Fluid Phase Equilibria, 95 (1994) 329-339

Elsevier Science B.V.

329

Virial coefficients of n-butane from pressure measurement only

B.D. Ababio, P.J. McElroy * and A.G. Williamson

Department of Chemical and Process Engineering, University of Canterbury, Christchurch (New Zealand)

(Received November 1, 1992; accepted in final form October 3, 1993)

ABSTRACT

Ababio, B.D., McElroy, P.J. and Williamson, A.G., 1994. Virial coefficients of n-butane from pressure measurement only. Fluid Phase Equilibria, 95: 329-339.

A new technique requiring pressure measurements only is used to determine accurately virial coefficients for n-butane in the temperature range 308.15-348.15 K at pressures up to 110 kPa. Second virial coefficients obtained are in excellent agreement with literature values, demonstrating the efficacy of the method. Third virial coefficients are in reasonable agree- ment with the only value reported in the literature.

Keywords: experiments, data, virial, hydrocarbon.

INTRODUCTION

The virial coefficients of gases are derived from the virial equation of state either in the pressure series form as

pv = n(RT + flp + yp* +. . .> (1)

or in the volume series form as

pJ&=RT[l +B(T)/I/,+C(T)/V~+...] (2)

Methods for the measurement and collation of p- V-T data have been extensively reviewed by Cox and Lawrenson (1973) and lately to some extent for high pressures by Holste et al. (1986).

The infinite series inversion of eqn. ( 1) into eqn. (2) leads to the relations

B(T) = B (3)

* Corresponding author.

0378-3812/94/$07.00 0 1994 - Elsevier Science B.V. All rights reserved SSDZ 0378-3812(93)02441-O

330 B.D. Ababio et al. 1 Fluid Phase Equilibria 95 (1994) 329-339

and

C(T) = p” + yRT (4)

Typically, in widely used standard techniques, the values of B and /I were determined from slopes and intercepts of linear plots of p V versus p or l/ V. These methods, which were mostly simple comparative expansion or compres- sion processes, depended on two accurately measured variables, pressure and volume, and were based on the assumption that the amount of gas, n, were constant throughout the sequence of observations. These approaches were open to errors as a result of adsorption effects which compromised the assumption of a constant n and were also subject to systematic errors in the volume calibrations. A linear fit to other PV versus P or P V versus l/ V plots assumes both C and y in eqn. (4) to be zero, a condition that cannot hold if B (or /3) is finite. This discrepancy has been observed and discussed in some detail by Scott and Dunlap (1962). The pressure range and the absolute pressures in most second virial coefficient studies were usually kept low to support the truncation of the virial equation at the second term, but the effect of C is complicated by the effects of eqn. (4) above and cannot be simply ignored. The method described in this work addresses this problem.

EXPERIMENTAL

The apparatus used in this investigation has been described elsewhere by An Xueqin et al. (1990), and the experimental technique by Ababio et al. (1992). It is schematically represented in Fig. 1 and the process is briefly explained below.

Fig. 1. Basic apparatus for the second virial coefficient measurements: V,, V’, operating vessel voltimes; AV, volume of enclosed piping ( V2 = V2A V).

Pressure Source

Dead Weight Gauge

B.D. Ababio et al. 1 Fluid Phase Equilibria 95 (1994) 329-339 331

For the same amount n of three different V,, i = 1,2,3, of gas sample, rearrangement of eqn. (1) truncated at the quadratic term with V, + V, = V, and after collecting terms gives

WPT’ -PZ’ -p;’ -PC’) = /3 + y(p, +p2 -pJ (5) The left-hand side of the above equation is referred to as the apparent second virial coefficient, japp :

Bapp = RT(p,’ -P;’ -PC’) (6)

This equation holds if n is constant, and when mass is conserved during the transfer of sample from Vi into V, and the final expansion into V,, the total volume. With measurements of pressures (p, ,p2,p3) associated with the valumes Vi, I’, and V, for a specified amount of sample, the calculation of an apparent second virial coefficient paPP is made possible, and for a series of such measurements at the same temperature, a plot of fiapp versus (pl +p2 -p3) extrapolated to zero pressure gives a limiting value fo Papp which should be the “true” infinite series value /I; that is

The limiting slope to this linear regression as described by eqn. (5) results in the pressure series third virial coefficient, y.

