J . Phys. Chem. 1990, 94, 2652-2651
2652
peaks for AgCl and I appear in TPD, we suppose that the surface is completely covered by Cl(a) and I(a) and that the remaining undecomposed CI2CO is now adsorbed over Cl(a) and I(a). (The surface is now probably AgCl and AgI.) Since the distance between C12C0 and metallic Ag is now much greater than without I(a) and Cl(a), the quenching rate is reduced significantly, which tends to increase the effective photolysis rate. Moreover, since these molecules do not directly couple with Ag, they are not subjected to red-shifted photolysis, just as for physisorbed C12C0 on Ag(ll1) and CH,Br on C / P t ( l l l).46 Within the framework of the direct excitation model, both these effects tend to bring the photolysis rate close to the gas-phase absorption cross section value. While it must be investigated further, the increased rate of photolysis found at long irradiation times (high halogen coverages) in Figure 13 is interesting from another perspective. Increasing the total halogen coverage, in the absence of CI,CO, will certainly increase the work function. Moreover, at halogen coverages high enough to give AgCl and AgI decomposition products in TPD, the local potential at an adsorbed ClzCO molecule must certainly be different from that for a clean metal. Confirming this, the TPD desorption energy drops slightly. Thus, even though the local electronic potential that substrate-excited electrons must surmount is higher, the rate is faster. This might be taken as an argument for the direct excitation model dominating under these conditions. To resolve these questions, initial rate measurements are needed on heavily iodized or chlorided Ag surfaces. 5. Summary Phosgene adsorbs molecularly on both clean and I(a)-covered Ag(l11) surfaces at 100 K. It desorbs reversibly without any thermal decomposition. Adsorption of 1 ML of phosgene lowers the surface work function by 0.5 eV on clean Ag( 11 1) and 0.7 eV on I/Ag( 1 11). Monolayer and submonolayer phosgene is readily photolyzed on both clean and I(a)-covered surfaces. The photolysis produces surface Cl(a) and gas-phase C O Cl(a) desorbs
as AgCl above 600 K in subsequent TPD and C O desorbs during photolysis. There is no photon-stimulated desorption of molecular phosgene. With or without I(a), the photolysis of the first monolayer, but not multilayers, on both surfaces shows a strong red shift from the gas-phase photodissociation. The threshold for the photolysis is 2.6-2.8 eV. Multilayer phosgene is photolyzed at a measurable rate only for photon energies above 4.1 eV. In general, the photolysis rate on I/Ag( 111) is slower than on clean Ag( 11 1). There is at least one exception: with the full arc, phosgene separated by a layer of I(a) and/or Cl(a) from the Ag surface is photolyzed more rapidly than phosgene adsorbed directly to Ag. The red shift is qualitatively explained by either a hot electron or a direct adsorbate-substrate complex absorption model. The latter model assumes that adsorbed phosgene molecules absorb photons directly and undergo an electronic transition from a nonbonding orbital of C1 to an antibonding orbital of C-C1, the latter involving a large Ag contribution. The interaction lowers the potential energy of the excited state of phosgene and, with hole screening, gives the red shift. The hot electron model assumes that photons produce energetic subvacuum electrons that attach to adsorbed CI2C0 to form a dissociative negative ion state. The slower photolysis rate on I/Ag( 11 1) (submonolayer I) than on clean Ag( 11 1) is explained in terms of faster quenching because of stronger coupling of phosgene to I/Ag( 1 11) and in terms of partial loss of the red shift. When the direct interaction between phosgene and Ag is weakened by placing a full layer or more of Cl(a) and I(a) between the two, the quenching rate is reduced, resulting in a more rapid photolysis rate. Acknowledgment. This work was supported in part by the U S . Army Research Office. We thank our colleagues, particularly Xiaoyang Zhu and Sohail Akhter, for many helpful discussions. Registry No. CI,CO, 75-44-5; Ag, 7440-22-4; 11, 7553-56-2; Clz, 7782-50-5; CO, 630-0870,
Effect of Hydrostatic Pressure and Salinity on the Stability of Gas Hydrated Y. Paul Handa Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K I A OR6 (Received: July 17, 1989)
The distribution of gas in the hydrate and the liquid phases for the two-phase systems methane hydrate-water and methane hydrate-seawater has been calculated as a function of hydrostatic pressure. It is found that at hydrostatic pressures higher than the three-phase hydrate-liquid-gas equilibrium pressure, a two-phase hydrate-liquid equilibrium exists in which the hydrate phase is increasingly enriched in and the liquid phase depleted of the gas as the hydrostatic pressure increases. At higher pressures, the methane content of water necessary to form and to stabilize the hydrate is much less than required for saturation with respect to the gas phase. Similar results are obtained for the methane hydrateice system where, at hydrostatic pressures above the three-phase hydrate-ice-gas equilibrium pressure, hydrates of increasingly higher cage occupancy are obtained at the expense of converting a small amount of hydrate to ice.
