Bureau of Standards

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Quantum Mechanical Calculations

Of the Second Virial Coefficients

For Hydrogen

Marjorie E. Boyd and Sigurd Y. Larsen

412

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U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

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THE NATIONAL BUREAU OF STANDARDS

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UNITED STATES DEPARTMENT OF COMMERCE • Alexander B. Trowbridge, Acting Secretary

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

NBS TECHNICAL NOTE 412

ISSUED APRIL 10, 1967

Quantum Mechanical Calculations

of the Second Virial Coefficients

for Hydrogen

Marjorie E. Boyd and Sigurd Y. Larsen

Institute for Basic Standards

National Bureau of Standards

Washington, D.C., 20234

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, Government Printing Office

Washington, D.C., 20402 - Price 15 cents
CONTENTS

Page

lo Introduction 1

2. Background--- 2

3. Calculations 4

4. References 10
Quantum Mechanical Calculations of the Second Virial

Coefficients for Hydrogen*

Marjorie E. Boyd and Sigurd Y. Larsen

Quantum mechanical second virial coefficients of dilute

ortho and para hydrogen have been calculated for the case

of a Lennard-Jones potential. These computations cover

the temperature range from °K to 80 °K. Special attention is paid to the behaviour of the Boltzmann and exchange contributions. The purely statistical contribution

to the ortho-para difference is shown to vanish to three

decimal places by about 10. 5 °K. Comparison with experimental data indicates that considerable improvement

in the potential function and inclusion of non-spherical

terms will be required at the low temperatures considered

here.

Key Words - Second Virial Coefficient,Quantum Mechanics,

Hydrogen, Equation of State, Statistical Mechanics,

Intermolecular Potential

1. INTRODUCTION

This note summarizes the initial calculations in a program aimed at

accurate computation of the quantum mechanical second virial coefficient of dilute hydrogen. Ultimately, it is planned to carry out the

computations over the complete range of temperatures from absolute zero

up to the classical region. The work reported here extends up to 80 °K.

Both the ortho and para forms are considered as well as their mixtures.

While present calculations are concerned with spherical potential fields

for hydrogen, it is hoped to extend them eventually to include the effects of non-spherical potential terms.

One of the aims of this work is to provide a sound theoretical basis

for the evaluation and correlation of existing data on hydrogen now being

carried out by the Cryogenic Data Center at the National Bureau of

Standards Boulder laboratory. Our studies would also aid in supplying

information in ranges where the data are limited in accuracy or nonexistent.

Supported in part by NBS, Boulder Laboratory, Cryogenic Data Center

under NASA Order R-45, entitled "Investigations and Studies of Liquid

Hydrogen and Other Cryogenic Fluids' , for the Office of Research Grants

and Contracts of the National Aeronautics and Space Administration.

1
These calculations can be considered as part of a larger program

aimed at searching for an optimum intermolecular potential function

suitable for determining not only the second virial coefficient but also

the transport properties of hydrogen. In such an undertaking accurate

calculations for the Lennard-Jones potential seem the logical starting

point due to its widespread use.

2. BACKGROUND

The best previous quantum mechanical calculation of the second

virial coefficient for H is to be found in the work of Cohen, Offerhaus,

van Leeuwen, Roos, and de Boer 'TM (which also contains calculations of

the various transport properties), hereafter referred to as COLRB. Their

work is based on a spherically symmetric potential field assumed the

same for all states of the molecule so that the para-para, ortho-ortho

and para-ortho interactions are all represented by the same molecular

field. The second virial coefficient is then expressed in terms of integrals which involve phase-shifts arising from the characterization of

the asymptotic behaviour of the wave function of relative motion of two

H molecules. They chose as the potential field the well-known

Lennard-Jones form

r 12 6 )

4?<r) = 4ej^ - (-2-)

J

where e is the depth of the potential well, and <j is the "size" of the

molecule (the radius at which the potential is zero).

The values used for these parameters were obtained previously from

high temperature data by de Boer' 2 ) who found

e/K = 37.00 °K

N a = 1 5. 12 cm /mol

from which he obtained, using the 1952 values of Dumond and Cohen for

Avogadro's number N, and the Boltzmann constant K, the values

€ = 31. 88 X 10~4

ev

o

CT = 2.928A

The Schroedinger equation for the relative motion of two particles in

this potential field must then be integrated numerically for each value of

energy (wave number k) and angular momentum (C) out to a sufficiently
large value of the radius to determine the asymptotic behaviour and thus

the phase shifts (77 (k)). The presence of bound states, if any, and

their values must be ascertained. In their calculations COLRB used

values for the phase shifts obtained by Van Wijngaarden and Duijvestijn

using an electronic computer and calculating to three decimal places.

