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Direct measurement and calculationof the second refractivity virialcoefficients of gasesR.C. Burns a b , C. Graham a & A.R.M. Weller aa Department of Physics , University of Natal ,Pietermaritzburg, South Africab De Beers Diamond Research Laboratory , P.O. Box 916,Johannesburg, 2000, South AfricaPublished online: 22 Aug 2006.

To cite this article: R.C. Burns , C. Graham & A.R.M. Weller (1986) Direct measurementand calculation of the second refractivity virial coefficients of gases, Molecular Physics: AnInternational Journal at the Interface Between Chemistry and Physics, 59:1, 41-64, DOI:10.1080/00268978600101901

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MOLECULAR PHYSICS, 1986, VOL. 59, NO. 1, 4 1 - 6 4

Direct m e a s u r e m e n t and ca lcu lat ion of the s e c o n d refractivity virial coef f ic ients of gases

by R. C. B U R N S ~ , C. G R A H A M and A. R . M. W E L L E R

D e p a r t m e n t of Phys ics , U n i v e r s i t y of Na ta l , P i e t e r m a r i t z b u r g , S o u t h Af r ica

(Received 18 ffuly 1984 ; accepted 4 May 1986)

Values of second refractivity virial coefficients BR, obtained by differen- tial measurement of the optical path in a gas when a sample is decompressed, are reported for Ne, Ar, Kr, Xe; CH4, CH3F , CH2F2, CHF3, CF4; CO 2, N 2 and SF6, at four wavelengths 632'8, 514'5, 488"0 and 457'9 nm. There is evidence of a possible wavelength dependence of B R . First refractivity virial coefficients A s are also reported for helium and the above gases at the same four wavelengths.

The classical statistical-mechanical theory of B R is briefly reviewed and extended, and the leading contributions to B R have been evaluated numeri- cally by computer for eleven of the twelve gases. Agreement between theory and experiment is often within the large experimental limits, but there is a tendency for the calculated values to exceed the observed.

1. INTRODUCTION

T h e vi r ia l expans ion

n 2 -- 1 Rm - n 2 + 2 p - 1 = .ZlR + BR p + CRp2 _~_ . . . (1.1)

of the m o l a r re f rac t ion R m as a p o w e r series in p, the dens i ty in moles pe r un i t vo lume , is conven ien t to use [1] in inves t iga t ions of the de ns i t y d e p e n d e n c e of the re f rac t iv i ty of gases. He re AR, BR, CR, . . . , are the first, second, th i rd , . . . , r e f rac t iv i ty v i r ia l coeff ic ients , i n d e p e n d e n t of p b u t func t ions of f r equency and t e m p e r a t u r e . In the absence of s h o r t - r a n g e m o l e c u l a r in te rac t ions (1.1) reduces to the L o r e n z - L o r e n t z equa t ion [2, 3]

n 2 - 1 p - 1 Ncr (1.2) n 2 + 2 3~0

so tha t A R = N~o/(3eo), where s 0 is the po la r i zab i l i t y of an i so la ted molecu le . T h e second re f rac t iv i ty v i r ia l coeff ic ient BR desc r ibes the excess c o n t r i b u t i o n to R m due to the in te rac t ions of pa i rs of molecu les , and has been the sub jec t of m a n y e x p e r i m e n t a l [ 4 -15 ] and theore t ica l [1, 10, 13] inves t iga t ions .

Present address: De Beers Diamond Research Laboratory, P.O. Box 916, Johannes- burg, 2000 South Africa.

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42 R . C . Burns et al.

In the early experiments [4-7] B R was deduced from measurements of the refractive index n as a function of the gas pressure, but since refractivity virial effects are generally very much smaller than density virial effects, moderate uncertainties in density virial coefficients used to deduce gas densities lead to large errors (-~100 per cent) in the values of BR. A considerable increase in precision has since been achieved by a new technique for the direct measurement of density effects, based on the differential measurement of total optical path when a gas is decompressed from one cell into a similar evacuated cell, proposed by Buckingham, Cole and Sutter [16], and used by Buckingham and Graham [9, 10], St-Arnaud and Bose [12, 14], Burns and Graham [13], and Coulon et al. [15]. This paper summarizes the results of an extensive set of direct differential measurements of B~ for the inert gases, fluorinated methanes, CO2, SF 6 and N 2 , carried out at four wavelengths 632-8 nm, 514"5 nm, 488"0nm and 457"9nm on the apparatus described by Buckingham and Graham [-9, 10]. A feature of the measurements is an apparent wavelength dependence of BR for some gases, which if confirmed independently, will be of considerable theoretical interest since present statistical-mechanical theories of B R do not predict any wavelength dependence beyond the small dispersion of the molecular polarizabilities which enter Be.

2. EXPERIMENTAL DETAILS

The apparatus and experimental techniques have previously been described in detail [9, 10]. A sensitive interferometer, internally compensated for the effects of mechanical vibrations, with a sensitivity and long-term stability of better than 10 -3 of a fringe, is used to compare the phases of a linearly polarized reference beam of wavelength 20 and a second coherent linearly polarized beam traversing two cells A and B, as nearly identical as possible. I f A is initially filled with gas under pressure and B is evacuated, and a valve connecting the cells is opened, the decompression for perfectly matched cells results in a change of phase

27rl c~a = -~0 [2(n2 - 1) -- (nl -- 1)] (2.1)

between the sample and reference beams, where n 1 and n 2 are the initial and final values of refractive index, and l is the length of each cell. We write

(n - - 1)p - t = A, + B , p + C, p2, + . . . , (2.2)

in which the refractive index second virial coefficient describes the density depen- dence of (n -- 1)p-a arising not only from molecular interactions, but from bulk polarization effects. Then for low initial gas densities Pl

2x l 6A . . . . B , P 2. (2.3)

2o 2

It is readily shown from (2.2) and (1.1) that

AR = ~A. (2.4)

and

BR 2B. 1 2 = - ~A., (2.5)

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Refractivity second virial coefficients 43

so tha t if ~/n is m e a s u r e d b y s t a n d a r d p rocedu re s , and Bn is m e a s u r e d by the expans ion t e chn ique de sc r ibed above, BR m a y be eva lua ted f rom equa t ion (2.5). In p rac t ice the effects of unavo idab l e m i s m a t c h in cell vo lume s are e l im ina t e d by ca r ry ing ou t two success ive decompres s ions , in wh ich we exp lo i t the change in s ign of the phase change due to cell defects as o p p o s e d to the un id i r ec t iona l changes due to t rue dens i t y effects [9].

3. EXPERIMENTAL RESULTS

3.1. Purity of the gases

T h e supp l i e r s and p u r i t y speci f ica t ions of ou r gas s a mp le s are given in tab le 1. T h e gases were all used w i thou t f u r t he r pur i f ica t ion , bu t were passed t h r o u g h 1 # filters to ensure f r e e d o m f rom dust . Since no p u r i t y speci f ica t ion of our C H 2 F 2 sample was s u p p l i e d b y the m a n u f a c t u r e r a m a s s - s p e c t r o m e t r i c analys is was u n d e r t a k e n and the c rack ing p a t t e r n s c o m p a r e d wi th those of M c C a r t h y [17]. C o n t a m i n a t i o n b y o the r ha logena t ed m e t h a n e s was assessed at less than 0"1 per cent , and the total level of c o n t a m i n a t i o n b y o the r o rgan ic and inorgan ic gases even less.

3.2. Measurements of As

O u r o b s e r v e d values of AR are s u m m a r i z e d in tab le 2 toge the r wi th l i t e ra tu re values for c o m p a r i s o n purposes . Values for H e are i nc luded even t h o u g h m e a - s u r e m e n t s of B R were no t poss ib le . F o r each gas at least th ree separa te m e a s u r e - m e n t s were m a d e at each wave leng th . In all cases the s t a n d a r d dev ia t ion was less

Table 1. Suppliers and minimum stated purity of the gases used.