The technique is thus usable when the pressure triplets (pI ,p2,p3) are known for a constant sample amount 12 and when the sample is easily transferable from vessel to vessel (for example, by liquid nitrogen distillation).

The technique can be extended to avoid material transfer and to be usable for substances not quantitatively transferable by liquid nitrogen distillation. In the modified method a series of expansions are made from V, into I’, generating the pressure pairs (pl ,p3,), and another independent series of expansions from I’, into V, generating the pressure pairs (p2, Pan). These two sets of observations are then functionally related as

Pl = T(P31) (7)

and

P2 = HP321 (8)

It is then possible, as illustrated in Fig. 2, to obtain the values “missing” from the triplets (pl, p2, p3) by imputation from eqns. (7) and (8) as

PI-talc = 4P32) (9)

and

PZ-talc = tic P3*> (10)

332 B.D. Ababio et al. / Fluid Phase Equilibria 95 (1994) 329-339

60000 -

60000 -

Q

40000 -

20000 -

0 ...“....‘....“...‘..” 0 10000 2oooop3 30000 40000 50000 ,Pa

Fig. 2. Relationship between pressure pairs (p,, p3,) and (~2. ~32): 0, PI; 0, PZ

The set of pressures (p, , p2, p3) required for use in eqn. (5) is now replaced

by [~(P& HP > P 1 3 , 3 , w h ere this is a function ofp, only and z(pg) and $(p3)

give the best estimates of p1 and p2, respectively. Equation (5) becomes

Papp = WPT’ - +(P3>-’ - T(PJ -‘I = P + YMP3) + $(P3) - P31 (12)

Most of the uncertainties encountered in this work are directly due to pressure measurements. The dead weight gauge (DWG) in use has the advantage of high relative linearity, so that the error in the pressure p due to the uncertainty in the DWG piston area causes only a small overall constant percentage systematic error in /I. An error analysis of the term (P? -P;’ -PC’) h s ows that the uncertainty in a single determination of ljapp from pressure measurements alone is

SP = 4RT6pl~kix

where pmax = pl.

The functional form of eqns. (7) and (8) also minimizes the random errors in the experimental data. Another source of uncertainty in the final determination of the virial coefficients is the effect of adsorption. If we write

PiVi =n,RT+n$p, (for i=l,2,3) (13)

and if ni = nT - Ani, where nT is the total amount of substance and Ani is the amount of substance adsorbed on vessel V,, then the uncertainty in the

B.D. Ababio et al. 1 Fluid Phase Equilibria 95 (1994) 329-339 333

determination of the second virial coefficient due to adsorption effects,

APAD, is given by

APAD = V,(An, - An,)/n2 + V,(An, - An2)/n2 (14)

RESULTS AND DISCUSSION

Raw data for the isotherms investigated in this work are presented in Table 1. Figure 2. shows the relationships between the pressure pairs

TABLE 1

Raw data for n-butane

T(K) PI (Pa) P31 (pa) T, W T3, WI P2 (pa) P32 (Pa) T2 WI T32 WI

318.15 99165 44093 318.151 318.147 76206 43115 318.151 318.147 87397 38798 318.149 318.147 67509 38160 318.147 318.147 76730 34012 318.144 318.151 59579 33652 318.147 318.151 64983 28756 318.147 318.147 50889 28717 318.147 318.151 54260 23980 318.151 318.151 41978 23668 318.149 318.149 43665 19270 318.147 318.151 33956 19128 318.151 318.151 33902 14939 318.151 318.147 24962 14049 318.153 318.151 21684 9542 318.151 318.151 16021 9005 318.147 318.151

328.15 98876 43892 328.151 328.136 76606 43304 328.143 328.146 90118 39966 328.143 328.154 69717 39385 328.154 328.146 76315 33786 328.146 328.154 58788 33178 328.154 328.154 62740 27729 328.146 328.154 49761 28062 328.146 328.154 49834 21993 328.150 328.148 39879 22468 328.146 328.150 39857 17567 328.154 328.154 30995 17449 328.146 328.150 30205 13299 328.150 328.143 23339 13130 328.150 328.150