They obtained these phase shifts for values of k*, the reduced wave

number (l?s ctv^lE , ^g- )

t from to 4. at intervals of 0.1, and

from 4. to 8. at intervals of 0. 2 and for values of the angular

momentum quantum number, £, from to 23.

The potential field they chose was found to contain two discrete

energy levels and these were determined to be as follows:

<t=o ; - 3.147 X10"^eV = -3.651°K

1=1: - 1.227 X 10 eV= -1.424°K

For these two values of the angular momentum the phase shifts go to the

value tt instead of zero as k* goes to zero.

The second virial coefficients, B* = B/Ncr were computed for parahydrogen (pH2) and ortho-hydrogen (0H2) in terms of virials for Bose and

Fermi statistics, using the following relations derived by consideration

of the degeneracies occuring in the two forms:

(la) B* = B*

PH

2

(lb, B* = Y

B

t

+ TB

a*

where .#3 ZA

s + ' id wi^n /Vy2 fr (ZM)r,i w

j

y iy

a tt T*"

I = even nos.

-t = odd nos. -372

t

*3/2 1_{ZI + i) (e * -d

l = even nos. I = odd nos.

3
in which the subscripts s or a and the sums over ^-even or t-odd refer

to Bose-Einstein (symmetric) or Fermi-Dirac (antisymmetric) statistics,

respectively. The various reduced quantities involved are: * . . A = h/CT Vme

T = T (K/c) * Ei

- Vf

y

2 = \z k

2 /mKT

The first term, B , is the second virial coefficient of an ideal

quantum gas and has the magnitude A / 16tt ' T

3' 2 . The second

term is the contribution of the continuum states of molecules interacting

through a potential, while the last term gives the contribution of the

bound states of energies E«.

For mixtures of para and ortho hydrogen these authors use the

expression

B*mix " *p V, + 2 *pXo VoH + *oB

oH,

where B* TT = 1/2 (B + B ) and the y's are the percentages of each

p-oH s a A

species in the mixture.

Results are given at temperatures of 1. 85*K and at intervals of 3. 7*

from 3. 7*K to 37*K. (See table I) Comparison with experiments indicated that their results were about 10% too large at low temperatures.

COLRB believed the discrepancy to be due largely to the lack of

knowledge of the extent of influence of non-spherical terms in the

potential over the temperature range involved. To investigate this

possibility further these authors applied the Wang Chang ' ' classical

non-spherical corrections to the high temperature theoretical curve. They then redetermined the Lennard-Jones parameters from the resulting curve. The low temperature discrepancy between theory and

experiment was then reduced to 3%. However, they now considered

the parameters to be of doubtful meaning and the procedure was

regarded as unsatisfactory since the non-spherical corrections should

also be applied at low T.

3. CALCULATIONS

Our initial calculations for the quantum mechanical second

virial coefficients are, in general, an amplification of the computations

of COLRB, with several alterations and improvements. We employ a 4
formulation in terms of the phase shifts, as they have done, but have

used a different arrangement of Eq. (1) in which we have separated the

contribution of a Boltzmann gas (B- direct) from the effects due to

quantum statistics (B-exchange). ^'

For a gas of spin S we may then write the virial as a sum of two

terms:

direct exch

where:

XT . 2

k

< 3

' B

direct= -2l/X P

U+1) 7" /

dk k V1 "

e

2 TT

and

(4) B___ = +- 2

1/2 NX

3

*£(V(2«)j^L /

, dkk exch (2S+1) 3

n V 2

D/ ^r25

Here X^ = h/ ^2irmKT , (3 = l/KT, and E , is the energy of the

th

n' n bound state for angular momentum £. In B the upper signs

refer to the symmetrical (Bose) statistics while the lower signs refer

to the Fermi or antisymmetric case.

