.Kfards r

NBS TECHNICAL NOTE

On the Selection of the

Intermolecular Potential Function:

Application of Statistical Mechanical

Theory to Experiment

360

.»-**»' »f c.

\

1«

o

^'?eAU of

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

r^ " / /
THE NATIONAL BUREAU OF STANDARDS

The National Bureau of Standards^ provides measurement and technical information services

essential to the efficiency and effectiveness of the work of the Nation's scientists and engineers. The

Bureau serves also as a focal point in the Federal Government for assuring maximum application of the physical and engineering sciences to the advancement of technology in industry and commerce. To

accomplish this mission, the Bureau is organized into three institutes covering broad program areas of research and services: THE INSTITUTE FOR BASIC STANDARDS . . . provides the central basis within the United

States for a complete and consistent system of physical measurements, coordinates that system with the measurement systems of other nations, and furnishes essential services leading to accurate and uniform

physical measurements throughout the Nation's scientific community, industry, and commerce. This

Institute comprises a series of divisions, each serving a classical subject matter area: —Applied Mathematics—Electricity—Metrology—Mechanics—Heat—Atomic Physics—Physical

Chemistry^Radiation Physics—Laboratory Astrophysics'—Radio Standards Laboratory,^ which

includes Radio Standards Physics and Radio Standards Engineering—Office of Standard Refer- ence Data. THE INSTITUTE FOR MATERIALS RESEARCH . . . conducts materials research and provides

associated materials services including mainly reference materials and data on the properties of materials. Beyond its direct interest to the Nation's scientists and engineers, this Institute yields services which are essential to the advancement of technology in industry and commerce. This Institute is or- ganized primarily by technical fields: —Analytical Chemistry—Metallurgy—Reactor Radiations—Polymers—Inorganic Materials—Cryogenics^—Materials Evaluation Laboratory—Office of Standard Reference Materials.

THE INSTITUTE FOR APPLIED TECHNOLOGY . . . provides technical services to promote the use of available technology and to facilitate technological innovation in industry and government. The

principal elements of this Institute are: —Building Research—Electronic Instrumentation—Textile and Apparel Technology Center

Technical Analysis—Center for Computer Sciences and Technology^—Office of Weights and Measures—Office of Engineering Standards Services—Office of Invention and Innovation—Clearinghouse for Federal Scientific and Technical Information.'

^ Headquarters and Laboratories at Gaithersburg, Maryland, unless otherwise noted; mailing address Washington, D. C,

20234.

2 Located at Boulder, Colorado, 80302.

3 Located at 5285 Port Royal Road, Springfield, Virginia, 2215L
UNITED STATES DEPARTMENT OF COMMERCE

Alexander B. Trowbridge, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

NBS TECHNICAL NOTE 360

ISSUED NOVEMBER 20, 1967

ON THE SELECTION OF THE INTERMOLECULAR

POTENTIAL FUNCTION: APPLICATION OF STATISTICAL

MECHANICAL THEORY TO EXPERIMENT

H. J. M. HANLEY

Cryogenics Division

Institute for Materials Research

National Bureau of Standards

Boulder, Colorado

MAX KLEIN

Heat Division

Institute of Basic Standards

National Bureau of Standards

Washington, D. C.

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.

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

Price 50 cents.

CONTENTS

ABSTRACT v

1. INTRODUCTION 1

2. THE INTERMOLECULAR POTENTIAL FUNCTION 5

3. THE STATISTICAL MECHANICAL EXPRESSIONS FOR

THE PROPERTIES 6

3. 1 Equilibrium properties - . . . . 6

3. 2 Transport properties 7

4. METHOD . 9

4. 1 Background 9

4. 2 Graphical procedure 13

4. 3 Computer procedure 14

4. 4 Effect of experimental error 17

5. RESULTS 20

5. 1 Exact fit to a property and its first derivative 20

5. 1. 1 General results 22

5. 2 Fit to a given property within assigned confidence

limits 26

5. 2. 1 Equilibrium properties 27

5. 2. 2 Transport properties 29

6. EXPERIMENTAL VERIFICATION 31

6. 1 Equivalence of potential functions 31

6. 2 The sensitivity of the simultaneous fit of the high and

low temperature viscosity data to the potential function . 35

6. 3 The insensitivity of intermediate range data to the

potential function 38

7. CONCLUSIONS 39

8. ACKNOWLEDGEMENT 41

9. REFERENCES 43

10. FIGURES 47

111

ABSTRACT

We have developed a method here to evaluate quantitatively the

relationship between model intermolecular potential fiinctions and

macroscopic experimental properties. Specifically, we have studied

the fionction families: m-6, Kihara, exp:6, and Morse, and the properties: viscosity coefficient, diffusion coefficient, second virial, and

Joule -Thomson coefficient. Our method is not restricted to these

fvmctions and properties; it is valid for any fionction and any property

provided the appropriate theoretical expressions are available.

The principal conclusions from this work are: 1. A temperature range exists (around room temperature for most substances) in

which data are completely insensitive to the potential function. This

range makes it possible to quantitatively define what is meant by "high"

or "low" temperatures. These definitions are most important when

extrapolation of data arises. 2. The function families studied are

essentially equivalent. 3. More than one function is required to represent data over a range of about 1000 °K. 4. We have estimated the

experimental error w^hich can be tolerated if experimental data are to

be used to select a potential function. The conclusions are experimentally verified for both transport and equilibrium properties.

Key words: Intermolecular potential function, experimental data,

theory, correlation, transport properties, equilibrium

properties.

1. INTRODUCTION

One describes a natural phenonaenon by methods that are appropriate to the level that one wants to \ander stand it. The levels of \anderstanding range from the most empirical to the most fundamental.

If we wished to classify methods in terms of levels we might

list, in order of rigor:

1. Curve fitting empirical methods which merely serve to correlate

experimental data. Examples are the least square fits to empirical

multi -parameter equations of state.

2. Macroscopic theories which are soundly based on verifiable macroscopic argioments and postulates. Thermodynamics and hydrodynamics

are examples.

3. Theories, such as statistical mechanics and kinetic theory, which

try to \anderstand and derive the laws of nature and the properties of

materials from molecular considerations. These theories explain the

fundamentals and results of the raacroscopic theories of the previous

level. We raust have, however, a know^ledge of the internnolecular

potential function to translate formal expressions into working equations,

4. The most rigorous theories which try to explain natural phenomena

from first principles. Quantum mechanics is the example here.

The choice of the level to work with is obviously a matter of

convenience. The pure scientist would be at horae at level 4, whereas

the laboratory worker might prefer levels 1 or 2.

One can argue that the theories at level 3 are the most rewarding in the long run. Quite apart from the academic satisfaction of

describing a natural process in terms of nnolecular properties, the

theories provide very powerful methods to correlate and predict experi-
ment. An empirical theory cannot do this. The rigorous theories, on

the other hand, involve much too much detail with which present-daycomputing and analytic tools cannot cope.

Naturally, application of the 3rd level theories is limited to

phenomena which can be described in terms of molecular properties.

But there is another limitation about which we are concerned here.

As we said above, the 3rd level requires the intermolecular potential

fionction to get results from experiraent. Unfortunately, quantum

mechanics has not yet derived a complete fionction from first principles.

One is therefore left with only model functions which approximate to

the real potential as far as it is knov/n. The testing and choice of the

most realistic model fxonction must come from experiment.

This point has long been realized and numerous authors have

tested model function w^ith experimental data. The explosive use of

the computer has enormously increased the output of this type of work.

Although most authors are aware of the overall relation between model

functions and experiment, most of the knowledge is intuitive rather

than quantitative. All too often the computer has been abused with the

result that much of the work published is at best repetitious and at

worst incorrect and misleading.

We shall examine here exactly what requirements have to be

satisfied if one is to obtain sensible conclusions when a given fionction

is inserted into statistical naechanical theories and the result compared

with experiment. In particular we will see what experimental properties

and what temperature ranges are nnost likely to give us information on

the f\inction. We will also investigate just how accurate experimental

data must be, so that this information is not obscured by random

error.
To emphasize the conclusions that come frora this study we

state them here in advance. However, first the following definitions

are required:

a) results are given in terms of reduced temperatures, T''\ A reduced

temperature is defined by the relation T^'' = T/(e/k), where T is the

absolute temperature, e is the value of the maximunn. energy of

attraction between two molecules for a given potential function, and

k is Boltzmann's constant. We most frequently refer to the temperature reduced with respect to the 12-6 (Lennard-Jones) potential,

Ti2_6 ') this choice is a matter of convenience because values of

(e/k)i2-6 are kno'wn for most coramon substances, at least to a first

app roximationt

.

b) We will use the terms two- and three-parameter potential functions.

A two-parameter function is a function described by e/k and R, with

R a distance parameter. A three-parameter function is described by

e/k and R together with a third parameter which designates the

family of which the function is a member. The functions are completely

specified when values are assigned to the parameters.

Keeping in mind the above definitions, the conclusions that

result from this work are as follows:

1. It is difficult to distinguish between one reasonable intermolecular potential function and another in the reduced temperature range

of about 2. < Ti3_6 < 5. 0.

t There is nothing magic about the choice of the 12-6 function here.

At this point in our discussion we are using the parameters for the

12-6 function merely as a convenient means for reducing certain of

the experimental quantities.
2. If a potential function in one three-parameter family correlates

data in a particular manner, it appears that a potential of another such

family can al-ways be chosen to correlate the data in a sinailar manner.

This is especially true for the temperature range 1. 5 < Tjs-e ^ 10.0.

This conclusion obviously includes the result of 1 (above).

3. It seems that it is impossible to find a three -parameter function

that will satisfactorily correlate experimental data over the temperature range of approximately 1.0 < Ti2_6 < 20.0. This is a very wide

range; for argon, for example, it is 125°K < T < 2500°K.

4. More information on the behavior of a potential function results

from a study of experimental transport data (e. g. , viscosity coefficients) than results from a study of experimental equilibrium data

(e.g., second virial coefficients).

5. Calculations have indicated that the possibility of distinguishing

between one function and another, which is already negligible between

2. < Ti2_6 < 5.0, is further reduced over a wider temperature range

if the data have a certain random error. For viscosity data, this

error is about 0. 5%. For second virial data the error is a constant

which is about 0.03 (bo)i2-6 [where (bo)i2-6 = §''^NRf2-6» N = Avogadros number, and R12-6 = the distance parameter for the 12-6

function].

We work with the properties: viscosity, diffusion, second

virial, and Joule -Thomson effect, and with the functions: the m-6,

the Kihara, the exp:6, and the Morse. We limit ourselves to these

properties and intermolecular potential functions only because they

are the most frequently discussed in the literature. In principle, the

procedure described can be applied to any property and fxonction.
Our discussion v/ill necessarily contain nnaterial covered in an

earlier publication [ij^ (hereafter referred to as I). The present work

contains a more complete and slightly more accurate discussion than

foxmd there, although the basic conclusions previously reached remain

unchanged.

2. THE INTERMOLECULAR POTENTIAL FUNCTION

The potential functions available at this time can be divided

into families, each with a family parameter. The discussion will be

res-tricted to the four most commonly used families of functions: the

m-6, the Kihara, the exp:6, and the Morse. These have all been

described in the literature so it will suffice to define these fixnctions

without explanation.

If U(r) is the interaction potential of two molecules separated

by a distance r and if e is the maximum energy of attraction (i. e. , the energy minimum), one writes the fiinctions as follows:

The m-6 [2, 3] :

(IT' - (~rT-'

where o is that value of r for which U(r) = 0. The family parameter

m denotes the repulsive exponent of the potential.

The Kihara [4, 5] : U(r) = w , r ^ a

. T/^ cr - a V2 ^ a - a ^^

, r >a

Here the finite size of the molecule is taken into accoiint by means of

a hard core diameter a; a is again that value of r for which

+ Nvunbers in brackets refer to references.
U(r) = 0, It is convenient to define a reduced core diameter by

Y

= a/a, -which is the family parameter.

The exp:6 [6] :

m

Here r is the value of r at the energy minimum and Q^ is the m

family parameter which represents the steepness of the repulsive

part of the potential.

