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Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at 255–320 K
Fluid Phase Equilibria 208 (2003) 223–238
Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at 255–320 K Yoshimori Miyano a,∗ , Koichiro Nakanishi a , Kenji Fukuchi b a
Department of Chemistry and Bioscience, Kurashiki University of Science and the Arts, 2640 Nishinoura, Tsurajimacho, Kurashiki 712-8505, Japan b Department of Chemical and Biological Engineering, Ube National College of Technology, 2-14-1 Tokiwadai, Ube 755-8555 Japan Received 18 December 2002; accepted 5 February 2003
Abstract The Henry’s constants and the infinite dilution activity coefficients of butane, isobutane, 1-butene and isobutene in methanol at 255–320 K are measured by a gas stripping method. The rigorous formula for evaluating the Henry’s constants from the gas stripping measurements is proposed for these highly volatile mixtures. By using this formula, a volume effect of vapor phase and the effect of nonideality of fluids are discussed. In the evaluations for activity coefficients the nonideality of solute was not negligible especially at higher temperatures. The values of Henry’s constants of butane are much different from those of isobutane, while the activity coefficients are not so different to each other. The activity coefficients of butane are about 2.5% greater than those of isobutane, and those of 1-butene are about 4% greater than those of isobutene. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Data; Mixture; Henry’s constant; Activity coefficient; Methanol; Gas stripping
1. Introduction A systematic study of gas solubilities like the Henry’s constants is useful in providing design data for absorption processes, as well as, indirectly, in aiding the analysis of molecular interactions in solutions. Butane gas is highly soluble in nonpolar, nonassociating solvents. Solubilities of nonreacting gases in nonpolar, nonassociating solvents generally follow an order of increasing solubility with increases in the normal boiling point temperature of liquefied gas. As the boiling point temperature of butane is higher than that of isobutane, it is expected that the solubilities of butane are greater than that of isobutane which means that the Henry’s constants of butane are smaller than those of isobutane. Similarly, it is expected that ∗
Corresponding author. Tel.: +81-86-440-1069; fax: +81-86-440-1062. E-mail address: [email protected] (Y. Miyano). 0378-3812/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-3812(03)00036-0
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the Henry’s constants of 1-butene are smaller than those of isobutene. While this approximate relationship exists, it is not sufficiently consistent to permit the prediction of gas solubility even for simple solutions. In polar and associating solvents, the degree of association in the solvent has a most profound effect on the solubility [1]. Although a large number of alkane, alkene + alcohol solubility data were published, few data are available for C4-gases like butane, isobutane, 1-butene and isobutene + alcohol mixtures. The solubility data will be useful to develop prediction methods. Especially for group contribution methods, it may be necessary to take into account the differences between isomers. For developing the molecular theory, on the other hand, the accurate intermolecular potential is necessary. The Henry’s constant is directly related to the residual chemical potential of solute at infinite dilution which is evaluated from the intermolecular potential between a solute molecule and a solvent molecule. Therefore, the Henry’s constant is a suitable macroscopic property for testing the intermolecular potential between different kinds of molecules. The gas stripping method originally proposed by Leroi et al. [2] has been usually used for measuring the activity coefficients at infinite dilution of solutes in nonvolatile solvents, and the vapor pressures of solutes were negligibly small in most cases [3,4]. In this work, however, the Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at temperatures from 255 to 320 K are measured by this method. For these highly volatile solutes and solvents, careful treatments for data reductions are necessary and the rigorous expression is derived for this purpose. 2. Theory The gas stripping method originally proposed by Leroi et al. [2] is based on the variation of vapor phase composition when the highly diluted solute of the liquid mixture is stripped from the solution by a constant flow of inert gas. They derived an approximated equation to express the relationship between the solute peak area measured by a gas chromatography and the gas volume flowing out of the equilibrium cell as follows: γ ∞ Pgsat S Pssat P ln = − 1 ln 1 − Dt (1) S0 Pssat P − Pssat ns,0 RT where D is the flow rate of inert gas, t the experimental time, and Dt is the total volume of inert gas flowing out of the equilibrium cell at time t. From Eq. (1) the activity coefficients of solute at infinite dilution can be obtained as the slope of the line in a log–log plot. When we use the dry gas volume, Dt, it is not possible to solve the differential equations which are based on the material balance and the thermodynamic relationships at equilibrium. Eq. (1), which was derived under some assumptions, therefore, may not be used for highly volatile mixtures. On the other hand, when we use the volume of the saturated gas flowing out of the equilibrium cell, we can get the rigorous solution as follows. In general, the pressure–volume–temperature relations of a fluid can be shown as follows: PV = ZnRT
(2)
Here, P is the total pressure, V the volume, Z the compressibility factor, n the number of moles, R the gas constant, and T the absolute temperature.
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If n is the total number of moles of solute (gas), solvent and the inert gas (helium) in the cell at time t, the quantities withdrawn from the cell during a very short time dt by the inert gas flow is: P −dn = dV (3) ZRT −dn = −dng − dns + dnHe (4) where dV is the volume of the saturated gas flowing out of the cell, ng and ns are, respectively, the total number of moles of solute (g) and solvent (s) in the cell, and dng and dns are, respectively, the change of moles of solute and solvent in the cell, and expressed as: dng = yg dn
(5)
dns = ys dn
(6)
where yg and ys are, respectively, the mole fraction of solute and solvent in vapor phase, dnHe is the number of moles of the inert gas flowing out of the cell and the same quantity of the inert gas is introduced into the cell at the same time. Therefore, the total number of moles of helium in the cell will be kept constant. Substituting Eq. (3) into Eqs. (5) and (6) yields Pyg dng = − dV (7) ZRT Pys dns = − dV (8) ZRT On the other hand, the following thermodynamic conditions should be satisfied at equilibrium: fgV = fgL ,
fsV = fsL ,
V L fHe = fHe
(9)
where f is the fugacity, the superscripts V and L are vapor and liquid phases, respectively. When we use a fugacity coefficient, the above equations can be rewritten as: ϕiV =
fi V Pyi
,
ϕiL =
fiL Pxi
fgV = ϕgV Pyg ,
fsV = ϕsV Pys ,
fgL = ϕgL Pxg ,
fsL = ϕsL Pxs ,
(10) V V fHe = ϕHe PyHe L L fHe = ϕHe PxHe
(11) (12)
where xg and xs are, respectively, the mole fraction of solute and solvent in liquid phase. If the solute is highly diluted in the solvent and if the solubility of helium in the liquid phase is negligible, the mole fraction and the activity coefficient of the solvent may be taken equal to 1, and if the existence of helium does not affect the equilibrium between solute and solvent, the following two equations must be satisfied: fgL = ϕgL Pxg = Hg xg = ϕgV Pyg ,
fsL = ϕsL Pxs = fsL,0 γs xs = fsL,0 = ϕsV Pys
(13)
where Hg is the Henry’s constant of solute, fsL,0 is the fugacity of pure solvent at the reference state (pure liquid at system temperature and pressure). Then Eqs. (7) and (8) become: dng = −
Hg nLg Hg xg dV = − dV ϕgV ZRT ϕgV ZRT nL
(14)
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dns = −
fsL,0 xs fsL,0 nLs dV = − dV ϕsV ZRT ϕsV ZRT nL
nL = nLg + nLs
(15) (16)
where nL is the total number of moles in the liquid phase, nLg and nLs are, respectively, the number of moles of solute and solvent in the liquid phase. As the mole fraction of solute in the liquid phase is very small (less than 10−3 ), nLg nL
≈
nLg
, nLs
nLs ≈1 nL
(17)
Then Eqs. (14) and (15) become: dng = −
Hg nLg dV ϕgV ZRT nLs
(18)
dns = −
fsL,0 dV ϕsV ZRT
(19)
Here we have to know the relationships between n and nL : ng = nLg + nV g
(20)
ns = nLs + nV s
(21)
When we define the volume of vapor phase in the cell at time t as VGP , nV g
nLg Hg VGP Hg VGP PVGP = yg = xg V = L ZRT ϕg ZRT ns + nLg ϕgV ZRT
Substituting Eqs. (17) and (22) into Eq. (20) yields: nLg Hg VGP 1 Hg VGP L L ng = ng + L V = ng 1 + L V ns ϕg ZRT ns ϕg ZRT
(22)
(23)
Similarly, Eq. (21) becomes: L ns = nLs + nV s = ns + ys
PVGP f L,0 VGP = nLs + sV ZRT ϕs ZRT
(24)
Integration of Eq. (19) yields the following result with approximation that all parameters on right-hand side of the equation are constants. It can be shown that in most cases this approximation is valid when the mole fraction of the solute is less than 10−3 : ns − ns,0 = −
fsL,0 V ϕsV ZRT
where ns,0 is the number of moles of solvent in the cell at the initial state.
