Indian Journal of Chemi stry Vol. 44A, July 2005, pp. 1339- 1344

Calculation of the second virial coefficients of non-spherical molecules in binary mixtures

Fariba S Hashemi"*, Ezat Keshavarzih & Parvaneh Maleki"

"Department of Chemistry, Faculty of Science. A lzahra University. 199389 1 167, Tehran. Iran hDepartment of Chemistry, Isfahan Uni versity of Technology, Isfahan , Iran

Email : shashemi @al zahra.ac.ir;shashemi46@yahoo.com

I?eceived 16 Novell/ber 2004; revised 3 May 2005

T he ~econd virial coefficients for simple molecul ar gases and their mi xtures of non-spheri ca l mo lecules have been ca lculated through a rev ised analysis of the effect of long-range non-spheri cal terms in the intermolecular potenti al functions. T he auxiliary functi ons in the pel1urbation terms for calculating the non-spherical contributions have been calculated numerically and tabul ated over a w ide range of temperatures. Hartree-Fock dispersion indi viduall y damped (HFDID I ) potential has been used as a core potential 10 calculate both the spherica l and non-spherical contributions 10 the second virial coefti cient of N2, O2, CO. and CO2, The cross-v i ri al coefficients in the mixtures of those molecules among themsel ves and with the noble gas (A r) are predicted with excellent agreement with experimental values. The results for pure gases show that the calculations are in good agreement w ith the experimental data. Examinati on of the dev iation plots show that the calculated values for the interacti on second virial coeffi cients are in good agreement with the experimental values over the whole r~ nge of temperatures and al so reported values .

The second virial coe ffici ent as a pure two-body interaction depends upo n interactions between pairs o f molecules. The evaluation of the second vi rial coefficient of a moderately dense gas is straightforward if the intermolecular pair potential is accurately known. The potential energy of interaction between two po lyatomic molecules IS usually assumed to consi st of a spherically symmetric component plus a contribution due to the asphericity of the molecular charge distribution . The interaction of two molecules at long range comes from the dispersion interactions. T he most important term varies as r 6 and describes the dipole-induced dipole interaction. The theoretical treatment of the short­range repulsive forces between mo./ecules has proved much more difficult than the analysis of the long­range forces. For short range, electronic exchange due to overlapping of molecular e lectronic shell is very essential I . Buckingham and Pople2 have reported a numerical method to calcl! !ate second virial coef ficients for potential functions conta1llll1g spherical and non-sphericdl terms. This method has been used successfully to obtain virial coefficients of the some pure molecular gases3

-5

.

The properties of mixtures at low densities can be expressed in terms of the properties of its single pure

components plus functional s corresponding o nly to the possible binary interactions in mixture . Very accurate combining rules for the noble gas potent ial s have been devi sed6 and extended to mo lecu lar systems7

. The molecular rules do not attempt to predict the entire anisotropic potential energy function , but only the effective spherical parameters C l2 and at 2 corresponding to the potential-well reg io n, which is all that is needed for use with the principle o f con'esponding states.

Calculation of the second virial coeffici ent are reported here for the technologically and industri a lly important linear molecular gases of CO2, CO, N2, O2

and binary equimolar mixtures of those among themselves and with Ar which are related to the functions proposed by Buckingham and Pople and Hartree-Fock di spers ion individually damped (HFDIDl) core potential S.

Methodology The second virial coefficient involves two body

interactions9 and the pertinent statistical mechanical expression for it read as t

:

1340 INDIAN J CHEM, SEC A, JULY 2005

where f dWI and f d~ are written for integrations over the angular coordinates of the molecules, r is the distance between centers, V is the intermolecular potenti al , NA is Avogadro 's number, k is the Boltzmann's constant, v is volume, and Q is the normalization factor for the angular integration over dw. The intermolecular potenti al can be written as a spherical part and a non-spherical contribution. The spherical part, Vo( r ) , depends on the di stance between the molecul ar centers of mass and the non-spheri cal part depends on the relative ori entation of molecules. The contributions to the non-spherical potential can be regarded as the electronic, induction, dispersion and shape effects. The effects of different contributions are reported alI·ead/.4 . Calculation of the non-spherical part is done by an expansion of a Taylor series as:

II (- I) j . . exp(-Vns / kT) = 1+ I -.,-evns / kT) l ... (2)

j =1 J.