The Morse [7, 8] :

U(r) = e |exp [ -^(r - r^)] -2 exp [

- ^ ( r - r^)]} ,

where r is again the value of r at the minimum and the family m

parameter c is related to the curvature of U(r) at r = r . na

3. THE STATISTICAL MECHANICAL EXPRESSIONS FOR

THE PROPERTIES

Statistical mechanics and kinetic theory provide microscopic

descriptions of equilibrium and nonequilibrium thermodynamics. As

such, their use generally results in expressions relating macroscopic

experimental properties to theoretical microscopic intermolecular

potentials. These expressions are central to our work.

3.1 Equilibrium Properties

According to the Ui sell -Mayer density expansion of the equation of state, the experimental second virial coefficient of KamerlinghOnnes is related to the intermolecular potential [9] by

B(T) = b^ B*(T*) = - bJ [ exp ( -^i^) - 1 r*^ dr* ,
where

3

b ^ AjLit^ , t" = k T/e, U" = U/e, and r* = r/R.

o 3

Here R is the characteristic length associated with the potential U(r)

(i. e. , R = a or R = r , as the case may be), N is Avogadro's

number, e the depth of the potential well, and k is Boltzmann's

constant.

With the aid of this same theory, one can derive the expression for the Joule -Thorason coefficient at zero density, |Jj , o b

(o) where C is the zero pressure molar heat capacity and

p , . b'* = T*dB'''/dT". We shall actually work with the quantity \x C^°'.

3. 2 Transport Properties

The Chapman-En skog solution of the Boltzmann equation provides

one with a formally complete solution of the pioblem of describing the

transport properties of dilute gases. In particular, one obtains expressions which relate the transport coefficients appearing in the experimental phenomonological flow equations to the intermolecular potential function. The expressions for the properties of particular interest

to us are, to a first approximation [9, 10 ] ;

Viscosity (1]):

1. ,^7 r^ 266.93 (MT)^ -i -i

10 Tj = i — g cm sec R^Q^^'^^*(T*)
Thermal Conductivity (X):

10'' X =

Self -diffusion coefficient (D)

10^ D =

,7 , 1989.1 (T/Mf

1

,4 ^ 26.28 (T^/M)^

J cm sec deg

^2 -1 cm sec

p R^ Q^i'^)*(T*)

Here M is the molecular weight, R the characteristic distance

associated with the potential, T the absolute temperature, p the

(Z, s

)

'"' gas pressure, and Q ' (T''') the reduced collision integral at the

reduced temperature T''~ = kT/e. As we will discuss in the next section,

we only require the collision integral part of the above expressions.

Thus, the expressions for \ and T\ are equivalent.

The intermolecular potential function is contained in the

reduced collision integrals ' (T''^). This quantity is given by

q(^'^)*(t*)

(s + 1)! T

>;- ( S + 2 ) J o

,.2 , >:= >M2s+3) ^(^)>:<_>:.^

exp (- g' /T ) g ^Q {g^dg'" ,

S^"' where Q (g ) is a scattering cross section at a reduced energy

g''' defined by

Q^^^* (g^^) -

1 -i (1 + (-1) r (1 - cos x) b''' db

''

1 + I

The scattering angle X(g''', b") for reduced energy g''' and

impact parameter b''' is related to the intermolecular potential function

U''~(r"~) through

X{g''\b''') = IT -2b'"

m

1 -

J * / '!=2 dr /r

wz g"^ J

Here r" is the reduced distance between a pair of molecules at the m ^ point of closest approach.
4. METHOD

4. 1 Background

The method which we have used to determine the sensitivity of a particular experimental property with respect to changes in the potential function has been described in I for the second virial coefficient.

For the sake of completeness we shall repeat the discussion, only this

time using the viscosity coefficient as an example. As in I, the

discussion will be limited to two -parameter potential functions. This

results in no loss of generality since a three -parameter function can

be considered to generate a family of two -parameter functions, one

function being associated with each value of the third parameter.

At a given experimental temperature, T, one obtains an

experimental value of the viscosity, T|(T), for the dilute gas in question. For a given potential function, which we shall designate by the

subscript 1, this value of the viscosity is predicted by the relation

T](T) = co^st. T

R^ Qp'2)*(Tf)

which, in terms of the experimental temperature, is

^(T) = ^°^^^- ^*

R^ Qp'^^*(kT/ei)

given the proper values of the potential parameters R and e /k. In

fact, given the experimental value at the temperature T and having

chosen the potential function this constitutes an equation in the two

unknowns R ande,/k. The loci of pairs of points (R , e /k) which

satisfy this equation lie on a curve in a plane determined by R, e /k

as axes. This is illustrated in Fig. 1 for the case of argon at 123°K [ll]

Had we chosen to use a second potential function, designated by

subscript 2, this same experimental value would have been predicted by
^, , const. T^

"HT =— —I . (1)

Our method is based on the requirement that the same value be predicted by both potentials. Thus these two expressions must be equated.

After so doing one obtains

But

T = —; = —:

so that we have

Ts Si

Thus, the requirement that the single experimental point T| { T) be

predicted simultaneously by both potential functions also requires that

the equation

^ ( 2 , 2 ) ^_!l_ r^f ^

Q o

( 2 , 2 ) >:= f rr^i^ \ S)V Ks

^

be satisfied. This relationship connects the two potential functions

through the collision integrals Cf^'^'''. Note that the experimental

T1(T) has disappeared. For each value of T', Eq (2) is a relationship between the parameter ratios s /e and R /R such that the

viscosity predicted using Q^ ' ' (T ') is exactly given by the viscosity

obtained using Q^ ^ ' ' (T'''). Clearly, having chosen a particular pair

of potential functions as 1 and 2, Eq (2) constitutes, for each value of

T'"*, a single equation in the two unknowns e /e , R /R^. A plot of

the loci of solutions to this equation will look like Fig. 1 with the axes

n

10
now being labeled e /e and (R /R ) . It should be noted that Eq (2)

can be written in terms of T' merely by substituting T = e^/e T''

Equation (2) has been derived here for the viscosity. It is easy

to verify that corresponding equations for the other properties are:

a) the second virial coefficient

(bo)i B? (Xf) =(bo)2 Bs (tJ) (2a)

b) the Joule-Thomson coefficient

(bo)i [Bf ' (T?) - B* {T{)] = (bo)2 [Bf ' (tJ) - bJ (t|)] (2b)

c) the self-diffusion coefficient

R? qP'^)* (Tr) = Ri 4'''^* (T?) . (2c)

Returning to Eq (2): This single equation in two unknowns

becomes two equations in two unknowns if one requires that, for some

property other than the viscosity, a single datum point also be predicted

simultaneously for the same pair of potentials. Suppose that we take as

this property the first derivative of the viscosity with respect to the

logarithm of the temperature. Now, in terms of unreduced collision

integrals, for potential 1 we have [9, lO]

It follows then that

T-^ =iT1 =

kT

kT

8 Qp'2) (T)

+ kT'

dT 8 q(2.2)(t) 8 [q(2.2) (T)f

dfi (2,2)

dT

din Q

(2,2) _, = T1(T)

|_ 1 +

It is easily shown that [l]

dJ^n Q^^*^^

_ _Q

dj^n T

~ Q

dj^nT

(2,2)

(2,2)

11
so that

(2,3)

dTi r ^

^ 3-1

In terms of reduced collision integrals this becomes

,(3.3)*(^*) 3 T-|^= T1(T) 4 ^i

^(2,2) (^yj

For potential 2, this is written

^ dT]

dT = T1(T)

^(2,3)>:.'' (tJ)

L Q,^^'2)- (t|)

]•

i]- (la)

Equating these two and remembering that we also require the equality

of the viscosity itself for the two potentials we see that we, in effect,

have to solve the equation

q(2,3)^ ^^y^ ^ p(2,2) ^^>:<j - p(2,2)>:< ^^|j •

p(2,3)>:= ^^|j

In terms of T''~ this becomes

Op'")- (Tf) " Qp'^)* (T? ei/e,) '

For each T and for a particular choice of potentials 1 and 2, this

is a single equation in the ratio e /e . This equation is solved as

previously described [l] . That is, one calculates for each function, a table of values of the function

?:= Q^^ '^)''' (T*)

The equation to be solved is then

Sj' (TH = S^ (T^) (3)

An equation (3) can be derived for each of the other properties

of interest. In each case, however, the function S"{T''') will have a different definition. These definitions are as follows:

12
a) the second virial coefficient

S"^ (T'~) = n^tT^^ ^T^ (3a) ., t" dB*

B'''(T"^) dT'

b) the Joule -Thomson coefficient

2 (T'")

.2d"B"-

dT-V^

s"'(T"~)= :^ (3b)

T--^-Bc) the self -diffusion coefficient

Q

( 1 , 2 ) >:^ S'MT-)=^^(TTTTir • (3c)

Our approach is equivalent to taking the properties calculated

for potential 1 as "experimental" data and attempting to fit them by

properties calculated for potential 2. In other words, we take our

'^experimental" system to be that system whose molecular interactions

are described by potential 1. Our entire discussion could have proceeded with no mention of experiment. In particular, the left hand side

of Eq (1) could contain the viscosity calculated for potential 1 instead

of that determined from experiment. The same description would apply

to Eq (la). Among other things, our method enables us to deal with

exact "experimental" data and to control the effect of error. In what

follows we shall use quotation marks to set off the words "experimental"

and "data" from their counterparts for a real laboratory system.

4. 2 Graphical Procedure

Our method can be illustrated graphically. This is most simply

done with the second virial coefficient and its first derivative as

described in I. We write Eq (2a) in terms of T''\

B? (Tf)=iM^ bJ {e,/e, T?) . (4)

(bo)i

13
Equation (4) is a single equation in the two ratios (bo )3 /(bo )i and

eg /si . For each value of T' , this is a relationship between the

parameter ratios Si /ss and (bo)i /(bo)s such that the second virial

coefficient predicted using B (T ') is exactly equal to the second virial

coefficient predicted using b''^(T!, ). Figure 2 contains the loci of

solutions to Eq (4) for several values of T''~ when the two potentials

are the 12-6 and the exp:6 for » = 12. That is, we denote function 1 as the m-6 with m = 12 and function 2 as the exp:6 with 0=12 and

write T' as Ti'a_s . For graphical purposes we define Psi -(b© )3 /(bo )i By equating the first derivatives of the second virial coefficient

one obtains the equation

^ ^-"-i^ ~ TT-r ^2 (Ti es/si) (5)

(bo)i

where B^'"^' - T'''dB''VdT''\ This is again a relationship connecting the

two parameter ratios. Figure 3 contains the loci of solutions to

Eq (5) for several reduced temperatures.

The two relationships hold simultaneously, at a given reduced

temperature, only for those parameter pairs which are simultaneously

on corresponding isotherms of Figs. 2 and 3 , i. e. , those parameter

pairs which are at the intersections of corresponding isotherms.

Figure 4 contains some of the curves of Figs. 2 and 3 with the intersection points indicated. Clearly, these intersection points define

the loci of solutions of Eq (3),

4. 3 Computer Procedure

There is obviously no point to solving these equations graphically

except for purpose of illustration as we have just done. It is far quicker

to give the entire task to the computer. Returning to the viscosity

example: In the general case, we solve Eq (3) by inverse interpolation.

14

1
That is, given T , one has a value S (T'') in the table associated

with function 1. The value of T' associated with this value is then

found in the S^(Tp table associated with function Z by interpolation.

The ratio e /e^ can then be calculated simply from

ei/es = tJ/tJ .

2 2 The corresponding value of the second unknown R /R can be calculated from the relation (2) corresponding to the same pair of values

T

' and T''~, e. g. , for the viscosity

rI _ Q^^'^^'' (T|)

r| ' Q^^'^)* (T?)

At a given T''\ one now has the pair of values, e, /s^ and (R /R-,) , such that Qi ' ^

' and its first derivative are exactly reproduced by

Q^3 ' ' and its first derivative for the second potential at the reduced

temperature T'''. It follows that the theoretical values of T1(T) and

TdTl/dT, predicted by potential 1 when the parameters s., R, are

used, will be exactly reproduced with potential 2 when the parameters

e^{= e^/{e^/e^)), R2(= R^ /(R^ /R^)) are used.