(25)
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On the other hand, from Eq. (24) we can get: ns − ns,0 = nLs − nLs,0 +
fsL,0 (VGP − VGP,0 ) ϕsV ZRT
(26)
Combining Eqs. (25) and (26) yields: nLs = nLs,0 −
fsL,0 (V + VGP − VGP,0 ) ϕsV ZRT
(27)
where VGP,0 is the initial volume of the vapor phase in the cell, while VGP slightly depends on the gas volume withdrawn from the cell and it is expressed as: VGP = VGP,0 + αV ,
α=
Pssat vsL,sat ZRT
(28)
where vsL,sat is the liquid molar volume of solvent at saturation. Then Eq. (27) becomes: nLs = nLs,0 −
(1 + α)fsL,0 V ϕsV ZRT
(29)
Substituting Eq. (23) into Eq. (18) yields: dng = −
Hg V ϕg ZRT
[nLs
ng dV + (Hg VGP /ϕgV ZRT)]
(30)
By substituting Eq. (29) into Eq. (30), dng = −
Hg V ϕg ZRT
[nLs,0
− {(1 +
ng L,0 V α)fs /ϕs ZRT}V
+ (Hg VGP /ϕgV ZRT)]
dV
(31)
Integration of this equation yields
[(1 + α)fsL,0 /ϕsV ] − (αHg /ϕgV ) Hg ϕsV ng = ln 1 − V ln ng,0 ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)] ϕgV (1 + α)fsL,0 − αϕsV Hg
(32)
where ng,0 is the number of moles of solute in the cell at the initial state. If the peak area, S, of the solute detected by a gas chromatography is proportional to the solute partial pressure, Hg Hg nLg S = KGC Pyg = KGC V xg ≈ KGC V L ϕg ϕg ns we can get the following relationship: nLg nLs,0 nLs,0 + (Hg VGP,0 /ϕgV ZRT) ng S = L = S0 ng,0 nLs + {Hg (VGP,0 + αV )/ϕgV ZRT} ng,0 nLs
(33)
(34)
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By inserting Eq. (29) into this equation,
S ln S0
[nLs,0 − {(1 + α)fsL,0 /ϕsV ZRT}V + {Hg (VGP,0 + αV )/ϕgV ZRT}] ng = ln − ln ng,0 nLs,0 + (Hg VGP,0 /ϕgV ZRT) [(1 + α)fsL,0 /ϕsV ] − (αHg /ϕgV ) ng = ln − ln 1 − V ng,0 ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)]
Combining Eqs. (32) and (34 ) yields the final result as follows: [(1 + α)fsL,0 /ϕsV ] − (αHg /ϕgV ) Hg ϕsV S ln = V − 1 ln 1 − S0 ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)] ϕgV (1 + α)fsL,0 − αϕsV Hg If the volume of vapor phase in the cell is negligible (VGP,0 = 0 and α = 0), Hg ϕsV S fsL,0 ln = − 1 ln 1 − V V S0 ϕs ZRTns,0 ϕgV fsL,0
(34 )
(35)
(36)
Furthermore, if the vapor phase can be treated as an ideal gas (ϕgV = ϕsV = Z = 1) and using an approximation of fsL,0 ≈ Pssat , we can get Hg S Pssat ln = − 1 ln 1 − V (37) S0 Pssat ns,0 RT This equation is similar to Eq. (1) proposed by Leroi et al. [2]. When the vapor pressure of solute is negligibly small, the PDt/(P − Pssat ) in Eq. (1) is approximately equal to the volume of the saturated vapor, V, and Hg ≈ γ ∞ Pgsat . Pisat is the vapor pressure of pure component i. The details for the calculation methods of each value of fugacity coefficients, etc. are described in Appendix A.
3. Experimental The experimental apparatus used for measuring the Henry’s constants is shown in Fig. 1. The inert gas (helium) flow rate was measured by a mass flow meter (Koflok Model 3300, maximum flow rate = 2 cm3 min−1 , accuracy = 1%) and the total volume was measured by an integrating unit (Kofloc CR-700). The flow rate of the inert gas was adjusted by a fine metering valve (double needles, Swagelock SS2-D). The equilibrium cell which volume is 44 cm3 is shown in Fig. 2. This cell contains an inner glass tube and a magnetic stirrer. The inner tube makes a good mixing of the whole solution, and also makes a counter flow of liquid against the rising bubbles. This means that the vapor–liquid contacting time becomes longer. The exit port of the inert gas in the equilibrium cell was made of a stainless tube with inner diameter of 0.14 mm and small size bubbles were made to achieve phase equilibrium within a short time. The mass flow meter was calibrated by a volumetric flow meter shown in Fig. 3. Ethylene glycol was used for the liquid in a burette because of its nonvolatility. The inert gas flowing out of the mass flow meter was introduced into the burette. The inner volume of the burette can be changed by moving a flask connected by a flexible tube. The pressure in the burette was kept to a constant value within ±2 Pa by
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Fig. 1. Experimental apparatus for measuring Henry’s constants.
using a pressure gage (Paroscientific, Digiquartz Pressure Transducer Model 215A and Sokken Digiquartz Pressure Computer Model 600S) and changing the level of the flask. First of all, about 43 cm3 of the solvent (methanol) was introduced into the equilibrium cell and the accurate quantity of the solvent was determined by mass. Then the equilibrium cell was immersed in a constant-temperature bath and connected to a supply of solute gas and inert gas. After evacuating the connecting lines, the lines were pressurized with a solute gas at an arbitrary pressure (absolute pressure of about 110 kPa), and then the solute gas was introduced into the equilibrium cell. The amount of the solute gas introduced into the cell was adjusted to keep the mole fraction lower than 3 × 10−4 in the solution. After evacuating again the connecting lines, it began to flow the inert gas into the equilibrium cell with
Fig. 2. Equilibrium cell for the gas stripping method.
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Fig. 3. Volumetric flow meter to calibrate the mass flow meter.
a constant flow rate of about 2 cm3 min−1 . After achievement of the steady state, the measurement of the inert gas volume by the mass flow meter was started and the sampling for the gas chromatography was done every 60 min. These samplings were automatically done by using an air actuation sampling valve (Valco, Model A4C6UWE) and a controller unit, and were continued for about 15 h. The pressure in the cell was approximated equal to the atmospheric pressure and measured by the Digiquartz Pressure Transducer. The atmospheric pressures, the temperatures of the bath and the integrated inert gas volumes were also measured every 60 min. The temperature of the constant-temperature bath (filled with ethylene glycol + water) was controlled within ±0.02 K, and measured by a quartz thermometer (Hewlett-Packard, Model 2804A) with accuracy of 0.01 K. The gas flowing out of the equilibrium cell was kept to a higher temperature than that of the bath to avoid any condensations, and introduced to a gas chromatograph (Hitachi, Model G-3000, with double FID detectors) for measuring the solute peak area. The butane, isobutane, 1-butene and isobutene were supplied by Takachiho Kagaku Kogyo at specified minimum purities of 99.8, 99, 99 and 99%, respectively. The methanol was of HPLC grade and supplied by Aldrich Chemicals, with a specified minimum purity of 99.9%. The accuracy of this measurement for the Henry’s constants may be considered within 2%, and it mostly depends on the accuracy of the mass flow meter. 4. Results and discussion Fig. 4 shows the temperature dependency of the Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at the temperature range of 255–320 K. For comparative purposes, this figure contains the literature values [7–10] which had been predicted from the experimentally obtained gas solubility data at 298.15 K and pressures under 102 kPa. The Henry’s constants measured by this gas stripping method agree well as shown in this figure except the data proposed by Kretschmer and Wiebe [10].