Introduction The temperature and pressure conditions prevailing at the sea bottom and under permafrost fall within the range of stability of hydrates of natural gas. The natural occurrence of hydrates on earth is now well established,' and they are also speculated to occur on other planets in the solar system2 In the natural environment on earth, water is generally present in excess, and thus the only requirement to be met before hydrates can form is the availability of gas. If gas is available, either in free form or dissolved in water above a certain critical concentration, then Issued as NRCC No. 3 103 1,
0022-3654/90/2094-2652$02.50/0
hydrates will form. However, there have been some misconceptions about the conditions necessary for the hydrates to form and to be stable under these geological environments. It has frequently been said that methane undergoes an abrupt fall in solubility in liquid water when hydrates begin to form, that hydrate former should be present in the gaseous state for the hydrates to form, and that liquid water should be supersaturated with gas before hydrates can form. None of these observations is thermodynamically correct. (1) Panayev, V . A. Int. Geol. Reo. 1987, 29, 596. (2) Lunine, J . I.; Stevenson, D. J. Astrophys. J . Suppl. 1985. 58, 493.
Published 1990 by the American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990 2653
Hydrostatic Pressure and Salinity Effects on Gas Hydrates Makogon3 reported that the solubility of methane in liquid water increased steadily with methane pressure and then fell, by a factor of 3-5, depending on the temperature, when the pressure corresponding to hydrate formation was reached. This, of course, is not possible because the hydrate cannot be in equilibrium with two aqueous solutions with different concentrations of methane. Nevertheless, this idea that the solubility of methane in liquid water was much smaller in the presence of hydrate than in its absence was incorporated in a number of models that attributed some large accumulations of free gas to its liberation from solution because of this fall in solubility. This misconception has kept recurring in the gas hydrate literature and as late as 1980 was incorporated in a paper by Du R o u ~ h e t . ~ A few treatments have been reported in the literature regarding these concerns. Miller5 showed that the presence of free gas is not necessary for hydrate formation or stability and calculated the effect of hydrostatic pressure on the solubility of methane in liquid water in equilibrium with the hydrate, without however allowing for a change of hydrate composition with pressure. Mokogon and Davidsod did allow for change of composition but not simultaneously with that of solubility; experimental values of the cage occupancies were not used, and occupancies of the small and the large cages were assumed to be the same; the treatment was limited to pure water only, and the effect of dissolved salts was not taken into account. A qualitative argument on the solubility behavior of gas when the gas hydrate-aqueous solution system is subjected to hydrostatic pressure has been presented by Barkan and Voronov.’ In the conditions under which gas hydrates are found in nature, either in equilibrium with ice as under the permafrost or in equilibrium with aqueous solution as at the sea bottom, there should not be any free gas present as long as water is present in excess. Since under these conditions the hydrostatic pressure is generally greater than the three-phase hydrate-liquid-gas or hydrate-ice-gas equilibrium gas pressure, the hydrostatic pressure is transmitted to any free gas present. This free gas under the excess pressure will then react with ice or liquid water to form hydrates until no free gas phase is left. It has been shown by Enns et a1.8 that the effect of hydrostatic pressure on a solution of gas in water is to squeeze the gas out of the solution and thus to decrease its solubility below that of the saturation solubility with respect to the free gas phase. Then, in analogy with the case of the aqueous solution of gas, at the disappearance of the gas phase the hydrates still should continue to form until the concentration of dissolved gas falls below its saturation value to a value corresponding to the hydrostatic pressure. In this paper, we shall consider the effect of hydrostatic pressure on the phase equilibria in the systems methane hydrate-water-gas, methane hydrate-seawater-gas, and methane hydrate-ice-gas, for the case where water or ice is present in excess. Some speculations on what may happen in a natural geological environment are also presented. Methane hydrate is chosen only as an example, the treatment presented below is applicable to any gas hydrate system containing a single gas.
