-6-8 potential for krypton and xenon in the same manner as described for argon. For both gases, dilute gas viscosity data and second virial data were simultaneously fitted using eq (13). Plots were obtained similar to figures 4 and 5. The data for viscosity were taken fronn references [14-16, 44] for krypton, and

references [14, 15, 45-47] for xenon. The second virial data for both gases were taken

fronn reference [17].

For both krypton and xenon there was little doubt that the use of an m-6-8 with

m=ll, Y=3.0 resulted in an e /k - b plot with the smallest area, this being our

criterion for the best solution of eq (13). In this way we obtained the best potential

parameters given in table 1.

Viscosity data for both high and low temperatures and second virial data for all temperatures were fitted separately for each gas. We confirnned that our choices of paranneters based on the simultaneous fits to all data were also valid for the individual

properties. Thus we found that the nn-6-8 potential can be used to satisfy requirements

(1) and (2) of section 1 for krypton and xenon, just as it could for argon.

Corresponding States: We have, incidently, thereby verified that, by our

criteria , the corresponding states principle is applicable to argon, krypton and xenon. The law of corresponding states postulates that for fluids with naolecules whose pairwise interactions obey a spherical potential $(r), one can write

§ (r) = e f (r/o)

where e and a are the energy and distance parameters respectively, and f is the

same function for all fluids. Since the only potential parameters which differ for each

of the gases argon, krypton, and xenon are a and e/k, corresponding states is valid

by this criterion.

17
The Coefficient C , : We have already pointed out for argon that the coefficient of

D the inverse sixth power term, when allowed to behave as a parameter in fitting data, was

found to be about 1 /2 - 1 /3 of the numerical value predicted by quantum mechanics. We

have found this to be also true for krypton and xenon: (theoretical values in brackets [42])

Kr, C, ~ 43 [~ 107] and Xe, C, ~ 100 [~ 236], all values times 10 . D D

6. 1 Deviation Curves for Krypton and Xenon

Figures 14 and 15 give deviation curves for the viscosity and second virial

coefficients of krypton and xenon, respectively. Figures 16 and 17 are curves for the

thermal conductivity coefficients determined by eq (14) is compared to experinnent

[31,35,48,49,50-52]. Figures 18 and 19 are deviation curves for the thermal diffusion

coefficients for the two gases [53,54]. In figure 20, we include a deviation curve for the

recently measured [55] self- diffusion coefficient of krypton.

7. Application to Polyatomic Molecules

The evidence presented in the previous sections indicates that the mi-6-8 potential

is satisfactory for the monatomic nnolecules. However, the function would be more

innportant as a practical tool if it could also be successfully applied to polyatomic

molecules. Unfortunately, this extension is not straightforward since two very basic

complications are introduced:

1. In principle, the spherically symmetric mi-6-8 potential (or any spherical

potential) can never properly depict the interaction between two polyatonnic nnolecules;

the potential for such molecules must be dependent on the relative orientation of the

colliding nnolecules.

2. Even if one has a potential which can account for orientation dependent

interactions, the statistical mechanical expressions should allow that collisions

between polyatonnic molecules will be inelastic, that is, the nnolecular internal degrees

of freedonn have to be considered.

The nnost common way to resolve these problenns is to ignore them unless it is very obvious that one cannot do so (when considering ions, for exannple). This oversinnplified approach can be justified in a way which we shall describe below. Let us, therefore, investigate the application of the nn-6-8 potential to the viscosity coefficient

and the second virial coefficient of the polyatomic gas, nitrogen, using the same expressions and procedure as given for the rare gases. The sources for nitrogen data are: viscosity, references [13-16,22,56-58] and second virial, reference [17],
7. 1 Simultaneous Fit of the Second Virial Coefficient and ViscosityCoefficient for Nitrogen

We followed the procedure described in section 3.2 by comparing the second virial

and viscosity coefficients for nitrogen to calculated values based on the m-6-8 using

eq (13). Recall again that we were seeking values for m and y which give values for

b and e/k which are essentially independent of temperature when allowance is made

for a reasonable amount of experimental scatter.

Partial success was achieved: it was possible to show that solutions for m s 13

did not seem likely, but we were unable to select that m-6-8 potential for m ^ 12 which

gave the smallest area in a b - e/k plot. A possible choice is m = 10, y = l-0»

o = 3.694 A, e/k - 87 K. Deviation plots for the viscosity and second virial coefficients

determined fronn these paranaeters are shown in figure 21.

These curves, although showing evidence of systematic error, are actually not unreasonable, i.e. , above 200 K both T] and B are fitted to within 2%. We have,

nevertheless, been unable to achieve results for nitrogen as satisfactory as were those

obtained for the rare gases. It thus appears that requirement (2) of section 1 is not

satisfied for the polyatonnic miolecule nitrogen.

7.2 Fits to Single Properties: The Viscosity Coefficient

and the Second Virial Coefficient

We next follow the procedure outlined in section 3. 1, i.e. , the correlation of the

viscosity coefficient individually. Here we do have a significant result: for nitrogen as

for argon, a single set of parameters was found which enables the viscosity coefficient

to be fitted simultaneously at low and at high temperatures. These parameters are m = 12, Y

= 2.0, a = 3.54 A, e/k = 118 K, see table 2 [59]. In other words, the m-6-8

potential can satisfy requirenaent (1) of section 3. Figure 22 gives the deviation curve

for the fit for viscosity and also shows the corresponding fit of the second virial

coefficient for these same parameters. The latter plot indicates highly systematic

deviations.

The second virial coefficient can also be fitted by itself and we have done this in

figure 23 using the parameters m = 12, y = 2.0, a - 3.60 A, e/k = 123 K. The

corresponding viscosity deviation curve is also shown in figure 23. Note the deviation

pattern which shows that the percentage deviation is essentially independent of temperature !

19
7.3 Fits for Methane, Oxygen and Carbon Dioxide

We found essentially the same results for these gases as we did for nitrogen; an

attempt to fit both second virial and viscosity data simultaneously gave parameters which

produced only a moderately good representation of such data. Just as for nitrogen it was

possible to fit the viscosity at both high and at low temperature with a single set of parameters, see table 2. The parameters required for such fits were found not to be

suitable for the second virial coefficients, however. The converse was also true,

parameters obtained from a best fit of the second virial data, were not satisfactory in the

corresponding fit for viscosity. In fact, deviation curves were obtained similar to figure

23 for nitrogen.

Deviation curves for the viscosity of these gases are shown in figures 24-26.

Sources of data: for oxygen, see the appendix and references [22,47, 56, 64-66], for

methane, references [56-60,61,63] and for carbon dioxide, references [16,56,62].

8. Kinetic Theory of Polyatomic Molecules

The two problems introduced when statistical mechanical expressions are applied

to polyatomic molecules have been discussed by Mason and his coworkers [67]. Let us simply list his conclusions and verify that our results with nitrogen, oxygen, naethane and

carbon dioxide are consistent with them.

8. 1 Inelastic Collisions

The effect of inelastic collisions on the statistical mechanical expressions are as

follows:

Viscosity Coefficient: The evidence indicates [98] that inelastic collisions

have little effect on the viscosity.

Diffusion Coefficient: Formal expressions for the diffusion coefficient of

polyatomic nnolecules are available [98] but it appears that they differ from the corresponding monatomic gas expression by a factor which is very close to unity.