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Fig. 4. Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at temperatures from 255 to 320 K.
The Henry’s constants of isobutane show higher values than others. This may be explained by the general relationship: because the boiling point temperature of isobutane (261.4 K) is the lowest among these gases. The boiling point temperatures of 1-butene (266.9 K) and isobutene (266.2 K) are similar and that of isobutene is little bit lower than that of 1-butene. According to the general relationship, it is expected that the Henry’s constants of isobutene may be greater than that of 1-butene. However, the Henry’s constants of isobutene show the smallest values in the entire temperature range. Furthermore, while the boiling point temperature of butane (272.7 K) is much higher than that of isobutene, the Henry’s constants of butane are much greater than that of isobutene. These facts show that the general relationship is not sufficiently consistent to permit the prediction of gas solubility even for simple gases. On the other hand, 1-butene and isobutene have dipole moments of 0.3 and 0.5 debyes (1 debye = 3.162 × 10−24 J1/2 m3/2 ), respectively. This may be the reason why the Henry’s constants of alkene are smaller than that of alkane in methanol which is a dipolar molecule. Fig. 5 shows the temperature dependency of the infinite dilution activity coefficients for the same systems. As shown in this figure, the infinite dilution activity coefficients between butane and isobutane show similar temperature dependencies and the differences in the values are very small, about 2.5%, and the activity coefficients of butane are little bit higher than that of the isomer. The same phenomenon can be observed between 1-butene and isobutene. The differences in the infinite dilution activity coefficients between 1-butene and isobutene are about 4%. The Henry’s constants and the infinite dilution activity coefficients measured in this work are numerically indicated in Table 1 with some other properties predicted for the data reductions. The effects of the volume of vapor phase in the cell and the nonideality of vapor are discussed in the following sub-sections.
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Fig. 5. The infinite dilution activity coefficients of butane, isobutane, 1-butene and isobutene in methanol at temperatures from 255 to 320 K.
4.1. Volume effect of vapor phase In general, the existence of vapor phase in the cell affects the evaluation of the Henry’s constants for the gas stripping method. When a solution contacts to a fresh inert gas, some amounts of solute and solvent in the solution will move into the inert gas bubbles to keep the equilibrium. If there is a space to keep the vapor in the cell, the solute can stay there for its residence time, and some amounts of them will resolve into the solution again. This will make the speed down for the gas stripping. Therefore, the evaluated value of the Henry’s constant without the correction of the volume effect of vapor phase will become smaller than the true value. For example, the initial volume of the vapor phase in the equilibrium cell was 2.3 cm3 for the system of isobutane+methanol at 320.02 K. The final volume of the vapor phase for this system after 16 h was about 5.5 cm3 (this volume change is expressed by Eq. (28)). The volume of vapor phase increased about 2.2 cm3 during this experiment. Then we obtained the result that the Henry’s constant of isobutane was 7320 kPa as indicated in Table 1. When the initial volume of the vapor phase was settled to zero, VGP,0 = 0, we obtained the smaller value of 7270 kPa. The difference was 0.7% and this may be within the experimental error of this measurement. For the settle of α = 0, the obtained result was 7310 kPa. The difference was 0.1%. For the settle of VGP,0 = α = 0, the result was 7260 kPa which was lower than that about 0.8%. This means that the Henry’s constant is more sensitive to the initial volume than α. The value of α is 0.00077, and this effect is very small for this system. The difference in the Henry’s constants is mainly proportional to the initial volume of the vapor phase in the cell. Therefore, the initial volume of the vapor
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Table 1 Henry’s constants and infinite dilution activity coefficients of solutes in methanol T (K)
Hg (kPa)
γ∞
ϕgV
ϕsV
fsL,0
ϕgsat
λPOY
Z
Butane 255.05 259.98 270.05 280.00 290.01 299.96 309.97 319.99
860 1040 1470 2040 2730 3510 4510 5600
17.9 17.4 16.6 16.1 15.5 14.8 14.4 13.8
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.19 1.69 3.31 6.13 10.9 18.4 30.0 47.3
0.976 0.972 0.963 0.953 0.941 0.928 0.914 0.899
1.002 1.002 1.000 0.999 0.996 0.994 0.990 0.986
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
Isobutane 255.00 260.04 269.99 280.01 290.04 300.01 310.03 320.02
1320 1570 2180 2910 3780 4830 6020 7320
17.5 17.1 16.5 15.8 15.1 14.6 14.1 13.6
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.18 1.69 3.30 6.13 10.9 18.4 30.1 47.3
0.966 0.961 0.950 0.938 0.924 0.909 0.893 0.876
1.001 1.000 0.998 0.996 0.993 0.989 0.984 0.978
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
1-Butene 255.06 260.03 270.00 279.98 290.01 300.02 309.98 320.09
620 750 1100 1500 2070 2730 3490 4400
10.2 10.1 10.0 9.7 9.6 9.5 9.3 9.1
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.19 1.69 3.30 6.12 10.9 18.4 30.0 47.5
0.973 0.968 0.959 0.948 0.935 0.922 0.907 0.891
1.002 1.001 1.000 0.997 0.995 0.992 0.988 0.983
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
Isobutene 255.02 259.98 270.03 280.01 290.00 300.08 310.04 320.02
600 730 1060 1480 2020 2670 3450 4360
9.8 9.7 9.6 9.4 9.3 9.2 9.0 8.9
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.18 1.69 3.31 6.13 10.8 18.5 30.1 47.3
0.972 0.968 0.958 0.947 0.935 0.921 0.906 0.890
1.002 1.001 0.999 0.997 0.995 0.992 0.988 0.983
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
phase should be made as small as possible. In addition, the volume effect will depend on the total volume of the cell and the flow rate of the inert gas. If the solute is nonvolatile, the volume effect will be reduced. 4.2. Effects of nonideality The values of the fugacity coefficients of solute and solvent in vapor phase at experimental conditions and the fugacity of solvent at reference state are also indicated in Table 1. The fugacity coefficients of
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solute in vapor phase were almost 1 as indicated in Table 1. On the other hand, the fugacity coefficients of solvent in vapor phase differ from 1 by about 2% at the highest temperature. For example, the Henry’s constant was 7320 kPa for isobutane at 320.02 K when we used the values of the fugacity coefficients listed in this table. When we settled these values (ϕgV , ϕsV , ϕgsat , λPOY , Z) to 1, we obtained the same value of the Henry’s constant. On the other hand, the value of the infinite dilution activity coefficient was changed from 13.6 to 11.7. The difference was about 14%. As shown in Eqs. (A.11) and (A.13), the ratio of the fugacity of solvent at the reference state and the fugacity coefficient of solvent in vapor phase becomes: fsL,0 = Pssat ϕsV This means that the ratio may not be affected by the nonideality and the Henry’s constant will depend only on the nonideality of the solute, ϕgV . For the systems indicated in Table 1, the values of the fugacity coefficients of the solute were mostly 1 for all systems, as the results of the Henry’s constants were not changed. This is the reason why the Henry’s constants were not changed even for the system with ϕsV = 0.978. On the other hand, the infinite dilution activity coefficient was changed from 13.6 to 11.7. As indicated by Eq. (A.2), the activity coefficients directly depend on the multiple of the fugacity coefficient and the Poynting correction factor. When the value of the Henry’s constant was kept to be constant, the difference on the multiple will be proportional to that of the activity coefficients. The thermodynamic properties for evaluating the Henry’s constant were calculated by the virial equation of state truncated after the second virial coefficient. For highly volatile solutes like butane, isobutane, 1-butene and isobutene, the vapor pressures at 320 K are about 600 kPa. To examine the availability of the virial equation of state at these moderate pressures, the Soave equation of state [11] was used for comparative purposes, and the fugacity coefficients of solute at saturation were calculated. The obtained value of it for isobutane at 320.02 K from the Soave equation of state was 0.878. This value agrees well to that calculated from the virial equation of state (0.876). 5. Conclusion The Henry’s constants and the infinite dilution activity coefficients have been obtained from the gas stripping measurements at temperatures from 255 to 320 K. The rigorous formula to evaluate the Henry’s constants from these experiments has been proposed. By using this formula, the effects of nonideality of fluids and the existence of gas phase in the cell have been discussed. In general, the Henry’s constants do not depend on the nonideality so much, on the other hand, the activity coefficients strongly depend on the nonideality of solute at the reference state. The infinite dilution activity coefficients show a simple tendency that its value for butane agrees with that for isomer within 2.5% and that for 1-butene agrees with that for isomer within 4% in the temperature range from 255 to 320 K. List of symbols a, b constants in Eq. (A.6) B the second virial coefficient D gas flow rate
Y. Miyano et al. / Fluid Phase Equilibria 208 (2003) 223–238
f H KGC n P R S t T v V VGP x y Z
235
fugacity Henry’s constant proportional constant for gas chromatography number of moles Pressure gas constant peak area detected by GC time absolute temperature molar volume volume volume of vapor phase in the cell mole fraction in liquid phase mole fraction in vapor phase compressibility factor
Greek letters α volume factor defined by Eq. (28) γ activity coefficient λPOY Poynting correction factor ϕ fugacity coefficient ω acentric factor Subscripts 0 initial value at time = 0 c critical property g solute (gas component) i component i j component j mix mixture s solvent Superscripts 0 property at a reference state ∞ property at infinite dilution L property in liquid phase sat property at saturation V property in vapor phase
Acknowledgements This paper reports part of the work supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (13650820), which is gratefully acknowledged.