The first term in the expansion gives the spherical contribution to the second virial coefficient and the others give the non-spherical contributions . By putring the series expansion into Eq.( I), the results for rad ial part are expressible as a set of dimensionless in tegrals, fiT' ), which are related to the proposed functions by Buckingham and Pople2 and are defined as:

J n -3( CYO)II-3 f= 2- 11 (-VO* )d =-- - x exp --*- x II 2 r T

11/ 0

... (3)

* ' d d h where T =kT!C, x =rk , Vo = VoIG, an £ an (J are t e potential well depth and the interparticle distance at which Vo=O, respecti vely. This integral is a function of temperature, which is usually evaluated nu merica llylo. Here, fiT' ) values have been calculated using HFDID 1 type core potentials8

, and are employed to calculate the non-spherical contri bution to the second virial coefficient of pure and binary mixtures of gases .

In the case of gas mi xtures, the virial coeffici ents Bmix(T, xi 1.12 are related to the gas composition and the intermolecular potenti al energy functions which characterize the different type of interactions which occur. For a binary mixture, the second virial

ff' . b' 11 3 coe IClent may e wntten as' :

. . . (4)

and ,

... (5)

... (6)

In these formulae, XI and X2 denote the mole fractions of the pure components in the mixture, and BI I and B22 their virial coefficient". BI2 is the interaction second vi ri al coefficient and it is re lated to the unlike interact ions in the mixture in the same way that BI I and B22 depend on the potential interactions. Also, TI 2 * =kT!c12 and C l2 and 0'1 2 are the mixed interaction parameters and (B"s\2 is reduced non­spherical contribution to the interaction second viri al coefficient. For the mixed-interaction functi onal (BO*)1 2' we have used the arithmetic mean of the functional s fo r gas I and 2 and the (B,,: ) I? is calcul ated explicitly according to theoreti cal form ul a for the various orientation-dependent interactions (e.g., dipole-dipole, dipole-quadrupole, etc.). The eq uations for the (Bo' ) and (B,,: )12 are already known4.5. IO. 13. 14 .

Results and Discussion HFDID 1 type has been used here as a core

potentials. Although, thi s potential has been developed for Ar, it is regarded as a reali sti c general model for Vo(r). The values of f n(T*) for thi s potential model are calculated by numerical integration for 0.5 ::; T*::; 10 and 6 ::; n::; 304

• These fu ncti ons are slowly varying, so that interpolation in the tables is easy. For T' > 10 the non-spherical contributions to the

Table I - T he parameters used for calcul ation o f the so.!cond viri al coeffic ients (Data from Ref. 13)

Gas fl O· a . co' K

CO2 0 0.85 0.0547 C. 268 1.89 CO 0.138 0.84 0.0404 0.090 1.86 N2 0 0.47 0.0357 0. 13 1 2. 18 O2 0 0. 14 0.0357 0.229 2. 27 A I' 0 0 0.0437 0 2.2 1

Table 2 - The e ffective spherica l parameters

Gas I:,jk (K) a'J (nm )

Ar-Ar 143.2 0.3350 O2-0 2 12 1. 1 0.3407 Nr N2 98.4 0.3652 CO-CO 98.4 0.3652 CO2-CO2 245.3 0.3769 COr Ar 182.2 0.3568 CO-Ar 117.6 0.3499 N2-Ar 11 7.7 0.3499 0 2-Ar 130.9 0.3378 COr 0 2 168.8 0.3596 Nr 0 2 108.8 0.3529 CO2-N2 155.0 0.3711 COr CO 155.2 0.37 11