(4) Analysis of the behaviour of the B , term as well as exch

numerical evaluations for various cases' ^> 6) demonstrates that the

quantum statistical effects embodied in it are sharply suppressed with

rising temperature whenever a strong repulsive core is present in the

potential. In fact, above a few degrees Kelvin the Lennard-Jones gas

may be regarded as effectively a Boltzmann gas.

This equation also has the advantage of displaying the very

simple dependence of the second virial coefficient on the spin. In

addition, the subtraction involved between the interacting gas and the

ideal quantum gas contributions in evaluating the B , term makes exch
the effect of numerical inaccuracies quite conspicuous. That is, they

cause the subtraction to be inexact as B , -» and erratic results

occur of the order of magnitude of the digit in error.

Turning our attention now to hydrogen we first consider the case

of the para modification. The spins of the two hydrogen atoms in the

molecule are opposed giving a net spin of zero and in its ground state

the rotational quantum number J is zero. We thus have no degeneracy

and only symmetric wave functions are allowed. That is, we have a spinless Bose gas and

<5> B „ = B^ f + B

B°S° (8=0) pH direct exch

where B " , is calculated setting S = and using the upper signs in _ ... exch

Eq. (4).

For ortho hydrogen the situation is more complicated. Since

the spins of the atoms are parallel they must be added, giving a spin

of 1 for the H molecule. Thus, we have a 3-fold degenerate state,

where the z component of spin, of interest in a collision of two

molecules, can be 1, 0, or -1. in addition, the ground state of the

ortho molecule has rotational quantum number J = 1, also three fold

degenerate. The result is thus 9 possible states for the molecule and

9 = 81 possibilities when two molecules interact. Considering

M = J + S for each molecule we have the possible M values of 2 (which

is 5-fold degenerate), 1 (3-fold degenerate), or 0. Collisions between

molecules of different M values will give only Boltzmann type terms,

so no contributions will occur to b , in these cases. For two exch o molecules of M = 2 (degeneracy 5) we get b L - 25 cases, between

those of M = 1, 3 2 = 9 cases, and between those of M = 0, 1 case.

Thus for the exchange term, we get, averaging over all the possible

cases:

B

. R

B0Se ii_ r 25 e 1 + qc—i + _i 1 exch, oH

2

exch l °

u; ^81

L (2S+l) g=2

^ (2S+l) g=1

(2S+1), J S=0i

b

Eos

I (s=o) :?_ exch

81

1
1 B

B°S

! (S=0)

9 exch

and the total second virial coefficient for ortho hydrogen will be

Bose ,„ „» (6) B TT = B, + 1 B 7 S=0 v

' oH direct —— exch

Numerical improvements over the previous work of COLRB are as

follows. First, the phase shifts are calculated to five figure accuracy

in most cases instead of the three obtained by the previous workers.

Second, instead of cutting off the sum over I at an arbitrary value (23

in the previous work), we have included the phase shifts for all values

of £ . This is done by finding for each k* the value of I at which the

phase shift computed by integration of the Schroedinger equation agrees

with the value obtained by the Born approximation to within one digit in

the 4th decimal place. The sum over the Born approximation values of

the phase shifts from this value of £ to t= »can then be evaluated

analytically.

The third numerical .improvement was to increase the range of k*

values and to decrease the interval size between k* values used in the

numerical integration of Eqs. (3) and (4). We have calculated phase

shifts up to k* = 16, twice the range of COLRB at intervals of 1/16, interpolating the sums over them to 1/64. This has the effect of improving

the accuracy of the virial coefficients obtained at the low temperatures

and of allowing their calculation at higher temperatures than previously

obtained. We give values to 80 °K.

The values of the energies of the bound states in this molecule

were obtained using a general computer program for calculating bound

states for a Lennard-Jones potential of any given parameters which was

written by J. E. Kilpatrick, M. Fraser and S. Y. Larsen( unpublished).

Using the de Boer parameters these energies were calculated to seven

digits, as contrasted to the four digits used by COLRB. This is

necessary in order to obtain more digits in the low temperature values

of the virial as can be easily seen by examining the individual terms

given in Table II. The lower bound state was determined to be at

-3.651296°K, agreeing with the COLRB value to the four digits they

used. The upper one we have computed is at -1.422061°K, which differs

from the COLRB value in the fourth digit. The behaviour of the exchange

term is especially sensitive to these values and displays conspicuously
the effect of an error. In this case, the erroneous digit caused B

to change sign when it became of the order of magnitude of the decimal

digit affected by the error.