These equations can be solved as functions of T '\ Having done

2 2

this, one has the ratios e /e and R /R as functions of the reduced

temperature T''\ With these ratios one can, for each T '\ determine

,., 2

the temperature T''~ and scale factor (R^) such that both T] and dT]/dT

as calculated for potential 1 will be duplicated for potential 2. In the

particular trivial case that occurs when we choose to take potential 2

identical with potential 1, each of these ratios is a constant equal to

unity for all T''\ In particular, the ratios are then independent of Ty

One expects that, in the more general case where potential 2 is taken

different from potential 1, these ratios will not be independent of T'j\

15
Suppose, however, that, after solving Eq (3) as a function of T with

potential 2 different from potential 1, the ratios are found to be

constant over a range of values of T''\ This would mean that potentials

1 and 2 are equivalent over that range of temperature , as far as the

viscosity is concerned, since both T] and TdT|/dT as obtained from one

of the functions (e. g, , potential 1 with the parameters e , R ) are

exactly reproduced by a calculation using the second function (e. g. , potential 2 with a single pair of parameters e , R^).

In this work we are concerned with solving Eq (3) for the ratios

e /e , (R /R ) as functions of T' for various pairs of potentials.

For siraplicity we take potential 1 to be the 12-6 function. In this

sense we take as our "experimental" system that system whose molecular interactions are described by the 12-6 function and try to fit

that system's properties with those predicted for other potentials.

This enables us to deal with exact properties and to control the effect

of error. A number of conclusions will result from this "experimental"

system. We show later that our results do not depend on this particular

choice for potential 1. Any function could have been taken to be potential

1. We will show that our results are valid for a real system.

Collision integrals and reduced second virial coefficients were

taken from the references indicated in Table I.

Table I

Function Collision integral refs. Second virial refs.

m-6 [2] [12]

Kihara [4] [ 12]

exp:6 [6,9] [ 12]

Morse [ 12] [ 12]

16
4. 4 Effect of Experiraental Error

It should be noted that our discussions have been based on the

requirement that the property (T1(T) in our example) predicted by one

potential be exactly equal to that predicted by a second potential!". The

same has been true for the first derivative. So Eq (3) represents, in

principle, precise equivalence between the two potentials as far as the

property being discussed is concerned. We cannot expect precise

equivalence when working with real experimental data. But as pointed

out in I, results which indicate a lack of uniqueness between two

potentials are thereby strengthened. Thus, if two potentials yield the

same property in an exact fit, they will, a fortiori, continue to do so when experimental error is introduced and one accepts a more approximate fit as exact. On the other hand, any ability to distinguish between

two potentials which we may have found applies strictly only to the

conditions of the calculation, namely, the exact equality of the property

for the two potentials. It is quite possible that this ability to distinguish

between potentials might disappear entirely on the introduction of any

uncertainty in the property. Clearly it disappears when one introduces

complete uncertainty.

An exact fit is obviously experimentally unrealistic. Before

we can relate our results to experimental situations we must introduce

t Actually, the exactness is limited by the accuracy of the tables used.

These are estimated as better than one part per thousand for the transport properties and one part per ten thousand for the equilibrium properties. These accuracies can be called exact when compared with the

precision of the corresponding experimental data. We note, in this

context, that the correction factors have not been included in Eq (2).

A calculation has shown that the correction factors f-n and f^ and their

derivatives can be neglected as contributing less than one part per

thousand in the transport part of the calculations.

17
a model of the experimental uncertainty into our calculation.

We only need to describe how experimental error affects the

relationship between a potential function and a macroscopic property.

It is not necessary in the discussion to include the effect of error on

the second equality, i. e. , the equality of the first derivatives for the

two potentials.

Suppose, for a given temperature, we estimate that an uncertainty ±A exists in the experimental property. That is, the experimental value can lie anywhere in the range T1(T) ±A. The left-hand

side of Eq (1) can be replaced by any value in the range TKT) ±A. Any

2

such value will have associated with it a curve in the (e/k, R ) plane

describing the loci of solutions to Eq (1) for that experimental value.

Let us consider the extreme values. Figure 5 contains curves for

fixed extreme values for a single datum point (the same one used for

Fig. 1) for a particular experinnental system. Clearly, the curves for

values intermediate to T1(T) ±A must lie within the area defined by the

pair of curves of Fig, 5. The shaded area of Fig. 5, therefore, corresponds to those pairs of values (e/k, R ) which satisfy Eq (1)

within the confidence limits assigned to the datum T](T).

This same method for introducing uncertainty can be applied

when, instead of an experimental datum point, the left-hand side

contains the value of this property predicted by a second potential

function. In this case the experimental temperature T in Eq (2)

would be replaced by T . Figure 5 will still apply except that the

2 2 2

axes will be labeled €,/£;> and R /R rather than e/k and R . In

the discussion to follow we shall always be considering this latter

case (the equality for two potentials).

18
Suppose now that we were to add to Fig. 5, the pair of curves

associated with a second value of T' This would add a second shaded

area to Fig. 5 as indicated in Fig. 6. The common intersection of the

two shaded areas now represents the loci of pairs of ratios

e. /e^, R I'R.y such that the values of the property for one potential

can be replaced by those for the other simultaneously for the two values

of T' within the confidence limits assigned. The addition of curves

corresponding to other values of T''' leads to the introduction of still

more such areas. In general, every T' curve added reduces the

connmon area of intersection. This is illustrated in Fig. 6 for three

temperatures. Potentials are entirely equivalent for a given property

for given assigned confidence limits when there remains an area of

overlap, no matter how many values of T ' are added in this way.

2 2

That is, there will be e /e^,. R, /R-, pairs so that, within the confidence limits assigned, the property predicted for one potential can

be predicted for the other simultaneously for all of the values of T

considered. The absence of a net area of overlap means that the

property under discussion can be used to distinguish between the two

potentials, provided the data covers the same range of T ' shown

and provided the confidence limits for the data are the same as that

used here.

We have used the method just described to introduce the effect

of experimental error. This has enabled us to get a good idea of the

accuracy required of the experimental properties considered in order

to distinguish between potential functions. The results will be given

below.

19
5. RESULTS

We have obtained results which are conveniently divided into

two categories. First, there are our results for the exact fit to an

individual property and its first derivative. Second, we have results

for the fit to the individual properties within assigned confidence limits.

5. 1 Exact Fit to a Property and Its First Derivative

As mentioned earlier, we have always taken the 12-6 inter

molecular potential function as the reference function. In what follows

we shall always use the subscript 1 to refer to the 12-6 function,

the subscript 2 designating the potential which is compared with it. We are thus taking the system whose molecular interactions are

described by the 12-6 function as our "experimental" system.

Representative results from our method are depicted by

Figs, 7 - 10, Figures 7 and 8 contain plots of the ratio e /e versus

the reduced temperature T^'^.e for some members of the exp:6

family for the viscosity and second virial coefficients. Figures 9 and

10 contain the corresponding plots of the ratio Pis. where

F'ls - (t>o)i /(t)o)s for equilibrium properties and Pjg = Ri /R9 for

transport properties. Similar plots were done for the other properties

and other families. These plots are fully described in I and do not

have to be discussed at length here.

In examining these curves one should remember that a potential

function is equivalent to the reference function if the ratios e /e and

P^g are independent of Ti'g.g . Independence of Ti3_g means that

the corresponding graphs are parallel to the abscissa. Examination

of Figs, 7-10 (and the other curves which are not reproduced here)

showed most of the curves to be essentially flat in the temperature

20
range of about 2. < T^a-e "^ 5. O"'". It therefore follows that, for the

properties considered, all potentials are equivalent in that temperature range. Conversely, one cannot possibly choose any of the

potential functions if one has data only in this range*. Such data are

insensitive to the potential function.

Such data are, therefore, entirely of no use for the selection of

one potential function over another.

We can define "high" and "low" temperature ranges according

to this sensitivity. We define Ti'a_g as a low temperature if

Ti'g_e < 2. and Tig_s as a high teinperature if Tip_s > 5. 0.

According to Figs. 7 - 10, however, the second virial coefficient

provides no discrimination over the larger temperature range

2, < Tig_g < 10. 0. Therefore the definitions of high and low temperatures need to be altered as follows:

a) for transport properties we define Ti'p_g as a low temperature if T]3_s < 2. 0, and Tig.g as a high temperature if Tia_g > 5. 0.

t For the transport properties (Figs. 7 and 9), the o' = 12 curve shows

a rapid variation between T'"' = 4. and 5. 0, indicating that a value

of 4. might be preferred over 5. 0. Experience with applying our

results to real experimental systems (see section 6 below) showed

the range 2. < Tis_s < 5. to be the best rule of thumb definition,

however.

* While the temperature range has a precise definition here, it should

be taken as only approximate when applied to a real experimental

system. Experimental error and uncertainty in the value of (e/k)i2_g

might alter it to, say, 1. 5 < Tf^.g < 4. 5 or 3. < Tis_g< 6. 0.

Considerable uncertainty in (e/k)ia_g can be introduced when the

12-6 function is far from adequate. This results in a considerable

degree of arbitrariness in the choice of a best fit to the data.

21
b) for equilibrium properties we define Tig_g as a low temperature if Ti'a_g < 2. 0, and Tjg.g as a high temperature if Tj^.g > 10. 0.

We note that the temperature range is beyond the reach of

experiment for most substances while Ti3_g = 5. is not. For this

reason the transport properties are much more useful as probes of the

potential function.

This result, i. e. , the proof of the existence of temperature

ranges in which the properties are insensitive to the potential function,

has a very important corollary. That is, one can always extrapolate

properties over the entire insensitive region (i. e. , 2. < T^g^g < 5.

for transport properties and 2. < Tig_g < 10. for equilibrium

properties) based on potential functions determined by data in a

sensitive region. This corollary is most usefully applied to substances

for which (e/k)ig_g is quite large. Thus, for example, one can

safely extrapolate the second virial coefficient of xenon ((e/k)ig_g ^ 224 ) to temperatures up to 2240°K based on fits to data which can be at

temperatures entirely below 448°K.

5. 1. 1 General Results

Our results can be presented much more concisely by plotting

the e ratios against the Pjg ratios. These curves contain all the

necessary information. Figure 11 contains curves for the exp:6

family of functions for the simultaneous equality of the second virial

coefficient and its first derivative. Note that the curves are made up

from the intersection points of corresponding isotherms for a property

and its first derivative. These are intersection points similar to

those of Fig. 4. In fact, the a- = 12 curve of Fig. 11 is precisely

that curve which follows from Fig. 4. (Note that the abscissae and

ordinates of Fig. 11 are the reciprocals of Fig. 4).

22
In each curve of Figs. 11 - 26, the reduced temperature

Tis-6 appears as a parameter signifying distance along the curve.

In other words, each point on a curve corresponds to a different value

of Ti'5_g. If ? represents the distance along the curve, then d5/dT'''

represents the rate of change along the curve. Clearly, then, both

the e and Pig ratios are slowly varying functions of Tia_6 near a point where d§/dT''~ is small. On the other hand, the property being

examined is sensitive to the difference between the two potential

functions when d§/dT' is large. With the manner of presentation of

Fig. 11, one needs to look at only one curve to determine sensitivity

with respect to the potential function. The advantage of a one curve

presentation is more apparent in Fig. IZ which contains curves for

the Kihara function. In Fig. 12, the Pxa ratios are seen to depend

much more strongly on Ti'g.g than do the e ratios at the low temperature end of the curves while the opposite is true at the highest temperatures. Thus, the curves are mainly vertical at low temperatures and

mainly horizontal at high temperatures. Both ratios are insensitive to

the potential function in the intermediate temperature region. For

this reason, a large temperature range is crowded into a small distance

along the curve in this intermediate region, making d§/dT' small.

Figure 11 illustrates, in an interesting way, another of the

results discussed in I, namely, that a three -parameter function is

not sufficiently flexible to replace a two -parameter function like the

12-6 as far as the thermodynamic properties predicted by each is

concerned. Consider, first, the data at low temperatures. At such

temperatures, the curves have opposite slopes for 0=13 and c = 15.