Thermodynamic Considerations Consider the following equilibrium at temperature T and pressure P (G-nH,O), e aqueous solution
+ gas
(1)
chemical potential of water in the hydrate phase &, is given by ph
pe
+ R T Z v j In (1 - 8 j ) i
(2)
where pe refers to the chemical potential of water in the hydrate lattice with all cages empty, Le., the empty lattice; Bi, i = 1, 2, to the degree of occupancy of the small and the large cages; and vi to the number of water molecules per cage type i. The empty lattice actually does not exist and thus serves as a hypothetical reference state. The chemical potential of water in the aqueous solution pl is given by p1 = p l o
+ R T In x I + R T In y 1
(3)
where plo refers to the chemical potential of water in the reference state, x1 to the mole fraction of water, and y I to its activity coefficient. In eqs 2 and 3, the reference state is defined at the system temperature and pressure. For ice or water in aqueous solution, the reference state is pure ice or water at T and P. At equilibrium, ph = pl,and so from eqs 2 and 3 R T C v i In (1 - Oi) = p l o - pe i
+ R T In x , + R T In y1 (4)
For a change of pressure at constant temperature, we then have
On rearrangement, eq 5 becomes
where V, is the volume of the hydrate lattice per mole of water, Vl is the partial molar volume of water in the solution at T and P,, and x2 is the mole fraction of dissolved gas on salt-free basis. The cage occupancy is given by9 (7) where Ci is the Langmuir constant for cage type i andf, is the fugacity of the gas. In the absence of a gas phase,fi represents the fugacity of the hydrate former in the liquid phase,&, and can be looked upon as the effective fugacity of the gas that would be in equilibrium with the dissolved methane and methane in the hydrate if a gas phase existed. This fugacity can be measured by subjecting the hydrate-aqueous solution system to hydrostatic pressure and measuring the gas pressure across a membrane permeable to gas or water vapors but not to liquid. Such a setup was used by Enns et aL8 for studying the effect of hydrostatic pressure on the solubility of gases in water. The fugacity fzis related to the Henry’s law constant H by the relation
fi = x2H exp(Vi(P - I ) / R T )
(8)
where V2is the partial molar volume of the gas in the aqueous solution assumed to be independent of pressure and x2 is its solubility at T and P. The Henry’s law constant H is equal to the fugacity of the gas at 1 bar divided by the solubility of the gas at its partial pressure of 1 bar. Differentiating eqs 7 and 8 with respect to pressure at constant temperature leads to the relation
where G.nH20 is the hydrate of stoichiometry n. In terms of the statistical thermodynamical model for clathrate hydrates? the (3) Makogon, Y. F. Hydrares of Narural Gas (English translation by W. J. Cieslewicz); Penn Well Books: Tulsa, OK, 1981; p 1 1 1. (4) Du Rouchet, J. Bull. Cent. Rech. Explor.-Prod. ElfAquitane 1980, 4, 119. (5) Miller, S. L. In Narural Gases in Marine Sediments; Kaplan, I. R., Ed.; Plenum: New York, 1974; p 151. (6) Makogon, Y.F.; Davidson, D. W. Gas Ind. 1983, 4 , 37. (7) Barkan, Y. S.; Voronov, A. N. Int. Geol. Rev. 1984, 26, 570. (8) Enns, T.; Scholander, P. F.; Bradstreet, E. D. J. Phys. Chem. 1965, 69. 389.
On integration, eq 9 gives
(9) van der Waals, J. H.; Platteeuw, J. C. Adu. Chem. Phys. 1959, 2, 1.
