Vapour-liquid equilibrium in the krypton-xenon system

Physica 117A (1983) 127-138 North-Holland Publishing Co.

VAPOUR-LIQUID EQUILIBRIUM

IN THE KRYPTON-XENON SYSTEM

Jorge C.G. CALADO*, Elaine CHANG and William B. STREETT

School of Chemical Engineering, Cornell University, Ithaca, New York 14853, USA

Received 4 May 1982

Isothermal vapour-liquid data were measured for the krypton-xenon system at ten temperatures between 165 and 270K and pressures to 6.7 MPa, using a vapour recirculating technique. The mixture critical line has been located in (P, T, x) space. Barker’s method of data reduction has been used to test the thermodynamic consistency of isotherms below the critical temperature of krypton (209.4 K) and the excess Gibbs energy was evaluated, at the same temperatures, as a function of composition. The results of the experiments have been compared with predictions of the Peng-Robinson equation of state. With interaction parameter calculated by fitting the isotherm of 200.64K, this equation predicts the liquid and vapour phase compositions to within about a few mole per cent over most of the experimental range.

1. Introduction

In the mid-seventies, Trappeniers and his collaborators undertook a systematic experimental investigation of the phase equilibria in binary mixtures composed of simple molecules’-‘). They were interested, from a theoretical point of view, in mixtures containing only spherical molecules, and thus chose the rare gases as the basic components. They published results for the neon-argon?, neon-krypton’) and argon-krypton’) systems but, as far as we know, no work was carried out on the equally important krypton-xenon system. Along with argon-krypton, the krypton-xenon system is the only rare gas system which is virtually free of quantum effects, and for this reason they both constitute a standard testing ground of statistical theories of liquid mixtures’). (The argon-xenon system is also of interest, however these two substances cannot be mixed as liquids because the critical temperature of argon lies below the triple point temperature of xenon.)

One of us had already been involved in a thermodynamic study of liquid mixtures of krypton and xenon but at one temperature only, namely the triple-point temperature of xenon, 161.39 K, the lowest temperature at which the whole composition range can be covered in the liquid state. Here we report liquid-vapour equilibrium data for ten temperatures between 165 and

* Permanent address: Laboratorio de Quimica Estrutural, Complexo I, Instituto Superior Ttcnico, 1096 Lisboa, Portugal.

037%4371/83/0000-0000/$03.00 @ 1983 North-Holland

128 J.C.G. CALADO et al.

I I 7 I I

9-

0-

P/MPa

7-

\

‘bD -

/

300 T/K

Fig. 1. Pressure-temperature diagram f6r the Kr-Xe system. AB and CD are the vapour pressure curves of krypton and xenon. BD is the critical line. The shaded area shows the pressure and temperature range covered in this study.

270 K. Four of these isotherms, at 165.60, 180.03, 190.03 and 200.64K lie below the krypton critical temperature (209.46 K) and the remaining six, at 216.61, 228.78, 240.42, 245.20, 257.85 and 268.69 K, fall between the critical temperatures of the two components (critical temperature of xenon: 289.74 K). Virtually, the whole domain of the liquid-vapour equilibrium in the krypton-xenon system, from the triple point of xenon almost to its critical point is covered in these two studies, as can be seen in fig. 1.

1. Experimental results

A detailed description of the apparatus and experimental technique has been published elsewhere7S8). Here we shall describe only the most important features of the method. The two-phase system is contained in a stainless steel cell of approximately 10 cm3 capacity. The vapour is recirculated in a closed loop by the action of a magnetic pump, and its bubbling through the liquid provides the agitation and intimate contact between the two phases which

VAPOUR-LIQUID EQUILIBRIUM IN KRYPTON-XENON 129

allow equilibrium to be reached quickly, usually within five to ten minutes. There are sampling lines for both the liquid and vapour phases, and these are analyzed by a thermal conductivity method. Calibrating mixtures of pre- determined composition were prepared by condensing known amounts of each gas into sampling vessels of about 25 cm3 capacity; for the P-V-T calculations necessary to extract the composition of the mixtures, the virial equation of state through the second virial term was used (the measured pressures remain always below 1.5 bar). We start by introducing into the cell a convenient amount of the less volatile component (xenon) and the pressure is raised by consecutive additions of the other component (krypton). The measurements are conducted isothermically and the cryostat is of the boiling- liquid type, the temperature being regulated by controlling the vapour pressure of the liquid through a Cartesian manostat. Cryogenic liquids used in this study were carbon tetrafluoride (in the 165-200 K range), carbon dioxide (215-240 K) and monochlorodifluoromethane (240-270 K).