Thermal Diffusion: Approximate calculations [91] indicate that the expression

for the isotopic thermal diffusion factor as given by eq (16) is not complete for polyatomic

molecules and should be modified. At the present tinne, it is not clear how to do this,

however, so that it is common practice to use eq (16) without alteration.

Thermial Conductivity : The thermal conductivity equation, eq (14), is definitely

incomplete and has to be modified for polyatomic gases. We discuss this in a separate

section.

20
8.2 Nonspherical Potentials

A nonspherical potential can be written as the sum of a spherical contribution,

which is independent of the relative orientations of the colliding molecules, and a nonspherical contribution, which depends on this relative orientation. In an exact calculation,

the relative orientations must be considered to change during a collision. Because of this, a proper determination of the second virial coefficient and collision integrals can become

very complicated. In practice, however, a sinnplification is made based essentially on

the fact that a statistical averaging must eventually be performed. Thus, one assumes

that the molecules collide with a fixed relative orientation and collision integrals are

calculated for each orientation independently. The final values of the collision integrals are then determined by a statistical averaging over all these possible orientations [102].

A potential of interest to us is the 12-6 with a term added to allow for quadrupolequadrupole interactions:

$ = $(12-6) + $(quadrupole - quadrupole) (23)

Collision integrals for eq (23) have been calculated [97] and were found to be very close

to the corresponding collision integrals for the 12-6 alone. [However, the combinations

of the collision integrals required for the thermal diffusion factor are significantly

different from their equivalents for the 12-6.] Second virial coefficients for several gases

have also been determined from eq (23). [Although, in this case, extra terms were added

to account for other nonspherical effects.] Unlike the collision integrals, the second

virial coefficients were found to be substantially different from their counterparts

calculated with the 12-6 alone.

8. 3 Summary

Returning to our fitting procedures: one would conclude from the above comments

that the viscosity could be fitted satisfactorily with a spherical m-6-8 potential but a simultaneous fit of the viscosity and second virial coefficients then obtained would not be

satisfactory. This is what was observed. Work is in progress to modify the m-6-8

potential to include nonspherical effects.

9. Thermal Conductivity of Polyatomic Molecules

It is known that the simple equation for the thermal conductivity, eq (14), is not

valid for polyatomic molecules. Several modifications have been proposed, but the most

satisfactory from a theoretical viewpoint is the relatively simple expression obtained by

Mason and Monchick who approximated the Wang- Change-de Boer theory for a dilute

21
polyatomic gas [67]. Mason and Monchick showed that the conductivity can effectively be

separated into two parts: a part depicting the transfer of thermal energy by the translational motion of the molecules, and a part depicting the transfer of energy by changes in

the internal energy of the molecules:

X = X' + X" (24)

where X' is given by eq (14). Splitting the specific heat into a translational piece and

a piece associated with internal degrees of freedom

c = c ' + c " (25) V V V

Mason and Monchick write:

^' = - Tl (— - Aj (26)

c"

^" = PD. ^ (^^ + a) (27) mt \ m /

were D. is a diffusion coefficient for internal energy and A is a term accounting for

the intercharge of internal and translational energy. According to Mason and Monchick,

A is given by

where Z is the collision number for rotation, related to the number of collisions required

for the interchange of rotational and translational energy.

If one assumes that D. can be replaced by the self- diffusion coefficient of

mt

eq (13) (It is not clear what difference this assunnption makes) then the formula for the

conductivity of a dilute polyatomic gas becomes

2m m TT\2 T\ / mZ

Although eq (29) is not new and has often been quoted, it has rarely been used to determine

numerical values of the thernnal conductivity. The reasons for not using it were uncertainty in the potential function and the lack of information for Z. Today, however, these reasons are much weaker: we have presented an argunnent as to why the nn.-6-8 potential can be used to evaluate the viscosity and diffusion coefficients for a simple polyatomic

22
gas; and several experimental determinations of Z have now been published. [Some of these Z values are based on acoustical measurements [68] and others on thermal

transpiration studies [69].]

We were unable to find reliable values of Z for carbon dioxide. There are,

however, values of Z given in reference [69] for nitrogen, oxygen, and nnethane. Using

these and the potential parameters for the viscosity, table 2, we computed the thermal

conductivity from eq (Z9) for these latter gases. Corresponding deviation curves are

given in figures 27-29. Sources of data: for nitrogen, references [27, 28, 32 , 35 , 70- 77];

for oxygen, references [27, 74, 75 , 78-85]; and for methane, references [27,32,34,81,86-88],

Values of c

" were taken from NBS Circular 564 [89] and reference [90].

Although there is some evidence of systematic error, we are generally satisfied

with these curves.

10. Thermal Diffusion of Polyatomic Molecules

Realizing that the equation for the thermal diffusion factor, a , may not be complete, and that the collision integral combinations from the m-6-8 potential nnay not be

adequate for polyatomic molecules, let us, nevertheless, consider a comparison of a with experiment for nitrogen, oxygen and carbon dioxide using the unmodified eq (16) and

the potential parameters obtained for viscosity. The experimental results are interesting

because they indicate that c^ can be negative for temperatures below critical. Curves

connparing theory with experiment are given in figures 30-32. Data: nitrogen [92],

oxygen [93], and carbon dioxide [94]. One sees that a is correctly predicted to go

through zero for oxygen but negative values are not obtained for carbon dioxide or

nitrogen. [See reference [95], however.] We do not draw conclusions from these figures:

In addition to the theoretical problems, the experimental data must not be regarded as

reliable [96].

11. Conclusion

The four-paranneter m-6-8 potential function has been derived and we have

dennonstrated how it represents the viscosity, thernnal conductivity, second virial and

diffusion coefficients, and the thermal diffusion factor for the gases argon, krypton,

xenon, nitrogen, oxygen, carbon dioxide and methane. It appears the m-6-8 potential is

the simplest model function which can represent the above mentioned properties for the rare gases with a single set of parameters used for each gas for all properties. It can

also represent the transport properties for the polyatomic gases listed, but is less

successful when simultaneously applied to the viscosity and second virial coefficient

(although the second virial can be fitted by itself). For the polyatomic gases, it will be

23
necessary to include nonspherical characteristics of the intermolecular interactions in the

potential. Work in this direction is in progress. Work is also in progress to modify the

m-6-8 potential to allow for quantum effects.

On the basis of a comparison between the repulsive part of the argon ra-6-8 (m = 11)

potential and molecular beam experiments, and between values we find for the dispersion

coefficients and the equivalent values estimated from quantum mechanics, it seemis that

the n:i-6-8 is too simiple to represent the argon-argon interaction fromi a theoretical

standpoint. However, we show that it is not clear that sinnply adding additional parameters

to the m-6-8 will resolve this.