Gas Supplier Grade

Stated minimum

purity % Nature of main impurities

He Matheson High purity

Ne Matheson Prepurified Ar Matheson Prepurified Kr Matheson Research

Xe Matheson Research

CH 4 Matheson Ultra high purity

CHaF Matheson Standard CHEF 2 du Pont de - -

Nemours C H F 3 Matheson Standard

CF 4 Matheson Standard

CO 2 Matheson Coleman instrument

N 2 Matheson Prepurified SF 6 Matheson Standard

99.995

99"99 99-998 99"995

99"995

99'97

99'0

98"0

99.7

99.99

99-997 99"9

H20 12p.p.m., Ne 14p.p.m., N 2 14 p.p.m.

Not stated 02 4-5 p.p.m., N 2 6-7p.p .m. Xe < 0-0025 per cent (by vol.), N 2 < 0"0025 per cent (by vol.) Kr 50 p.p.m. (by vol.), hydrocarbons

10p.p.m. (by vol.) N 2 40-50p.p.m., C2H 6 20-30p.p.m.,

CO 2 40-50 p.p.m. SiF4, (CH3)20

Other halogenocarbons 0-9 per cent (by wt.), H20 25 p.p.m.

H20 15p.p.m. (by wt.) max., air 1�89 per cent (by vol.) max.

Not stated

0 2 0"0008 per cent, Ar 0.0010 per cent Air 0"04 per cent, CF 4 0'05 per cent

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L m l ~ - t i m l ~ ~ - ~

~oo ~ o ~ o ~ ~o

~ . ~ ~J1 ~ m l O o L m l L . ~ v

~ Oo CIO ~ 1

v ~ v

C ~

ol

s~

0

s~

(IQ

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I+ '~' I+ I+

I+ I+ I+ -- I+ ~ 1 + I+

I+ I+ I+ I+

~ m N ~ ~ N N FF I+ I+ L+

I I

b. l

I+ I+ FF I+

I I I I

l-b I+ I+ H

[F I-F IF ~, I+ H-

I+ I+ I+ l+

~ e a

I+ I+ I+ ~ I+ ~ 1 + I+

9 9 9 ~

I+ I+ I+ ~ 1 + I+

d) ,=

db o

0

t~ 0

0

<:

t~

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46 R . C . Burns et al.

than 0"1 per cent. Several hundred fringes were counted and the phase changes measured to bet ter than 10 -2 of a fringe. Pressures were measured with a M K S Baratron capacitance manometer to an accuracy of +0"05 per cent and the tem- perature of the water jacket surrounding the cell was measured to + 0"05 K. Gas densities were derived with the aid of second virial coefficients compiled by D y m o n d and Smith [-18].

3.3. Measurements of BR

Our values for B R are summarized in table 3 together with l i terature values in cases where earlier measurements have been reported. As in the measurements of Buckingham and Graham [9] the quoted uncertaint ies are obtained by taking the sum of three uncertainties, namely the s tandard deviation, the variat ion in B R due to a + 30 per cent error in the calculated correction for the effects of cell dis tor- tion due to gas pressure [-9, 10], and an uncer ta inty due to drift in the end-po in t after a decompression. In most cases the s tandard deviation accounts for roughly half the quoted uncertainty.

4. DISCUSSION OF THE MEASUREMENTS

In figure 1 the continuous lines are a plot of the observed values of B R as a function of wavelength, and the broken lines represent calculated Values in which allowance has been made for the dispersion in the polarizabi l i ty ct of these mol- ecules, but the dispersion in the f ield-gradient quadrupolar izabi l i ty tensor C~ar~ , in t roduced later, is unknown and not included. Figures 2 and 3 show the wave- length dependence of BR for N2, CO2, SF6 and the fluorinated methanes. The errors, indicated by error-bars , are large, but for many of the gases investigated there is evidence of a wavelength dependence of B R beyond the limits of est imated

! ~t

N r - t

" lO

Ar

tm~al tna~.h / r l

Figure 1. Continuous lines show the observed values of B R as a function of wavelength for Ar, Kr, and Xe. The broken lines represent calculated values in which allowance is made for the dispersion in ~ but not in C.

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Refractivity second virial coefficients 47

Figure 2.

B

N

O

30

10

50 500 550 600

wovQl ~ngt~ /ns

Wavelength dependence of B R for N 2 , C O 2 a n d S F 6 .

error. This is surprising, and not expected on the basis of simple statistical- mechanical theories of BR [1] which do not predict any wavelength dependence beyond the small dispersion of the molecular polarizabilities which enter B a. This is evident from figure 1.

When these apparent trends were initially observed the apparatus was thor- oughly checked. In particular, our quarter-wave retarders were all examined and found to be precise; and when the measurements were repeated on a number of gases the trends were found to persist. On the other hand no anomalies were observed when the same apparatus was used to measure A~. A similar unexpected wavelength dependence of B R was reported by Beaume and Coulon [7] for ammonia, as shown in table 4. Independent confirmation of the anomalous dis- persion of BR would be valuable, and of considerable theoretical interest.

Figure 3.

'

wavel~n9th Irm

Wavelength dependence of B s for the fluorinated methanes.

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48

Table 4.

R. C. Burns et al.

Wavelength dependence of /~R for NH 3 reported by Beaume and Coulon I-7]. All observations were made at 298.2 K.

2/nm 1012B~m 6 mole - 2

667.8 220 + 100 587-6 450 +__ 100 501.5 680 + 100 447.1 1130 _ 100

5. C L A S S I C A L S T A T I S T I C A L - M E C H A N I C A L EXPRESSIONS FOR BR T h e first refractivity virial coefficient A R which accounts for all but a very

small fraction of R m is directly proport ional to the polarizability s 0 of an isolated molecule; and B R which describes the excess contr ibution to R m due to inter- acting pairs of molecules is given by the classical statistical-mechanical average [1] over the relative configuration z of two molecules 1, 2

BR = (NE/3e0 f~) f [�89 - - S 0 ] exp [ - -U12( '~) /kT j dz (5.1)

of the difference [{s12(z ) - s0] between the mean polarizability {Sa2(Z) of a mol- ecule when participating in a binary collision and s o . Here Ux2(z ) is the potential energy of the pair 1,2 in the relative configuration z, and

f dz = V m . (5.2) t~

In this work we express [�89 ) -- s0] and U12(z) in terms of multipole moments , polarizabilities and hyperpolarizabilit ies of an isolated molecule, and evaluate the integral in (5.1) numerically by computer . The procedure is rigorously correct only for intermolecular separations where the interacting charge distributions do not overlap, but models of this type often work very well despite some charge overlap in a collision and are widely used [29, 30, 31] when the alternative, namely an ab initio quantum-mechanica l calculation for the interaction [32], is prohibit ively complicated, especially for many-elect ron atoms.

5.1. Discrete-molecule expressions for [�89 - sol

We write

F 0/~1) ] (5.3) - S o ] = e , - s o

and for o) /~, , the total oscillating dipole induced in molecule 1 by the light-wave field go in the presence of molecule 2, write

~(1)(~X~O) [ . ( 1 ) l.~, ~,(1) l~,( 1 ) = + + + +--.)(g0# +

s ~ ( 1 ) + 3 ~ a O r ~ " # r ' ( 5 . 4 )

where the sum in the first pair of parentheses is the effective or differential polarizability of molecule 1 when the powerful electric field Ft~ 1), and field gra- dient Ft~ ) at molecule 1 due to permanent multipole moment s of molecule 2 force the response of molecule 1 into non-linear regions. Also, ~ 1 ) is the oscillating

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Refractivity second virial coefficients 49

field at 1 in excess of the applied field go due to the p rox imi ty of molecules 1 and 2. Fo r example, the light wave m a y induce in molecule 1 an oscillating dipole, which in turn gives rise to an oscillating field at 2, result ing in an addit ional induced oscillating dipole on 2 which now cont r ibutes an oscillating field at 1 in excess of g0 ~o~-(x) is the oscillating field gradient at molecule 1 due to the p rox im- �9 ~ # y

ity of molecule 2. A powerfu l aid in the derivat ion of explicit expressions for F~r 1), F~r~), ff~a) . . . .

is the set of T- tensors [33]. For example, the field E~, x) and field gradient E~,~ ) at the origin of molecule 1 due to a dipole m o m e n t #~2) on molecule 2 are s imply wri t ten

_1A(2) q~(I) --, = ~,#~'# - -3v#y- ,#~ and = -- (5.5)

where, if R is a vector f rom the origin of 1 to 2,

1 1 T ~ ) = 4ze0 V , V a R - 1 - - 4roe0 (3R, R a -- R26~a)R -5 (5.6)

and 1

T ~ = -- - - V,VaV ~ R - x 4rr%

3 = 4rce---~o [5R~R~Ry -- R2(R~,6#~ + R # ~ , + R~6~,p)]R -7. (5.7)

T h e field and field gradient E~ ) and Et~ ) at molecule 2 are similarly evaluated with tensors T ~ ) and T~)~ where T~)~... = V~V#Vy . . . R -1. Since the vector f rom 1 to 2 is minus the vector f rom 2 to 1 T ") = ( - - 1 ) n T TM where n is the order of the tensor [33].