Presumably, d§/dT'' approaches zero over the entire low temperature range for a value of a between these. The high temperature ends

of the curves, on the other hand, change very little in this same 0"

23
range. At the high temperature end of the curves, a similar change

takes place between » = 1 5 and 0" = 1 7 in which range the low temperature behavior is not much affected. Thus, while there exists a value

of a such that the second virial coefficient for the 12-6 potential

can be replaced by the exp:6 at low temperatures and there exists

another value of a which allows a similar replacement at high

temperatures, a single value of a does not exist for which this can

be done simultaneously at high and low temperatures. This makes the

requirement of the simultaneous fit of high and low temperature data

(high and low being quantitatively defined) a potentially sensitive probe

of the potential function. It should be noted (Fig. 11) that the high

temperature direction change occurs at T^t-s-e - 10.0. Thus, unfortunately, the high temperature part of this sensitivity for the second

virial coefficient occurs at temperatures which are beyond the laboratory range for many substances (i. e. , 2240°K for xenon and 1200°K

for argon).

Figures 13 and 14 contain results for the second virial coefficient and its first derivative for the Morse and m-6 families,

respectively. In each case, the behavior is similar to that observed

for the exp:6 family of functions. Thus, for the Morse family there

is a change in the direction of the curves at low temperature between

c = 4. and 5. 0. Presumably, there is a value of c in that interval

for which d§/dT'' ^ 0, for the low temperature range. The high

temperature part of the curve changes very little in that region. A

similar change of direction occurs at high temperatures between

c = 6, and 8. in which interval the low temperature data change

very little.

Figure 14 is particularly instructive. Here an exact fit occurs

for m = 12. In this case, as expected, the curves change direction

24
as m goes through 12 at both high and low temperatures simultaneously. The sensitivity to the repulsive exponent, m, seen at low

temperatures in Fig. 14 seems to contradict the argument which states

that only the long range part of the potential is important at low

temperatures.

Figures 15-18 contain similar curves for the viscosity. Results

obtained for these properties are essentially the same as obtained for

the second virial coefficient except that the temperature scale is somewhat different. Here again, the low temperature end of the curve

changes direction between an = IZ and Of = 15 where the high temperature

end of the curve remains unchanged. The high temperature end, on the

other hand, changes direction between a = 1 5 and a = 17. In this

latter range the low temperature end of the curve does not change

character. This high temperature sensitivity occurs at a reduced

temperature T^^.g = 5. as can be seen in Fig. 15 where the direction

change occurs at Ti'g.g = 5.0. This is an accessible experimental

temperature for most substances. The application of this result to

viscosity data for argon will be discussed below.

We have also examined the Joule -Thomson and diffusion coefficients. Results for the Joule-Thomson coefficient are essentially the

same as those for the second virial coefficient while those for the

diffusion coefficient repeat our results for viscosity. The Joule -Thomson

coefficient results are presented in Figs. 19 - 22 while the results for

the diffusion coefficients are shown in Figs, 23 - 26. It should be noted

that these properties (the Joule-Thomson and diffusion coefficients)

are slightly more sensitive than their counterparts (second virial and

viscosity coefficients).

25
The figures contain important additional information. We can

learn from them that these three parameter functions are equivalent.

All figures show the same reduction in sensitivity as the third potential

parameter approaches the value associated with the best representation

of the 12-6 function. Hence, this best representation for one familyis equivalent to that for another family. This is actually a general

result since the choice of the 12-6 function for the reference function

was entirely arbitrary. The choice of another reference function would

have led to a best fit for some other member of each family and so to

another set of equivalent functions. Proceeding in this manner, from

one reference function to another, one could "map out" the equivalence

of entire families.

This result can also be illustrated in a different manner. One

need not concentrate on the best fits to the reference function. Since,

for a given property, all of the figures have essentially the same

character, one can find a curve for each family all of which fit the

"data" in essentially the same way. This is quite apart fronn the best

fit mentioned above. The potentials corresponding to these curves are

equivalent. We can illustrate this for the second virial coefficient

(Figs. 11 - 14). The curves for the 15-6, y = 0. 1, and c = 8. are

very similar when superimposed. We have since found these to behave

equivalently when fitting real data.

5. 2 Fit to a Given Property Within

Assigned Confidence Limits

Our second group of results have to do with the fits of individual

properties within assigned confidence limits.

26
5. 2. 1 Equilibrium properties

We will work with the second virial coefficient and with the

Ck" = 12 member of the exp:6 family representing potential 2.

Since our purpose in this section is to emulate the effect of

experimental error, the method of assigning the confidence limits

must have some relation to the usual error associated with experimental

data. Experience indicates that the uncertainty in the second virial

coefficient should be essentially independent of temperature. Therefore, we have replaced the second virial coefficient for the 12-6

function (our "experimental" system) by Bjg.g + A, where A is a constant equal to a percentage of the B{'3_s value for T' = 2. 0.

This was done for A = 0, 1, 5, and 10% of the T* = 2. value.

When A = 0, as illustrated by Fig. 2, there is only a single

curve associated with each reduced temperature. Two potentials are

equivalent in this case if all curves have a single common intersection point. Now, according to Fig. 2, there is no common inter

section point for the temperature range 1. < Ti3_6 < 20. 0. The

curves associated with the isotherms T" = 1.0 and 20. do not intersect near the area where the others intersect. This illustrates the

sensitivity of precise data to the potential function at high and low

temperatures. Let us now examine the accuracy with which data must

be taken in order to preserve this sensitivity. Figure 27 contains

results for the case A = 10% of the Bia_s value at Ti'a_g = 2.0.

There being a net area of overlap, it is clear that the sensitivity

available with A = cannot be depended on, given an experimental

uncertainty which is this large. Figure 28, on the other hand, contains

results for A = ± 1% of the Bis-e value at Ti3_6 = 2.0. There being

no net area of overlap, it is clear that one can use second virial coefficient data with this amount of experimental uncertainty to distinguish

27
between potential functions. Figure 29 contains the data for A = ± 5%

of the B{3_6 value at Ti's.s = 2. Oo Since there is just barely a net

overlap, it is clear that this case represents an upper limit on the

error allowed.

Naturally, the precision required increases as one approaches

the best fit with respect to Of. Thus, the 5% obtained here represents

an upper limit only when the potential function being examined is as far

off from the best choice as is the Of = IZ from that value of 01" which

gives the best fit to the 12-6. Clearly, the error allowed decreases for

a function which is closer to the real function.

These error limits can very easily be converted into experimental error for a given system as soon as the value of bg for that

system is known. Thus,

so that if

it follows that

B(T) = bo B*(T^'') ,

b''''(T''') - Bo (T''^) + A ,

B(T) = boBj(T*) +boA . Now, if A = aBi3_s(T*) for Ti'^.g =2.0 then, since Bj^.g = - .62763

for T" = 2. [ 12] , it follows that A = - . 62763 abo . For argon, bo

is approximately 50 cm^/mole for the 12-6 function so that

A = -(31. 4a) cm^/mole. If we take 4% of the Tis_s =2.0 value as

the appropriate upper limit of the uncertainty, it follows that one can

have an experimental uncertainty of approximately 1, 4 cm^/mole

and still distinguish between potential functions, to the extent that the

cy = 12 function is not an appropriate representation of the 12-6

28
function as far as the second virial is concerned. The statistical

uncertainty in the argon measurements of Levelt, for example [ 1 3] , are well within these limits.

We have also found the limiting experimental error allowed for

the zero density Joule -Thomson coefficient to determine the potential

function. We again worked with the exp:6, cy = 12 function. Our

experience indicates that, when the Joule-Thomson coefficient is

obtained from experimental P-V-T data, its uncertainty is independent

of temperature and is given approximately by five times the uncertainty

in the second virial coefficient. Our experience with more direct

measurement of the Joule-Thomson coefficient is somewhat limited

mainly as a result of the paucity of such data. In this latter case we

have used a percentage of the value as a model of the error.

Error calculations based on the P-V-T experimients show that the

added uncertainty (i. e. , the factor 5) over that for the second virial

coefficient more than overcomes the extra sensitivity of the Joule

Thomson coefficients as a probe of the potential function. Thus, the

increase in sensitivity apparent when Figs. 19 - 22 are compared

with Figs. 11 - 14 is not attainable in an actual experimental situation

when the Joule-Thomson coefficients are obtained from P-V-T data.

For the direct measurements we found that an error of 1% was the

limiting error required to mask the sensitivity.

5. 2. 2 Transport properties

There are two sensitive regions to consider for the transport

properties as opposed to effectively one region for the equilibrium

properties. Therefore three kinds of error limits need to be described

here. These are: a) the experimental uncertainty needed to mask

the sensitivity of the property at the low temperatures (i. e. , Tjs-s '^ 2. 0),

29
b) that needed to mask it at high temperatures (i. e. , T'' > 5. 0), and

c) that needed for removing the sensitivity simultaneously at both high

and low temperatures. We shall first describe these for the viscosity

and as before, shall use the viscosity calculated for the 12-6 function

as our "experimental" system. How close these data fit to that

calculated for the (7=12 member of the exp:6 will again serve as

the criterion for determining the limits of experimental uncertainty.

We found that an uncertainty of 0. 5% in the low temperature

data will mask the sensitivity at low temperatures. In the temperature range 5. < Ti'g.g < 20. 0, on the other hand, an uncertainty of

approximately 3% was required. In order to lose complete sensitivity

over the entire range 1, < T^'g.g < 20. it was necessary to introduce

an uncertainty of 5%*. It is thus clear that, given viscosity data

sinaultaneously at both high (Ti^.g > 5. 0) and low {Ti-s.g < 2. 0)

temperatures, high and low being quantitatively defined, one can learn

something about the intermolecular potential function even when such

data are highly inaccurate. This is a powerful result. Generally

speaking, the same experiment will not cover both of these ranges

because of the different temperature control techniques required. One

is, therefore, almost always required to use data from different

laboratories.

t The juxtaposition of the values 0, 5% and 3% can be misleading. It will be recalled (see Fig. 23 for diffusion) that the high temperature

range is best fit by a value of a equal approximately to 16 while the

low temperature range is best fit by a- equal approximately to 13. 5. The error limits of the fit to the function en = IZ therefore do not have

the same meaning at the two ends of the temperature scale considered

separately . The 5% limit placed on the simultaneous fit to both ends of

the temperature range retains its full meaning, however.

30
Our results show that such data can be highly inconsistent and

still yield a moderate amount of information on the potential function.

As already stated, this sensitivity in the simultaneous fit of

high and low temperature data is not available for the second virial

coefficient because the high temperature region (Tjg.g > 10. 0) is not

amenable to experiment for most systems.

6. EXPERIMENTAL VERIFICATION

As stated earlier, our approach has been, in effect, to take an

idealized system with a known intermolecular potential (i. e. , the 12-6)

as our "experimental" system and fit its properties to those calculated

for other potential functions. It might be argued that the behavior

which we have found should not be typical of a real experimental system

simply because the 12-6 function is not a suitable function for real

systems. We have, therefore, investigated some real experimental

systems, looking specifically for the behavior which we found in our

idealized "experimental" system,

6. 1 Equivalence of Potential Functions

We examined the equivalence of three -parameter families of

functions by fitting the experimental second virial coefficient data for

argon to five families of potential functions. These fits are presented

in Table II. The method was to determine, for each potential, the pair

of parameters e/k, b^ , such that the mean square deviation of

experiment from theory was a minimum. This was done to four figures.

Such precision is reasonable for our purposes since we are interested

in obtaining the same degree of fit to all potential functions. The

resulting parameters should not, however, be taken as the precise

parameters for argon since other pairs near them would also fit the

data within experimental precision.

31
TABLE II

Best fits, including mean least square derivations (A), for the second

virial of Argon. Data frora Ref. 13.