338.15 102141 45312 338.150 338.146 80040 45226 338.148 338.148 91015 40325 338.150 338.150 71160 40182 338.148 338.148 79483 35171 338.146 338.150 62538 35286 338.150 338.146 68370 30215 338.146 338.150 53725 30293 338.150 338.150 56784 25063 338.148 338.145 44567 25109 338.150 338.146 45213 19932 338.146 338.154 35197 19815 338.150 338.146 34221 15067 338.150 338.150 27184 15295 338.154 338.146 22500 9896 338.147 338.150 17832 10023 338.146 338.146

348.15 102045 45221 348.153 348.153 80556 4549 1 348.146 348.151 90594 40099 348.151 348.151 71436 40313 348.155 348.146 79350 35083 348.151 348.151 62611 35311 348.151 348.151 68650 30320 348.151 348.151 53018 29878 348.151 348.151 57410 25328 348.146 348.151 44618 25129 348.153 348.151 45305 19963 348.146 348.146 34123 19203 348.149 348.146 33944 14940 348.149 348.155 26841 15097 348.149 348.146 23174 10191 348.151 348.146 17902 10061 348.149 348.155

TABLE 2

Coefficients for the functions z(p,,) and $(pjg) as defined by eqns. (15) and ( 16)

Q3,)lPa = c( + 5p3, + ZP:, $(p32)/Pa = A + w32 + v&2 Range of p3

T(K) a Pa) 5 1077c (Pa-‘) 1 (Pa) p 10’~ (Pa-‘) minp, (Pa) maxp3 (Pa)

318.15 - 10.65666 2.2803 1 -7.05313 8.97059 1.78067 -3.10479 9274 43604

328.15 - 7.39200 2.28017 - 6.25003 2.31697 1.78101 -2.77255 13214 43599

338.15 - 10.30869 2.28041 - 5.73441 6.97153 1.78079 -2.47171 9960 45269

348.15 - 8.65667 2.28020 -5.18100 6.82388 1.78090 -2.25057 10125 45356

B.D. Ababio et al. 1 Fluid Phase Equilibria 95 (1994) 329-339 335

(pl,pjl) and (p*,~+~) as experimentally determined. The functional forms resulting from Fig. 2 and as defined by eqns. (7) and (8) are taken as quadratic, generally represented as

r(P3i)IPa = a + lp:p31 + ~PL (15)

and

1c/(P32) Ipa = A + w32 + d2 (16) where the numerical values of the coefficients (a, t,rr, A, ,U and r) are shown in Table 2. The deviations of the experimentally measured pressures from the smooth curves given by eqns. (15) and (16) are shown in Fig. 3. Results for a typical run at 328.15 K are presented in Table 3. Figure 4 shows the

Plot of Papp versus [z(pg) + $(p3) -p3]. It should be noted that the “data” for this procedure do not scatter since eqns. (15) and ( 16) are used to impute pl-calc and p2-calc. The resultant “true” second virial coefficient p, the infinite series value of Papp, is obtained by extrapolating to zero pressure for the various temperatures with least-squares straight lines fitted to smooth data at equality spaced points and weighted inversely as the estimated uncertainties. These values are summarized in Table 4 with their estimated uncertainties, together with the values of the limiting slopes which represent the pressure series third virial coefficient, y. The results of this work as shown in Fig. 5 agree very well with those of other workers as reported by

3 . . ..I'...I'..'I....I...'I.'..I....I.."I....I....

z- x Y

.

.

tz . 1: x

h Y u. JO-

QI Y 0 4-1 - .

II

z- x .

v -2 1 . .

-3 -

x

-4 . . ..'....'..."'...'.".'...."...'...."...'.... 0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000

Pa JPa

Fig. 3. Deviations plot (6p, = p, - P_,,J vs. pxz : x , &I, =p, =JI_~,~; 0, 6p, =pz -P~_._~,~.