24
TABLE 1 Values of the parameters of the in-6-8 potential function

which fit both transport and equilibrium properties

gas m Y o e/k

A [s 10" ^°m] K

Ar 11 3 3.292 153

Kr 11 3 3.509 216

Xe 11 3 3.841 295

TABLE 2 Values of the parameters of the m-6-8

potential function which fit transport data

gas m y a e/k

N^ 12 2 3.54 118

CH, 11 3 3.68 168

4

0^ 10 1 3.437 113

CO^ 14 1 3.68 282

25
(f>A

0.5

-0.5-

-1.0-

0.5

-0.5

-I.Oh

-1.5

(a) rl8-6,x =

—

1 / ^12-6,7 =

^

(b)

rl2-6, 7 = 2.0

—

1 / V,2-6,y =

—

1 1

1/

1 1 1 1 1 0.8 1.6

r/cT

2.4 3.2

Figure 1. Plots of the m-6-8 potential computed from eq (10) to illustrate how the

attractive part varies with (a) an increase in m at fixed y> (t>) an increase

in Y at fixed m.

26
m;

riV (Ti o dJ a >

rt

Oil

n j:2

CD

u > J2 QC iJ -* en H (U3

Tl r-H

r! rt

to > ^ ^

CI

1)

13u

a.

0)

x:

t^ TJ J :n C

•n

fl

u

QJ <u n, a CD s cu 13C <i; iH <i ^

-i-> o u c

x:oau

xlo B

a;aeo

(D x:

u• r-4

27
66

64

62

60

58

56

b„ 52

50

48

46-

44-

42

40

v-;

9-6

o

12-6

17-6

T,K

D 129.56

• 143.16

V 148.14

O 157.41

A 203.15

O 248.15

298.15

A 373.15

T 448.15

573.15

• D

A

o

24-6

80 84 88 i<r

18

M-, ±

50 154 ^^ ± ±

176 180 184 188

Figure 3. Solutions of eq(13) for argon at several experimental temperatures for the

m-6 potential function. Note the "turn around',' with respect to temperature,

of the solutions at 129.56 K and 143.16 K, circled in the figure.

28
O

CD ro

1 II — >s._ m ~

11 T

»-

co" O

II

— — ^

K •% -

CO

1

ID yv -~^—

- ^i < II ^ —

CD

1 ^

- -

< ^?P*

1 1 CO

in

o

lO

00 CD

O

CD

CD

LO

00

o

-sJCDro

03

C\J

Vl/

c

•H

to , td o

en ro

OJ

>-4

II 3

4-1 >~

cfl M 42

tu 4J & H

a 15 cu 4J 00 M

1

n) u iH C -H

<Ue tu •H 43

)-l 4-J 01a >.

XI 4=1 QJ c

.H (U

tfl >

Vj •H

tu bO >

tu CtJ V) (U

S-i 4-1 rt

tfl .H c rH o to 60 B ^ to

CtJ p^ M S-l O tu

M-l >

,_^ tu ro 4d

rH 4J •^

cr tU

tu iJo

4-1 z

o

in , C CO o

•H (U

4-1 M

3 3

tH bO o •H

CO M-l

-*

(U^J3M

•H

fe

29
CM ^ 11 ^

(U 5

to o

T3C3OM

cdC3

4J

0)o

2:

0)MD

00o

0)

Cou I n o

0)u3

00

•H

30
0|D0 _dX9g

00

I ^ B CoM

U

u

o

•H

S-l •H>

c

oo

c

0)S

•H

*-i

X

0)

01ao

u

CO

OJ 13

01

60 II c

•H >- c

O r-l a. r-l w

OJ II s-l ^

o ^;j O £:

0)M

3

00

•H

T3

(U

>-i (0exEo

o

31
001 X

^'°\-^^^(t

UJ

<

UJ

Q_

c

0) •HOH

14-1 M-l0)O

a

CO60

00

I

I e

e

o

c

o

•HM

cd •H>

0) Xl

0)60

n) u

C

(Uo

OJ

0)cH

em

OJ T3

a

•rl

4-1 0)o

0)

II

OJ

CMCM

II

en

CM

CM

Csl

x;m

•H

60C

•HU

CO •H

"73

O —

(U

t-l

cfl a

eo

co

60U

cfl

o

o

g vO

cd •H

4-1 c

0) 4Jo

ft

0)

60

32
dxd

001 X

X

0|D0 6X9

01

<

LU

Q.

UJ

>. M —

'

•H>

•HU

O

13C

o

o

g ^ !:i -^

tN

in

0-)

•H

T3

00

I —• ro

x; i—

I

a*

cu

T3

CuB

o

c

cu •HO

•H

M-l

OJOO

CO

>

0) T3

CU60

cd 4JC

CUo

cu

cuu

p

60

•H

co

60

cfl

O

M-l

T3C

o

CJ

cuen X)C

c« o

33
001 X

OIDO^ dX9^

a- a

o S QLO

on

compare<

the

viscosit

is

relative

t(

o

o f

arg s

for [37]

'si- ^ snts

o

tial

a

from

point.

•» O (U rt <u liJ

•-* -" tJ " « o t^ Cl O

O

ro

or

iffusion

CO

:he

m-6-8

Note:

the

this

refer <

XI '-'

, -d . o cr r-( in o nt o UJ

CVJ a.

f

axperime

led

from

eq

[36],

D[37

e

calculate S

jj OO — plot

o

;ermir

ent

A

sed

th

ation

ss

det

eff

ici

We

u

>-3 S . -?13=^

CD ^ -2 in

CJ) (tl ^

'-' /-N S .li > 00* O S^-gss^ „ (-• ''^ —

o^

<ou

(JO fe

34
¥

\J.^

THERMAL DIFFUSION

OF ARGON

0.4 ^

''"^ •

0.3

•

^0 02

0.1

-0.1

- V»

1 1 1 1 100 200 300 400

TEMPERATURE, K

500

Figure 10. Experimental and theoretical [eq(16)] thermal diffusion factors for argon.

Data from [39]. We use the same m-6-8 potential as for the other transport

coefficients

.

35
100

10

*(r),eV 10-

0.1-

.01

Figure 11. The repulsive potential at small values of r. We compare the 11-6-8 (y = 3.0)

potential, found to be satisfactory for the dilute gas properties of argon,

with the potential estimated from high energy scattering of beams of neutral

argon atoms [41]. Also shown, as a matter of interest, are the curves for the

well-known 12-6 and 18-6 potentials. [This latter represents the best threeparameter potential fit for argon at low temperatures]

36
400 800 1200 1600 2000 2400

TEMPERATURE. K

6

o

o 4

X 2

o

o Q. o X

-2

X

<D

f^ -4

-6

ARGON

oO°°o°°°°ooooooQ'OOqOO

ooooooooooo

1 1

400 800 1200 1600 2000 2400

TEMPERATURE,

K

Figure 12. Deviation plots for the second virial and viscosity coefficients of argon

based on the m-6-8 potential with the inverse sixth power coefficient

fixed at ~ 60 X 10-^0 ergs/cm^: m = 1 1 , y = 2. 0, a =3.356 A, e/k = 137 K. Data not distinguished.

37
vu

sC

4-1

u

U

U](U ^ S (Uo

iH • 0) o

<U 4J Ji 4-> o

4^ 1—

1) ni d ^K .^

Cfl o ^ u

r-l o > u o a X!