In general the dipole, quadrupo le and octopole m o m e n t s # , , 0,# and f~,#y induced in a molecule by an electric field Ea, field gradient E~r and gradient of field gradient E~ra may be wri t ten

I t 11~ ~t?- #~ = %# E# +-~A,#~E#y + 15a,,~qffy~#y~ -]- . . . , ( 5 . 8 )

0,# = Ay,t~ Er + C,#ra E'ra + . . . , (5.9)

fl,#~ = B~#~ E~ + . . . . (5.10)

In the calculat ions which follow the molecular d imens ions are cons idered to be m u c h less than the wavelength of the light, so that f ield-gradient effects in the l ight-wave fields are disregarded. On the o ther hand, gradients in the intermolecu- lar fields may be significant and are considered. T h e differential polarizabil i ty ~'~ in t roduced in (5.4) is used for c~# in lower-order terms, bu t in the h igher -o rder terms non- l inear effects p roduced by powerful in termolecular fields o f pe rmanen t mul t ipoles are neglected.

Wi th the use of the above equations, and the aid of the T- tensors we write

~ ( 1 ) ,/~ 1)~(2)ff rp(1)/~(2) ]b-~2)# 1 "/~(1)~,{2) ]~(2) ]~(2)~_

rp(1)ev(2)q~(2)N(1) _~ lr/~1) ZI(2) "/X2) /v(1)A:

r/-(1)~(2),'p(2)~(1)qr~(1)~(2)~2 l q ' ~ l ) Zl(2) ~_ + a bY ~ Y ~ x ~r ~ 4 ~ ~ q~t~rt2 ' t~02 - - 3 a # 7 ~ ~ x e 7 6 ~ ' 0 ~

_.I,~(11 A(2),~(2),~(1)~ 1,/,(1) /'-,(2) r~2) N(1)A2

i q~Dt~(2) 7,(2) ~(1)# _2Lq,(D ~(2)

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50 R . C . Burns e t a l .

and ~# !rp(1) d(2) rr(2) ~O)P_

h~-(1) = 1) #(2)# ..[_ r/~l) ,v(2) "/'~2),v(1).~_ + 3 ~t B3,a z~tae~b ~ e~bq ~T/2 u'02 ~" B3, 3,a ~ae t~Oe ~ B3,a ~ae .z e4, ~4~/'t~ OT/

!,at,(1) A(2)~0~ I,~(1) A(2)'T~2)N(D# - 3 - #3,~e ~'~ae - 3 - B3,ae ~-~ae" ~2 ~'o~. (5.12)

Subs t i tu t ion of (5.11) and (5.12) into (5.4) and (5.3) and averaging over all orien- tat ions of the light wave yields

a 8 ~ e~ -- o~ o = (~(i) _ O~o) + !~(~)~r~l)M2) !,,(1)~r(1)R(2) ~(2)

!A,(I )-p(1)~,(2) ]~(2) J7(2) !,v(I) -T~(1),v(2) ,'/~2),v(I) -l- 6~# ~ #3' ~'3,~ae za ~ e + 3~# ~ B~' ~,a ~ ~e ~e~

_l_ _i N(1) "T,(1)M(2) "/~2)M(I) qr,(l ),v(2 )

_I~(1)'/~(1) ZI(2) ,T~(2) ~(1) IN(1)"/~1) ,9(2) q~2)#(1) "at" 9~'~B ~ B3, z'3,ae .t 6er u~b~ - - 9~tB ~t p3,a zxe3,a" e~ ~

I_.N(1),'~(1) n(2) IN(1)"/~l) /"(2) "~2) N(1) - - 9 t ~ f l ~ P3,a z'ta~,6 - - 9~atP ~t Pra ~-'3,ae~b ~ c O 2 ~ 2 ~

1.,v(1),T,(t) i~(2) ,/',(2) ,. , ,(1) 1,,,,(1),-i~1 ) g],(2) dr. 4 5 ~ B .[ P3, x"3,ae4) .t as4)). ~2a "q- 4 5 ~ a p " P3,ae aJ3,ae~

! ,q(1) lp(1) 1/'~1) ]~(1)q.(1),,,,,(2) + 3/"~3, " 3' "t- 3 P a B 3 , .,t. 3, ~t pa ~a~ + �9 �9 �9

1_.~,(1) ]~",(1)]~(1) _14,(1) ]b-'~(1) "l- 6r ~-a ~ 3~/"a~ya ~" 3,a

_1 ,d(1) ,'/'~1) ,..,(2) 1 , d ( l ) ,'ir~l) ,v(2)'T,(2),.v(1) "1- 9.~,-a#3' ~ Bya '.",a~ -I.- 9.~.~#~, �9 #3,a '.',ae ~ e4~ ~'~,~

_:L z/(1) ,T~I) j ( 2 ) qr~(2) ~(1) __1A(1) "T'(1) z$(2) -F 27~.%tp3, ~t B3,a zJ'ae4," e4,~ ~,t~ -- 27w~By ~ B3,~e zx~zae

J1 zl(1 ) ,7,(1) z l (2 ) r l " (2) . ( l ) (5.13) -- 27~a~B3, a B3,ae xatbar. ~ d p A ~ ; ~ "

To proceed it is necessary to introduce the explicit forms of parameters like ~l)~B, ~#3,,~) F(~ x) and /~). Equation (5.13) simplifies considerably for molecules of high symmetry requiring fewer independent tensor components to describe their properties. The spherically symmetric inert gas molecules in their ground state possess no zero-field mul t ipole m o m e n t s of any order [34] so that the b-~ ) and F(~ in (5.13) are identically zero. T h e third rank polar tensor A~#r has no non-zero

~0) is isotropic, so that componen t s , and the polarizabil i ty tensor ~,~#

ot i) = o~(i) ~o, p (5.14) ~B where

r iN( i )

Also, for these spherically symmet r i c molecules the tensor C ~ has only one independen t componen t , and we write

C(2) = 2.}C(r ($a~, + 63,,~ ~ae) -- �89 ~Se~, (5.15) 3,ae4,

With these simplifications equat ion (5.13) reduces to

( ~ t ~ ) 20~ (1)3 2~X (1)4 100~(1)2C

de0 e~ - ct o = (0t (~) -- ~t0) + (4/teo)2R6 + (4~Zeo)aR9 + ( 4 ~ t e o ) 2 R S . (5.16)

Unti l recently only the second term in (5.16) p ropor t iona l to ~3R-6 has been included in calculations of Be for spherical and quasi-spherical molecules. T h e third te rm in R - s conta in ing the h igher -o rde r four th rank C- tensor descr ibing field gradient effects in the in termolecular fields was first derived and discussed by Buckingham, Mart in , and Wat t s [35] ; and Logan and M a d d e n [36] were first to demons t ra te that its inclusion in Be led to a substantial addit ional contr ibut ion.