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Appendix A The relationship between the Henry’s constants and the activity coefficient at infinite dilution is expressed as: Hg = fg0 γ ∞ = fgsat λPOY γ ∞ = ϕgsat Pgsat λPOY γ ∞ γ∞ =
Hg sat ϕg Pgsat λPOY
where λPOY is the Poynting correction factor, and well approximated as: (P − Pgsat )vgL,sat λPOY ≈ exp RT
(A.1) (A.2)
(A.3)
where vgL,sat is the liquid molar volume of the saturated pure solute at the system temperature. For the evaluation of the nonideality of vapor in the cell, we adopted the virial equation of state. This equation is valid if the vapor pressures of solute and solvent are not so high: Z =1+
Bmix vmix
The second virial coefficient of gas mixtures is expressed as: Bmix = yi yj Bij i
(A.4)
(A.5)
j
For pure gases, we used the correlation methods proposed by Van Ness and Abbott [5] and Tsonopoulos [6] to evaluate the second virial coefficients. This correlation was very useful because there were no data for methanol at lower temperatures: BPc 0.422 0.172 a b = 0.083 − 1.6 + ω 0.139 − 4.2 + 6 − 8 (A.6) RTc Tr Tr Tr Tr where Tc and Pc are the critical values, Tr = T /Tc , ω the acentric factor, a and b are constants for polar substances. The critical properties, acentric factors and vapor pressures for pure substances were evaluated from the Data Bank in the literature [12]. The saturated liquid densities for pure substances were also evaluated from the data book [13]. For helium, however, the acentric factor was adjusted as ω = −0.17 to give good agreements with literature values [14] for the second virial coefficients. Furthermore, we used the following values for methanol which were determined by fitting to the available experimental second virial coefficients at temperatures from 320 to 350 K [14]. a = 0.0878 and b = 0.057 (for methanol) a = b = 0 (for hydrocarbons and helium) For the evaluation of the cross-second virial coefficients, we used the following mixing rule: Bij Pc,ij aij bij 0.172 0.422 = 0.083 − 1.6 + ωij 0.139 − 4.2 + 6 − 8 RTc,ij Tr,ij Tr,ij Tr,ij Tr,ij
(A.7)
Y. Miyano et al. / Fluid Phase Equilibria 208 (2003) 223–238
where Tc,ij
= Tc,i Tc,j ,
ωij = 21 (ωi + ωj ),
Tr,ij
T = , Tc,ij
Tc,ij = Pc,ij
aij = 21 (ai + aj ),
(Tc,i /Pc,i )1/3 + (Tc,j /Pc,j )1/3 2
237
3
bij = 21 (bi + bj )
Then the fugacity coefficient of component k in vapor phase at a pressure of P is expressed as: P V ϕk = exp 2 yi Bki − Bmix RT i
(A.8) (A.9)
(A.10)
The variable ϕgV was evaluated from this equation; ϕsV , however, was evaluated from a method based on the liquid fugacity as follows to avoid the uncertainties of the estimated cross-second virial coefficient:
(P − Pssat )vsL,sat V sat ϕs = ϕs exp (A.11) RT ϕssat and ϕgsat , which are the fugacity coefficient of pure solvent and pure solute at saturations, respectively, were evaluated as follows: sat Pk Bk ϕksat = exp (A.12) RT The fugacity of solvent at the reference state was similarly calculated as follows:
(P − Pssat )vsL,sat L,0 sat sat fs = ϕs Ps exp RT
(A.13)
A.1. Evaluation of saturated gas volume For each component: Pssat dV = −ZRT dns Pyg dV =
(A.14)
S dV = −ZRT dng KGC
(P − Pssat − Pyg ) dV = ZRT dnHe
(A.15) (A.16)
KGC is the proportional constant between the peak area of solute and its partial pressure. The values of KGC for all solutes were determined by using pure solutes. Combining above three equations yields P dV = ZRT dnHe + Pssat dV +
S dV KGC
(A.17)
On the other hand, Eq. (35) is rewritten as: S = (1 − C2 V )C1 S0
(A.18)
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where C1 =
Hg ϕsV ϕgV (1 + α)fsL,0 − αϕsV Hg
− 1,
C2 =
[(1 + α)fsL,0 /ϕsV ] − [αHg /ϕgV ] ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)]
(A.19)
Integration of Eq. (A.17) combined with Eq. (A.18) yields (P − Pssat )V +
S0 [(1 − C2 V )C1 +1 − 1] = ZRTnHe KGC C2 (C1 + 1)
(A.20)
V can be obtained from this equation. If the mass of solute in vapor phase is negligible, the following approximation can be used: V ≈
ZRTnHe P − Pssat
(A.21)
nHe is the number of moles of helium flowing out of the cell and it was measured by the mass flow meter every 60 min. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
W. Hayduk, H. Laudie, AIChE J. 19 (1973) 1233–1238. J.-C. Leroi, J.-C. Masson, H. Renon, J.-F. Fabries, H. Sannier, Ind. Eng. Chem., Process Des. Dev. 16 (1977) 139–144. K. Fukuchi, K. Miyoshi, T. Watanabe, S. Yonezawa, Y. Arai, Fluid Phase Equilib. 182 (2001) 257–263. K. Fukuchi, K. Miyoshi, T. Watanabe, S. Yonezawa, Y. Arai, Fluid Phase Equilib. 194–197 (2002) 937–945. H.C. Van Ness, M.M. Abbott, Classical Thermodynamics of Nonelectrolyte Solutions, McGraw-Hill, New York, 1982. C. Tsonopoulos, AIChE J. 21 (1975) 827–829. Y. Miyano, W. Hayduk, Fluid Phase Equilib. 135 (1997) 259–268. Y. Miyano, K. Nakanishi, Fluid Phase Equilib. 204 (2003) 131–142. Y. Miyano, K. Nakanishi, J. Chem. Thermodyn. 35 (3) (2003) 519–528. C.B. Kretschmer, R. Wiebe, J. Am. Chem. Soc. 74 (1952) 1276–1277. G. Soave, Chem. Eng. Sci. 27 (1972) 1197–1203. R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liquids, McGraw-Hill, New York, 1987. B.D. Smith, R. Srivastava, Thermodynamic Data for Pure Compounds. Parts A and B, Elsevier, Amsterdam, 1986. J.H. Dymond, E.B. Smith, The Virial Coefficients of Pure Gases and Mixtures, Clarendon Press, Oxford, 1980.
Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at 255–320 K Yoshimori Miyano a,∗ , Koichiro Nakanishi a , Kenji Fukuchi b a
Department of Chemistry and Bioscience, Kurashiki University of Science and the Arts, 2640 Nishinoura, Tsurajimacho, Kurashiki 712-8505, Japan b Department of Chemical and Biological Engineering, Ube National College of Technology, 2-14-1 Tokiwadai, Ube 755-8555 Japan Received 18 December 2002; accepted 5 February 2003
Abstract The Henry’s constants and the infinite dilution activity coefficients of butane, isobutane, 1-butene and isobutene in methanol at 255–320 K are measured by a gas stripping method. The rigorous formula for evaluating the Henry’s constants from the gas stripping measurements is proposed for these highly volatile mixtures. By using this formula, a volume effect of vapor phase and the effect of nonideality of fluids are discussed. In the evaluations for activity coefficients the nonideality of solute was not negligible especially at higher temperatures. The values of Henry’s constants of butane are much different from those of isobutane, while the activity coefficients are not so different to each other. The activity coefficients of butane are about 2.5% greater than those of isobutane, and those of 1-butene are about 4% greater than those of isobutene. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Data; Mixture; Henry’s constant; Activity coefficient; Methanol; Gas stripping
1. Introduction A systematic study of gas solubilities like the Henry’s constants is useful in providing design data for absorption processes, as well as, indirectly, in aiding the analysis of molecular interactions in solutions. Butane gas is highly soluble in nonpolar, nonassociating solvents. Solubilities of nonreacting gases in nonpolar, nonassociating solvents generally follow an order of increasing solubility with increases in the normal boiling point temperature of liquefied gas. As the boiling point temperature of butane is higher than that of isobutane, it is expected that the solubilities of butane are greater than that of isobutane which means that the Henry’s constants of butane are smaller than those of isobutane. Similarly, it is expected that ∗
Corresponding author. Tel.: +81-86-440-1069; fax: +81-86-440-1062. E-mail address: [email protected] (Y. Miyano). 0378-3812/03/$ – see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0378-3812(03)00036-0
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the Henry’s constants of 1-butene are smaller than those of isobutene. While this approximate relationship exists, it is not sufficiently consistent to permit the prediction of gas solubility even for simple solutions. In polar and associating solvents, the degree of association in the solvent has a most profound effect on the solubility [1]. Although a large number of alkane, alkene + alcohol solubility data were published, few data are available for C4-gases like butane, isobutane, 1-butene and isobutene + alcohol mixtures. The solubility data will be useful to develop prediction methods. Especially for group contribution methods, it may be necessary to take into account the differences between isomers. For developing the molecular theory, on the other hand, the accurate intermolecular potential is necessary. The Henry’s constant is directly related to the residual chemical potential of solute at infinite dilution which is evaluated from the intermolecular potential between a solute molecule and a solvent molecule. Therefore, the Henry’s constant is a suitable macroscopic property for testing the intermolecular potential between different kinds of molecules. The gas stripping method originally proposed by Leroi et al. [2] has been usually used for measuring the activity coefficients at infinite dilution of solutes in nonvolatile solvents, and the vapor pressures of solutes were negligibly small in most cases [3,4]. In this work, however, the Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at temperatures from 255 to 320 K are measured by this method. For these highly volatile solutes and solvents, careful treatments for data reductions are necessary and the rigorous expression is derived for this purpose. 2. Theory The gas stripping method originally proposed by Leroi et al. [2] is based on the variation of vapor phase composition when the highly diluted solute of the liquid mixture is stripped from the solution by a constant flow of inert gas. They derived an approximated equation to express the relationship between the solute peak area measured by a gas chromatography and the gas volume flowing out of the equilibrium cell as follows: γ ∞ Pgsat S Pssat P ln = − 1 ln 1 − Dt (1) S0 Pssat P − Pssat ns,0 RT where D is the flow rate of inert gas, t the experimental time, and Dt is the total volume of inert gas flowing out of the equilibrium cell at time t. From Eq. (1) the activity coefficients of solute at infinite dilution can be obtained as the slope of the line in a log–log plot. When we use the dry gas volume, Dt, it is not possible to solve the differential equations which are based on the material balance and the thermodynamic relationships at equilibrium. Eq. (1), which was derived under some assumptions, therefore, may not be used for highly volatile mixtures. On the other hand, when we use the volume of the saturated gas flowing out of the equilibrium cell, we can get the rigorous solution as follows. In general, the pressure–volume–temperature relations of a fluid can be shown as follows: PV = ZnRT
(2)
Here, P is the total pressure, V the volume, Z the compressibility factor, n the number of moles, R the gas constant, and T the absolute temperature.
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225
If n is the total number of moles of solute (gas), solvent and the inert gas (helium) in the cell at time t, the quantities withdrawn from the cell during a very short time dt by the inert gas flow is: P −dn = dV (3) ZRT −dn = −dng − dns + dnHe (4) where dV is the volume of the saturated gas flowing out of the cell, ng and ns are, respectively, the total number of moles of solute (g) and solvent (s) in the cell, and dng and dns are, respectively, the change of moles of solute and solvent in the cell, and expressed as: dng = yg dn
(5)
dns = ys dn
(6)
where yg and ys are, respectively, the mole fraction of solute and solvent in vapor phase, dnHe is the number of moles of the inert gas flowing out of the cell and the same quantity of the inert gas is introduced into the cell at the same time. Therefore, the total number of moles of helium in the cell will be kept constant. Substituting Eq. (3) into Eqs. (5) and (6) yields Pyg dng = − dV (7) ZRT Pys dns = − dV (8) ZRT On the other hand, the following thermodynamic conditions should be satisfied at equilibrium: fgV = fgL ,
fsV = fsL ,
V L fHe = fHe
(9)
where f is the fugacity, the superscripts V and L are vapor and liquid phases, respectively. When we use a fugacity coefficient, the above equations can be rewritten as: ϕiV =
fi V Pyi
,
ϕiL =
fiL Pxi
fgV = ϕgV Pyg ,
fsV = ϕsV Pys ,
fgL = ϕgL Pxg ,
fsL = ϕsL Pxs ,
(10) V V fHe = ϕHe PyHe L L fHe = ϕHe PxHe
(11) (12)
where xg and xs are, respectively, the mole fraction of solute and solvent in liquid phase. If the solute is highly diluted in the solvent and if the solubility of helium in the liquid phase is negligible, the mole fraction and the activity coefficient of the solvent may be taken equal to 1, and if the existence of helium does not affect the equilibrium between solute and solvent, the following two equations must be satisfied: fgL = ϕgL Pxg = Hg xg = ϕgV Pyg ,
fsL = ϕsL Pxs = fsL,0 γs xs = fsL,0 = ϕsV Pys
(13)
where Hg is the Henry’s constant of solute, fsL,0 is the fugacity of pure solvent at the reference state (pure liquid at system temperature and pressure). Then Eqs. (7) and (8) become: dng = −
Hg nLg Hg xg dV = − dV ϕgV ZRT ϕgV ZRT nL
(14)
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dns = −
fsL,0 xs fsL,0 nLs dV = − dV ϕsV ZRT ϕsV ZRT nL
nL = nLg + nLs
(15) (16)
where nL is the total number of moles in the liquid phase, nLg and nLs are, respectively, the number of moles of solute and solvent in the liquid phase. As the mole fraction of solute in the liquid phase is very small (less than 10−3 ), nLg nL
≈
nLg
, nLs
nLs ≈1 nL
(17)
Then Eqs. (14) and (15) become: dng = −
Hg nLg dV ϕgV ZRT nLs
(18)
dns = −
fsL,0 dV ϕsV ZRT
(19)
Here we have to know the relationships between n and nL : ng = nLg + nV g
(20)
ns = nLs + nV s
(21)
When we define the volume of vapor phase in the cell at time t as VGP , nV g
nLg Hg VGP Hg VGP PVGP = yg = xg V = L ZRT ϕg ZRT ns + nLg ϕgV ZRT
Substituting Eqs. (17) and (22) into Eq. (20) yields: nLg Hg VGP 1 Hg VGP L L ng = ng + L V = ng 1 + L V ns ϕg ZRT ns ϕg ZRT
(22)
(23)
Similarly, Eq. (21) becomes: L ns = nLs + nV s = ns + ys
PVGP f L,0 VGP = nLs + sV ZRT ϕs ZRT
(24)
Integration of Eq. (19) yields the following result with approximation that all parameters on right-hand side of the equation are constants. It can be shown that in most cases this approximation is valid when the mole fraction of the solute is less than 10−3 : ns − ns,0 = −
fsL,0 V ϕsV ZRT
where ns,0 is the number of moles of solvent in the cell at the initial state.