4-1 <u

O

O

U 5 o 0) a CO U

d

(U (U

en XI ^

to IS c o o H

ni

a

4-) CO

W

CO

CO

ni

ru

ro (0 0) vD

t—

•-' Si

U > XJ

II s:

(U Sh

Q

M ^ O

OJ <+H Id

w x:

4J ^ 4_j d

T3 o (U Sh

C! -Q

u <+H rt 4J

^4 H-) 1 MH <+-! (^ >< -=*<

tn o

(U

0)

Sh

II o O b !-l O <4H a o

«« CL 0) o rt ^

f-H2

<M c >^ 4-> ni I—

d

II <j (U (U

4-1 4-1 >- J2 ni o o £

Q. *

xi ..-1 Sh 1—

^

><o

d)

u

CU

u

a,

0)B £ -.-4 ni a

ni

(tt Sh

4-> to ni x: XI (U a,

.-1 o _> 4-> ^ (U QC

r—

1—1 >

JD £

.c;

O

0)

d (U j:l ^ ^ ^ o 4J T3 (0 0-. ni d

00 4-1 nic d 1 o o

-d

(U -d

d

4-> o

niuo

"4-1

COO

(0

Ha < • CO

»—

(U^c

?)bo h

38
200 400 600 800

TEMPERATURE,

1000

o

o

Q.X

0>

10

o

o— 51-

&0

-5

-10

KRYPTON

h, ^A^

D D D n P

^^A ^A^A A A A A A A A

400 800 1200 1600

TEMPERATURE. K

Figure 14. Deviation curves for the viscosity and second virial coefficients of krypton.

Data for viscosity, [14], A [15], V [16], n[44]. Theoretical values

determined for eqs(ll) and (12) with the m-6-8 potential, parameters as given

in table 1.

39
10

5-

o

o

"?

Q.

0)

CD -5-

-10

XENON

.oo tgoggSS^^^ o

c>.

1

200 400 600 800

TEMPERATURE, K

1000

400 800 1200 1600 1800 2400

TEMPERATURE. K

Figure 15. Deviation curves for the viscosity and second virial coefficients of xenon. Data for viscosity, [14], A [15], d[45], V [46,47]. See table 1.

40
IT) LO

dX9

001

X

0|D0 dX9

o

o

o

o

2 i

o

^^ h- o

Q. MIM >• D o

oc D

^ D o

a o

° • ^

••• o

D«

< t >

1 1

o

o

CD

O w O -^

(\i ~ LU

o

o

UJ

a.

o LU o

o

oH

4J

0)M

O

CU H

o

4-1 p.>^U

o

o

T3a

o

0)

(0 •H>

(I) Q

(U

bO

S-l CO

13

0)

o

41
o

o

o

o —

o

o

z

UJ

X

o

o

Oo

o

o

°

—

O o

o

o Q. t

>

>

1 1 o lO o

dX8

lO

001 X X

0|D0 dxa

o

o

~ UJ

a:

o

o

UJ

8

^

00<r

CO '—

0)3 o

.H> —

cfl a

•H

4J

(UM

O

<U

O

3

X)C

o

o

0)

01 J2

u

00

•H

0)

(-1

0)

ex

0)

X)

o

42
0.5

0.4

0.3 -

THERMAL DIFFUSION

OF KRYPTON

a.

0.2

0.1

200 400 600

TEMPERATURE, K

800

Figure 18. The isotopic thermal diffusion factor for krypton calculated from eq(16),

using parameters given in table 1, compared to experiment [53].

43
0.5

0.4

0.3

a,

0.2

0.1 -

THERMAL DIFFUSION

OF XENON

1

200 400 600

TEMPERATURE, K

800

Figure 19. The isotopic thermal diffusion factor for xenon calculated from eq(16), using

parameter given in table 1, compared to experiment [54].

44
r

¥ o

o

c

o

1 1

1

1

1

1

1

o

CD CVJ

001 X

o

o

CVJ

o

o

o LlI

^

a:

o

O

o

CO

o

o

o

O CJ

1 ^ CD

1 dX9^

<

cr.

UJ

Q_

UJ

0)

u

M-lc

O

•ua.>.

S-i

O u^

C

01 •Ha

•H

M-l

M-lQJO

H

T3

01

o

CM

(U

•H

s

0)B

•H

)-l 0)ftX

O)

01

0|D0Q_dX9^

45
o

o 2

o

I Q.X

(D

CD

-2

-6

NITROGEN

— p/^"9

o

c

o

g»3 u

-

o

-

8

1

1 1 1 1 400 800 1200 1600 2000 2400

TEMPERATURE. K

6

o

o 4

X

i ? o

o Cl

o X

1 Q.

a> -2

X

Q>

P-

"-4

-6

NITROGEN

\-(9

o

oooopOoq-

^ o o o

1 1

400 800 1200 1600 2000 2400

TEMPERATURE. K

Figure 21. Deviation curves for the viscosity and second virial coefficients of nitrogen

o based on a simultaneous fit. Parameters: m = 10, y = 1.0, a = 3.694 A and

e/k = 87.0 K. Data (not distinguished): viscosity [13-16], second virial

coefficient [17].

46
400 800 1200 1600 2000 2400

TEMPERATURE. K

o

o

o

o

o

I Q.X

6

4

2

-4

-6

Q.X

a>

3>

1

NITROGEN

pjl^ o o ^/\r^^^ v^twwv^t^^

P^ DnDnnnnnnn

1 1 1 1 400 800 1200 1600 2000 2400

TEMPERATURE, K

figure 22. Fits of the viscosity and second virial coef f f icients of nitrogen^based on a

fit of the viscosity data. Parameters m = 12, y = 2.0, o = 3.54 A, e/k =

118.0 K. Viscosity data, V [13], A [14], d[15], [16], .[22], O [56],

A [57], #[58].

47
o

o

CD

a,X

0)

GQ

o

NITROGEN

4

—

o

o °

o

-4 —

-8 1 1 1 1 200 400 600 800

TEMPERATURE. K

1000

o

o

o

o

I Q.X

<1>

6

4

2

^

X

2

-4

-6

o^

o o ooPOo°°°OooooooOooo°°

^ooooooooo

NITROGEN

400 800 1200 1600 2000 2400

TEMPERATURE, K

Figure 23. Deviation curves for the viscosity and second virial coefficients of nitrogen

based on a fit of the second virial coefficient alone. Parameters, m = 12,

Y

= 2.0, o = 3.60 A, e/k = 118.0 K. Note (a) the unusual deviation pattern

for the viscosity, (b) that the potentials used here and in figure 22 differ

only by a small amount in a.

48
001

X

0|D0^_dX9^

00

II

O o<i!

QJC

<uBo

01

<uu

3

•H

00

o

0)

0)

0)

!-l 3

4-1 n)

1-1 QJO13 —

•

(30

O

J3

01

01>

:3

o

49
001

31 D3. dxe.

a- Ll

H

O

u-io

4=

O U)

X)

en 43 <^ CM

>. II O D

O

CJ

CO

•rH>

QJ 42

•HX> VO C

a. ^D

vO

CM

0)

&0 H

Co

XI

CO

13

Cb

T3

0)

>

42

dJ 4=

0)B

o

w <r o

2 O

XI

0)

X)

in

50
UJQ

X

o D

Q D

2 D O

GO D

or

< D

o > a

> P

•I

i

CO (M

dX9

001 X

L

3103/ dX3;

a- a

o

o

O

o

o

CVJ

8 UJ

CD CC

CO

o

o

o

o

CX)

o

o

o

<

cr

LlI

QUJ

CM00

C CO o ^o

0)W II nj J3 D

Q) XI o

•HO

•H II XI

>- c

oU rH

n3O II

M-l g

•1-1 5-1 •H 0)

to iJ O 0)

CO S

•H ^

> cfl p-l

in

>

01

CM

(Un

3

cm

•H

51
o

o

(X)

001 X

o

o

o|DOy_dxay

Ld

CL

LU

In)U

toex

0) u

c

o

Xl

01

COu

0) -p

0)s

CO

J^

COft

CO •H

O

o

o i^ gen b

Poten OJ •» o LxJ CN cc

o a0) ^ u CO •H -H > > O H 4-J X) O CU 3 iJ O < X) CO C ^

O 3 00 ct: al

c

Calc

LTl

CU

CU

CO •H>

CU P

CUu

a

00

•H

O

tj-l

X)c

3

O

52
^<

<

<•

«l

<

"^

i LlI <

O

>- <

X

o

.