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Refractivi ty second virial coefficients 51

5 . 1 . 1 . [�89 12('L') - - 0~0" ] for axially symmetric molecules

T h e relative configuration z of two axially symmetr ic molecules may be speci- fied by the four parameters 01, 0z, q~ and R [33] shown in figure 4. Here R is the distance between centres, 0 t and 02 are the angles between the line of centres and the dipole axes of molecules 1 and 2, and ~b is the angle between the planes formed by the molecular axes and the line of centres. Also, if l~ i) is a unit vector along the axis of molecule (i), and 2~ is a unit vector along the line of centres directed f rom 1 to 2

and

l~1)2~ = c o s 0 t , l~2)2~ = - c o s 02 (5.17)

l(1)/(2) = COS 012 = - -COS 01 COS 02 "~- s i n 01 sin 02 cos ~b. (5.18)

I t has been shown by Buckingham [33] that for axially symmetr ic molecules the proper ty tensors may be expressed in terms of/~). For example

where

(5.I9)

or,) • and ~c=(~g~ - ~ ~ 1 7 6 ~11j /~ . (5.20)

Use of (5.19) together with other results listed by Buckingham [33] in equation (5.13) and substitution of the explicit forms f rom [33]

and

F(czl ) qr~(1),,(2) _.1 ,-p(1) /3(2)

]~(1)-~- qr~l) ,,(2) l_.qr,(l) /3(2) ~fl'/Py - - 3zcp3,6uy6

(5.21)

( 5 . 2 2 )

for the field and field gradient at 1 due to dipoles and quadrupoles on molecule 2, lead after lengthy but simple manipulat ion to the result

~ r e~ - Oto = n ~ l A . , (5.23)

/ / y

R

Figure 4. The coordinates R, 01, 02 , and ~ describing the configuration of two axially symmetric molecules.

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52 R . C . B u r n s et al.

w i t h

A1 = 0~(1) -- ~o;

0~ 2 A 2 -'l~(1)r/"(1)ev(2) - - ~ R - 3 { K ( 1 - - x)(3 COS 2 0 t "3 L 3 COS 2 02 - - 2)

= 3w,,atfl ~t f ly ~yo~ - - 4rteo

+ 3x2(2 cos 2 01 cos 2 02 -- s in 01 cos 01 sin 02 cos 02 cos ~b

- - sin 2 01 sin 2 02 COS 2 (~)} ;

~3 A 3 1N(1)'T'(1)N(2)qr'(2)'v(l) R - 6 { 2 ( 1 -- ~c) a + to(2 + to)(1 -- x)

= 3 ~ a f l ~ f ly u~y 6 x ~ t w e a (47~e0)2

X (3 cos 2 01 + 1) + tO(1 --/<)2(3 cos 2 02 + 1) + 3/r -F K)

x (2 cos 01 cos 02 + sin 01 s in 02 cos (~)2};

A4 1N( 1 ) r/~ 1),.v(2) ,/~(2 ),..( 1} qr~(1 )~(2) ~ - 3 ~ . a f .t f ly ~ .?~ .t be ~'etp z ~b2~.Aa

O~ 4

- (4n%)3 R - 9 { 2 + 6x(3 cos 2 01 + 3 cos 2 02 -- 2)

n t- K3154(COS 2 01 -'[- COS 2 02 - - COS 4" 01 - - COS 4 02)

- 324 cos 2 01 COS 2 02 "[- 243(COS 2 01 COS 4 02 + COS 4 01 CO$ 2 02)

- 8 + 81 cos 012(cos 01 cos 3 02 + cos 3 01 cos 02)

- 108 cos 01 cos 02 cos 012 - 12 cos 2 012 ]

+ r4127(cos 4 01 + cos 4 02) - 18(cos 2 01 + cos 2 02)

- - 243(cos 2 01 CO$ 4 02 + COS 4 01 COS 2 02 - - CO82 01 COS 2 02)

-- 729 cos 3 01 cos 3 02 cos 012

- - 81 cos 012(cos 01 CO$ 3 02 "3V CO$3 01 COS 02)

+ 945 cos 01 cos 02 cos 012 -- 729 cos 2 0t cos 2 02 cos 2 012

+ 24 cos 2 012 -- 243 cos 01 cos 02 cos 3 012 -- 27 cos 4 012]};

As • Ft:) = 3/Jaa~,

5 #fl R _ 3 { 2 cos 01 cos 02 + sin 0 x sin 02 cos ~b} 5 0fl 9 4rCeo 6 4he0

• R - 4 { 3 cos 2 02 cos 01 + 2 cos 02 s in 01 s in 02 cos q~ -- cos 01};

1 #~ ..L,.,,(1) ,"p(1) R( 2 ) ~[7(2) A6 = 3 ~ f * f~ 'yae - e = 3 (4he0) 2

x R - 6 { f l • -- x)(3 cos 01 cos 02 -- c o s 012 )

+ 9x(6 cos 3 01 cos 02 + cos 2 01 cos 012

- - 2 cos 01 c o s 02 - - c o s 012) '] --]- (]~ll - - 3fl•

x [(1 -- x)(3 cos 2 02 -- 1)(3 cos 01 cos 02 + cos 012)

+ 3 tc ( - -9 cos 2 01 cos 2 02 cos 012 -- 6 cos 01 cos 02 cos 2 012

- c o s 3 0 1 2 ) ] } ;

(5.24)

(5.25)

(5.26)

(5.27)

(5.28)

(5.29)

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Refract iv i ty second virial coefficients 53

A7 IR(D ~(*)9~(*),,(2) = I pCt

= 3~'~t~)' ~) ' ~ #~ ~ 3 (4he0) 2

x R-6{f l • - x)(3 cos 01 cos 02 - co s 012)

+ 9 x ( 6 cos a 02 cos 01 + cos 2 02 cos 012

- 2 cos 01 cos 02 - cos 0 , 2 ) ]

+ (fill - 3fl• - x)(3 cos 2 01 - 1)(3 cos 01 cos 02 + cos 012)

+ 3 t c ( - 9 cos 2 01 cos 2 02 cos 012 - 6 cos 01 cos 02 cos 2 012

- cos 3 0~2)]} ;

1 ~t A8 _ *_A(1) -r(1) ,,(2)

- 9~-~#)'-#)'~ ~ - 3 (4=eo) R-4{(~A II - 2A•

x [(1 -- x) (5 cos a 01 -- 3 co s 0 t ) + 3 x ( - - 5 cos 2 01 cos 02 COS 012

"-~ COS 02 COS 012 - - 2 cos 01 cos 2 012),1

+ 6~cA• cos 01 cos 2 02 - cos 01 + 2 cos 02 cos 012)};

1 ct A9 1,v(1)q~l) ,~(2) .~-- R - 4 { ( a A I I - 2 A •

= - 9~,~tfl Jt//),6 zz'/~)'o 3 (4xe0)

x [(1 - ~ ) ( - - 5 cos 3 02 + 3 cos 02) + 3x(5 cos 01 cos 2 02 cos 012

- - COS 01 COS 012 -~- 2 cos 02 cos 2 012)'1

+ 6 x A • cos 2 01 cos 02 + cos 02 -- 2 co s 01 cos 012)};

2 ~2 A10 -_2~ t l )~ l ) a t2 ) , r t2 )0c~1) = 3 (4~eo) 2 R-7{(3AII

- - 9t~ctfl " 1 3 ) ' ~ ) ' 6 e * 6e4, -- 2 A •

x [ - - 4 ( 1 - x) 2 cos a 02 + 3x (2 + r ) ( - 1 5 cos 2 01 cos a 02

+ 3 cos 2 0 t cos 02 -- 11 cos 01 cos 2 02 cos 012

+ cos 01 cos 012 - 2 co s 02 cos 2 012)]

+ 6 A • - ~c) 2 c o s 0 2 At- x(2 + x)

x ( - - 4 cos 2 01 cos 02 - - c o s 01 cos 012 - - c o s 02)'1};

1 ct 2 A 11 I_ zJ(1) q'~l) ~(2)r/~2)~,(1) R-7{(2~AII - 2 A •

x [(1 - x )24 cos 3 0 t + 3x(1 - x ) (4 cos 3 01

+ 15 cos 3 0 a cos 2 02 + 11 cos 2 0 t co s 02 cos 0 t2 - 3 co s 0 , cos 2 02

+ 2 cos 01 cos 2 012 - cos 02 cos 012 ) + 9x2(15 cos 3 01 cos 2 02

+ 11 cos 2 01 cos 02 cos 012 - 3 cos 01 cos 2 02

+ 2 cos 01 cos 2 012 - cos 02 cos 012),1

+ 2 A . [ 3 x ( 1 - x ) (4 cos 3 01 + 4 cos 01 cos 2 02

+ cos 02 cos 0 t2 + cos 0 t ) + 9x2(15 cos 3 01 cos 2 02

+ 11 cos 2 01 cos 02 cos 0 ,2 - 3 co s 01 cos 2 02

+ 2 cos 01 cos 2 012 - cos 02 cos 012) + 6(1 - x) 2 co s 0t-I} ;

(5.3o)

(5.31)

(5 .32)

(5 .33)

(5 .34)

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54 R . C . B u r n s et al.