Temperatures were measured on the IPTS-68 with a NBS calibrated pla- tinum resistance thermometer and Mueller bridge (Leeds & Northrup, model G2). Temperature regulation within 0.01 K is sufficient for the measurement of phase equilibria, except near the critical region where a stabilization within a few milikelvin is necessary. The pressure was measured with an Autoclave Engineers digital pressure gauge (model DPS/0021 PM) which was calibrated by measuring the vapor pressure of liquid krypton as a function of temperature and checking the results against the published values of Theeuwes and Bearman?. After calibration, the gauge is accurate to 0.25% of the measured values, as shown in the good agreement between the experimental and calculated”‘) values of the vapour pressure of xenon (for instance, at 228.78 K the measured vapour pressure was 1.368 MPa and the calculated value 1.369 MPa). Uncertainties in the phase compositions are due mainly to errors associated with sampling, and are estimated to be a few tenths of one mole percent, although they may be an order of magnitude higher near the critical line where the equilibrium compositions are very sensitive to small changes in pressure and temperature. For this reason, our results near the critical region can be considered only exploratory, although the simple behaviour of the system makes extrapolating easy and helps with the definition of the mixture critical line.

Matheson research grade krypton of stated minimum purity 99.99% and xenon, also from Matheson, of both CP and research grades were used in this work. The CP grade xenon which is at least 99.9% pure, was further purified by

fractionation in a low-temperature column”): the final purity of all the xenon samples used in the experiments was 99.99%.

Vapour and liquid compositions have been measured at ten temperatures


TABLE I Vapour-liquid equilibrium data for the krypton-xenon system. xi is the Kr mole fraction in the liquid phase and yr is the corresponding value in the

vapour phase

0.106 o.oooo o.oooo 0.315 0.2021 0.723* 0.392 0.2489 0.7860 0.483 0.3364 0.8359 0.553 0.4052 0.8610 0.630 0.4581 0.8829

0.222 0.0000 0.0000 0.350 0.0692 0.3412 0.574 0.1757 0.637 1 0.65 1 0.2175 0.6802

0.749 0.2763 0.7288 0.924 0.3583 0.7959 1.169 0.4938 0.8568 1.322 0.5866 0.8835

0.370 0.0000 0.0000

0.65 1 0.1196 0.5092

0.931 0.2387 0.6616 1.141 0.3163 0.7296 1.659 0.5128 0.8422

1.820 0.5816 0.8622 2.030 0.6579 0.8880

0.542 0.0000 o.oooo 0.707 0.0649 0.2234 0.867 0.1188 0.3638

1.190 0.2076 0.5665 1.470 0.288 1 0.6483 1.830 0.3900 0.7456 2.072 0.4632 0.7866 2.415 0.5660 0.8353

0.936 0.0000 0.0000

1.148 0.0472 0.1482

1.505 0.1241 0.3489

1.827 0.1886 0.4739 1.952 0.2143 0.5173 2.450 0.3182 0.6310 2.786 0.3874 0.6856

T = 165.60 K 0.700 0.770 0.860 0.952 1.309

T = 180.03 K 1.484 1.694 1.770 1.841 1.917 1.931 2.241

T = 190.03 K 2.268 2.463 2.659 2.694 2.785 3.117

T =200&K 2.849 3.024 3.115 3.311 3.528 3.716 4.296

T = 216.61 K 3.787 4.186 4.606 4.718 5.886 5.235 5.396

0.5189 0.9011 0.5681 0.9160 0.6459 0.9339 0.7310 0.9462 1.0000 1.0000

0.6600 0.9107 0.7563 0.9363 0.790* 0.9428 0.8163 0.955* 0.847* 0.9625 0.8564 0.967* 1.0000 1.0000