HASH EM I el al.: SECOND VI RIAL COEFFICIENTS OF NON-SPHERICAL MOLECULES IN BINARY MIXTURES 134 1

second viri al coefficient are usually negligible compared to the corresponding spherical contribution . Also, below T* = 0.5, the convergence of series expansion, Eq . (2), is poor. Table I lists the material parameters needed for the calculation of the non­spherical contribution to the second virial coefficienes.16. Table 2 shows the effective spherical scaling parameters e;jk and a ij 13. Employing the values of JIl(T*) and the equations, as reported alread/, we have calculated the second virial coeffic ient for the pure dilute gases of N2, O2, CO, and CO2, The calculated values of the second virial coefficient for CO2 in comparison with the experimental results by Dymond and Smith l7 and with the previous calculations by Boushehri et al. 18 are shown in Fig. 1. The results are also compared with

W,---------------------------------,

~ ·w E \) .!:!. a:J" -100

-l W

-200 200 400 600 800

T(K)

Fig. I - Comparison of calculated second virial coefficients of CO2 in the present work ( _ ) with experimental ( 0 )17, previous calculations by Boushehri et al. ( /I,. )1 8 and ab initio calculation ( • )19.

4,------------------------------------.

o E "i) 0 u iii' <.0

·2

o 0 .. • •

• • • 00

• • o • o .

-4+--------.---------.--------.-----__ ~ o 200 400 800 800

T(K)

Fig. 2 - Plot of the deviations DB of the second vi rial coeffic ient for N2 [Ref. 17 {( . ) No.1 , (0) No.2, (_) No. 16, (0) No.6, ( A ) No. 7, (/I,.) No. 10, (0) No.14}].

the recent results that obtained from new intermolecular potential energy function for CO2 by ab initio calculations 19. Deviation plots of 6B = Bexp(D- Bcal(D versus temperature for CO, N2, and O2

are shown in Figs 2-4. Analyses of the results in Figs 1-4 show that our calculations are 111 good agreement with experimental data and other methods 18.20.

Our principal goal is to demonstrate the ability of HFD type potential and this theory2 to cope with binary mixtures like pure components. In the case of the mixture, we have used the values of In(T*), and the arithmetic mean for the spherical contribution of interaction virial coefficient (BO*)1 2 and the values of (Bns* )1 2 are calculated from the formulae, as reported earlier 13

.

.. .. .. ~ . . . !) u 0 u .;, .. #' 0 iii' • I!I i IX)

<.0 0

• '0 . • -4

o · ••

00

·8 0 150 300 450

T(KI

Fig. 3 - Same as Fig. 2 for O2 [Ref. 17 {( • ) No.3, ( 0 ) No.5, ( A ) No.7, (/I,.) No.6, ( T) No.9. (0) No.1, (_) No.2, (0) No.6b}].

0 E I:l u iii' <.0

5.0 ,------------------------------------,

2.5

0.0

-2.5

o o

o •••• 0 A 0

A 00

6 A 6 •••

o o

o

og 0 &

o

-5.0 +--------.--------...,---------.----------1 o 200 400 600 800

T(KI

Fig. 4 - Same as Fig. 2 for CO [Ref. 17 {( 0 ) No.5b, ( 0 ) No.7, (0) No.8, (/I,.) No. 10, ( . ) No.5 a}].

1342 INDI AN J CHEM, SEC A, JUL Y 2005

Tab le 3 - T he contribu tion of different terms to the second vi ri al coe lTicielll of CO2-CO at 303.2 and 6 13.25 K

Contr ibut ion A t T=303.2 K At T=6 13.25 K

B' ns(O 0)12 - 0. 1286 - 0.0343

B'"lp 0)1 2 - 1.68 x IO·.l -4.02 X 10-4

B*"Jp. indp)1 2 - I.74x I0-8 -4.78 x 10-9

S'"lO, illd 11)1 1 - 1.20 x 10-4 -3.92x I0-5

B"",(p 0, ind P)l l - S.94 x IO') - 1.76 x IO')