•

• i

:>< ^•^ 4 • ^^^

^%>

> Ocri^

1 1 o LO

o

o

CD

O

O LU

o

o

8

^

§ LlI

tnoa

en •H>

0) 43

C

O

T)

CUM

nj Xic

0)60X

o

CM00

O00

I 00

o —

m

>, • i^ H cyv o

H

O

T3C

o

o

XT

0)

•H>

dX9

0) 43

0)>

do

c

o

•H

4J

n) •H>Q

00

0)u

•H

00

(0 -U r 1 e (fl in

00

OJ

-p

u

<u

4-1 0)

u

to

001 X

0|DD dX9^

53
001

0) 00

^3

3O

O

d

B

•H>

•H

4J

CJ

=) -dC

o

o

QJ

cu

>

3o

c

o

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cd •H>

0) Q

0)

1-1

•ri I*

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(J

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42

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54
\J.^

THERMAL

OF NITROGEN

DIFFUSION

^.'^^^

^^

0.4 — >/

1

0.3

Oo 0.2

/ < »

0.1

(

)

-0.1 ^ 1 1 1 100 200 300 400

TEMPERATURE. K

500

Figure 30. Isotopic thermal diffusion factor for nitrogen calculated from eq(16) with the

potential parameters used for viscosity. Data from reference [92].

55
.50 THERMAL DIFFUSION

OXYGEN

200 300

TEMPERATURE, K

500

Figure 31. Isotopic thermal diffusion factor for oxygen calculated from eq(16) with the

viscosity potential parameters. Data from reference [93].

56
0.5

0.4

0.3

Oto 0-2

0.1

-0.1

THERMAL DIFFUSION

OF CARBON DIOXIDE

200 300 400 500

TEMPERATURE. K

600

Figure 32. Isotopic thermal diffusion factor for carbon dioxide calculated from eq(16)

with the viscosity potential parameters. Data from reference [94].

57
12. References

1. Hanley, H.J.M. and Klein, M. , On the Selection of the Intermolecular Potential

Function: Application of Statistical Mechanical Theory to Experiment, Nat. Bur. Stand., Tech. Note 360 (Nov. 1967)

2. Klein, Max and Hanley, H.J.M., Selection of the Intermolecular Potential, Part 2, From

Data of State and Transport Properties Taken in Pairs, Trans. Faraday Soc. 64 , 2927-2938 (1968)

3. Hanley, H.J.M, and Klein, M. , Selection of the Intermolecular Potential Function, III; From the Isotopic Thermal Diffusion Factor, J. Chem. Phys. 50, 4765-4770 (1969).

4. The particular choice of reducing T is made for convenience only; there is no special significance in the use of the 12-6 potential.

5. Barker, J. A. and Pompe, A., Atomic Interactions in Argon, Aust. J. Chem. 21, 1683-1694

(1968). See also reference 43.

6. For example: Mason, E.A. , Munn, R.J. and Smith, F.J., Recent Work on the Determination of the Intermolecular Potential Function, Discussions Faraday Soc. 40, 27-34

(1965); Munn, R.J. and Klein, M.L., Interaction Potential of the Inert Gases, II, J. Chem. Phys. 43, 3998-4002 (1965).

7. Dymond, J.H. and Alder, B.J., Pair Potential for Argon, J. Chem. Phys, 51, 309-20

(1969).

8. Klein, Max, Smith, F.J., Hanley, H.J.M. and Holland, P.M., NBS-WSRDS Monograph, In

press, 1972.

9. For example, about 1% precision is required for viscosity, thermal conductivity and

diffusion, approximately 10% for thermal diffusion, and about 0.5% for the second

virial coefficient.

10. Hirschfelder, J.O., Curtiss, C.F. and Bird, R.B., Molecular Theory of Gases and

Liquids, John Wiley & Sons, New York (1964), Second Printing.

11. We have verified that our conclusions obtained by studying the relationship of the

potential with the transport properties are not affected by taking the statistical

mechanical expressions to the first approximation only.

12. Hanley, H.J.M. and Childs, G.E., Discrepancies Between Viscosity Data for Simple

Gases, Science 1_59, 1114-1116 (1968).

13. Guevara, F.A. , Mclnteer, B.B. and Wageman, W.E., High Temperature Viscosity Ratios

for Hydrogen, Helium, Argon, and Nitrogen, Phys. Fluids 12, 2493-2502 (1969).
14. Clark, A.G. and Smith, E.B., Low Temperature Viscosities of Argon, Krypton and Xenon,

J. Chem. Phys. 48, 3988-3991 (1968),

15. Dawe, R.A. and Smith, E.B., Viscosities of the Inert Gases at High Temperatures,

J. Chem. Phys. 52, 693-703 (1970).

16. Kestin, J., Wakeham, W. and Watanabe, K. , Viscosity, Thermal Conductivity and

Diffusion Coefficient Ar-Ne, and Ar-Kr, Gaseous Mixtures in the Temperature Range

25-700°C, J. Chem. Phys. _53, 3773-3750 (1970); Dipippo, R. and Kestin, J., the

Viscosity of Seven Gases up to 500°C and its Statistical Interpretation, Brown

University Report NSF-GK 1305 (1967).

17. Senger, J.M.H. Levelt, Klein, Max and Gallagher, J.S., to appear in the American

Institute of Physics Handbook, McGraw-Hill Book Company, New York. Also available as Report AEDC-TR-71-39, Arnold Engineering and Development Center, Tullahoma, Tennessee.

This report can be obtained from the Clearing House for Federal Scientific and

Technical Information (CFSTI), Springfield, Virginia 22151 as AD 719749.

18. This "turn around" was predicted in [2] from a study of the comparison of the 12-6

potential with the other m-6 potentials but was not observed experimentally up to that time. We have used more recent viscosity data for our examination here, however

[14].

19. Johnston, H.L. and Grilly, E.R., Viscosities of Carbon Monoxide, Helium, Neon, and

Argon Between 80° and 300° K, Coefficients of Viscosity, J. Phys. Chem. 46, 948-963

(1942).

20. Gracki, J.A. , Flynn, G.P. and Ross, J., Viscosity of Nitrogen, Helium, Hydrogen and

Argon from -100° to 25°C up to 150-250 atm. , J. Chem. Phys. 51, 3856-3863 (1969).

21. Timrot, D.L., Sevednitskaya, M.A. and Traktueva, S.A., Teploenergetika J^, 83-84

(1969).