A~ 2 __t a (1 ) ,T,(1) ,~(2) ,/,(2),,,(1) = - - 2 7 " ' a # r " / D & ' ~ & " r = 9 ( 4 ~ 0 ) 2 R - s { [ ( 1 + 2x)

x (~AII -- 2A• 2 + 3K(3AII A • -- 4 . 4 2 ) ] ( - - 1 0 5 cos a 01 cos 3 02

+ 15 COS 3 01 COS 02 "~- 15 COS 01 COS 3 02

- - 95 cos 2 01 cos 2 02 cos 012 - 26 cos 0t cos 02 cos 2 012

- - 3 cos 01 cos 02 + 5 cos 2 01 cos 012 + 5 cos 2 02 cos 012

-- 2 cos 3 012 -- cos 012) + [(1 + 2x)(aAII A • - 4 A 2) + 1 2 x A 2]

x ( - 2 5 cos 3 01 cos 02 -- 9 cos 2 01 cos 012 -- 3 cos 01 cos 02

- - 3 cos 012 ) + (1 -- ~c)(3.Z/ll A • - 4 A 2 ) ( - 2 5 cos 01 cos 3 02

- 9 cos 2 02 cos 012 -- 3 cos 01 cos 02 -- 3 cos 012)

+ 4(1 - x ) A 2 ( - 3 6 cos 01 cos 02 -- 12 cos 012)};

1 ~ t A t 3 l z l ( 1 ) ,/N1) zl(2) qr~2) N(1) = 27"~a%tpy "t p),6za-6e~ J" 8dp2 ~'2ot =

3 (4r~80) 2

x R - 8 { [ ( 1 + 2~)(3AII -- 2.4• 2 + 3x(3AII .4• -- 4 .42)]

• ( - - 2 5 cos 3 01 cos 3 02 + 5 cos a 01 cos 02 + 5 cos 01 cos a 02

- 20 cos 2 01 cos 2 02 COS 012 - - 4 cos 01 cos 02 cos 2 012

- - cos 01 cos 02 + 2 cos 2 01 cos 012 + 2 cos 2 02 cos 012)

+ [(1 + 21c)(3AII .4• - - 4 . 4 2) + 12A2 ~]

X ( - - 5 COS 3 01 COS 02 - - COS 2 01 COS 012 + COS 01 COS 02

+ cos 012) + (1 -- x)(3AII A • - 4 A 2)

X ( - - 5 COS 01 COS 3 02 - - COS 2 02 COS 012 "~- COS 01 COS 02 COS 012 )

+ 4.42(1 -- x ) ( - - 4 cos 01 cos 02 + 2 cos 012)};

1 R - 5

AI 4 = I_La(t) ,'/"(1) zJ(2) {(.~AII _ 2A• - - 2 7 2 x ~ f l y ~ fly6e z~t~tdie = 9 (4he0)

x (35 cos 2 01 cos 2 02 COS 012 - - 5 COS 2 01 COS 012

- - 5 cos 2 02 cos 012 + 20 cos 01 cos 02 cos 2 012 + 2 cos 3 012

+ cos 012 ) + (3.411Ax -- 4 . 4 2 ) ( - - 3 5 cos 3 01 cos 02

-- 35 cos 01 cos 3 02 -- 15 cos 2 01 cos 012 -- 15 cos 2 02 COS 012

+ 30 cos 01 cos 02 + 6 cos 012)};

Ct 2 A15 J,.,v(1)qN1) /'-,(2) ,'/~(2) ,v(1) __ R - 8 { ( C 3 3 3 3 ..p 8C1313

+ 8Cl111)1-1 + 2~c 2 + 2/c(2 + to)(3 cos 2 01 -- 1)]

"~ ~8(5C3333 "4- 4C1313 - 8 C l l t t ) [ 1 6 ( 1 + 2tc2)(3 cos 2 02 - - 1)

+ x(2 + x){4(3 cos 2 02 -- 1)(6 cos 2 01 -- 1)

(5.35)

(5.36)

(5.37)

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Refractivity second virial coefficients 55

+ 12 sin 2 01(3 sin 2 02 cos 2 ~b -- 1) + 18 sin 201 sin 202 cos qS}]

+ ~(2C3333 - 4C1313 + Cl111)[5(1 + 2~2)(35 cos 4 02

-- 30 cos 2 02 + 3) + •(2 + K){(27 cos 2 01 - 5)

x (35 cos 4 02 -- 30 cos 2 02 + 3) + 12(cos 2 01 -- 1)

x (5 cos z 02 -- 1) + 45 sin 201 sin 202 cos ~b(7 cos 2 02 - 3)

+ 60 sin 2 01 sin 2 02 cos 2 ~b(7 cos 2 02 - 1)}]}; (5.38)

= - - ~(~3333 "k- 273311) } g/,,~6 12 (4/~eo) 2 R-6{Y -- 1

8 #2 • {1 + 3 cos ~ o} ~ 5 ( 4 ~ o ) 2 R - " { ~ + ~ 1 1 - ~1111}

• {(cos 01 cos 02 + �89 sin 01 sin 02 cos ~b)2}; (5.39)

A17 iA(1) = 3w,z,~,,~ F(~g )

- - ~ R-' :I(~D3333 + 2(~1133 - - (J)1111 - - (J)2211 - - (J)3311) 41re 0

x (3 cos 2 01 cos 02 + 2 cos 01 sin 01 sin 02 cos ~ - cos 02). (5.40)

In the de r iva t ion of equa t ions (5.25) to (5.40) t e rms of up to o r d e r four t een in the t enso r suffixes have been re ta ined . T h e t e r m Alo is the s u m of two t e rms , n a m e l y i 'v(1) 'T'(1)d(2) ' / ' (2) ,v(1) and !,~(1)-p(1) zl(2)q-~2)N(1) 9~p ~ ~ asr ~r - - 9 ~ p * ~ '*~v~" *~*~ in (5.13), and s ince these have the ident ica l expl ic i t form, Al0 is twice the c o n t r i b u t i o n of e i ther t e rm.

T e r m s con ta in ing the B - t e n s o r have been omi t t ed , s ince no e x p e r i m e n t a l or ca lcu la ted values of the B - t e n s o r c o m p o n e n t s are avai lable for use in the ca lcu- la t ions.

In the work which fol lows the in t r ins ic po l a r i zab i l i t y of a mo lecu le is a s sume d to be i n d e p e n d e n t of m o l e c u l a r sepa ra t ion even at shor t range, so tha t A 1 = ~(1) _ _ 0( 0 is neglec ted .