Examination of the results obtained indicates general agreement

with the results of COLRB, as one would expect. Table I gives our

results for para and ortho H compared to COLRB' s calculations. At

the lowest temperature shown, 1. 85°K, our result for para H is about

10 cm /mole lower (only about 0.1%) than the earlier calculations, while

at the upper end of their range, 37°K, the change is about 1. 6cm3/mole

or about 3% change in the direction of the experimental values, which

are some 7% to 8% lower still. It is interesting to note that 7% is

about the amount of improvement that was made in the calculations of

COLRB by-employing the classical non-spherical correction at high T.

Thus, if applied to our curve, agreement with experiment within about

a per cent or so would result, but this must be regarded as rather

fortuitous at the present stage of computation. Table III gives our

results for B ,. ^, B^ose and the ortho-para difference up to 80 °K.

direct exch

The part of the ortho-para difference which is due to statistics alone

(thst is, the potential function is assumed the same for both) depends

only on B , which can be seen to have vanished to the accuracy we

exch

have calculated by about 10. 5°K. We emphasize, however, that if the

potential function does differ for the different rotational states, as is

the case in reality, the ortho-para difference would persist to higher

temperatures.

It should be noted here that at least two re-determinations of the

Lennard-Jones parameters for H have been made since the calculations

of COLRB. Michels, De Graaff and Ten SeldamC) made a new fitting of

these parameters based on experimental data obtained in their laboratory' 8 ' for the temperature range -175°C to +150°C. It was not possible to find a set of values for the parameters which would enable the

fitting of the virials within the experimental accuracy over the whole

temperature range involved, even with the introduction of high temperature corrections for the non-spherical effects. However, the best set

of values was given as

e/K = 36.7°K

3 3 Nor =15. 60 cm / mol

It should be noted that the expression for the second virial coefficient

at high T used by these authors is incorrect, containing a term for the

ideal gas (statistical) contribution which can be shown to have been

cancelled out by the two-body exchange effects at this temperature^ 4 ).

8
It is not likely that this would account for the whole difficulty in

obtaining a fit over the range of temperatures involved, but the parameter values given by these authors must be considered somewhat unsatisfactory.

(9) More recently, Diller and Mason fitted viscosity data and

obtained the values

c/K = 37.2°k

a = 2.97A

using mainly the date of Coremans, et. al.

,

and Johnston and

McCloskey '^' but also measurements of van Itterbeek, et. al., ' ' and Becker and Misenta U^), It has been known for some time, for

example in the case of He, that the best parameters or even forms of

potentials for fitting the second virial are not the best for the

viscosity, U^) so it is not surprising that the values of Diller and Mason

are different from those of Michels, et. al. The main conclusion about

them which can be drawn at present is that the question of optimum

potential parameters for H is still an open one. In fact, more

recently H. Hanley and M. Klein of NBS (unpublished work) have examined the possibility of fitting the data over a large temperature

range in the classical region by means of various three -para meter

potentials and concluded that it was not possible to do so with any

of them.

In order to extend our calculations over the complete range of

temperatures we have recently computed new phase shifts extending

out to a value of kcr = 42. 5. These should allow the computation of

the second virial coefficient up to the classical region,,

The computer programs used in these calculations are an

adaption and elaboration of codes originally developed by Kilpatrick,

Keller, Hammel and Metropolis 17 who were interested in helium.

These new codes, which we have used for the calculations reported

in this paper, were developed by the authors in collaboration with

Professor Kilpatrick and the support of the Los Alamos Scientific

Laboratory. We are especially grateful to Malcolm Fraser,

Roger Lazarus and Kathleen Witte of LASL for much of the programming

and the extensive testing of numerical methods as well as for their

enthusiastic aid and cooperation.
4. REFERENCES

(1) E. G. D. Cohen, M. J. Offerhaus, J. M. J. van Leeuwen, B. W.

Roos and J. de Boer, Physica .22, 791 (1956).

(2) de Boer, J., Rep. on Progr. in Phys. J2, 305 (1949).

(3) C. S. Wang Chang, Thesis, Michigan (1944).

(4) S. Larsen, J. Kilpatrick, E. Lieb, H. Jordan, Phys. Rev. 140,

A129 (1965).