22. Wobser, R. and Muller, F. , Die innere Reinbung von Gases and und Dampfen und ihre

Messung in Hoppler-Viskosimeter (The Viscosity of Gases and Vapors and Their Measurement with the Hoppler Viscometer), Kolloid-Beih. _52, 165-276 (1941).

23. Michels, A., Botzen, A. and Schuurman, W. , The Viscosity of Argon at Pressures up to 2000 Atmospheres, Physica 20. 1141-1148 (1954).

24. Keyes, E.G. and Vines, R.G., The Thermal Conductivity of Nitrogen and Argon, J. Heat

Transfer 87, No. 2, 177-183 (1965).

25. Michels, A., Sengers, J.V. and Van de Klundert, L.J.M., The Thermal Conductivity

of Argon at Elevated Densities, Physica 2^, 149-160 (1963).

59
26. Uhlir, A., Thermal Conductivity of Fluid Argon and Nitrogen, J. Chem. Phys. 20 , 463-472 (1952).

27. Eucken, A., Uber die Temperaturabhangigkeit der Warmeleitf ahigkeit einiger Case

(Concerning the Temperature Dependence of Thermal Conductivity of Several Gases)

,

Physik Z. 12, 1101-1117 (1911).

28. Ziebland, H. and Burton, J.T.A., The Thermal Conductivity of Nitrogen and Argon in the

Liquid and Gaseous States, Brit. J. Appl. Phys. 9^, 52-59 (1958).

29. Ikenberry, L.D. and Rice, S.A., Kinetic Theory of Dense Fluids, XIV. Experimental and

Theoretical Studies of Thermal Conductivity in Liquid Ar , Kr, Xe and CH , J. Chem.

Phys. 39, 1561-1571 (1963).

30. Rothman, A.J., Thermal Conductivity of Gases at High Temperatures, U.S. Atomic Energy

Coram. Report UCRL 2339, 8-114 (1953).

31. Kannuluik, W.G. and Carman, E.H., The Thermal Conductivity of Rare Gases, Proc. Phys,

Soc. (London) B65, 701-709 (1952).

32. Le Neindre, B. , Tufeu, R. , Bury, P., Johannin, P. and Vodar, B., Thermal Conductivity

Measurements of Some Noble Gases, Proc. 8 Conference on Thermal Conductivity, Purdue,

Univ., Lafayette, Ind. (1968); Le Neindre, B. , Thesis; Univ. of Paris 1970; Le Neindre,

B. , private communication.

33. Vargaftik, N.B. and Zimina, N.Kh. , Heat Conduction of Argon at High Temperatures,

Teplofiz. Vysok. Temp. _2, 716-724 (1964).

34. Van Dael, W. and Cauwenbergh, Measurements of the Thermal Conductivity of Gases. I, Physica 4_0, 165-172 (1968). ^

35. Haarmann, J.W. , Thesis, Delft, The Netherlands (1969).

36. Hutchinson, F. , Self-Diffusion in Argon, J. Chem. Phys. 1^, No. 11, 1081-1086 (1949);

Winn, E.B., The Temperature Dependence of the Self-Diffusion Coefficients of Argon

Neon, Nitrogen, Oxygen, Carbon Dioxide and Methane, Phys. Rev. ^, 1024-1027 (1950).

37. Vugts, H.F., Boerboom, A.J.H. and Los, J., Measurements of Relative Diffusion

Coefficients of Argon, Physica 4^, 219-226 (1969). Note, these values are relative

measurements, relative to 298. 15K.

38. Paz, M.De., Turi, B. and Klein, M.L., New Self-Diffusion Measurements in Argon Gas,

Physica 34, 678-890 (1967).

39. Paul, R. , Howard, A.J. and Watson, W.W. , Isotopic Thermal Diffusion Factor of Argon,

J. Chem. Phys. ^, 3053-3056 (1963).

60
40. See, for example, Copeland, D.A. and Kestner, N.R., Accurate "Effective" Intermolecular

Pair Potential in Gaseous Argon, J. Chem. Phys. _49, 5214-5222 (1968).

41. I. Amdur and J.E. Jordan In Molecular Beams (Advances in Chemical Physics Vol. 10),

J. Ross, editor, Interscience, New York, 1966, Chapter 2.

42. Margenau, H. and Kestner, N.R. , Theory of Intermolecular Forces, Pergamon, New York

(1969), page 32. See also the appropriate references quoted therein.

43. Bobetic, M.V. and Barker, J.A. , Lattice Dynamics with Three Body Forces: Argon,

Phys. Rev. B2, 4169-4175 (1970).

44. Goldblatt, M. , Guevara, F.A. and Mclnteer, B.B., High temperature viscosity ratios

for krypton, Phys. Fluids 13, 2873-74 (1970).

45. Goldblatt, M. , and Wageman, W.E., High Temperature Viscosity Ratios for Xenon, Phys.

Fluids 14, 1024-25 (1971).

46. Rankine, A.O. , Uber die Anderung der inneren Reibung der Gase der Argongruppe mit

der Temperatur (On the Variation with Temperature of the Viscosities of the Gases of

the Argon Group), Physik. Z. _11, 745-752 (1910).

47. Kestin, J. and Leidenfrost, W. , An Absolute Determination of the Viscosity of Eleven

Gases Over a Range of Pressures, Physica ^, 1033-1062 (1959).

48. Saxena, V.K. and Saxena, S.C, Thermal Conductivity of Krypton and Xenon in the

Temperature Range 350-1500 K, J. Chem. Phys. 51, 3361-3368 (1969).

49. Curie, M. and Lepape, A., Conductivilite Thermique des Gaz Rares (Thermal Conductivity

of Rare Gases), J. Phys. Radium^, 392-397 (1931).

50. Keyes, E.G., Thermal Conductivity of Gases, Trans. ASME _7Z' 1395-1396 (1955).

51. Zaitseva, L.S., An Experimental Investigation of the Heat Conductivity of Monatomic

Gases Over Wide Temperature Intervals, Soviet Phys. Tech. Phys. 4_, 444-450 (1959).

52. Thornton, E. , Viscosity and Thermal Conductivity of Binary Gas Mixtures: XenonKrypton, Xenon-Argon, Xenon-Neon and Xenon-Helium, Proc. Phys. Soc. (London) 76 , 104-112 (1960).

53. Paul, R. and Watson, W.W. , Isotopic Therman-Diff usion Factor for Krypton, J. Chem,

Phys. h5, 4132-4134 (1966).

54. Paul, R. , Howard, A.J. and Watson, W.W. , Isotopic Thermal-Diffusion Factor for

Xenon, J. Chem. Phys. 43, 1890-1894 (1965).

61
55. Weissman, S. and Dubro, G.A. , Self-Diffusion Coefficient for Krypton, Phys. Fluids

13, 2689-2692 (1970).

56. Johnston, H.L. and Mc Closkey, K.E. , Viscosities of Several Common Gases Between 90°

K and Room Temperature, J. Phys. Chem. _44, 1038-1058 (1940).

57. Johnston, H.L., Mattox, W.R. and Powers, R.W. , Viscosities of Air and Nitrogen at Low

Pressures, Nat'l Advisory Comm, Aeronaut. Tech. Note No. 2456 (1951).