5.2. Classical expressions for the intermolecular potential energy Ui 2(z)

T h e genera l fo rm of the i n t e r m o l e c u l a r po ten t i a l ene rgy U12(z) used in our ca lcu la t ions is [33, 37]

(5.41) u12(0 = uLj + u , . , + u,.o + uo.o + u , . i . , , + uo.i . , , + us,a,.

in wh ich ULS is the fami l ia r L e n n a r d - J o n e s 6 : 12 po ten t i a l

F/Ro\ 12 (5.42)

Uu, u, Uu, o and Uo, o are the e lec t ros ta t ic d i p o l e - d i p o l e , d i p o l e - q u a d r u p o l e and q u a d r u p o l e - q u a d r u p o l e in te rac t ion energ ies of the p e r m a n e n t m o m e n t s of the two mo lecu l e s ; Uu, indu and Uo, l.du are d i p o l e - - i n d u c e d - d i p o l e and q u a d r u p o l e - i n d u c e d - d i p o l e in te rac t ion energies . T h e an i so t ropy of r epu l s ive forces is rep-

r e sen ted by Ushape.

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56 R . C . Burns et al.

For interacting linear polar molecules in the coordinate system shown in figure 4 [33],

l Uu.~ = 4rce---~ {#2R-3(2 cos 01 cos 02 + sin 01 sin 02 cos ~b)}; (5.43)

1 Uu'~ = 4ne-----~ {3#0R-4[c~ 01(3 c~ 02 - - 1) + cos 02(3 cos 2 01 -- 1)

+ E sin 01sin 02 cos 02 c o s ~ b + E s i n 0 1 cos 01 s in0zcos~b]} ; (5.44)

1 U~176 = 4zre---o {~02R-5(1 - 5 cos 2 01 - - 5 COS 2 02 + 17 COS 2 01 COS 2 02

+ 2 sin 2 01 sin E 02cos 2 ~b+ 16 sin 01 cos01 sin 02 cos 02 cos ~b)}, (5.45)

1 U,.ina, - (4~re0)2 {--�89 cos 2 01 - 1) + (3 cos 2 02 -- 1)]}; (5.46)

and

Uo, ind # - - - - 1

t'r I "ne0 "z { - ~ s 0 Z R - S ( 4 c~ 0, + 4 cos 4 02 + sin g 01 + sin g 02)}.

(5.47)

The static polarizability cq is used in U~, i n d # and Uo, ind # to describe the quasi- static interaction. In equation (5.46) U t t , ind u has been written so that its unweighted orientational average is zero, and the or ientat ion-independent part is assumed to be incorporated in the R -6 te rm of the central-force potential ULj [37]. The shape potential is [38]

/ / R o ' ~ 12 Ushap e = 4De~--~) (3 cos 2 01 + 3 cos E 02 - 2), (5.48)

where D is a shape factor which can vary between --0"25 and +0"5 in order to ensure that the R-12 te rm is always repulsive at short range. A positive D corre- sponds to a rod-like molecule and a negative D to a plate-like molecule. For spherically symmetr ic molecules D = 0.

The intermolecular potential energy expressions given in equations (5.43) to (5.48) are directly applicable to pair interactions of linear dipolar molecules, but may also be used for non-polar linear and spherical molecules by setting the relevant multipole moments equal to zero. In our calculations we follow earlier workers [29, 37, 39] by treating CH3F and CHF3 as linear molecules with the dipole lying along the threefold rotation axis. Equations (5.24) to (5.40) and (5.43) to (5.47) are not strictly applicable to CHEF E which belongs to the CEv symmet ry point group, and no calculations have been a t tempted for this gas.

6 . E V A L U A T I O N O F BR B Y N U M E R I C A L I N T E G R A T I O N

6.1. Introduction Where experimental or calculated molecular propert ies are available the

expressions (5.16) or (5.24) to (5.40) for contr ibutions to [�89 - a0] have been substi tuted into the general equation (5.1) together with appropriate forms for

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Refractivity second virial coefficients 57

Ulz(z ) and va lues of B R at 632-8 n m have been ca lcu la ted b y n u m e r i c a l in tegra - t ion on a H e w l e t t - P a c k a r d 1 0 0 0 M c o m p u t e r for e leven of the twelve gases we e x a m i n e d expe r imen ta l l y . A lack of m o l e c u l a r data , as well as the c ompl i c a t i ons

ar i s ing f rom its low (C2v) s y m m e t r y p r e c l u d e d work on C H 2 F 2 . T h e ca lcu la t ions r e p o r t e d here were ca r r i ed ou t by G a u s s i a n q u a d r a t u r e wi th

the ranges of 01, 0 2 , and ~b each d i v i d e d into 10 in te rva l s ; and wi th a range for R of 0"1 to 3"0nm d iv ided into 64 in tervals . T h e e s t ima ted p rec i s ion is be t t e r than 0.3 pe r cent . In a c o m p r e h e n s i v e inves t iga t ion where k n o w n values o f in tegra l s wh ich cou ld be eva lua ted ana ly t i ca l ly were ca lcu la ted b y S i m p s o n ' s rule and by G a u s s i a n q u a d r a t u r e , we found tha t for func t ions of the type cons ide red in th is work, and for a g iven prec is ion , t yp i ca l ly ha l f the n u m b e r of subd iv i s ions pe r var iab le w o u l d suffice in the gauss ian p rocedu re , l ead ing to a saving in c o m p u t e r t ime b y at least a fac tor of 16 for f ou r -va r i ab l e in tegra t ions .

F o r the l inear and quas i - l i nea r molecu les , p a r a m e t e r s were unava i l ab le for the eva lua t ion of m a n y of the t e rms l i s ted in (5.24) to (5.40). H o w e v e r , to exp lo re the re la t ive m a g n i t u d e s of these c o n t r i b u t i o n s we have c ons ide r e d HCI , for wh ich the re is a c o m p r e h e n s i v e range of ca lcu la ted and o b s e r v e d m o l e c u l a r data. T h e s e resu l t s are d i scussed in w 6.3.

6.2. The spherical and quasi-spherical molecules

F o r the spher ica l iner t gas mo lecu le s and the quas i - sphe r i ca l mo lecu le s C H 4 , C F 4 and S F 6 it fol lows f rom (5.16) and (5.1) tha t

B R 4 N5 [ 2~ (')3 2~(x)4 10~(1)2C ] _ 3e0 (~(x) _ ~0) + (4~ze0)ZR6 + (4~ze0)3R9 + ( ~ j

x exp { - U 1 2 ( R ) / k T } R 2 dR

in wh ich we r ep re sen t UI2(R) by the L e n n a r d - J o n e s 6 : 12 po ten t i a l g iven in (5.42). H e r e the four c o n t r i b u t i o n s to B R are spec ia l ized fo rms of Ax, A3, A 4 and Ax5, and are r e fe r red to b y these labels .

As s ta ted ear l ie r it is b e y o n d the scope of th is work to cons ide r changes in the in t r ins ic po la r i zab i l i ty , and A 1 is neglec ted . M o l e c u l a r p a r a m e t e r s used in the eva lua t ion of the r e m a i n i n g t e r m s are given in tab le 5, and the ca lcu la ted and o b s e r v e d va lues are c o m p a r e d in tab le 6. In the cases of C F 4 and S F 6 the con t r i - b u t i o n A15 due to the C - t e n s o r cou ld not be ca lcu la ted owing to the lack of da ta

Table 5. Molecular parameters used to calculate B R of the inert gases, CH4, CF4, and SF 6 .

10*~ 106~ e/k R o Molecule C 2 m 2 j - 1 C 2 m 4 j - 1 K nm

Ne 0'4401 [13] 0"178 [-44] 35"7 [40] 0.2789 [-40] Ar 1"85 [-13] 1.213 [-45] 124.0 [-40] 0'3418 [-40] Kr 2'806 [,13] 1'819 [-46] 190'0 [-40] 0-3610 [-40] Xe 4-569 [-13] 2-960 [46] 229.0 [-40] 0"4055 [-40] CH 4 2"897 [13] 2"316 [-47] 184"5 [41] 0-362 [41] C F 4 3" 164 [13] - - 153-0 [-42] 0'47 [42] SF 6 4"983 [,13] - - 259-0 [-43] 0'5005 [--43]

t ~(v) values are optical polarizabilities at 632"8 nm.

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58

Table 6.

R. C. Bu rns et at.

A comparison of calculated and observed values of B R for spherical and quasi- spherical molecules for 2 = 632.8 nm and at room temperature (298'2 K).