0.7446 0.9141 0.8029 0.9310 0.8679 0.9579 0.8722 0.960* 0.9002 0.9668

1.0000 1.0000

0.6816 0.8829 0.7307 0.8962 0.75 19 0.908* 0.8001 0.9214 0.8519 0.9398 0.8925 0.9518 1.0000 1.0000

0.5192 0.7967 0.6506 0.8299 0.7230 0.8572 0.7501 0.8662 0.7682 0.8757 0.8273 0.9960 0.8509 0.9065

VAPOUR-LIQUID EQUILIBRIUM IN KRYPTON-XENON

TABLE I (continued)

131

Vapour-liquid equilibrium data for the krypton-xenon system. XI is the Kr mole fraction in the liquid phase and y, is the corresponding value in the vapour phase

P

OfPa) XI Yl &Pa) XI Y1

2.933 0.4178 0.6980 5.598 0.8833 0.9176

3.087 0.4558 0.7281 5.676 0.8960 0.9200

3.458 0.5192 0.7675 5.732 0.8996 0.9221

1.369 1.491

1.680 1.799 2.044 2.303 2.743

3.080 3.304 3.983

O.oooO 0.023 1 0.0560 0.0756 -

0.1658 0.2414 0.2956 0.3368 0.4350

0.0000 0.0741 0.1669 0.2322 0.3174 0.3953 0.5013 0.5613 0.5964 0.6764

T = 228.78 K 4.263 4.592 4.899 5.200 5.598

5.865 6.019 6.096 6.152 6.215

0.4744 0.7030 0.5335 0.7282 0.5660 0.7496 0.6197 0.7696 0.6779 0.7895 0.7197 0.8015 0.7392 0.8046 0.7541 0.8081 0.7751 0.8075 0.7855 0.8081

1.892 0.0000 0.0000 2.380 0.0630 0.1830 2.660 0.1002 0.2649 3.094 0.1622 0.3707 3.535 0.2188 0.4397 4.137 0.3084 0.5320

T = 240.42 K 4.759 5.291 5.738 6.103 6.215

0.3857 0.5907 0.4637 0.6315 0.5320 0.6610 0.5798 0.6823 0.5936 0.6818

2.552 0.0000 0.0000 5.739 0.4512 0.5958 5.893 0.4708 0.6057 6.173 0.5026 0.6173 6.326 0.5331 0.6222

T = 245.20 K 6.466 6.537 6.614 6.579 6.596

0.5479 0.6222 0.5642 0.6197 - 0.6098 0.5730 0.6162 - 0.6134

2.932 0.0000 0.0000 3.115 0.0194 0.0533 3.262 0.0361 0.0852 3.380 0.0494 0.1162 3.604 0.0754 0.1646 3.808 0.0970 0.1967 3.884 0.1023 0.2084

T = 257.85 K 4.025 4.304 4.585 4.746 5.088 5.487 6.236

0.1185 0.1474 0.1768 0.1932 0.2294 0.2742 0.3641

0.2333 0.2773 -

0.3426 0.3735 0.4130 0.4619

3.768 0.0000 0.0000 4.033 0.0316 0.0562 4.357 0.0664 0.1156 4.544 0.0858 0.1471 4.885 0.1191 0.1950

T = 268.69 K 5.030 5.250 5.492 5.816 6.088

0.1328 0.2141 0.1562 0.2403 0.1805 0.2661 0.2129 0.2919 0.2419 0.3107

* Interpolated values


from 165 to 270 K and pressures to 6.7 MPa. The experimental results are recorded in table I and the isotherms are plotted on a pressure-composition diagram in fig. 2. The internal consistency of the data can be better observed in fig. 3 with the aid of the K-value plots which tend to exaggerate scatter in

the results (Ki = yi/Xi, where Xi is the mole fraction of the component i in the liquid phase and yi is the corresponding mole fraction in the vapour phase). These plots were also used for a better definition of the mixture critical line since above the critical temperature of krypton the two branches converge and exhibit a vertical tangent at the mixture critical point where K = 1. Data on the mixture critical points are recorded in table II.