B' "J C6 aniS)l l - 6.5 I x I0·3 - 1.86 x 10-3

B' ",(p. ind fl x f) , ind 11)1 1 - 1.97 X 10.6 -S .89x I0·7

B' ",(0 () X C6 an is)' 2 + 9.52x IO·3 +2.5 1 x 10-3

S' nJJI. ind/l x C6 anis) 12 -6.68x I0·() - 1.87 x 10·6

B""5(0, ind p x C(, all is)l l - 1.24 x 10-3 - 3.7.1 X l OA

B' "s(tota l)1 2 - 0. 1286 - 0.0345

(B' O) l l - 0.4784 + 0. 1322

50

• -50

·100 +-----~----,..-----~,_---___j

a 200 400 600 800

T(K)

Fig. 5 - Comparison of 8 mi x for the equi molar mixtures of COr Nl in the present work ( _ ) wi th experimental ( . )17, prev ious calcu lat ions by Bzowski el al. (0 )1.1.

T 0

o 0 0

0 0 o • 0

0 . . 0

0 E U a ~ a:i

00 «l .. 0

-2

-4+----.,----.,----~---~---~

100 200 300 400 500

T(K)

Fig. 6 - Dev iat ion plots for the interaction second virial coeffi cients of CO2 - Ar IRef. 17 I ( 0 ) No. 1, c-) No.3, (0 ) No.4, ( 0 ) No.6. (t.) No. 7}, (. ) Ref. 2 1).

0

0

0

0 ..E u u -;:. a:i «l

-1

-2

-3

a 100 200 300 400 500

TIK)

Fig. 7 - Same as Fig. 6 for COr Ol IRef. 17 I (0) No.2, (0 ) No.3, (t.) No.4 }].

5.0 .,------------- --------,

•

2.5 • 0 E U 0.0 ~ a:i «l ..

-2.5

-5.0 100 200 300 400 500

T(K)

Fig. 8 - Same as Fig. 6 for COr CO [( 0 ) Ref. 17].

0

0 0

0

0 0

0 0

E u ~ a:i «l

-2

-4 +----.,----~---,..__----~---_4

a 100 200 3eo 400 500

T(K)

Fig. 9 - Same as Fig. 6 for Ar - CO fRef. 17 {C. ) No. I, (0 ) No.2 1 ].

HASH EM I er al.: SECOND VI RIAL COEFFICIENTS OF NON-SPHERlCAL MOLECULES IN BINARY MIXTURES 1343

3.0 -,----- -----------------,

. . ~ 0

1.5

o E U ~ 0.0

al <0

·1.5

·3.0 +----~----~---~-------j o 200 400 600 800

T(K)

Fig. 10 - Same as Fi g. 6 for AI' - N2 [Rer. 17 {( e ) No . 2, ( 0 )

0.3. (D) No.4, ( ... ) No.5}, C- ) Ref. 20].

3.0 ,--------------------,

e

1.5 e

• • • • • 0 0.0 E U .!:,. al <0

·1 .5

·3.0 +------~-----_r_------I

200 400 600

T(K)

Fi g. II - Same as Fig. 6 for AI' - O2 reO ) Ref. 17, (e ) Ref. 20].

Table 3 gives the non-spherical and spherical contributi on to the second vi rial coefficient of COr CO as typical example at two temperatures, a very low temperature and a relatively high temperature. The results 5how that the dominant contribution to the non-spherical part is due to e lectrostatic interaction (q uadrupole-q uad rupo le), and when the temperature increases the non-spherical contribution decreases. Figure. 5 shows comparison of Brni x for cqu imolar mixtures of COz-N2, within the temperature range, for whi ch experimental data are availab le. These results are also compared with the previous calculations by Bzowski et cll l 3

. It seems best to show the interaction coefficient BI 2 rather than B rnix for the whole mixture. The coefficient BI 2

contributes to Brni x with a weight factor of at most Y2,

for an equimolar binary mixture. The interaction coeffic ients. BI 2 depend ent irely on the unlike

3.0

o

1.5

0 E U 0.0 .!:,. al <0

·1.5

·3.0 200 400 600

T(K)