1012BR

1012B~ 3 1 0 1 2 B ~ " 1012B~ Is m6mo1-2

Molecule m 6 mol- 2 m 6 mol- 2 m 6 mol- 2 Calc. Obs.

Ne 0"052 0"00065 0"010 0"063 -0 -14 4- 0.14 Ar 2"10 0-053 0.40 2-55 1'57 ___ 0-58 Kr 6-77 0.21 1-13 8"11 6"23 _ 1.55 Xe 22'08 0-79 2'93 25-8 25"50 +__ 2.85 CH 4 7"33 0.23 1.50 9"06 7-76 + 1.32 CF 4 4'17 0"068 - - 4"24 4"27 + 1-38 SF 6 16'05 0"34 - - 16"39 27"28 + 5"18

for these molecules. Moreover , it is po in ted out by Logan and M a d d e n that the C-values for argon, krypton, and xenon should be regarded as est imates rather than accurate values.

T h e r e is very little unce r t a in ty in the values of a, bu t repor ted values of the L e n n a r d - J o n e s parameters R 0 and e vary considerably . T h e values used in this work are bel ieved to be der ived f rom the most reliable data, usual ly viscosity

m e a s u r e m e n t s [37], bu t to i l lustrate the sensi t ivi ty of B R to the L e n n a r d - J o n e s parameters , BR values calculated for some of the gases us ing different pairs of repor ted R 0 and e are shown in table 7. A s t r iking feature of table 6 is that the h ighe r .o rde r t e rm Ax5 (propor t ional to the C- tensor which describes the excess field at molecule 1 due to a quad rupo le i nduced on 2 by the field grad ien t of a

dipole induced on 1 by the l ight wave) makes a con t r i bu t i on to BR typical ly one-f i f th the size of the famil iar K i rkwood f luctuat ion t e rm which was c o m m o n l y

presen ted as the only signif icant con t r ibu t ion . Logan and M a d d e n [36] were the first to recognise the impor tance of the C - t e r m in their t r ea tmen t of B, for the

Table 7. Sensitivity to the Lennard-Jones parameters of the calculated B R values.

1012BR m 6 tool- 2

e/k R o

Molecule K nm Ref. Calc. Obs.

Ne 35"7 0'2789 [40] 0"063 -0"14 +_ 0.14 60.9 0.2648 [48] 0.071

Ar 124.0 0'3418 [40] 2"55 1-57 + 0-58 152.8 0.3292 [48] 3.00

Kr 190.0 0.3610 [40] 8"11 6"23 ___ 1'55 206-4 0.3522 [48] 9.01

CH 4 184.5 0.362 [41] 9.06 7.76 + 1.32 137.0 0.388 [40] 6.74 144.0 0-380 [49] 7.30

SF6t 259"0 0-5005 [43] 16-39 27"28 + 5"18 200.9 0.551 [403 11.00

t Calculation excludes contribution of C-term.

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Refractivity second virial coefficients 59

Table 8. Comparison of observed values of B a for spherical and quasi-spherical mol- ecules with those calculated using a modified (7 : 28) and normal (6 : 12) Lennard- Jones potential. Modified Lennard-Jones parameters are from I-51].

1012Ba

m 6 m o l - 2 ~/kt Rot

Molecule K nm Calc. (7 : 28) Obs. Calc. (6 : 12)

Ar 240 0-315 3'38 1"57 + 0"58 2"55 Xe 470 0-363 49'75 25"50 ___ 2"85 25"8 CH 4 310 0"340 10'88 7"76 + 1-32 9.06 CF4t 315 0"433 6"41 4-27 + 1"38 4'24 SF6t 414 0"503 18'72 27"28 + 5'18 16"39

t Calculations exclude contribution of C-term.

iner t gases. O u r da ta in tab le 6 conf i rms the i r conc lus ion tha t ca lcu la ted values of Ba cons i s t en t ly exceed the o b s e r v e d values b y an a m o u n t wh ich t ends to be lower as one p roceeds f rom neon to xenon. T h i s t r e n d cou ld be due to nega t ive non- classical s h o r t - r a n g e c o n t r i b u t i o n s to B a (neg lec ted in th is work) w h i c h b e c o m e p rog res s ive ly greater , the smal le r the molecu le .

N o C - t e n s o r va lues were avai lable for S F 6 , and the Ax5 t e r m has been omi t t ed . H o w e v e r , even if one es t ima tes a C - t e r m c o n t r i b u t i o n a m o u n t i n g to the typ ica l 20 per cent of the K i r k w o o d f luc tua t ion t e rm, the B a value (wi th R 0 = 0"5005 n m and e/k = 259 K) w o u l d be 19"60 • 10-12 m 6 m o l - 2 which

is stil l well ou t s ide the e x p e r i m e n t a l l imi ts of the o b s e r v e d value, 27"28 + 5"18 x 1 0 - 1 2 m 6 m o 1 - 2 .

In the i r t r e a t m e n t of the second K e r r vir ia l coeff ic ient and the p r e s su re - i n d u c e d depo l a r i zed l ight sca t te r ing in t ens i ty of S F 6 , B u c k i n g h a m and Clarke H u n t [50] r e p o r t e d i m p r o v e d a g r e e m e n t be twe e n e x p e r i m e n t and t heo ry wi th the use of a mod i f i ed L e n n a r d - J o n e s po ten t ia l . O u r ca lcu la ted resul t s wi th a 7 : 28 mod i f i ed L e n n a r d - J o n e s po ten t i a l due to M c C o u b r e y and S ingh [51] are shown in tab le 8. T h e K i r k w o o d f luc tua t ion t e r m rises f rom 16"39 x 1 0 - 6 m 6 m o 1 - 2 to 18"72 x 1 0 - 6 m 6 m o 1 - 2 and an e s t ima ted C - t e r m c o n t r i b u t i o n of 20 pe r cen t of this t e r m yie lds a Ba of 22"8 x 1 0 - 6 m 6 m o 1 - 2 , wh ich is now wi th in the exper i - men ta l l imi ts . F o r Ar , Xe, C H 4 and C F 4 , however , the mod i f i ed po ten t i a l leads to poo re r a g r e e m e n t be tween t h e o r y and e x p e r i m e n t .

6.3. Linear and quasi-linear molecules

In c e n t r o - s y m m e t r i c l inear mo lecu le s on ly the t e rms A1, A 2 , A 3 , A 4 and A15 c o n t r i b u t e to BR, b u t for po l a r l inear molecu les all t e rms A 1 to Ax7 con t r ibu te . N o C - t e n s o r c o m p o n e n t s are k n o w n for any of the l inear molecu les inves t iga ted , and AI5 cou ld not be eva lua ted . L ikewise , the unava i l ab i l i t y of va lues for the A - t e n s o r c o m p o n e n t s p r e c l u d e d eva lua t ion of the c o n t r i b u t i o n s f rom A 8 to A14 to BR for the po la r molecules . H o w e v e r , in o r d e r to inves t iga te the re la t ive m a g n i - tudes of as m a n y t e rms as poss ib le , a de ta i l ed s t u d y was m a d e of HC1 for wh ich there is a c o m p r e h e n s i v e range of ca lcu la ted and o b s e r v e d m o l e c u l a r da ta ( s u m m a r i z e d in table 9). Values of the r e l evan t C - t e n s o r c o m p o n e n t s of H C I were not avai lable , and these were e s t ima ted by a scal ing of the values quo t ed b y Rivai l and Car t i e r [52] for H F in p r o p o r t i o n to the re la t ive va lues of the ~- and A -

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60 R . C . B u r n s et al.