58. Lazarre, F. and Vodar, B. , Measurement of the Viscosity of Compressed Nitrogen up

to 3000 Atmospheres, Proc. Conf. Thermodynamic and Transport Properties of Fluids,

London, July 1957, Inst. Mech. Engrs , 159-162 (1958).

59. Also the 12-6-8 potential with Y = 2.0 found for nitrogen is similar to the 11-6-6

vith Y = 3-0 found for the rare gases. Thus it seems that nitrogen does not deviate

much from a simple law of corresponding states as far as the viscosity is concerned.

60. De Rocco, A. and Halford, J.O., Intermolecular Potential of Argon, Methane and

Ethane, J. Chem. Phys. _28, 1152-1154 (1958).

61. Giddings, J.G., Kao, J.T.F. and Kobayashi, R. , Development of a High Pressure

Capillary-Tube Viscometer and its Application to Methane, Propane and their Mixtures

in the Gaseous and Liquid Regions, J. Chem. Phys. 43, 578-586 (1966).

62. Maitland, G.C. and Smith, E.B., La Viscosite CO Entre 293 K et 1500 K (The Viscosity

of CO2 Between 293 K and 1500 K) , J. Chim. Phys, 67^, 631-632 (1970).

63. Dave, R.A. , Maitland, G.C, Rigby, M. and Smith, E.B., High Temperature Viscosities

and Intermolecular Forces of Quasi-Spherical Molecules, Trans. Faraday Soc. 66 , 1955-1965 (1970).

64. Trautz, M. and Melster, A. , Die Reibung, Warmeleitung und Diffusion in Gasmishungen,

XI. Die Reibung von H„, N , CO, C , H , und ihren Binaren Gemischen (The Viscosity

Heat Conduction, and Diffusion in Gas Mixtures. XI. The Viscosities of H„, N , CO,

C H , and their Binary Mixtures), Ann. Physik 2. 409-426 (1930); Trautz, M. and

Zink, R. , Die Reibung, Warmeleitung und Diffusion in Gasmischungen. XII. Gasreibung

bei hoheren Temperaturen (The Viscosity, Heat Conduction and Diffusion of Gas

Mixtures. XII. The Viscosity of Gases at High Temperatures), Ann. Physik _7> 427-452

(1930).

65. Andrussow, L. , Cpnductibilite Thermique, Viscosite et Diffusion en Phase Gaseuse.

Memoire 10. Relation entre les Coefficients de ces Phenomones et 1' Equation de

Maxwell (Thermal Conductivity, Viscosity and Diffusion in the Gas Phase. Report 10. Relation Between the Coefficients of These Phenomena and the Equation of Maxwell), J. 62
Chem. Phys. 52, 295-306 (1955).

66. von Lierde, J., Metinger over de Thermo-Dif f usie en de Inwendige Wrijving in Enkele

Gosmengsels bij Lage en Zeer Lage Temperaturen, (Measurements of Thermal Diffusion

and Viscosity of Certain Gas Mixtures at Low and Very Low Temperatures), Verhandel.

Koninkl. Vlaam. Acad. Wetenschap. Belg. Kl. Wetenschap. (24) _9» 7-78 (1947).

67. For details and references see the article by E.A. Mason in Proc. 4th Symp. on Thermophys. Prop., (ASME, New York, 1968), page 21.

68. Carnevale, E.H. , Carey, C. and Larson, G. , Ultrasonic Determination of Rotational

Collision Numbers and Vibration Relaxation Times of Polyatomic Molecules at High

Temperatures, J. Chem. Phys. _47, 2829-2835 (1967).

69. Ganzi, G. and Sandler, S.I., Determination of Thermal Transport Properties from

Thermal Transpiration Measurements, J. Chem. Phys. 55. 132-140 (1971).

70. Mlchels, A. and Botzen, A. , The Thermal Conductivity of Nitrogen at Pressures up to 2500 Atmospheres, Physica j^, 585-598 (1953).

71. Golubev, I.F. and Kal'sina, M.V., Thermoconductivity of Nitrogen and Hydrogen at Temperatures from 20 to -195°C and Pressures from 1 to 500 Atmospheres, Gaz. Prom.

_9, 41-43 (1964), Translation available from SLA translation center. No. LA-TR-65-1.

72. Vargaftik, N.B. and Zimina, N.Kh., Thermal Conductivity of Nitrogen at High

Temperatures, High Temp. 1_, ld,2-19Q (1964).

73. Westenberg, A. A. and de Haas, N. , Gas Thermal Conductivity Studies at High Temperature.

Line-Source Technique and Results in N , C0„, and N -C0„ Mixtures, Phys. Fluids _5, 266-273 (1962).

74. Geier, H. and Schafer, K. , Warraeleitfahigkeit von Reinen Gasen und Gasgemischen

Zwischen 0° und 1200°C (Thermal Conductivity of Pure Gases and Gas Mixtures Between

0° and 1200°C), Allgem, Warmetech. 10, 70-75 (1961).

75. Frank, E.U., Thermal Conduction in Highly Compressed Gases, Chem. Eng. Tech. 25 , 238-244 (1953).

76. Vines, R.G. , Measurements of the Thermal Conductivities of Gases at High Temperatures,

Mass. Inst, of Tech., Tech. Report MIT-20-P (Sept. 1958), DDC AD 205 694.

77. Nuttall, R.L. and Ginnings, D.C., Thermal Conductivity of Nitrogen from 50° to 500°C

and 1 to 100 Atmospheres, J. Res. NBS _58, 271-278 (1957).

63
78. Gregory, H. and Marshall, S., Thermal Conductivities of Oxygen and Nitrogen, Proc.

Roy. Soc. (London) A118 , 594-607 (1928).

79. Kannuluik, W.G. and Martin, L.H., Thermal Conductivity of Some Gases at 0°C, Proc.

Roy. Soc. (London) Al 4 4 , 496-513 (1934).

80. Nothdurf t, W. , Zur Absolutmessung des Warmeleitvermogens von Gasen, (The Absolute

Measurements of the Heat Conductivities of Gases), Ann. Physik _28, 137-156 (1937).

81. Johnston, H.L. and Grilly, E.R., The Thermal Conductivities of Eight Common Gases

Between 80° and 380° K, J. Chem. Phys. 14, 233-238 (1946).

82. Tsederberg, N.V. and Timrot, D.L., An Experimental Determination of the Thermal

Conductivity of Liquid Oxygen, Soviet Phys. Tech. Phys. 1, 1791-1797 (1956),

Translation from Zhur, Tekh. Fiz. 2^, 1849-1956 (1956).

83. Cheung, H. , Thermal Conductivity and Viscosity of Gas Mixtures, Calif. Univ., Lawrence

Radiation Lab., Berkeley, Report No. UCRL-8230 (April 1958).

84. Pereira, A.N.G. and Raw, C.J.G., Heat Conductivities of Polyatomic Gases and their

Binary Mixtures, Phys. Fluids ^, 1091-1096 (Aug. 1963).

85. Westenberg, A. A. and de Haas, N. , Gas Thermal Conductivity Studies at High Temperature.

II. Results for and -HO Mixtures, Phys. Fluids _6, 617-620 (1963).

86. Mann, W.B. and Dickins, B.C., The Thermal Conductivities of the Saturated Hydrocarbons

in the Gaseous State, Proc. Roy. Soc. A134 , 77-96 (1931).