336 B.D. Ababio et al. / Fluid Phase Equilibria 95 (1994) 329-339

TABLE 3

Typical results for n-butane at 328.15 K

r(p3) Pa) W3) (Pa) p31 (Pa) (03) + ti(p3) -p3) (Pa) Bapp (1O-6 m3 mol-‘1

30014 23489 13214 40289 - 588.4 34597 27081 15240 46438 - 584.8 39175 30670 17266 52579 - 582.3 43747 34257 19291 58713 - 580.6 48315 37842 21317 64840 - 579.3 52877 41425 23343 70959 - 578.4 57434 45005 25368 77070 - 577.7 61986 48583 27394 83175 - 577.2 66533 52159 29419 89272 - 576.8 71074 55732 31445 95361 - 576.5 75611 59303 33471 101443 - 576.4 80142 62872 35496 107518 - 576.2 84668 66439 37522 113585 -576.1 89189 70003 39548 119645 -576.1 93705 73566 41573 125698 -576.1 98216 77125 43599 131743 -576.1

-530

-540

-550

- -560

z -570 ------------____________________ E . -

m. -580 . .

- - l ______---’ _

E . .

(D -590 T

0 r -600 r I=:

E -610 r

s” -620 :

-630 :

-640 ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ’ * ’ ’ ’ 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

(T(P)+\IJ(P) -P )/kPa

Fig. 4. A weighted least-squares extrapolation for the imputed data: 0, imputed data; __, maximum possible error; -- ~ - - -, probable error.

B.D. Ababio et al. / Fluid Phase Equilibria 95 (1994) 329-339 337

TABLE 4

Summary of results for n-butane

T(K) /I ( 1O-6 m3 mol-‘) y (lo-” m3 mol-’ Pa-‘)

318.15 -632.5 f 1.2 10.9 f 2.8 328.15 -581.2 k 1.4 42.0 f 10.0 338.15 -539.3 f 1.2 11.3 f 3.7 348.15 -502.3 f 1.5 3.9 + 1.5

-2oo:.“. . . ** I. 7. .‘..‘. 1 “. . . . . * . .

-300 r

-400 : )(, a+

0 x.1

_ -500 : z y OF- O=

'; .a+ a -600 :

0

E @l-b=

"E -700 : ii@

IO 9 -I-

O -600 : +

: -900 r +A

I

+

-1000 : :+

-1100 :

-1200f. * *. * .'. . ' . . . . . ** ’ ” . . . . .' . . 250 300 350 400

T /K

Fig. 5. Comparison of n-butane results with those of other workers: +, Dymond et al. (1986); 0, McGlashan and Potter (1962); W, Bottomley and Spurling (1964); x , Strein et al. (1971); A, Tripp and Dunlap (1962); 0, this work.

Dymond and Smith (1980) and Dymond et al. (1986) in the temperature range investigated.

The uncertainties in the second virial coefficient due to adsorption effects may be regarded as minimal. Bottomley and Nairn (1977) reported the number of moles of per unit area required to form a monolayer for n-butane at lo-’ mol mP2 on glass in the temperature range 293-573 K and at subatmospheric pressures. We have estimated this value for the glass vessels used in this study as 4.5 x lop6 mol mP2 at typical temperatures and pressures, and using 0.37 nm2 (37 A’) as the area occupied by a molecule of n-butane. Using this value in the BET isotherm developed by Brunauer et al. (1938), the uncertainty introduced in the estimation of the second virial

338 B.D. Ababio et al. /Fluid Phase Equilibria 9.5 (1994) 329-339

coefficient estimated using eqn. (14) averages 0.2 x lop6 m3 mol-‘. Statisti- cal analysis of the experimental data gave standard deviations of the estimated second virial coefficients of less than 0.4 x lop6 m3 mall’. The total error in the estimation of the second virial coefficient is thus less than 10e6 m3 mol-‘. Attempts to correct the results for adsorption effects are therefore not considered necessary.