The experimental results for the three lower isotherms (where both components are subcritical) were also checked for thermodynamic consistency. Van Ness and his co-workers, who have assessed the different data reduction methods and other tests of thermodynamic consistency*2.‘3) have come out in favour of Barker’s method”), among other reasons because it is a single-step

PIMPa

0 0.2 0.4 XI

0.6 0.8 1.0

Fig. 2. Liquid-vapour equilibrium isotherms for the Kr-Xe system. (a) 165.60 K; (b) 180.03 K; (c) 190.03 K; (d) 200.64 K; (e) 216.61 K; (f) 228.78 K; (g) 240.42 K; (h) 245.20 K; (i) 257.85 K; (j)

268.69 K.


0.3

log Ki

0.2

Fig. 3. Ki vs. pressure plots for the Kr-Xe system: (a) 216.61 K; (b) 228.78 K; qc) 245.20 K; (d)

268.69 K.

analytical fitting technique that makes direct use of the measured data”). According to this method, the parameters in an expression relating the excess Gibbs energy GE to composition (usually the Redlich-Kister equation (1) are fitted by a least squares procedure that minimizes the pressure residuals, i.e. the differences between the experimental and calculated pressures, using only

TABLEII

Vapour-liquid critical line for the system krypton-xenon

209.46 (5.49) 1.000 216.61 5.77 0.922 228.78 6.25 0.800 240.42 6.55 0.678 245.20 6.62 0.608 257.85 6.66 0.450 268.69 6.51 0.307 289.74 (5.84) 0.000


(P-x) data. The parameters in the Redlich-Kister equation,

GE/RT = x,xJA + B(x, - x2) + C(x, - x2)* + . - -1, (1)

are then used to evaluate (P-y) data, via the Gibbs-Duhem equation, and the results are compared with the experimental measurements. It is assumed that the vapour phase can be described by a virial equation of state (the pressure remains below 3 MPa),

PV=RT+BP+CP’, (2)

so that the activity coefficient of component 1, f,, evaluated at zero pressure is given by”)

RTlnf,=RTlnE + [B,, -W&,1@” - PO,) + Pay: 01 I

+;cm(P*- Pi,) + (V,, - V,),=oP. (3)

GE then follows from the expression

GE = RT[x, In f, + x2 In f2]. (4)

Although we take C to be a linear function of the composition of the mixture (an assumption justified by the smallness of the corresponding term and the lack of reliable data), we retain the usual quadratic expression for B,

B(x) = B,,Y:+ ~BI~YIYZ+ B22~:> (9

with

S = 2B12- B,, - B2*. (6)

PO,, Vo, and V, are the saturation vapour pressure, orthobaric molar volume and partial molar volume of component 1. If we consider the excess molar volume VE to be given by a simple quadratic expression,

VE = &X,X*, (7)

which happens to be the case in many simple mixtures, then the last term in (3) can be written simply as -ax2P. VE data are only available for 161 K6.16) but usually VE is found to be fairly insensitive to temperature change. Thermodynamic data necessary to perform the calculations were taken from refs. 9 and 17 for krypton, and refs. 10 and 18 for xenon. For second virials, including the cross-virial coefficients, were used the compilation of Dymond and Smith’?. Third virial coefficients were estimated from a corresponding states correlation of Chueh and Prausnitz*@). The differences between the experimental vapour compositions and those calculated from Barker’s method are never larger than 1 to 1.5 mole per cent, which is acceptable given the


TABLE III

Redlich-Kister coefficients and GE values for the equimolar mixture at different temperatures, for the krypton-

xenon system

GE A B c (J mol-‘)

161.396) 0.3413 0.0207 0.0237 114.5 180.03 0.245 0.006 0.086 92 190.03 0.159 0.071 0 63 200.64 0.236 0.141 -0.093 98

uncertainties in the sampling process and the ability of a virial equation of state to describe the behaviour of the vapour phase. The values of parameters A, B and C in eq. (l), which enable us to calculate GE at zero pressure as a function of composition, are given in table III, together with GE values for the equimolar mixture. The familiar trend is observed, i.e. GE decreases with increasing temperature, reaches a minimum and rises again as the system approaches the critical temperature of krypton (209.46 K); GE also becomes a less symmetric function of composition as the temperature increases.