(5) M. E. Boyd, S. Y. Larsen and J. E. Kilpatrick, J. Chem. Phys. 45,

499 (1966).

(6) S. Y. Larsen, K. Witte, J. E. Kilpatrick, J. Chem. Phys. 44,_ 213

(1966).

(7) A. Michels, W. de Graaff and C. A. Ten Seldam, Physica 26,

393 (I960).

(8) A. Michels, W. de Graaff, T. Wassanaar, J. M. H. Levelt and

P. Louwerse, Physica .25, 25(1959).

(9) D. E. Diller and E. A. Mason, J. Chem. Phys. 44, 2604 (1966).

(10) J. M. J. Coremans, A. van Itterbeek, J. J. M. Beenakker,

H. F. P. Knapp, and P. Zandbergen, Physica 24, 557, 1102(1958).

(11) H. L. Johnston and K. E. McCloskey, J. Phys. Chem. 44, 1038

(1940).

(12) A. van Itterbeek and A. Claes, Physica 5, 938 (1938); A. van

Itterbeek and O. van Paemel, Physical, 265 (1940).

(13) E. W. Becker and R. Misenta, Z. Physikl40, 535(1955).

(14) W. E. Keller, Phys. Rev. 105. 41(1957).

(15) F. H. Varekamp and J. J. M. Beenakker, Physica .25, 889(1959).

(16) R. D. Goodwin, D. E. Diller, H. M.. Roder, L. A. Weber,

J. Res. N.B.S. 68A, 121(1964).

(17) J. E. Kilpatrick, W. E. Keller, E. F. Hammel, and N. Metropolis,

Phys. Rev. 94, 1103 (1954).

10
s

EH

CJo

•H

-P

a) oo LT\ LTN 3

CM CO on

H

H o C\J VO H CO r-{ t— o oH ao VO ON CM t— ITN vo a ao C— VO LTN ON O _=r u ao

CO CO ITN OO OO CM

CO 1 | 1 1 1 1 | •H£

^^ c a) CD H cd

ttD

O

G o 6 O u •H

T* ^^ •P

J>» a) a m H VO ON S 3 • • • • o o o c— H t— H

J2 ^_^ rH CO c— -3-

O

-P cd CO c— ITN OO M o oo H o CM

£Q 1 1 1 1 TJ o « c cq ^ cd oo

cdm

<jj PL.

<+Ho ao

0} •H P -P VO oo o

fl a) l/N ITN On CM rH

O 0) H ir\ t— ON co H t— o •H 3 O a VO CM ON CM C— LTN vo

•H

H On ON VO LTN ON

O -3- <M cd On t— CO LTN OO OO CM

«H o ON

H

CO 1 1 1 | 1 1 1 o CO o •H H Eh

cd •HH

•H>

T3 , , cd c co

G o H O o O •H

0) 6 -P

cd

-=1- o\ 3 • • • • oo o VO vo c— H & H CO CO -h; O

CJ cd ON t— LTN OO ^^ o

cq

ON1 H1 1 1

C\J

K

M ^

P< o

cqu

LTN l/N LTN LTN Nd CO C— LTN _* CM H ON

Eh

H oo LTN c— ON

H CMH * ir\ o LTN O LTN O LfN Eh o H H CM CM OO OO • • • # • • • oc

<um

CU «m

CU «aOH

Cm

cd

11
Table lb

Second Virial Coefficients of Hydrogen - Boltzmann Region

IJI* T°K

B (cm3 /mole

)

C0LRB Calculation This Calculation Experimental

.1*0 lU.8 -201.1 -20U.il -237

a M 16.65 -173.17 -191

a

.50 18.5 -11+6.8 -11+9.1+2 -171* a

.55 20.35 -130.65 -152 a

.60 22.-2 -113.2 -115.M -127.

b

.65 2k, 05 -102.886 -112.5

.70 25.9 - 90.3 - 92.31*8 -100.9

.75 27.75 - 83.383 - 90.90

.80 29.6 - 73.8 - 75.666 - 82.1*1

.85 31.1+5 - 68.956 - 75.03

.90 33.3 - 61. h - 63.070 - 68.52

.95 35.15 - 57.866 - 62.72

1.00 37.0 - 51.6 - 53.233 - 57.51+

1.081 1*0.0 - U6.716 - 50.32

1.216 1+5.0 - 37.950 - 1+0.80

1.351 50.0 - 31.08 - 33.39

1.1*86 55.0 - 25.56 - 27.1+8°

1.622 60.0 - 21.02 - 22.70

1.757 65.O - 17.2 - 18.6U

I.892 70.0 - ll+.O - 15.22

2.027 75.0 - 11. - 12.1*2

2.162 80.0 - 9. - 9.88

a)

b)

c)

Values interpolated from data in Reference 15.