Family

m - 6 m =

Kihara Y

exp : 6 Q"

12-n n =

Morse c =

Potential e/k bo A

9 84. 16 66.36 .392

12 115.06 54.76 , 223

15 140. 10 47.76 . 110

16 147. 50 45.96 . 103

17 154. 53 •44.34 . 118

18 161.22 42.87 . 145

21 179. 57 39. 19 . 252

24 195.79 36.28 .363

27 210.25 33. 93 .471

30 223. 14 32.04 . 576

40 254.82 28.35 .895

. 1 140.76 47.34 . 102

.15 154.72 43. 74 . 142

.2 169.67 40. 28 .245

.25 185.82 36. 78 .377

.3 203.31 33.35 . 539

.4 238.05 27.93 .985

.5 262.36 25.82 1. 589

.6 279.96 25. 11 2.252

12 99.90 61. 15 .271

13 110. 71 56.99 .220

14 120.44 53. 73 . 173

15 129.35 51.05 . 134

17 145.33 46. 78 . 102

19 159.46 43.47 . 144

4 36.84 90.03 .723

5 76.71 68. 83 .499

6 115.06 54.76 .223

7 153. 58 43. 50 .166

8 195.16 33. 75 .539

4 91.77 64.97 . 207

5 152.03 42. 53 .270

6 269.73 27.34 1. 125

7 300.04 24. 64 2. 554

8 349.24 24.00 3.996

10 504.80 23. 19 6.421

32
Figure 30 dramatically illustrates the equivalence of the

various families of potential functions used. In this diagram, we have

taken e/k for the abscissa and b^ for the ordinate, each best fit

then corresponding to a point on the graph. Figure 30 contains our

results for five families of functions. It is seen immediately that all

points fall on essentially the same curve . Given a point on this curve,

one can, for each family of functions, find a member of that family

which corresponds to it. All families, therefore, appear to be

equivalent as far as the second virial coefficient is concerned. It should be noted that the 12-n family has been added to the four

families previously described. This was done in order to study the

sensitivity of the experimental data to the exponent of the dispersion

term in the potential function.

In order to make certain that these produce the same fits, we

have plotted the root mean square deviation of theory from experiment

versus the parameter e/k for each member of each family. This is

presented in Fig. 31. The data for the m-6, Kihara, 12-n, and exp:6

families all fall on essentially the same curve. While deviations do

occur for large values of e/k these are less striking than are the

similarities between the curves. The deviations of the Morse function

data from the rest are not understood at this time, particularly in

light of the behavior of the Morse function data in Fig. 30 and in the

light of the behavior of the 12-n family of functions. In both Fig. 30

and 31, the 12-n family shows no peculiar behavior with respect to

those functions having a 1/r^ attraction term. It would seem, then,

that the absence of such a term in the Morse function is not responsible

for the deviations in Fig. 31.

t We have also verified that a corresponding curve exists for viscosity

data.

33
We can see the equivalence from Figs. 12 and 14 for example.

From these figures we predicted that, for the second virial coefficient,

the 15-6 function should be essentially equivalent to the y = 0. 1 function of the Kihara family. We then calculated, for these two functions, the second virial coefficients and the deviations of these calculated coefficients from the experimental argon data. The results are

contained in Fig. 32 where the equivalence of the two potentials with

respect to the second virial coefficient is strikingly clear. We have

also included deviation plots for the 12-6 and 18-6 functions in

Fig. 32 in order to establish the sensitivity of the deviation plot to the

potential function. It is clear that a small change in the potential

function results in a rather large change in the deviation plot so that

the equivalence of the 15-6 and y = 0. 1 functions, as demonstrated

in Fig. 32, is a very real thing. This equivalence is, in fact, much

stronger than that demonstrated by the essential superposition in

Fig. 30 of the points associated with these functions. Fig. 30 is based

on comparisons of the sums of the deviations of theory from experiment. The requirement of equivalence based on suras of deviations

is much less stringent than that based on detailed deviations such as

shown in Fig. 32.

The same approach was used to construct Fig. 33 for the

viscosity coefficient of argon. In this case. Figs. 16 and 18 were

examined and the 15-6 predicted to be equivalent to the Y = 0. 1. Figure 33 contains the corresponding deviation plots. Here, again,

the equivalence is striking.

This experimental verification of the equivalence of the threeparameter functions is very important. It experimentally verifies two

results:

1. That we may choose any of the functions as a reference; there is

nothing special about the 12-6.

34
2. It verifies our conclusions that if a member of one three -parameter

family fits data in a certain manner, then a member of another

three -parameter family can be chosen to fit the data in the same

manner.

6. 2 The sensitivity of the simultaneous fit of the

high and low temperature viscosity data to the potential function.

We examine the sensitivity of the fit of high and low tennperature

viscosity data to the potential function by fitting experimental data for

argon and nitrogen. Sufficient data exist at both extremes of temperature for both substances. We used the very sensitive method of plotting e/k versus T at constant a as our method of determining the

best fit [ 22] *. Parameters thus obtained were then used to calculate

theoretical viscosities and the deviations of these from experiment

determined. These deviations are plotted in Figs. 34 and 35. The

m-6 family was used exclusively.

Results for argon are contained in Fig. 34. These were

obtained as follows. The argon viscosity data of the several workers

were combined. These were then divided into three groups according

t The method of plotting e/k versus T at constant a was rendered

somewhat more sensitive as a result of our understanding of Fig. 13.

Note that the direction of the change of e/k with T* changes in going

fronn 0"= 12 and a = 1 5 with the best fit lying between these values.

A similar direction change in the fit of the low temperature experimental

data for argon was found in the e/k versus T''^ at constant a plots

for m = 18 and 21 indicating a best fit for m approximately equal to

20. Similar behavior was seen in the low temperature fit of nitrogen.

This method for the determination of the potential thus becomes a most

sensitive one. Similar behavior was not found in the high temperature

fits to the published data although, according to Fig. 13, it should have

been present.

Note added while in press: It is encouraging that such behavior is

found at high tenaperatures if we use some new unpublished data. See

note after acknowledgement, page 42.

35
to temperature. All low temperature data (Ti'g.s < 2. i. e., T < 240° K)

were placed in one group, all intermediate temperature data

(2. < Tis_s < 5. 0, i. e., 240°K < T < 600°K) in a second group, and

the high temperature data (Tia.g > 5. 0, i. e., T > 600°K) placed in a

third group. Plots of e/k versus T at constant a were done for

each of the m-6 potential functions for both the high and low temperature data, the middle temperature data being ignored completely. The

choice of a best value of m turned out to be a relatively simple matter

at both high and low temperatures. The parameters obtained are

listed in Table III. These parameters, along with the appropriate

collision integrals, were used to calculate a theoretical viscosity at

each experimental data point.

According to Fig. 34a, the low temperature parameters give an

excellent representation of the data at both low and intermediate temperatures even though the data at the interraediate temperatures were not

included in the parameter determination. The fit is, in fact, good even

to a reduced temperature Ti3_s = 10. 0, indicating that our definition

for the beginning of the high temperature range might need to be

revised upward for a real system. The theoretical viscosities calculated at high temperatures using these low temperature parameters

deviate from experiment. These deviations are outside the range of

estimated experimental accuracy and show a tendency to deviate further

from experiment as the temperature increases.

Figure 34b is based on the use of parameters obtained from

the fit of the high temperature data only. There is an excellent fit

over both the high and intermediate teinperature ranges even though

the latter data were not included in the determination of the parameters .

In this case, the fit at low temperatures is very poor and again is

36
TABLE III

Best values of m, e/k and a which fit high and low

temperature viscosity data.

Gas m e/k °K

Low Temperature

a A

Ar 21 170.5 3.26

N. 18 113.2 3.56

Ar

High Temperature

40

50

224. 1 189.6

3. 15

3.34

37
outside the range of estimated accuracy. That these two fits are

irreconcilable is apparent in Table III. Radically different values of

m were required in each case. From Fig. 35 we see that we get essentially the same results

for nitrogen. The same remarks apply to nitrogen as to argon.

6. 3 The Insensitivity of Intermediate Range

Data to the Potential Function

According to Figs. 34 and 35, the intermediate range data are

fitted quite well using either of the radically different high and low

temperature potentials. Clearly, data in this middle range of

temperatures cannot contribute any sensitivity in a determination of

the potential function. Since they do contribute the experimental

uncertainty associated with their measurement, these data can only

serve to mask the choice of a potential. Obviously, such data are less

than useless in this respect and should be ignored in the initial stages

of a determination of the intermolecular potential function as we have

done.

This insensitivity has been present, implicitly, in previous

work, for both transport and equilibrium properties. For example,

it was quite obvious in previous correlations of argon [ 34] and oxygen

and nitrogen [ 35] . We reproduce Fig. 2 of reference 34 as Fig. 36

here.

Figures 34 and 35 also very clearly demonstrate the corollary

which we attached to the existence of insensitive temperature ranges.

Thus, we stated that it is always safe to extrapolate a potential function from a sensitive temperature range over the entire insensitive

range. The potential functions of Figs, 34a and 34b, each, more

than adequately describe the data in the intermediate range even

38
though the intermediate range data were not included in the determination of the parameters. This extrapolation is good in spite of the fact

that the two potentials are radically different.

7. CONCLUSIONS

We have made a study of the information on the potential

function which one obtains from fitting theoretically calculated viscosity,

diffusion, second virial, and Joule-Thomson coefficient data to experimental data. In order to avoid the masking effect of experimental

error, we took an ideal "experimental" system whose interaction

potential is known and drew our conclusions from it. This also

enabled us to study the effect of experimental error in a carefully

controlled fashion. Conclusions resulting from this study of our

idealized system were then specifically sought and found in the

behavior of actual experimental systems. We have thus substantiated

the points made in the introduction.

We have found that it is impossible to distinguish between one

reasonable intermolecular potential function and another if one uses

experimental viscosity data in the reduced temperature range of about

2, < Ti3_e < 5. 0, the upper limit being increased to 10. for the

second virial coefficient. It follows, therefore, that potential functions

chosen as a result of the correlation of data entirely in that temperature range should be used with extreme caution in applications other

than the data correlation itself*. Furthermore, where data exist

both inside and outside the temperature range 2. < T'{^^s < 5.

t Unfortunately the literature is full of such applications. One

example is the use of the 12-6 potential and parameters for argon

based on the correlation of Michels 1948 data [ 36] .

39
(the latter being replaced by 10. for the second virial coefficient),

the data inside this range should be removed and not used until the

final correlation. Such data, as we have stated, do not provide information on the potential function but do add to the overall statistical

uncertainty, thus spuriously reducing the difference in standard

deviation between the best fit and other fits. By reversing these

arguments we concluded that it is perfectly safe to cover the insensitive temperature range by means of an extrapolation based on a

potential function determined from data outside such a range.

We have also seen that all three-parameter families are

essentially equivalent in fitting experimental data for the quantities

studied here. This means that the introduction of a new three

parameter function will rarely (if at all) result in an improvement

in the fit of such data. Such fitting to additional functions is then

unnecessary once a fit has been made to a reasonable three-parameter

function.

The simultaneous fit of high (Tjg.g > 5. 0) and low (Tip_g < 2. 0)

temperature viscosity data has been found to be potentially a sensitive

probe of the potential function. A corollary of these results is that

potentials determined for systems for which only low or (in rare

cases) only high temperature data exist should be used with caution

in applications other than the correlation itself since such functions,

in general, would not even fit data at the other end of the temperature

scale if such data existed.

The main part of our effort has gone into the study of "exact"

fits of data, i. e. , we have dealt with precise data from an ideal

"system". Our results might easily be masked by error in the

application to real systems. By studying the effect of controlled

40
amounts of error, we have found, however, that present day experimental errors will not mask these results. In particular it was found

that the sensitivity at low temperatures will remain, provided viscosity

data have a precision of better than 0. 5% and second virial data of

better than 4% of the value of the second virial coefficient at

Tig_s =2.0 (this latter is equivalent to 0. 03 (bo)ia_s ^^^ each systenn).

The sensitivity of viscosity data to the potential function in the simultaneous fit of high and low temperatures was found to remain even

when errors of 3% were introduced.

We have shown our idealized experimental system to emulate

real experimental systems sufficiently so that the conclusions drawn

from its study can be expected to be found in experimental systems.

Further use of our method is, therefore, clearly indicated. In

progress are studies of the thermal diffusion factor and the simultaneous fit of pairs of the properties studied here.

We have also shown that systems for which high and low temperature viscosity data exist should be studied with an aim to finding a four- (or more) parameter potential function which fits such data. In

particular, our results would appear to indicate that the inability

to properly fit such data in the past is not only due to systematic errors

at high temperature as has been suggested by some workers.