Fig. 12 - Same as Fig 6 for N2 - 0 2 fCe) Ref. 20, (0) Ref. 17].

interactions in the mixture, and thus supply a direct test of the combination ru les. The HFDID 1 potential and Buckingham method2 also reproduce very well experimental second virial coeffic ient for the binary mixtures . Deviation plots for the cross second vi rial coefficients for the seven binary mixtures for wh ich BI 2ex p(T) values are available in literature, i.e., CO2-A r , COr O2, COrCO, Ar- CO, Ar-N2, Ar-02 and N2-02,

are shown in Figs 6-12, respectively. Most of the experimental data of second viri a l coefficients is taken from Ref. 17. For Ar-N2, Ar-02 and Nr 0 2• we have also compared the results with the approximate non-conformal theory (ANC)20. The results for CO2-

Ar are also compared with the recent results that obtai ned from new intermolecular potential energy function2 1

• Examination of the deviation plots shows that our calculated values for the interaction second virial coefficients with the present method are in good agreement with experimental data over the whole range of temperatures and another works. Furthermore, the deviation BI 2 for most of the pairs is comparable of BII and B22 for the sing le components in this work .

Conclusions In spite of the si mplicity and also eas ier calculation

with HFDIDI in comparison with ab initio potentials , nearly the same results are obtained. The HFDID I potential presents a unique mathematical formula for all of the examined compounds.

Acknowledgement We acknowledge the help rendered by the

Research Committee of the Department of Chemistry, Alzahra University, Iran .

1344 INDIAN J CHEM, SEC A, JULY 2005

References I Maitland G C, Rigby M, Smith E B & Wakeham W A,

lnlermolecular Forces (Clarendon Press, Oxford) 1987. 2 Buckingham A D & Pople J A. Trail s Faraday Soc, 51

(1955) 1173. 3 Boushehri A. Mason E A & Kestin J, lilt J Therlllophys, 7

(1986) 111 5. 4 Hashemi F S & Boushehri A. J Sci I R lrall , 10 (1999) 43. 5 Eslami H. Mozaffari F & Boushehri A, Inl J Theml Sci, 40

(200 1) 999. 6 Tang K T & Tonnies J P, Z Phys, D I (1986) 91. 7 Bzowski J. Mason E A & Kestin J, lilt J Thermophys, 9

( 1988) 131. 8 Aziz R A, J Chem Phys, 99 (1993) 4518. 9 McQuarrie M C, StatisTical ThermodYllamic (Harper and

Row. New York) 1976, pp. 223. 10 Absard i J M. Boushehri A & Mason E A, Int J Therlllophys,

II (1990) 508. II Reed T M & Gubbins K E, Applied Statistical Mechallics,

ThermodYllamic alld Trallsport Properties of Fluids (McGraw-Hili, New York) 1973, pp. 219.

12 Mason E A & Spurling T H, The Virial Equatioll of StaTe (Pergamon Press, Oxford) 1965. pp. 56.

13 Bzowski J, Kestin J, Mason E A & Uribe F J, J Phys Chem Ref 2 Data, 19 (1990) 11 79.

14 Boushehri A & Maghari A, High Temp-High Press, 24 (1992) 443.

15 Maryott A A & Buckley F, Natl Bur STand Circ. 537 ( 1953) 29.

16 Bridge N J & Buckingham A D. Proc Roy Soc. 295A. ( 1966) 334.

17 Dymond J H & Smith E B. The Vi rial Coe/ficienfs of Pure Gases alld Mixtures a Critical Compilalioll (Clarendon. Oxford) 1980.

18 Boushehri A. Bzowski J. Kestin J & Mason E A. J Phys Chem Ref Data, 16 (1987) 445.

19 Bocks S, Bich E & Vogel E, Chem Phys. 25i (2000) 147. 20 Del Rio F. Ramos J E & McLure I A. Phys Chem Chem Phys

1 (1999)4937. 21 Huston J M. Ernesti A. Law M M & Roche C F. J Chelll

Phys, 105 (1996) 9130.