(25)
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227
On the other hand, from Eq. (24) we can get: ns − ns,0 = nLs − nLs,0 +
fsL,0 (VGP − VGP,0 ) ϕsV ZRT
(26)
Combining Eqs. (25) and (26) yields: nLs = nLs,0 −
fsL,0 (V + VGP − VGP,0 ) ϕsV ZRT
(27)
where VGP,0 is the initial volume of the vapor phase in the cell, while VGP slightly depends on the gas volume withdrawn from the cell and it is expressed as: VGP = VGP,0 + αV ,
α=
Pssat vsL,sat ZRT
(28)
where vsL,sat is the liquid molar volume of solvent at saturation. Then Eq. (27) becomes: nLs = nLs,0 −
(1 + α)fsL,0 V ϕsV ZRT
(29)
Substituting Eq. (23) into Eq. (18) yields: dng = −
Hg V ϕg ZRT
[nLs
ng dV + (Hg VGP /ϕgV ZRT)]
(30)
By substituting Eq. (29) into Eq. (30), dng = −
Hg V ϕg ZRT
[nLs,0
− {(1 +
ng L,0 V α)fs /ϕs ZRT}V
+ (Hg VGP /ϕgV ZRT)]
dV
(31)
Integration of this equation yields
[(1 + α)fsL,0 /ϕsV ] − (αHg /ϕgV ) Hg ϕsV ng = ln 1 − V ln ng,0 ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)] ϕgV (1 + α)fsL,0 − αϕsV Hg
(32)
where ng,0 is the number of moles of solute in the cell at the initial state. If the peak area, S, of the solute detected by a gas chromatography is proportional to the solute partial pressure, Hg Hg nLg S = KGC Pyg = KGC V xg ≈ KGC V L ϕg ϕg ns we can get the following relationship: nLg nLs,0 nLs,0 + (Hg VGP,0 /ϕgV ZRT) ng S = L = S0 ng,0 nLs + {Hg (VGP,0 + αV )/ϕgV ZRT} ng,0 nLs
(33)
(34)
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By inserting Eq. (29) into this equation,
S ln S0
[nLs,0 − {(1 + α)fsL,0 /ϕsV ZRT}V + {Hg (VGP,0 + αV )/ϕgV ZRT}] ng = ln − ln ng,0 nLs,0 + (Hg VGP,0 /ϕgV ZRT) [(1 + α)fsL,0 /ϕsV ] − (αHg /ϕgV ) ng = ln − ln 1 − V ng,0 ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)]
Combining Eqs. (32) and (34 ) yields the final result as follows: [(1 + α)fsL,0 /ϕsV ] − (αHg /ϕgV ) Hg ϕsV S ln = V − 1 ln 1 − S0 ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)] ϕgV (1 + α)fsL,0 − αϕsV Hg If the volume of vapor phase in the cell is negligible (VGP,0 = 0 and α = 0), Hg ϕsV S fsL,0 ln = − 1 ln 1 − V V S0 ϕs ZRTns,0 ϕgV fsL,0
(34 )
(35)
(36)
Furthermore, if the vapor phase can be treated as an ideal gas (ϕgV = ϕsV = Z = 1) and using an approximation of fsL,0 ≈ Pssat , we can get Hg S Pssat ln = − 1 ln 1 − V (37) S0 Pssat ns,0 RT This equation is similar to Eq. (1) proposed by Leroi et al. [2]. When the vapor pressure of solute is negligibly small, the PDt/(P − Pssat ) in Eq. (1) is approximately equal to the volume of the saturated vapor, V, and Hg ≈ γ ∞ Pgsat . Pisat is the vapor pressure of pure component i. The details for the calculation methods of each value of fugacity coefficients, etc. are described in Appendix A.
3. Experimental The experimental apparatus used for measuring the Henry’s constants is shown in Fig. 1. The inert gas (helium) flow rate was measured by a mass flow meter (Koflok Model 3300, maximum flow rate = 2 cm3 min−1 , accuracy = 1%) and the total volume was measured by an integrating unit (Kofloc CR-700). The flow rate of the inert gas was adjusted by a fine metering valve (double needles, Swagelock SS2-D). The equilibrium cell which volume is 44 cm3 is shown in Fig. 2. This cell contains an inner glass tube and a magnetic stirrer. The inner tube makes a good mixing of the whole solution, and also makes a counter flow of liquid against the rising bubbles. This means that the vapor–liquid contacting time becomes longer. The exit port of the inert gas in the equilibrium cell was made of a stainless tube with inner diameter of 0.14 mm and small size bubbles were made to achieve phase equilibrium within a short time. The mass flow meter was calibrated by a volumetric flow meter shown in Fig. 3. Ethylene glycol was used for the liquid in a burette because of its nonvolatility. The inert gas flowing out of the mass flow meter was introduced into the burette. The inner volume of the burette can be changed by moving a flask connected by a flexible tube. The pressure in the burette was kept to a constant value within ±2 Pa by
Y. Miyano et al. / Fluid Phase Equilibria 208 (2003) 223–238
229
Fig. 1. Experimental apparatus for measuring Henry’s constants.
using a pressure gage (Paroscientific, Digiquartz Pressure Transducer Model 215A and Sokken Digiquartz Pressure Computer Model 600S) and changing the level of the flask. First of all, about 43 cm3 of the solvent (methanol) was introduced into the equilibrium cell and the accurate quantity of the solvent was determined by mass. Then the equilibrium cell was immersed in a constant-temperature bath and connected to a supply of solute gas and inert gas. After evacuating the connecting lines, the lines were pressurized with a solute gas at an arbitrary pressure (absolute pressure of about 110 kPa), and then the solute gas was introduced into the equilibrium cell. The amount of the solute gas introduced into the cell was adjusted to keep the mole fraction lower than 3 × 10−4 in the solution. After evacuating again the connecting lines, it began to flow the inert gas into the equilibrium cell with
Fig. 2. Equilibrium cell for the gas stripping method.