8. ACKNOWLEDGEMENT

We acknowledge the grants supporting this work: (For

H. J. M. H, ) The National Aeronautics and Space Administration,

Contract Number R-06-006 -046, and (for M. K. ) the Air Force

Systems Command, Arnold Engineering Development Center,

Tullahoma, Tenn. , Delivery Order Number (40-600) 66-22 Program

Element 61445014, AF Project 8951.

41
We are also grateful to G. E. Childs for his assistance with

the computing, and to R. D. Weekley and Mrs. M. Birchfield for

their help with the preparation of the manuscript.

NOTE

While this manuscript was in preparation for publication, we

reassessed the high temperature viscosity data. We now feel that

the data largely represented by the work of Trautz and Vasilesco

are most probably erroneous. Recent measurements for argon [ 37]

for example, lead to a value of m = 13 for the high temperature

correlation [ 38] . This value is more compatible with independent

scattering experiments than m = 40.

We emphasize, however, that all conclusions noted here are

upheld with the new data.

42
9. REFERENCES

1. M. Klein, Determination of Intermolecular Potential Functions from.

Macroscopic Measurements, J. Res, Natl. Bur, Std. 70A, No, 3,

259-69 (1966),

~

2. M, Klein and F, J, Smith, to be published in the J. Natl. Bur. Std.

(1968).

3. J. H. Dymond, M. Rigby and E. B. Smith, Two-Parameter Intermolecular Potential Energy Functions for Simple Molecules, Phys.

Fluids 9_, 1222-9 (1966).

4. J. P. O'Connell and J. M. Prausnitz, Applications of the Kihara

Potential to Thermodynamic and Transport Properties of Gases,

ADVANCES IN THERMOPHYSICAL PROPERTIES AT EXTREME

TEMPERATURES AND PRESSURES, 19-31, Am. Soc, Mech,

Engr. , New York (1965).

5. J. A. Barker, W. Fock, and F. Smith, Calculation of Gas Transport Properties and the Interaction of Argon Atoms, Phys. Fluids

T, 897-903 (1964).

6. E. A. Mason and W. E. Rice, The Intermolecular Potentials for

Some Simple Non-polar Molecules, J. Chem. Phys. 22, 843-51

(1954).

7. S. E. Liovell and J. O. Hirschfelder. Tables of Collision Integrals

for Gases Obeying the Morse Potential, University of Wisconsin

Theoretical Chemistry Laboratory, WIS-AF-21 (June 1962).

8. D. D. Konowalow and S. Carra, Determination and Assessment of

Morse Potential Functions for Some Nonpolar Gases, Phys. Fluids

8, 1585-9 (1965).

9. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, MOLECULAR

THEORY OF GASES AND LIQUIDS, John Wiley & Sons, New York

(1964) Second Printing.

10. S. Chapman and T. G. Cowling, THE MATHEMATICAL THEORY

OF NON UNIFORM GASES, Cambridge (1964) 7th Printing.

43
11. H. L. Johnston and E. R. Grilly, Viscosities of Carbon Monoxide,

Helium, Neon, and Argon Between 80° and 300°K Coefficients of

Viscosity, J. Phys. Chem. 46, 948-63 (1942),

12. M. Klein, Unpublished data. We remark that the second virial

coefficients of Klein hardly differ from those from other sources.

13. A, Michels, J, M, Levelt and W. De Graaff, Compressibility

Isotherms of Argon at Temperatures Between - 25° C and - 155° C

and at Densities up to 640 Amagats, Physica 24 , 659 (1958).

14. A. L. Gosman, Thermodynamic Properties of Argon in the Liquid

and Gaseous State for Temperatures from the Triple Point to 300°K

and Pressures to 1000 Atmospheres, State Univ. of Iowa, Iowa

City, Ph. D. Thesis (1965).

15. A. Michels, A. Botzen, and W. Schuurman, The Viscosity of

Argon at Pressures up to 2000 Atmospheres, Physica 20, 1141-

48 (1954).

16. A. Van Itterbeek and O. Van Paemel, Measurements on the

Viscosity of Argon Gas at Room Temperature and Between 90° and

55°K, Physica 5, 1009-11(1938).

17. M. Trautz and R. Zink, 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 Higher Temperatures), Ann, Physik

]_, 427-52 (1930).

18. V. Vasilesco, Recherches Experimentales sur la Viscosite des

Gaz aux Temperatures Elevees, (Experimental Research on the

Viscosity of Gas at Elevated Temperatures), Ann. Phys. (Paris)

20_, 137-76 (1945).

19. G. P. Filippova and I. P. Ishkin, The Viscosity of Air, Nitrogen,

and Argon at Low Temperatures and at Pressures up to 150 Atmospheres, Inzh. Fiz. Zh. Akad. Nauk Belorussk SSR 4, 105-9(1961).

20. R. Wobser and F. Miiller, Die innere Reibung von Gasen und Dampfen und ihre Messung im Hoppler-Viskosimeter, (The Viscosity of

Gases and Vapors and their Measurenn.ent with the Hoppler Viscometer), Kolloid-Beih. 52, 165-276 (1941),

44
21. F. G. Keyes, The Heat Conductivity, Viscosity, Specific Heat

and Prandtl Numbers for Thirteen Gases, Mass. Inst, of Techol, , Cambridge, Proj, Squid, Tech. Rept. No. 37(1952), DDCATI167

173.

22. H. J. M. Hanley, Comparison of the Lennard-Jones, Exp: 6, and

Kihara Potential Functions from Viscosity Data of Dilute Argon,

J. Chem. Phys. 44, 4219-22 (1966).

23. C. F. Bonilla, S. J. Wang and H. Weiner, The Viscosity of Steam,

Heavy -water Vapor, and Argon at Atmospheric Pressure up to

High Temperatures, Trans. ASME 78, 1285-9(1956).

24. K - L. Yen, An Absolute Determination of the Coefficients of

Viscosity of Hydrogen, Nitrogen, and Oxygen, Phil. Mag. (6)

38, 582-96 (1919).

25. M. Trautz and P, B. Baumann, Die Reibung, Warmeleitung und

Diffusion in Gasmischungen. n. Die Reibung von Hg - Ng und

Hg - CO - Gemischen, (The Viscosity, Heat Conductivity, and

Diffusion in Gas Mixtures. II. The Viscosities of Hg - Ng and

Ha - CO Mixtures), Ann. Physik 2, 733-36 (1929).

26. M. Trautz and A. Melster, Die Reibung, Warmeleitung und

Diffusion in Gasmischungen, XI. Die Reibung von Hg , Ng , CO,

CgH4, Og , und ihren binaren Gemischen, (The Viscosity, Heat

Conduction, and Diffusion in Gas Mixtures. XI. The Viscosities

of Hg , Ng , CO, CgH4, Og , and their Binary Mixtures), Ann.

Physik 1, 409-26 (May 1930).

27. M. Trautz and R. Heberling, Die Reibung, Warmeleitung und

Diffusion in Gasmischungen. XVII. Die Reibung von NHg und

seinen Gemischen mit Hg , Ng , Og , CgH4, (The Viscosity, Thermal

Conductivity, and Diffusion of Gas Mixtures. XVII. Viscosity of NH3

and its Mixtures with Hg , Ng , Og , CgH4), Ann. Physik 10, 155-

77 (1931).

28. P. J. Rigden, Viscosity of Air, Oxygen, and Nitrogen, Phil. Mag.

(7) 25_, 961-81 (1938).

29. H. L. Johnston and K. E. McCloskey, Viscosity of Several Common

Gases between 90°K and Room Temperature, J. Phys. Chem. 44,

1038-58 (1940).

45
30. H. L. Johnston, W. R. Mattox, and R. W. Powers, Viscosities of

Air and Nitrogen at Low Pressures, Natl. Advisory Comm.

Aeronaut. Tech. Note No. 2546 (1951), NASA N62 54546.

31. C. F. Bonilla, R. D. Brooks, and P. L. Walker, Jr., The

Viscosity of Steam and Nitrogen at Atmospheric Pressure and High

Temperatures, Proc. General Discussion on Heat Transfer,

London, Sept. 1951, Inst. Mech. Engrs. , 167-73(1952).

32. L. Andrussow, Conductibilite Thermique, Viscosite et Diffusion

en Phase Gazeuse. Memoire 10. -Relation entre les Coefficients

de ces phenomenes et I'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. Chim. Phys. 52_, 295-306(1955).

33. F. Lazarre and B. Vodar, Measurement of the Viscosity of

Compressed Nitrogen up to 3000 Atmospheres, Proc. Conf.

Thermodynamic and Transport Properties Fluids, London,

July 1957, Inst. Mech. Engrs., 159-62(1958).

34. H. J. M. Hanley, The Viscosity and Thermal Conductivity Coefficients of Dilute Argon between 100 and 2000°K, Natl. Bur. Std.

Tech. Note No. 333 (March 1966).

35. G. E. Childs and H. J. M. Hanley, The Viscosity and Thermal

Conductivity Coefficients of Dilute Nitrogen and Oxygen, Natl.

Bur. Std. Tech. Note No. 350 (October 1966).

36. A. Michels, Hub. Wijker and HK. Wijker, Isotherms of Argon

Between 0°C and 150° C and Pressures up to 2900 Atmospheres,

Physica 1^, 627-33 (1949).

37. F. A. Guevara, B. B. Mclnteer and W. E. Wageman, Private

Commniunication: R. DiPippo, An Absolute Determination of the

Viscosity of Seven Gases to High Temperatures, Brown Univ. , Providence, Ph. D. Thesis (1966).

38. H. J. M. Hanley, Discrepancies Between Viscosity Data for

Simple Gases, to be published. An account of this work will

also be published in the Proc. of the 7th Conference on Thermal

Conductivity, Natl. Bur. Stds. (1967).

46
COOCO

CD

00

CM

<£»

IT) ^

rO o

CD

O

O

CO oo o

00 o

to o

rt

><<u

fl

rt

(U>

•iH

Clfl ^o

•I-l ^^

fl

-.-1

-!-> o

fls

<^-l f—1

(i •H

4->a

<u -Mo

0.1 (M

r—1 . N^

0)

O ^ CO

-l-> (\J u

f—1 o II <+-!

0)

H

u

<u +j

nJ

a

O

nJ W) ^ s-l c^ nJ 0^

"+-1 r—( o

n5

•i-l

•iH

(U CO

4-> oo

a. CD o >

•I-l 0) u j:: o

4J

r-l o

(D -(-> ^ -(-> H

•w

t4

47
CVJ cJ

<M

o

CO

cr>

CM CL

00

<D

IT)

CM CF> 00

VV

VU

esj

Fig. 2. The loci of potential parameter ratios which give a precise

conversion of several values of the second virial coefficient

from the 12-6 potential function to the exp:6 function with

Of = 12.

48
00

cvj

CD

CJ

CU

cvi

O

cJ

CD

CD

C\J

(Vl

(\J cr> 00 1^

vu

vy

CJ

Fig. 3. The loci of potential parameter ratios which give a precise

conversion of several values of the first derivative of the

second virial coefficient from the 12-6 potential function

to the exp:6 function with a = IZ.

49
o

CO

CD

CO

- a."

CD

LO

'^

in o ID O in o

O o CD CD 00 00

VU

\u

CM

Fig. 4. The loci of potential parameter ratios which give a precise

conversion from the 12-6 potential function to the exp:6

function with cv = 12. Both the second virial and its first

derivative are included. Each intersection point gives the

ratio for the simultaneous conversion of both properties.

The solid curve represents B', the broken curve B "'

.

The following Tig_s are included D , 1.0; A • 2. 0;

O 5.0; A 10.0; 20.0.

50
o

<T>

00

I^

M

cr

_ CD

IT)

ro

O

CD

O

O

C\J

OO o

CD o

CD

.2^

<u 5M•iH <*H ^o

•rH ^^^ao

•iH

-(-> • u <u C u d CI

<+H nJ •—t u

rt 4)

•iH 1—1 ^ O

0) T) o (U a< fl

bX) vO COrn

(VJ

<Tl •—{ C!

<u nJ :S •i-i ^ ^-i-> u •H

V-( ^

•« CO

t=£^

<1) o

-l-> ro

<U fNJ a

^H

«j II ;^oJ H

O.