3. Discussion

As expected, the phase diagram of the krypton-xenon system is very similar to that of its twin system, argon-krypton’); they both belong to class I in the Scott and Van Konynenburg classification*‘). The mixture critical line reaches a maximum pressure of about 6.69 MPa for a mixture whose krypton mole fraction is x = 0.51 (to be compared with x = 0.55 in the argon-krypton system).

In their paper on the argon-krypton system, Schouten et al. have shown that the van der Waals equation of state cannot predict quantitatively the mixture critical line’). This is not surprising since the repulsive term in that equation is a very poor approximation of the equation of state for hard spheres. Much better results are obtained if that term is replaced by the Carnahan and Starling equatior?). Another van der Waals-type equation which has been successfully used in the prediction of liquid-vapour equilibria is the Peng and Robinson equation (8)23).

RT a(T) -- lJ = V-b V(V+b)+b(V-b)’ (8)


where parameter a is temperature dependent and also a function of Pitzer’s accentric factor o (which has a zero value for atoms) and b can be estimated from the critical constants. Both the Carnahan and Starling and the Peng and Robinson equations have built into them a good approximation for the hard-sphere equation of state, as can be seen by expanding PVIRT into the virial series. For a mixture, the parameters a and b are assumed to have a quadratic dependence on composition (van der Waals mixing rules),

Cl(X) = 2 &jXiXj, i.i

b(x) = 2 biixixi. i.j

(9)

(10)

As for the cross parameters al2 and b12, while we allow for departures from the Berthelot rule,

a u = (1 - k13(alla23”*,

the Lorentz rule of additivity of the volumes is usually retained,

b,2 = (b,, + b&/2.

(11)

(12)

Strictly speaking, the arithmetic mean rule (12) should be written in terms of diameters and not of co-volumes, as above, but the calculations are insensitive to the change. The only adjustable parameter is thus k,, and this is determined by trial and error, starting with k,* = 0, so that the chemical potentials for the two components in the mixture satisfy the phase equilibrium constraint, for an isotherm near the middle of the experimental temperature range. The calculated value of k12 is then used to predict the behaviour of the other isotherms. For the 200.64K isotherm, the best fit is obtained with k,* = -0.0051, and a comparison of the experimental and calculated isotherms is shown in fig. 4. The agreement is very good for the subcritical isotherms, and fair for those above the critical temperature of krypton. The Peng and Robinson equation of state predicts the right shape for these isotherms, but fails somewhat in the estimation of the mixture critical points. It should be noted, however, that it is difficult for any analytical equation of state (or any theory, for that matter) to predict correctly a mixture critical line.

For simple systems, k,* is usually positive and an order of magnitude higher than the value obtained above; it is the smallness, rather than the sign of ki2 which is striking here, for it means that the system obeys the geometric mean rule as far as the Peng and Robinson equation of state is concerned. Computer simulation experiments suggest, however, a value k12 = 0.019 for the mixtures of krypton and xenor?).

In a more sophisticated treatment of the mixture critical line allowance


P/MPa

6- ’

XI

Fig. 4. Comparison between calculation and experiment: 0, experimental points; - calculated isotherm from Peng-Robinson equation of state; H, experimental mixture critical points; --- calculated critical line.

should be made for departures from the arithmetic mean rule (12), especially for molecules which differ appreciably in size. If the two molecules have diameters (+,, and (+22, then the modified equation (12) becomes

u12 = (1 + jlz)(all + ~~312. (13)

Soares et a1.25) found j,2= +0.0006 and k,2= 0.021 (the latter in good agreement with the value of McDonald and Singe?) by fitting the experimental values of the excess properties VE and GE for an equimolar mixture of krypton and xenon at 161.3 K, with their calculated values from conformal solutions theory26), using the empirical equation of state of Bellemans et al.“). A more realistic approach consists of using the Gosman et al. equation of state for argon%). Then a much higher value of j12 (j12 = 0.060) is needed to force agreement between the experimental and calculated values of the excess molar volume VE for the equimolar mixture. With k12 = 0.019 and ji2 = 0.060, the predicted mixture critical line agrees very well with experiment for temperatures above 240 K, but the agreement is less good for lower temperatures (high density region), as can be seen in fig. 4. Since the


conformal solutions theory is based on the law of corresponding states, these discrepancies may well be due to small but systematic departures from that law. There is evidence that even the rare gases fail to conform to the law of corresponding states, in particular under conditions when 3-body and higher- body potentials become important’?.