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Fig. 3. Volumetric flow meter to calibrate the mass flow meter.
a constant flow rate of about 2 cm3 min−1 . After achievement of the steady state, the measurement of the inert gas volume by the mass flow meter was started and the sampling for the gas chromatography was done every 60 min. These samplings were automatically done by using an air actuation sampling valve (Valco, Model A4C6UWE) and a controller unit, and were continued for about 15 h. The pressure in the cell was approximated equal to the atmospheric pressure and measured by the Digiquartz Pressure Transducer. The atmospheric pressures, the temperatures of the bath and the integrated inert gas volumes were also measured every 60 min. The temperature of the constant-temperature bath (filled with ethylene glycol + water) was controlled within ±0.02 K, and measured by a quartz thermometer (Hewlett-Packard, Model 2804A) with accuracy of 0.01 K. The gas flowing out of the equilibrium cell was kept to a higher temperature than that of the bath to avoid any condensations, and introduced to a gas chromatograph (Hitachi, Model G-3000, with double FID detectors) for measuring the solute peak area. The butane, isobutane, 1-butene and isobutene were supplied by Takachiho Kagaku Kogyo at specified minimum purities of 99.8, 99, 99 and 99%, respectively. The methanol was of HPLC grade and supplied by Aldrich Chemicals, with a specified minimum purity of 99.9%. The accuracy of this measurement for the Henry’s constants may be considered within 2%, and it mostly depends on the accuracy of the mass flow meter. 4. Results and discussion Fig. 4 shows the temperature dependency of the Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at the temperature range of 255–320 K. For comparative purposes, this figure contains the literature values [7–10] which had been predicted from the experimentally obtained gas solubility data at 298.15 K and pressures under 102 kPa. The Henry’s constants measured by this gas stripping method agree well as shown in this figure except the data proposed by Kretschmer and Wiebe [10].
Y. Miyano et al. / Fluid Phase Equilibria 208 (2003) 223–238
231
Fig. 4. Henry’s constants of butane, isobutane, 1-butene and isobutene in methanol at temperatures from 255 to 320 K.
The Henry’s constants of isobutane show higher values than others. This may be explained by the general relationship: because the boiling point temperature of isobutane (261.4 K) is the lowest among these gases. The boiling point temperatures of 1-butene (266.9 K) and isobutene (266.2 K) are similar and that of isobutene is little bit lower than that of 1-butene. According to the general relationship, it is expected that the Henry’s constants of isobutene may be greater than that of 1-butene. However, the Henry’s constants of isobutene show the smallest values in the entire temperature range. Furthermore, while the boiling point temperature of butane (272.7 K) is much higher than that of isobutene, the Henry’s constants of butane are much greater than that of isobutene. These facts show that the general relationship is not sufficiently consistent to permit the prediction of gas solubility even for simple gases. On the other hand, 1-butene and isobutene have dipole moments of 0.3 and 0.5 debyes (1 debye = 3.162 × 10−24 J1/2 m3/2 ), respectively. This may be the reason why the Henry’s constants of alkene are smaller than that of alkane in methanol which is a dipolar molecule. Fig. 5 shows the temperature dependency of the infinite dilution activity coefficients for the same systems. As shown in this figure, the infinite dilution activity coefficients between butane and isobutane show similar temperature dependencies and the differences in the values are very small, about 2.5%, and the activity coefficients of butane are little bit higher than that of the isomer. The same phenomenon can be observed between 1-butene and isobutene. The differences in the infinite dilution activity coefficients between 1-butene and isobutene are about 4%. The Henry’s constants and the infinite dilution activity coefficients measured in this work are numerically indicated in Table 1 with some other properties predicted for the data reductions. The effects of the volume of vapor phase in the cell and the nonideality of vapor are discussed in the following sub-sections.
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Fig. 5. The infinite dilution activity coefficients of butane, isobutane, 1-butene and isobutene in methanol at temperatures from 255 to 320 K.
4.1. Volume effect of vapor phase In general, the existence of vapor phase in the cell affects the evaluation of the Henry’s constants for the gas stripping method. When a solution contacts to a fresh inert gas, some amounts of solute and solvent in the solution will move into the inert gas bubbles to keep the equilibrium. If there is a space to keep the vapor in the cell, the solute can stay there for its residence time, and some amounts of them will resolve into the solution again. This will make the speed down for the gas stripping. Therefore, the evaluated value of the Henry’s constant without the correction of the volume effect of vapor phase will become smaller than the true value. For example, the initial volume of the vapor phase in the equilibrium cell was 2.3 cm3 for the system of isobutane+methanol at 320.02 K. The final volume of the vapor phase for this system after 16 h was about 5.5 cm3 (this volume change is expressed by Eq. (28)). The volume of vapor phase increased about 2.2 cm3 during this experiment. Then we obtained the result that the Henry’s constant of isobutane was 7320 kPa as indicated in Table 1. When the initial volume of the vapor phase was settled to zero, VGP,0 = 0, we obtained the smaller value of 7270 kPa. The difference was 0.7% and this may be within the experimental error of this measurement. For the settle of α = 0, the obtained result was 7310 kPa. The difference was 0.1%. For the settle of VGP,0 = α = 0, the result was 7260 kPa which was lower than that about 0.8%. This means that the Henry’s constant is more sensitive to the initial volume than α. The value of α is 0.00077, and this effect is very small for this system. The difference in the Henry’s constants is mainly proportional to the initial volume of the vapor phase in the cell. Therefore, the initial volume of the vapor
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Table 1 Henry’s constants and infinite dilution activity coefficients of solutes in methanol T (K)
Hg (kPa)
γ∞
ϕgV
ϕsV
fsL,0
ϕgsat
λPOY
Z
Butane 255.05 259.98 270.05 280.00 290.01 299.96 309.97 319.99
860 1040 1470 2040 2730 3510 4510 5600
17.9 17.4 16.6 16.1 15.5 14.8 14.4 13.8
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.19 1.69 3.31 6.13 10.9 18.4 30.0 47.3
0.976 0.972 0.963 0.953 0.941 0.928 0.914 0.899
1.002 1.002 1.000 0.999 0.996 0.994 0.990 0.986
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
Isobutane 255.00 260.04 269.99 280.01 290.04 300.01 310.03 320.02
1320 1570 2180 2910 3780 4830 6020 7320
17.5 17.1 16.5 15.8 15.1 14.6 14.1 13.6
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.18 1.69 3.30 6.13 10.9 18.4 30.1 47.3
0.966 0.961 0.950 0.938 0.924 0.909 0.893 0.876
1.001 1.000 0.998 0.996 0.993 0.989 0.984 0.978
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
1-Butene 255.06 260.03 270.00 279.98 290.01 300.02 309.98 320.09
620 750 1100 1500 2070 2730 3490 4400
10.2 10.1 10.0 9.7 9.6 9.5 9.3 9.1
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.19 1.69 3.30 6.12 10.9 18.4 30.0 47.5
0.973 0.968 0.959 0.948 0.935 0.922 0.907 0.891
1.002 1.001 1.000 0.997 0.995 0.992 0.988 0.983
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
Isobutene 255.02 259.98 270.03 280.01 290.00 300.08 310.04 320.02
600 730 1060 1480 2020 2670 3450 4360
9.8 9.7 9.6 9.4 9.3 9.2 9.0 8.9
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
0.998 0.997 0.996 0.993 0.990 0.987 0.983 0.978
1.18 1.69 3.31 6.13 10.8 18.5 30.1 47.3
0.972 0.968 0.958 0.947 0.935 0.921 0.906 0.890
1.002 1.001 0.999 0.997 0.995 0.992 0.988 0.983
1.000 1.000 1.000 1.000 0.999 0.998 0.995 0.990
phase should be made as small as possible. In addition, the volume effect will depend on the total volume of the cell and the flow rate of the inert gas. If the solute is nonvolatile, the volume effect will be reduced. 4.2. Effects of nonideality The values of the fugacity coefficients of solute and solvent in vapor phase at experimental conditions and the fugacity of solvent at reference state are also indicated in Table 1. The fugacity coefficients of