1—1 -M

oi nJC c O

0) ao

+->

;-i ni

!+-( VH o

•i-l o H

CO

1—1

a;

O

Ji

H

H >

ID

51
1.5

1.4

1.3

1.2 -

2 I

1.0 -

.8

.7 1.0 I.I 1.2 1.4 1.5 1.6

21

Fig. 6. The loci of potential parameter ratios which, within assigned

tolerances, convert the second virial coefficients of the

12-6 potential function into those for the exp:6 function

with ot - 12 at three temperatures.

52
I

cJ

O

CO 00 CD CMc \

CO CD

CM o

ro

CO

CM

CD

CM

CM

CM

CMO

CM

CD

CD

H

CM

— 00 — CD

CMO

CM ^ <D o U

<+H

a

4J

COo o (;) +J 1 1 flj (U d u

fl CJ n r-i <u

(TJ r| ^ U -i-> u tJ (-> 1

(tJ c\J Xi Jl

+j +j

m >^ <u U ^ Mo CO o Z

rt •iH

(1) •(-» ^ s u o

!>^

m nn ^ •i-( [n

(D • •H

<+H 1 H a

(-> o

CVI o

0^ (C 1 CO

CO vO

4) o u (M :S

f\]*

01 1—(

11 >

<u ao . ^ i% t^J («

(-> ^H

<l>

u ~^ 60

m

.-1

4->

CO 1

CO fTl (Cj .—1

CJ 1—<

^1

CO

0)

T) cti <U

r! CJ X! •F-l 1—1

•—1 s.H -^^ 0) 4-> u ^

•t-t CO . iS O

CJ

CO

I-! o

•I-l

COO

rH f )

t) o

CO > u o 4->u

1)

9)

1—1 ro 4) u

0)

-id

53
1 1

1 ro

(U 1 cj oS

1 — 00

n

repla

t

calcul t

these

/ — CD

one

ca

by

tha'

ote

tha

/ ^ 43 ri ^

/

(\j

whic nctio 10.

/

/ (NJ O '+H

/

C\J

CO Ti "-I

/ o 32 S o

/

CO

_ / _ CD

<C ' 0)

/

1 ^ ^

^ a; 5 — / — CD co^l .

/ * CO n ° .

/

^ •^ 0) o

" ^ ni 1

(^ p " 7.- ^

/ CO -- _u

1 (u H

\

/ W 3 (U (U -^ > - / — o

. i2 H

CO -^ ^

a, t! !^ -I-'

/ ^\ T) <+H '13 CTJ ^ >4-l -^ ^J — / h- — 00 ,-1 0) y CO ^ n fi r! CM

II a II a

II

CD

o

of

the

we

nd

virial

c(

he

exp:6

fu

re

quite

coi

^

B

rati

seco for

t

ios

a: J \\

- CM M +j +j !-i

r^

00 ^ (NJ O 00 CD

•iH

CM

54
^ a o

0)

(1) •M

fl ^ u d

Xi o u nJ X! u ^

(->

<+H 43 u -K

CO l>^ C rD

(1) rt a O

•J-l >s u

rO t^ ^1 1—<

*I1! H el md o.

to

^H sO

<u>

f—1

•. a (U Ph

j:3 -M

^1

•» o UJ >+-l u

(I) 73

+-» (U

(1)

M

fi ^

frt

d

^1

u

1—1

1—I

iH

4)

OO

H o CO M

Oh •iH C^ > w O

•i-i +J vO

0) —1

<u

CVJ a> 00 CD in K) c\j

130

<M

55
CD 00 N- CD IT) *

flrto >N

(U Xt

fl fl o o

•rH ^ -»-> () o

•iH ^

ji j3 ^

M-l ^H

M-t

frt

O •rH -M m fl

PI (U M

-(->

(1)

u

(1 u,

^-^

0)

j:1

H

4J

CO

;:3 "H

to T3 u Q)

(I) H-> > ^d n o

rH 1—<

Ph rtt

to CO . ^ -M i-l 1) fl o

H-> a

^H

0)

•rHo

•rH

•rHo

<+H

<+H

rt o vO

Q. o

Oh

•—t 1—1

1) oJ tti •iH •rH

+->

H->

;h •rH>

0)

O TJ ^

Ph fl

n

OO

<+H

+J

en 0) M-l +j o ^d

rHu

H-1 1) 1—1 nJ t) (tJ ^^

(tt CJ r—1

Ph

(U

t-> I

•rH

CVJ

56
CO

lO

ro

OJ— CM - \u

2 yyen — 00 — f^ — CO

lO

C\J — OCT^COI^COID ro CO

CM

CL

Fig. 11. e ratios versus P^g ratios for the replacement of the

second virial coefficients calculated for the 12-6 potential function by that calculated for the exp:6 function.

Note that each point on a given curve refers to a different

temperature. The following values of Tig.g are

included D , 1. 0; A , 2. 0; O , 5. 0; A , 10.0;

, 15.0; • , 20. 0; , 50.0.

57
o

1^

CD 00 I^ CD in ro OJ — O

C\J

* — q

CVI

d

^. vu~

II X

00

1^

CD

IT)

CD

CM CL

Fig. 12, e ratios versus P^g ratios for the replacement of the

second virial coefficients calculated for the 12-6 potential function by that calculated for the Kihara function.

The key to temperatures along the curve is as in Fig. 11

Note that an exact fit reduces a curve to a point as

demonstrated by the point at (1.0, 1.0) for Y = 0.

58
CO O

CO

CD 00 h- (D in ro CJ — o C7) 00

ro

CJ

O

CD — CO _ h- — C£> — LO

ro — CO

o

CM w

vu

CM

Fig. 13. e ratios versus Pjg ratios for the replacement of the

second virial coefficients calculated for the 12-6 potential function by that calculated for the Morse function.

The key to temperatures along the curve is as in Fig. 11,

59
O CD 00 (^ (X)

cvi of

Fig. 14. e ratios versus P^g ratios for the replacement of the

second virial coefficients calculated for the 12-6 potential function by that calculated for the m-6 potential

function. The key to temperatures along the curve is as

in Fig. 11.

(\j

3

1

60
QOJ

CD 5 -

Op o u i

r;^. <U ^ fl

nJ -y r3 ft to S <u nJ ^ calc Exc e

is

>^ t ID

to o U u (U ^.

<U vX) fl K) j:! "

+^ ft •-; ^ X ^

<U (0

C\J 4J (U <U

5^ -c! JH

<u +3

d S !h rt CM <U O ^H

— VL>

U '+H (U \ repla

lated temp 9 w"

0^

u

O i) o ^ -^ ^ Oj 0) CO O

^"^ -^^

h-: ^ a

h o TM

TM ••-< " " !-> +J CD P^^ § 3 tJ CD <« 3 -^ ^

in

<D ^

'^ rai +-> nj rt ^ u

Jh 1 ""^ ro ',13

— CVJ

'd- ro cu

— CJ> CX) h- CD lO

in

.—I

•H

CM

61
1 1 1 *

/i

1

1// 1

\

1

o i\

^o V

6

II

IT) 6

II ^\

X

yz

^

o

II

1 1 1 1

(T> 00

.<M cl"

— o

— (T> — 00 — 1^

CD CM

LO vi;~

^

ro — CO

r^cDio^rooo — oo^

Fig. 16. e ratios versus P^g ratios for the replacement of the

viscosity calculated for the 12-6 potential function by

that calculated for the Kihara potential. The key to

temperatures along the curve is as in Fig. 11.

62
a>

!2— 00

0) ^+j IS ^4 —

o ^

<+H

O

+-> '^ .. tp (U >^

+J (1) 1—rt ^1 d a; ID u ^

-^5^ u

^.— •H

O

CD >-< o tJ ro

,, <u CN 2^ w M » ^ s —— y 4) ' S^ -S • d) +3 ^H s ^

--

O . '^ <U o ^ VL> U >^-i bO \ -^ ^ h

(J) Vl/ ^ 3 s

00 +" «J

M

;,

U .w

O 4-. to f^ "U IS m 5 > O u

•iJ >^

d

rt -^ "

CO ^ PI a; O A

2"-^ ^

in P^^ ^ d

to <4-l —1 d ^ «* ^. M Oj to ^ a u > (u d m +^ -^^ ro 5? O Oj .2 f^ fH i> <u «j vo 9^ cu ^ ' a f^j s « rH +J

J o

r- CD in ro C\J CD 00 1^

CM

63
o>

00. r—1

^^ >—1

-M , 1^ Uo

Ti

ept

wh

as

in ^ +->

ctJ I—<

13 ^. o1—1 O CO — rt W CO o (U ^. • iH o ^ O ••-» o O (U f^ m rt Xi ._ c* d

-^ > '^ bO

(U « fl CJ j3 ^ o 1 r-l

CO

H 4) <U : fl -iJ ^

(U +3

d

a

U (tJ

O CNJ (U

O ^^ - vu o <+H 0)

a. lated temp

f—351

CO

U (U o ^ ^

<+H nJ (u 1^

too

tp

u

.2 t cv -M 4-> lO

fC fl o

0} <+-(

;3CO

'sf:

u 4-> (U>

ro

COO

4J T3 O (D

+-> <d

CM

b> ^^ 1.!h

CD in ro CJ

—

O cn 00

00

CM

64
lO

ro

OJ

O

0) \u — 00 — 1^

CD — ID

CM

ro CO — Oa>(X)i^cDio^. ro

CJ Q_

Fig. 19. e ratios versus P^g ratios for the replacement of the

zero density Joule -Thomson coefficient calculated for the

12-6 potential function by that calculated for the exp:6

function. The key to temperatures along the curves is as

in Fig. 11, unless otherwise indicated.

65
O) CX)

* -

1^ (X> in ro OJ —

CNJ

— (T> — 00 «^ VU

ro

<\J

O

CO

CD

— 1^ vl/ — ^

— in

— ^. — ro

Q_

Fig. 20. e ratios versus P^g ratios for the replacement of the

zero density Joule-Thonason coefficient calculated for the

12-6 potential function by that calculated for the Kihara

function. Unless otherwise indicated, the key to temperatures along the curves is as in Fig. 11,

41

66

II
ro

OJ

^

^J^ — 00 W — 1^ — CD

lO

^.

ro CD in ro oo — O CT> cx) 1^ CD

CM

Fig. 21. e ratios versus Pjg ratios for the replacement of the

zero density Joule -Thomson coefficient calculated for

the 12-6 potential function by that calculated for the

Morse function. Unless otherwise indicated, the key to

the temperatures along the curves is as in Fig. 11.

67
00 1^ <x» LO to C\i o 0^ 00 r-

(M

CL

Fig. 22. e ratios versus Pjg ratios for the replacement of the

zero density Joule-Thomson coefficient calculated for the

12-6 potential function by that calculated for the m-6

function. Unless otherwise indicated, the key to the

temperatures along the curves is as in Fig. 11.

68
QCO

0>

00

IS

tQ

in

ro

CM

— O)

00

1^

iD — lO

ro — CJ

rocM — ocDoof^cDiO'sr ro

CM of

CVI - - SiJ Q vu

0)

nJ C M o

-4-> D

•rH

c

•rH fl

• I-* O O to

•i-l •iH m 4-» (\J «*-! o

D CO

•iH CJ d (0

fl

OJ o

sO >

•rH

CO fi

^1 d >< CJ Mh <u

•i-l X)

IP M

CD +j ao ^-!-> ;h fl

^<H o o•—

<

ri

C! (D

to

s

o3 ^H 1—

1

4) o t\J O 1—1 ^

nJ t;J 1) CJ cx

0) I-l 4J X (U

4-> ^-!-> o c O

<M o

•iH

!->

CO -)-> >. o o (U

4-> cd X

Rj <+-!

(L) h 1—

1

^

C^ +J

ro

•iH cC 0) 4J M

CO o c;j

::J CI, o

•w

•X} enu o d (U 1 •I-l >

I—

1

<u

to CO o

•rt (U

•T-i u

j3

M

+->uo

-!-> (l> v^ o

• CO

fsj

tab

t-i fH

69
CM

Fig. 24. e ratios versus P^g ratios for the replacement of the

diffusion coefficient calculated for the 12-6 potential

function by that calculated for the Kihara function. Unless

otherwise indicated, the key to the temperatures along

the curves is as in Fis. 11.