In their work on the thermophysical properties of n-alkanes, Ewing et al. (1988) showed that the second virial coefficient they obtained could be represented by a square-well potential. The equation for the third virial coefficient consistent with the square-well fit to the second virial coefficients was obtained by these workers as

C/b; = 0.625 - 1.3956x + 2.0188x2 - 0.6636x3 (17)

where x = exp(382.5 K/T) - l] and b. = 2.75 x lop4 m3 mall’, and with the coefficients of the powers of x in eqn. ( 17) corresponding to a well depth of 0.4Ocr. The only pressure series third virial coefficient result in the litera- ture on n-butane was estimated by Scott and Dunlap (1962) as y = (380 + 200) x lo-l2 m3 mol-’ Pa-‘. This value, together with the results from this work, and values from eqn. ( 17) representing the work of Ewing et al. (1988), converted into pressure series third virial coefficients by use of eqn. (4), are illustrated in Fig. 6, which shows the results from this work agree fairly well with those of Ewing et al. (1988).

200

100

0 'al a

-100

z E

0 -200

E N : -300

0

5

x -400

310 320 330 340 350 360

T /K

Fig. 6. Third virial coefficients for n-butane: 0, Scott and Dunlap (1962); -, square-well reduction (Ewing et al., 1988); 0, this work.

B.D. Ababio et al. / Fluid Phase Equilibria 95 (1994) 329-339 339

REFERENCES

Ababio, B.D., McElroy, P.J., Salt, B and Williamson, A.G., 1991. On the determination of second virial coefficients from pressure measurements only. Chem. Eng. J. (Loughbor- ough, Engl.), 47: 113.

An Xueqin, McElroy, P.J., Malhotra, R., Shen Weiguo and Williamson, A.G., 1990. Accurate second virial coefficients of n-alkanes. J. Chem. Thermodyn. 22: 487.

Bottomley, G.A. and Nairn, D.B., 1977. Second virial coefficients at 300-500 K for butane, tetramethylsilane and ‘Freon 114’. Aust. J. Chem., 30: 1645.

Bottomley, G.A. and Spurling, T.H., 1964. Measurement of the temperature variation of virial coefficients. Aust. J. Chem., 17: 501.

Brunauer, S., Emmett, P. H. and Teller, E., 1938. Adsorption of gases in multimolecular layers. J. Am. Chem. Sot., 60: 309.

Cox, J.D. and Lawrenson, I.J., 1973. Specialist Periodical Report, Chemical Thermodynam- ics. In: M.L. McGlashan, The Chemical Society, London, Vol. 1, p. 162.

Dymond, J.H. and Smith, E.B., 1980. The Virial Coefficients of Pure Gases and Mixtures. Clarendon Press, Oxford, pp. 114-118.

Dymond, J. H., Cholinski, J. A., Szafranski, A. and Wyrzykowska-Stankiewicz, D., 1986. Second virial coefficients for n-alkanes: Recommendations and predictions. Fluid Phase Equilibra, 27: 1.

Ewing, M.B., Goodwin, A.R.H., McGlashan, M.L. and Tusler, J.P.M., 1988. Thermophysi- cal properties of alkanes from the speeds of sound determined using a spherical resonator 2. n-Butane. J. Chem. Thermodyn., 20: 243.

Holste, J.C., Hall, K.R., Eubank, P.T. and Marsh, K.N., 1986. High pressure PVT measurements. Fluid Phase Equilibria, 29: 161.

McGlashan, M.L. and Potter, D.J.B., 1962. An apparatus for the measurement of second virial coefficients of vapours: The second virial coefficients of some n-alkanes and some mixtures of n-alkanes. Proc. R. Sot., Ser. A., 267: 478.

Scott, R.L. and Dunlap, R.D., 1962. On the determination of second virial coefficients. J. Phys. Chem., 66: 629.

Strein, K., Lichtenthaler, R.N., Schramm, B. and Schafer, K.I., 1971. Measurement of the second virial coefficient of some saturated hydrocarbons from 300-500 K. Ber. Bunsenges. Phys. Chem., 75: 1308.

Tripp, T.B. and Dunlap, R.D., 1962. Second virial coefficients for the systems: n-bu- tane + pertluoro-n-butane and dimethyl ether + l-hydropertluoropropane. J. Phys. Chem., 66: 635.