Data from this temperature up from Reference 16.

Data from Reference 1.

12
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§

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CD

a;oo

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aJ •H

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co

CJ

a; co

0) X!P

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-p

(0a

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13

£>

•H

fU 13oo

to

cu t3

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CO

•Hoo

CU

COo

pq

CUa»

ct3 X!o

X

3}

-Po

CU •HQ

VO C— O (^J- lALT\lAOHiAt-f-0\0\WHHC\l 00\aOlAHt^OO^O\VOOO\C\ICOVi)VOCOOJ-oO

c\j oj co o\C\i j- j-oo n ooo irs^t oj i-h o o\ o\<o f- =OHOO\VD-=roOOJOJC\JHrHHHrHiH

-J- t- OJ

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VOCO IPi l/-\_3" t— OJ CO MD -=* OJ O 0\CO CO t— VO \Q LT\ U"\ \OOU)4 OJ WW WHHHHH H

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

CO

-p

CO

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II

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VQ_3-_3- OJ CO OOvoOJ OJVDCO

VO <X> OJ On O He— ONt— OnoOVQO U*\ t— l/N OH lAlAr-H H H HOW03 0\c0t-hOOt- ao O\iAOJvoo\nvDJ-J--?H0OO\cOJOJcO0OJ- (\lt-OJJ(Mf-01f-J0\HO\0rlO0\lAC0in H<D(OOHH\OC04 IT\_3" O0\O4 CT\-4" H CO \Q

oiAOiovoiAonvofti o\N\OJ-^f mcj cm aj hh t— OJHOLnoOOJHH

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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

VOCO t— 0\-X \Q H co m t~- J- voh m ir\ o\^t \o \o oooj o o\-3- o\ oj t— J- CO IACO t-COH C- t— t— 0\0JH0\C0-3-\00JO 00 0\0 0\JO^t-tM4-4 0\^Wr|04-0 o\omo ovo J t— oo co co h omj-co h h v£> vo CO H IAJ IACO H IAJ-\D f- J- JVO O IAC\J OWO J

OJO\ir\0-^-oooJLr\iHaOvoLr\_*ooroOJOJrHHH O J1 CM O lA m OJ H H CO t— OJ H

t—

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0)COo

•H

=q

0\H HVO LT\CO PO C- PO 0\ PO_3" C— t— IT\ t— -3" lAt- t— ir\ h -3- OJCOJ- t— o t— vo aj j- cr\H ojco h OOJOOOJOO-^- lAt-JVOIAHCOOOHMCOOJ -*00\0\NlAr|cOVOLr\OHOMr>U)\Dt-4'OJcO

ir\ CO -3" MD CO -=f 0\[-4COt-mcOW40\OJOOO\ 0\0J-4- MOO Lf\4 h-HCOlAO\t^OlAOJCMniA[- vocohco o m oj _=r o\-3- hcovo iamcm h o o\co

COHlAO\[-lA4rOWCMC\JHHHrH.rHH

t— OJ h

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

o

EH

1A O IA O IA O IAO IAO IAO l/NO IA O LT\ O IA O 30 NLA4 CM H 0\CO\0 lArOCMOONK^J-COHO

H GO l/\ t— ON rH OJ -4" VO CO O OJ-4" lf\ C— 0\ r-\ PO LT\ t~- HHHHHCMCMCMCMCMCMmrOCOM

13
Table III

Direct and Exchange Second Virial Coefficients for H,

T(°K) direct (cm /mole) Bose

gxch

(B0H

2 " V,'

1.85 -871+9.6 -12U6.9 +1108.3

3.70 -177^.61 - 18. Ik + 16.13

5.55 - 869.086 .870 + 0.773

l.k - 552.828 .065 + O.O58

9.25 - 397. II1* .006 + 0.005

11.1 - 305.707 .000 0.000

»U.S. GOVERNMENT PRINTING OFFICE: 1967 —251-025/88

14

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