70
09

IS

ID

ro

00

Q cvj — w

CT) \U

00 — 1^ — CD — 10 — ^

— ro _ CO

o

I—

I

u

-t-1

4)

1—! •'-<

0) u

o o O "^^ Iti

0)

U ,—I j_i <5 '^ 0) ^^

m -t-> J-'-'

o g o

O ^ 0) -M r-J -*

+-> 4-1 to O nJ

CO

(U I > fNJ

I—

I

toO ,,

N(DL^'^^roc^J — 0(T>cr) l^

o

<i3 m

IT)

•iH

CJ

71
(D

00

I^

(X)

ID

ro

CMO c\j

CD \U

CX)

1^

— (S) — in

'^r

ro

cu

00 h- CD IT) ro CJ

—

O CD 00

t3

(U -M

nJ

.—1

:3u

r-lfio

4-> s

0) ,

•1-1 fi u

(1) oO O C! f ) d

M-l

a n) o rn

rn

u

'^ M-l s •iH Tl <u

UJ ^-l-> ^+-> (h

M-l "4-1 OM XI

-1 <u (> u 1—1

nJ rt >—1

r) Ph

(1) +-> u 53 XI

CD -M

41!o fl

"4-1 o

•H

rn

M

O o

PI

•(-> '^ rt MH

^1 1—1

(Xt («

•I-l pC

-!-> rn o d 0^

CO

^1 vO

OJ 1 > CNJ

i-H CD

r-l d)

4-> jH

(Ti +3

^^ ^O W m• (Mm•Hh

(M CL

72
CM

Fig. 27. The loci of potential parameter ratios which convert

several values of the second virial coefficient from the

12-6 potential function to the exp:6 function for » = 12

within an uncertainty given by 10% of the value of the

second virial coefficient at Tjg.g = 2. 0. The common

area of overlap is the loci of all pairs which simultaneously

fit all the values within the assigned tolerance.

73
CM QFig. 28. The loci of potential parameter ratios which convert

several values of the second virial coefficient froin the

12-6 potential function to the exp:6 function for oc = 12

within an uncertainty of 1% of the value of the second

virial coefficient at Ti3_g = 2. 0. The absence of a net

area of overlap shows that the two potentials are not

equivalent within this tolerance over the temperature

range included.

74
CO CD 00

CD

CO

CD

cJ

CO

CO

CO

o

CO

CD

CD

CM

Q

I^

CJ CL

vuM

VU

Fig. 29. The loci of potential parameter ratios which convert

several values of the second virial coefficient from the

12-6 potential function to the exp:6 function for 01 = IZ

within an uncertainty of 5% of the value of the second virial

coefficient at Tis_g = 2. 0. The small size of the area

of overlap indicates that 5% is close to the minimum

uncertainty allowed for a simultaneous fit to all temperatures.

75
ooOIT) COOO

C\J O

ID

vu ooo

If) o oo o

00 o

CD o o

(M

O

M-l CO o

0) •rH

-t-» )-> fl U

(D fi •rH d O <HH

•iH

«4-l r-H

M-( nJ

0) •rH o

-M

T—i (-> nJ o

•iH Ok u

'T^ <+H >

T3 CO

fl 0) O •rH 1—1 O •rHa

(U ^ (D ^-!-> > > m •H =1 O O ^

<+H CO O

CO

a

CO

^H CO a j:3

0) (D -!-> ^ a

!>^

M .—<

r-l o

(U

c;J

-l-> rC •rH 4J M

bO fl a M OJ • iH o CO tJ <+H CO a T) (U o CD Oh -M d

CO nS o

(U r-l ^ S

1—1

1— o o

•—<

(\5 CJ u CO

CO (U

0)

•rH 0) CO rH

1—t > O

•rHa

43

+J rH o t—1

J2 " (T3

T3 1—1

(^5 ri 1—1

43

nj C!

)-> o 0) ^ -t-> o

<i> (ti ^ o

ro

ei • Hh

76
ooro COo

-M

O T* -M U 1 a

m 1 -H O

r-H

C CO

O CO _- IT) S ., 53 CO

0) o

box: .2

^1 Ph -^^

"S • 3 O CD <*-! M fl ni d O m

.2i - -H U O

O effii fun eM o

C\J o ':: -^ ^ '+f ^

nJ £^ ° "go > ft X3 T3 «*H Ph 5 o (u

O ^ eco: ies

exc

thes

famil

h

the

-U -iH ^_^ iH m ^ CO

"+-1 ^, 4J Q 0)

CO ^ mH

(u CO :7l o CU G o— 5i:

ogSS

C! ^,

M fl

O ?^ fi ctJ i^ ^ 2 ?:

.13M Sii > 5 "S <u o

in

'^ TJ

O

H

iJ rt nS

d

i5 ':i -ci

u ^ 3 (V) -H

+-> (^ ri nj W u .3 fi, o o in o lO o

CO

<]

77
.Ad 9|0UJ/ 00 • g 9

Fig. 32. The equivalence of the second virial coefficients calculated

for the m = 15 member of the m-6 family and the

Y = 0, 1 member of the Kihara family. The experimental

data are taken from [ 14] , Note that there is essentially

a point by point equivalence over the entire temperature

range. The deviation plots for the m = 12 and 18 members

of the m-6 family are also included to illustrate the

sensitivity of the plot. Parameters: m = 15, a = 3. 37A,

e/k = 138. 5°K; Y = 0. 1, a = 3. 35A, e/k = 139. 8°K.

78
<

>

UJo

or

IxJ

400 600 800

TEMPERATURE ,°K

1200

Fig. 33. Deviation curve, 'ie X p ~ "He »1 X 100, for the viscosity

''1c al c of argon for two functions: a) The Kihara with Y = 0. 1,

b) The m-6 with m = 15. In both cases a = 3. 35 A,

e/k = 139. 8°K. Curves (a) and (b) clearly indicate the

equivalence between these two functions. Experimental

data points were taken from Ref. 11, and Refs. 15 - 21.

Key:• [11] A [17] D [20]

A [15] S 1—

,00, X [21]

® [16] O [19]

79
1 1 1

1 1

1 1 1 1 1

1

I \2 \2

1 o o _

a Z

<u <a

cr>

1 a> c 1 £= o

1 ^ >

CD 1 _ CD — \_ O ^ o

3 3

a 4 o <

mper — - ^ Li ,0) xo —

O) h- H

JZ 5o

_l r - i E

;,0

r liO

c^O ^«>

t

X :

;0«>

n '< Dx

<

< .4

^1

^

1 1 1 1

LlI (r

H

<

LU

a.

UJ

N0llVIA3a lN30d3d

Fig. 34. Deviation curves for argon with the parameter values of

Table III. The key for the data is given in Fig. 33 except

for the addition of <^ [ 23] . Note that the high temperature

points are not fit properly if the potential appropriate to the low temperature data is used [ curve (a)] and that the

low temperature points are very poorly fit when the potential appropriate to the high temperature data is used

[ curve (b)] . The fact that the intermediate temperature

points, 250°K < T < 1000°K (i. e. , 2. < Tfg.g < 8. 0),

are properly fit in either case is of considerable inaportance.

80
--

1

—

" - 0^

> —

liJ ~ UJ o o

0^ < #^

q: X

'^ ir ^ 1

ATURE X

i X

UJ

1-<

;mper 4 ~

LiJ Q.

UJ x^ <• —

LU

1- 0^121 H 0^

'^n, -4 X ><2I ^ < ^ o — o

-J — X

•4 X^x* < _ '^

X J4

%^

X 4

8^ »

®

'^ T

<» — X ki« 8 — r — X n <» X n <»

LjL E [3 Q x^T«B ^yS •] — ^* - ^ * o<? ^°c^

1

o ,«

1 1

1 i 1

O X

rO

O I 1

I-<

o (r

S LlJ

UJ

CJ — o — CM CVJ — o — CJ ro ID u> NOIIVIABQ lN30a3d

Fig. 35. Deviation curves for nitrogen in the manner of Fig. 34 for

argon. The data are taken from Refs. 17, 18, 21 and 24-33.

The results are essentially as for argon.

Key:

A [17] • [25]

S [18] ^ [26] [30]

D [20] A [27] [31]

X [21] ^ [28] A [32]

A [24] O [29] H [33]

81
o

H 3

<

>

O 2

tz. I UJo

(£

UJ

Q.

-I

-2

2

I

-I

-2

LENNARD- JONES

A S,

EXP: 6

2 " ^

X

KIHARA

X ^-x-iJ" X AXQ

x A Q X,

200 400 600 800

TEMPERATURE, °K

1000 1200

Fig. 36. Deviation plots for the viscosity coefficients for three

farailies of function. These are taken from [ 34] . The

key for the data is given in Fig. 33. Note that the intermediate temperature region is fit equally well by each

function.

82

GPO 840 -620
NBS TECHNICAL PUBLICATIONS

i I

PERIODICALS

JOURNAL OF RESEARCH reports National

Bureau of Standards research and development in hysics, mathematics, chemistry, and engineering.

Comprehensive scientific papers give complete details of the work, including laboratory data, experimental

procedures, and theoretical and mathematical analyses. Illustrated with photographs, drawings, and

charts.

Published in three sections, available separately:

Physics and Chemistry

NONPERIODICALS

Applied Mathematics Series.

tables, manuals, and studies. Mathematical

Papers of interest primarily to scientists working in these fields. This section covers a broad range of

physical and chemical research, with major emphasis on standards of physical measurement, fundamental

constants, and properties of matter. Issued six times

a year. Annual subscription: Domestic, 35.00; for- eign, $em*.

^St

• Mathematical Sciences

Studies and compilations designed mainly for the

' athematician and theoretical physicist. Topics in piiiathematical statistics, theory of experiment design,

numerical analysis, theoretical physics and chemistry, logical design and programming of computers

and computer systems. Short numerical tables. Issued quarterly. Annual subscription: Domestic,

".2.25; foreign, 32.75*.

Engineering and Instrumentation

Reporting results of interest chiefly to the engineer

md the applied scientist. This section includes many

3f the new developments in instrumentation resulting

from the Bureau's work in physical measurement,

lata processing, and development of test methods.

It will also cover some of the work in acoustics,

ipplied mechanics, building research, and cryogenic

jngineering. Issued quarterly. Annual subscription:

)omestic, 32.75; foreign, 33.50*. TECHNICAL NEWS BULLETIN

The best single source of information concerning

le Bureau's research, developmental, cooperative

md publication activities, this monthly publication

fis designed for the industry-oriented individual whose

laily work involves intimate contact with science

ind technology

for engineers, chemists, physicists, research managers, product-development managers, and

company executives. Annual subscription: Domestic,

11.50; foreign, 32.25*.

'Difference in price is due to extra cost of foreign mailing.

Building Science Series. Research results, test methods, and performance criteria of building materials, components, systems, and structures. Handbooks. Recommended codes of engineering

and industrial practice (including safety codes) developed in cooperation with interested industries,

professional organizations, and regulatory bodies.

Special Publications. Proceedings of NBS conferences, bibliographies, annual reports, wall charts,

pamphlets, etc. Monographs. Major contributions to the technical literature on various subjects related to the Bureau's scientific and technical activities.

National Standard Reference Data Series.

NSRDS provides quantitative data on the physical

and chemical properties of materials, compiled from

the world's literature and critically evaluated.

Product Standards. Provide requirements for

sizes, types, quality and methods for testing various

industrial products. These standards are developed

cooperatively with interested Government and in- dustry groups and provide the basis for common

understanding of product characteristics for both

buyers and sellers. Their use is voluntary.

Technical Notes. This series consists of communications and reports (covering both other agency

and NBS-sponsored work) of limited or transitory

interest. CLEARINGHOUSE

The Clearinghouse for Federal Scientific and

Technical Information, operated by NBS, supplies

unclassified information related to Governmentgenerated science and technology in defense, space, atomic energy, and other national programs. For

further information on Clearinghouse services, write:

Clearinghouse

U.S. Department of Commerce

Springfield, Virginia 22151

Order NBS publications from:

Superintendent of Documents

Government Printing Office Washington, D.C. 20402
:i^m:m

U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAIO|

U.S. DEPARTMENT OF COMMBficr

OFFICIAL BUSINESS