Table 9. Summary of molecular propert ies of HC1.

a s = 2"867 x 1 0 - 4 ~ - 1 . . [53] re(v) 2"893 x 1 0 - 4 ~

~A~x(v) 0"346 x 1 0 - 4 ~ - ~ [54] lc = 0-04

fill = 0'009 X 1 0 - 5 ~ -2 [533 f l i = 0 - 0 1 5 • 1 0 - 5 ~ -2 [53]

fl = 3(2fl• + fill) . 0"0234 x 1 0 - 5 ~ -2

/~ = 3-646 x 1 0 - S ~ [55] 0 = 12 -4+ 0.4 x 10-4~ C m 2 [553

All = 1-16 x 1 0 - S ~ 3 [53] A• = 0'133 x 1 0 - 5 ~ a [53]

3~Cl111 = 0"8144 x 1 0 - 6 ~ -1 ~C3333 = 1"0504 x 1 0 - 6 ~ -1 3~C1313 = 0'6588 x 1 0 - 6 ~ -1

D = 0"01 (arbitrarily chosen) elk = 191.4K )

[56] R o 0.3641 nm

2 = 632.8 nm. Est imated values by scaling of values quoted for H F by Rivail and Cartier [52].

t e n s o r c o m p o n e n t s for HC1 a n d H F . T h e s h a p e - f a c t o r D was a r b i t r a r i l y c h o s e n

s u b j e c t to t he c o n s t r a i n t s l i s t ed a f te r e q u a t i o n (5.48).

T a b l e 10 l ists t he c o n t r i b u t i o n s to BR for HC1 f r o m t e r m s w h e r e ava i l ab le da t a

p e r m i t t e d ca l cu l a t i ons . T h e l a rges t c o n t r i b u t i o n ar ises f r o m the K i r k w o o d f luc-

t u a t i o n t e r m A 3 , and we n o t e t h a t a s ign i f i can t c o n t r i b u t i o n (nea r ly 10% of t h e

c a l c u l a t e d va lue o f BR) aga in ar ises f r o m A15 w h i c h c o n t a i n s t h e C - t e n s o r d e s c r i b -

i ng f i e l d - g r a d i e n t effects . T e r m A2, p r o p o r t i o n a l to ~ZR-3 is n e x t in o r d e r o f

i m p o r t a n c e , a n d we n o t e t h a t t h e h i g h - o r d e r t e r m A4 , p r o p o r t i o n a l to 0c4R -9 ,

o m i t t e d in ea r l i e r c a l c u l a t i o n s , m a k e s a s i gn i f i c an t c o n t r i b u t i o n o f n e a r l y 3 p e r

cent .

Table 10. Contr ibut ions to B R for HCI f rom terms in equations (5.25) to (5.40).

1012BR(An) Percentage

T e r m m 6 m o l - 2 contr ibut ion

A z 0"63 4"7 A 3 10'3 76'64 A 4 0"37 2.75 A 5 0.18 1"34 A 6 0'0013 0"0097 A 7 0.0011 0-0082 A s 0-20 1 "48 A 9 0'22 1 "64 A10 0"18 1'34 Al l 0'088 0-65 Ax2 -0 -0034 --0-025 A13 0.0014 0"010 A14 - 0 ' 0 0 0 9 3 --0"0069 Als 1-27 9-45

B R = 13.44 x 10 -12m6mo1-2

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Refractivity second virial coefficients 61

The established importance of field-gradient effects described by the C-tensor leads to the question whether field-gradient effects described by the `4-tensor in polar linear molecules play a role of similar significance. Terms As, A 9 and A10 are each linear in .4, and each give rise to contributions of just over 1 per cent, whilst All, A12 , Ala, each quadratic in A, make very small, almost negligible contributions. Clearly, definitive calculations of B R for linear molecules will be possible only when reliable values of C-tensor and A-tensor components are available, a situation which does not at present prevail.

Since the leading terms describing hyperpolarizability effects, As, A 6 and A 7 are relatively small, it was considered unnecessary to use the perturbed, or differ- ential polarizability in higher-order interactions thus avoiding considerable pro- liferation in the number of terms beyond A17.

Table 11 contains the molecular parameters used in our calculations of B R for CO2, N2, CHaF , and C H F a .

Values of the C-tensors for these molecules are not available and for CHaF and C H F 3 where the `4-tensor components contribute, these are also unavailable.

Experimentally determined values of 0 are available [57] for N 2 and CO 2 but not for CHaF or C H F a. The quadrupoles and shape factors D of CHaF and C H F 3 used in this work are the best-fit parameters from an iterative calculation of Bp by Copeland and Cole [37]. They comment that 0 = 7"7 x 1 0 - 4 ~ 2 for CHaF is reasonable while the value of 15"0 x 1 0 - 4 ~ 2 for C H F 3 is possibly large. The values of D, viz. 0"254 for CH3F and - 0 - 1 for CHF3 , are consistent with the condition that 0 < D < 0"5 for rod-like molecules (CHaF) and -0"25 < D < 0 for plate-like molecules (CHFs).

A shape factor of 0"2 for CO 2 was determined by Datta and Singh [41] in their calculation of multipole moments from viscosity and second pressure virial coefficients. There is no reported shape factor for N 2 and the value of 0"2 used here was arbitrarily chosen according to the above constraint for rod-like mol- ecules.

Calculated contributions of the A 2 , A 3 , A4, and A 5 terms are shown in table 12 together with the total calculated values of B a . Also included in this table for comparison are our experimental values.

If D for nitrogen is assigned the values 0"1 and 0-4 the corresponding calcu- lated values of B R are 1"74 • 10 -12 and 0"71 x 10-12 m6 mole -2, the latter being in closer agreement with the experimental value of 0-74 x 10-12 m 6 mole-2

For carbon dioxide the calculated and observed values agree within the experi- mental limits, and agreement will probably be retained when the C-tensor contri- bution is added. However, there is a tendency in the remaining three gases for the calculated values to exceed the observed values by an amount significantly in excess of the large experimental limits, and the disagreement will probably be aggravated by the addition of the C-term contribution. We note from table 13, which lists a number of trial values of the relevant C-tensor components (estimated by scaling the corresponding components of HC1 in proportion to the polarizability tensors a) that the C-tensor contributions BR als for CHaF and C H F a were positive and substantial. Note, however, that CHaF and C H F 3 have only three-fold rotation axes, and therefore have more non-zero components of the fourth rank C-tensor than the linear molecules for which our equations have been derived ; so that some care must be exercised in the interpretation of these results.

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62 R . C . Burns e t a l .

r~

" 0

Z

i

@

I m

~,T

i,~ " - ' i ~

r 1 6 2 t " , l v ~

6 6 6 6 I

k . . ~ k . . ~ k . . d k . . ~

I

k . . d t . . ~

t " ~ t - - - ~ t " ~ t - - - ~

t ' q r t . . ~ k . . ~

eq

r~

o o r

e~

t~ tq

4 - - + +

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Table 12.

Refract ivi ty second virial coefficients 63

A comparison of calculated and observed values of B R for CO2, N2, CHaF and CHF 3 for ~. = 632'8 nm and at room temperature (298.2 K).

Molecule

1012B~2 1 0 1 Z B ~ 3 1012B~ lO~2B~ ~

1012BR m 6 mol- 2

m6mo1-2 m6mo1-2 m6mo1-2 m6mo1-2 Calc. Obs.

CO2 N2 CH3F CHF 3

--2-64 6-89 0"13 0 4-38 4'75 _+ 1"30 --0"59 2"05 0"04 0 1'50 0-74 _+ 0"65 --0'62 10"79 0"36 --0"87 9'66 4-32 + 1"87 --0"85 6'45 0-14 --0.52 5-22 2'54 + 1-35

Table 13. Estimated C-tensor components and corresponding calculated values of the C-tensor contribution Als to B R .

CH3F CHF 3 CHF 3

Cl111 x 106~ -1 0'85 1'10 0'90 C3333 x 106~ -1 1'06 0"900 1'10 C1313 x 106~ -1 0-67 0"75 0"75 B~ Is x 1012/m6mo1-2 1"30 0-548 0"570

We grateful ly acknowledge financial assistance f rom the South African C.S . I .R . We wish to thank Professor R. E. Raab for helpful comments . T h e appara tus on which our measuremen t s were made was designed and built at the Cambr idge Univers i ty Chemical Laborator ies , unde r the supervis ion of Professor A. D. Buckingham, who kindly allowed its t ransfer to the Univers i ty of Natal.

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