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solute in vapor phase were almost 1 as indicated in Table 1. On the other hand, the fugacity coefficients of solvent in vapor phase differ from 1 by about 2% at the highest temperature. For example, the Henry’s constant was 7320 kPa for isobutane at 320.02 K when we used the values of the fugacity coefficients listed in this table. When we settled these values (ϕgV , ϕsV , ϕgsat , λPOY , Z) to 1, we obtained the same value of the Henry’s constant. On the other hand, the value of the infinite dilution activity coefficient was changed from 13.6 to 11.7. The difference was about 14%. As shown in Eqs. (A.11) and (A.13), the ratio of the fugacity of solvent at the reference state and the fugacity coefficient of solvent in vapor phase becomes: fsL,0 = Pssat ϕsV This means that the ratio may not be affected by the nonideality and the Henry’s constant will depend only on the nonideality of the solute, ϕgV . For the systems indicated in Table 1, the values of the fugacity coefficients of the solute were mostly 1 for all systems, as the results of the Henry’s constants were not changed. This is the reason why the Henry’s constants were not changed even for the system with ϕsV = 0.978. On the other hand, the infinite dilution activity coefficient was changed from 13.6 to 11.7. As indicated by Eq. (A.2), the activity coefficients directly depend on the multiple of the fugacity coefficient and the Poynting correction factor. When the value of the Henry’s constant was kept to be constant, the difference on the multiple will be proportional to that of the activity coefficients. The thermodynamic properties for evaluating the Henry’s constant were calculated by the virial equation of state truncated after the second virial coefficient. For highly volatile solutes like butane, isobutane, 1-butene and isobutene, the vapor pressures at 320 K are about 600 kPa. To examine the availability of the virial equation of state at these moderate pressures, the Soave equation of state [11] was used for comparative purposes, and the fugacity coefficients of solute at saturation were calculated. The obtained value of it for isobutane at 320.02 K from the Soave equation of state was 0.878. This value agrees well to that calculated from the virial equation of state (0.876). 5. Conclusion The Henry’s constants and the infinite dilution activity coefficients have been obtained from the gas stripping measurements at temperatures from 255 to 320 K. The rigorous formula to evaluate the Henry’s constants from these experiments has been proposed. By using this formula, the effects of nonideality of fluids and the existence of gas phase in the cell have been discussed. In general, the Henry’s constants do not depend on the nonideality so much, on the other hand, the activity coefficients strongly depend on the nonideality of solute at the reference state. The infinite dilution activity coefficients show a simple tendency that its value for butane agrees with that for isomer within 2.5% and that for 1-butene agrees with that for isomer within 4% in the temperature range from 255 to 320 K. List of symbols a, b constants in Eq. (A.6) B the second virial coefficient D gas flow rate
Y. Miyano et al. / Fluid Phase Equilibria 208 (2003) 223–238
f H KGC n P R S t T v V VGP x y Z
235
fugacity Henry’s constant proportional constant for gas chromatography number of moles Pressure gas constant peak area detected by GC time absolute temperature molar volume volume volume of vapor phase in the cell mole fraction in liquid phase mole fraction in vapor phase compressibility factor
Greek letters α volume factor defined by Eq. (28) γ activity coefficient λPOY Poynting correction factor ϕ fugacity coefficient ω acentric factor Subscripts 0 initial value at time = 0 c critical property g solute (gas component) i component i j component j mix mixture s solvent Superscripts 0 property at a reference state ∞ property at infinite dilution L property in liquid phase sat property at saturation V property in vapor phase
Acknowledgements This paper reports part of the work supported by Grant-in-Aid for Scientific Research from Japan Society for the Promotion of Science (13650820), which is gratefully acknowledged.
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Appendix A The relationship between the Henry’s constants and the activity coefficient at infinite dilution is expressed as: Hg = fg0 γ ∞ = fgsat λPOY γ ∞ = ϕgsat Pgsat λPOY γ ∞ γ∞ =
Hg sat ϕg Pgsat λPOY
where λPOY is the Poynting correction factor, and well approximated as: (P − Pgsat )vgL,sat λPOY ≈ exp RT
(A.1) (A.2)
(A.3)
where vgL,sat is the liquid molar volume of the saturated pure solute at the system temperature. For the evaluation of the nonideality of vapor in the cell, we adopted the virial equation of state. This equation is valid if the vapor pressures of solute and solvent are not so high: Z =1+
Bmix vmix
The second virial coefficient of gas mixtures is expressed as: Bmix = yi yj Bij i
(A.4)
(A.5)
j
For pure gases, we used the correlation methods proposed by Van Ness and Abbott [5] and Tsonopoulos [6] to evaluate the second virial coefficients. This correlation was very useful because there were no data for methanol at lower temperatures: BPc 0.422 0.172 a b = 0.083 − 1.6 + ω 0.139 − 4.2 + 6 − 8 (A.6) RTc Tr Tr Tr Tr where Tc and Pc are the critical values, Tr = T /Tc , ω the acentric factor, a and b are constants for polar substances. The critical properties, acentric factors and vapor pressures for pure substances were evaluated from the Data Bank in the literature [12]. The saturated liquid densities for pure substances were also evaluated from the data book [13]. For helium, however, the acentric factor was adjusted as ω = −0.17 to give good agreements with literature values [14] for the second virial coefficients. Furthermore, we used the following values for methanol which were determined by fitting to the available experimental second virial coefficients at temperatures from 320 to 350 K [14]. a = 0.0878 and b = 0.057 (for methanol) a = b = 0 (for hydrocarbons and helium) For the evaluation of the cross-second virial coefficients, we used the following mixing rule: Bij Pc,ij aij bij 0.172 0.422 = 0.083 − 1.6 + ωij 0.139 − 4.2 + 6 − 8 RTc,ij Tr,ij Tr,ij Tr,ij Tr,ij
(A.7)
Y. Miyano et al. / Fluid Phase Equilibria 208 (2003) 223–238
where Tc,ij
= Tc,i Tc,j ,
ωij = 21 (ωi + ωj ),
Tr,ij
T = , Tc,ij
Tc,ij = Pc,ij
aij = 21 (ai + aj ),
(Tc,i /Pc,i )1/3 + (Tc,j /Pc,j )1/3 2
237
3
bij = 21 (bi + bj )
Then the fugacity coefficient of component k in vapor phase at a pressure of P is expressed as: P V ϕk = exp 2 yi Bki − Bmix RT i
(A.8) (A.9)
(A.10)
The variable ϕgV was evaluated from this equation; ϕsV , however, was evaluated from a method based on the liquid fugacity as follows to avoid the uncertainties of the estimated cross-second virial coefficient:
(P − Pssat )vsL,sat V sat ϕs = ϕs exp (A.11) RT ϕssat and ϕgsat , which are the fugacity coefficient of pure solvent and pure solute at saturations, respectively, were evaluated as follows: sat Pk Bk ϕksat = exp (A.12) RT The fugacity of solvent at the reference state was similarly calculated as follows:
(P − Pssat )vsL,sat L,0 sat sat fs = ϕs Ps exp RT
(A.13)
A.1. Evaluation of saturated gas volume For each component: Pssat dV = −ZRT dns Pyg dV =
(A.14)
S dV = −ZRT dng KGC
(P − Pssat − Pyg ) dV = ZRT dnHe
(A.15) (A.16)
KGC is the proportional constant between the peak area of solute and its partial pressure. The values of KGC for all solutes were determined by using pure solutes. Combining above three equations yields P dV = ZRT dnHe + Pssat dV +
S dV KGC
(A.17)
On the other hand, Eq. (35) is rewritten as: S = (1 − C2 V )C1 S0
(A.18)
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Y. Miyano et al. / Fluid Phase Equilibria 208 (2003) 223–238
where C1 =
Hg ϕsV ϕgV (1 + α)fsL,0 − αϕsV Hg
− 1,
C2 =
[(1 + α)fsL,0 /ϕsV ] − [αHg /ϕgV ] ZRT[nLs,0 + (Hg VGP,0 /ϕgV ZRT)]
(A.19)
Integration of Eq. (A.17) combined with Eq. (A.18) yields (P − Pssat )V +
S0 [(1 − C2 V )C1 +1 − 1] = ZRTnHe KGC C2 (C1 + 1)
(A.20)
V can be obtained from this equation. If the mass of solute in vapor phase is negligible, the following approximation can be used: V ≈
ZRTnHe P − Pssat
(A.21)
nHe is the number of moles of helium flowing out of the cell and it was measured by the mass flow meter every 60 min. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
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