87. Baker, C.E. and Brokaw, R.S., Thermal Conductivities of Ordinary and Isotopically

Substituted Polar Gases and their Equimolar Mixtures, J. Chem. Phys. ^, 3519-3528

(1965).

88. Golubev, I.F. , A Bicalorimeter for Measuring the Thermal Conductivity of Gases and

Liquids at High Pressures and Various Temperatures, Teploenergetika 10^, 78-82 (1963).

89. Hilsenrath, J., Beckett, C.W., Benedict, W.S., Fano, L. , Hoge, H. J. , Masi, J.F.,

Nuttall, R.L., Touloukian, Y.S. and Woolley, H. , Tables of Thermal Properties of

Gases, NBS, Circ. No. 564 (Nov. 1955).

90. McDowell, R.S. and Kruse, F.H., Thermodynamic Functions of Methane, J. Chem, Eng,

Data 8, 547-548 (1963).

91. Monchick, L. , Munn, R.J. and Mason, E.A. , Thermal Diffusion in Polyatomic Gases: A

Generalized Stefan-Maxwell Diffusion Equation, J. Chem, Phys. ^, 3051-3058 (1966).

64
92. Raman, S., Mathur, B.P., Howard, A.J., Champlin, J.W. and Watson, W.W. , Thermal

Diffusion in Polyatomic Gases: Isotopic ^%2~'^^N2, J. Chem. Phys. 49, 4877-4879

(1968).

93. Mathur, B.P. and Watson, W.W,, Isotopic Thermal Diffusion Factor ^^02-^^02, J. Chem.

Phys. 51, 2210-2214 (1969).

94. Mathur, B.P. and Watson, W.W. , Isotopic Thermal . Diffusion Factor for CO , J. Chem.

Phys. 51, 5623-5625 (1969).

95. We would obtain negative values of a, however, with potential parameters selected

by fitting the second virial and viscosity coefficient simultaneously.

96. The possibility that dimerization occurs at low temperatures is, further, also

considered a complicating factor in assessing results for a . See [92-94] and

W. Wakeham, private communication.

97. Smith, F.J., Munn, R.J. and Mason, E.A. , Transport Properties of Quadrupole Gases,

J. Chem. Phys. _46, 317-321 (1967); Spurling, T.H. and Mason, E.A. , Determination of

Molecular Quadrupole Moments from Viscosities and Second Virial Coefficients, Ibid,

322-326 (1967).

98. See the article by E.A. Mason and T.R. Marrero in Advances in Atomic and Molecular

Physics, Academic Press, New York (1970) Ed. D.R. Bates and I. Estermann.

99. While this manuscript was in preparation, we have carried out further tests on the

validity of the 11-6-8 argon potential: H.M.H. Hanley, J. A. Barker, J.M. Parson,

Max Klein, and Y.T. Lee, Comments on the interatomic potential for argon, Mol. Phys.

(inpress)

.

100. W. M. Haynes of the Cryogenics Division, NBS , Boulder, has recently measured the

viscosity of oxygen. Preliminary dilute gas values are very close to our estimates.

101. G. Starkschall and R.G. Gordon, Improved Error Bounds for the long-range forces

between atoms, J. Chem. Phys. 54 663 (1971).

102. L. Monchick and E.A. Mason, J. Chem. Phys. _35, 1676 (1961).

65
APPENDIX

Viscosity of Oxygen

Although oxygen is a conannon and important fluid, there are very few reliable sets

of viscosity data for it. In fact, in our opinion, the reliable data lie in the limited temperature range of about 250 to 400 K. We reject data outside the range for the following

reasons:

a) Above 400 K: Data above 400 K are available but are due largely to one source - Trautz and co-workers - measured 40 years ago [64]. But

Trautz measured the viscosity for many gases and, for gases other than

oxygen [Ar, N , He, air and so on], modern experiments give results

which differ from his. The results differ by about 1-5% for tennperatur es up to 1000 K -- the error increasing with temperature. Since there are sound reasons to indicate that the more recent work is nnore likely to be

correct for these other gases [12], we have also to suppose that the old

data for oxygen may not be correct.

b) Below 250 K: Values reported below 250 K were measured 30 years ago

by Johnston and co-workers [56]. There appears to be no more recent

work [100]. But, the viscosities of Johnston for gases other than oxygen,

differ slightly from more recent data. Since, here again, we feel the

more recent data for these gases to be nnore reliable, we have to suspect

the earlier data for oxygen.

Discrepancies between the older and the nnore recent viscosities seenn to depend

only on the tennperature and not on the gas (at least to a first approxinnation) . We checked

this by constructing a deviation curve between old and nnodern data for argon, neon,

nitrogen, krypton, xenon, air, nnethane and heliunn. The curve is shown schennatically as

figure Al. On the basis of this figure we feel justified to adjust the viscosity results for

oxygen [56,64] according to the percentage discrepancy indicated at the appropriate

tennperatures. Adjusted oxygen data are given in table Al.

The adjusted data, together with data fronn references [22,47,65,66] were then

fitted to the nn-6-8 potential and the result is shown as a deviation curve, figure 25, in the

nnain text.

66
TABLE Al

Adjusted Experimental Viscosities for Oxygen

Temperature Viscosity

3 K 10 g/cm-s

90.3 0.0679

118.8 0.0890

131.3 0.0979

144.9 0.108

158.5 0.117

172.6 0.128

184.6 0.137

400.8 0.258

500.1 0.305

550.1 0.327

556.1 0.328

675.1 0.377

769.1 0.411

881.1 0.450

963.1 0.477

1102.1 0.521

67
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(I) >

4^d s

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4-)

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4NBS-114A (1-71)

U.S. DEPT. OF COMM.

IBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS-TN-628

2. Gov't Acccssiwo No.

ITLE AND SUBTITLE

On The Utility of the m-6-8 Potential Function

3. Recipient's Accession No.

5. Publication Date November 1972

6. Perfotming Organization Code

UTHOR(S)

H. J. M. Hanley and Max Klein

8. Performing Organization

10. Project/Task/Work Unit No. 2750124/2210143

ERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS. Boulder Labs

DEPARTMENT OF COMMERCE

Boulder, Colorado 80302

11. Contract/Grant No.

Iponsoring Organization Name and Address

Office of Standard Reference Data

National Bureau of Standards

Washington, D. C. 20234

13. Type of Report & Period

Covered

14. Sponsoring Agency Code

UPPLEMENTARY NOTES

iBSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

ibliography or literature survey, mention it here.) e calculation of transport and equilibrium properties of simple gases with the

6-8 potential function is discussed. Properties referred to specifically are the

cosity, diffusion, thermal conductivity and second virial coefficients, and the

Irmal diffusion factor. Gases referred to are argon, krypton, xenon, nitrogen,

rgen, carbon dioxide and methane. We also discuss in detail the general behavior

a model potential with respect to the fitting of data and the selection of potential

il:anaeters. The m-6-8 appears to be the simplest model potential which can

jisfactorily be used to fit data over a wide temperature range. The relationship of

\ potential to independent theory is briefly examined.

KEY WORDS (Alphabetical order, separated by semicolons) Dispersion coefficients; m-6-8 potential;

rameter selection; second virial coefficients; transport properties.

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