Acknowledgements

We are grateful to Dr. P. Clancy for help in the calculations of the mixture critical line.

This work was supported, in part, by the National Science Foundation, under grants CPE-78-23537 and CPE 79-09168, by a grant from the donors of Petroleum Research Fund of the American Chemical Society, and by the North Atlantic Treaty Organization (Research Grant No. 1373).

References

1) N.J. Trappeniers and J.A. Schouten, Physica 73A (1974) 527. 2) N.J. Trappeniers and J.A. Schouten, Physica 73A (1974) 539. 3) N.J. Trappeniers and J.A. Schouten, Physica 73A (1974) 546. 4) J.A. Schouten, A. Deerenberg and N.J. Trappeniers, Physica SlA (1975) 151. 5) J.K. Lee, D. Henderson and J.A. Barker, Mol. Phys. 29 (1975) 429. 6) J.C.G. Calado and L.A.K. Staveley, Trans. Far. Sot. 67 (1971) 289. 7) W.B. Streett and J.C.G. Calado, J. Chem. Thermodym. 10 (1978) 1089. 8) J.C.G. Calado, U. Deiters and W.B. Streett, J.C.S. Faraday I. 77 (1981) 2503. 9) F. Theeuwes and R.J. Bearman, J. Chem. Thermodyn. 2 (1970) 179.

10) F. Theeuwes and R.J. Bearman, J. Chem. Thermodyn. 2 (1970) 507. 11) R.H. Davies, A.G. Duncan, G. Saville and L.A.K. Staveley, Trans. Far. Sot. 63 (1967) 855. 12) H.C. Van Ness, S. M. Byer and R.E. Gibbs, AIChE Journal 19 (1973) 238. 13) H.M. Abbott and H.C. Van Ness, AIChE Journal 21 (1975) 62. 14) J.A. Barker, Aust. J. Chem. 6 (1953) 207. 15) J.C.G. Calado, Tecnica 34 (1972) 273. 16) C. H. Chui and F. B. Canfield, Trans. Far. Sot. 67 (1971) 2933. 17) W.B. Streett and L.A.K. Staveley, J. Chem. Phys. 55 (1971) 2495. 18) W.B. Streett, L.S. Sagan and L.A.K. Staveley, J. Chem. Thermodyn. 5 (1973) 633. 19) J.H. Dymond and E.B. Smith, The Virial Coefficients of Pure Gases and Mixtures (Claren-

don, Oxford, 1980). 20) P.L. Chueh and J.M. Prausnitz, AIChE Journal 13 (1967) 896. 21) R.L. Scott and P.H. van Konynenburg, Disc. Far. Sot. 49 (1970) 87. 22) N.F. Carnahan and K.F. Starling, J. C&m. Phys. 51 (1969). 23) D.Y. Peng and D.B. Robinson, Ind. Eng. Chem. (Fund) 15 (1976) 159. 24) I.R. McDonald and K. Singer, Mol. Phys. 23 (1972) 29. 25) V.A.M. Soares, C.A. Nieto de Castro and J.C.G. Calado, Rev. Port. Quim. 14 (1972) 117. 26) T.W. Leland, J.S. Rowlinson and G.A. Sather, Trans. Far. Sot. 64 (1968) 1447. 27) A. Bellemans, V. Mathot and V. Simon, Advan. Chem. Phys. 11 (1967) 117. 28) A.L. Gosman, R.D. McCarty and J.G. Hust, Nat. Stand. Ref. Data Ser. Nat. Bur. Stand. 27

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Vapour-liquid equilibrium in the krypton-xenon system