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The Journal of Physical Chemistry, Vol. 94, No. 6, 1990
Handa
TABLE I: Solubilities and Cage - Occupancies in Methane Hydratewater and Methane Hydrate-Seawater Svstems at 273.15 K Pibar 1
26.29 28.95 50 100 I50 200 250 300 350 400 450 500
fJbar 0.9977 24.73 27.09 44.48 79.36 107.9 133.6 159.0 185.4 213.6 243.9 276.8 312.5
I 03x7,~ 0.046 67 1.1 13
I 03xy,,
0;
0;
1.113
0.8929
0.9748
1.931 3.194 4.024 4.6 19 5.095 5.506 5.879 6.222 6.545 6.849
1.105 1.087 1.071 1.056 1.041 1.027 1.013 1.000 0.9883 0.9767
0.8963 0.903 1 0.9096 0.9156 0.9213 0.9267 0.93 I7 0.9363 0.9407 0.9448
0.9757 0.9774 0.9790 0.9805 0.9819 0.9832 0.9844 0.9856 0.9866 0.9876
which gives the required dependence of composition, Le., cage occupancy on pressure. Eliminating (8 In Bi,/dP), from eqs 6 and 9, we get
(T)
1
V, -
VI- V 2 / n
= -RT l / n - x 2 / ( l - x2)
(11)
where n = l/CuiOi. The first term in the denominator of the right-hand side of eq I 1 gives the ratio of methane to water in the hydrate phase and the second term the ratio of methane to water in the liquid phase. Equation 1 1 gives the required dependence of the solubility of dissolved gas on pressure. The right-hand side of eq 1 1 contains n and x2, which themselves are functions of pressure. The functional form of eq 1 1 is not convenient for integration either numerically or analytically. However, the right-hand side of eq 11 changes only by about 7% for a pressure change of 500 bar. Thus, for a small pressure change, 10 bar, the right-hand side can be assumed to be nearly constant. Equation 11 can then be integrated in small steps and the integrated form solved simultaneously with eq 10 to establish values of Bi (and hence n ) and x2 at the end point of each step and thus to obtain values of Oi and x2 above the three-phase equilibrium line. In derivation of eq 9, Cihas been assumed to be independent of pressure, thereby implying that the effect of hydrostatic pressure on the occupancy of each cage type is the same. The Langmuir constant depends on the intermolecular interactions between the encaged gas molecule and the water molecules forming the cage and thus on the size and shape of the cage relative to that of the gas molecule. If the gas molecules, like methane, occupy both the small and the large cages, then the effect of mechanical pressure on the Langmuir constant for the small cage will be larger than that for the large cage, assuming both cages change in size by the same fractional amount. However, the compressibility of the hydrates is similar to that of ice and so is quite small. For the relatively low pressures to be considered here, the cage dimensions are not expected to change significantly and the assumption of constancy of ciwith pressure is justified.
Methane-Water and Methaneseawater Equilibria The unit cell size a of structure I xenon hydrate is 11.84 A at 110 K.l0 The van der Waals diameter of methane is 4.58 A, which is about the same as the value of 4.57 A for xenon. Thus, the unit cell size of structure I methane hydrate can be taken to be I 1.84 A at 1 I O K. The thermal expansion of structure I ethylene oxide hydrate has been reported recently over a wide temperature range." The values are a = 11.86 8, at I10 K and a = 12.00 8, at 273 K. Assuming the thermal expansion of methane hydrate to be the same as that of ethylene oxide hydrate, we get a = 1 1.98 8, for methane hydrate in the temperature range 273-278 K. Thus, V, can be taken to be 22.5 cm3 mol-' at 273 and 278 K.
I0 3 q 0.035 87
1034;
1034~~ 0.035 87
0.933 4 1.484 2.455 3.093 3.550 3.916 4.232 4.5 19 4.783 5.03 I 5.264
0.9334 0.927 I 0.9124 0.8985 0.8854 0.8728 0.8609 0.8495 0.8386 0.8283 0.8185
0.03505 0.03447 0.033 12 0.03 1 82 0.030 58 0.029 38 0.028 23 0.027 13 0.026 06 0.025 04 0.024 06
6w 0.9013 0.9041 0.9105 0.9165 0.9221 0.9274 0.9324 0.9370 0.941 3 0.9454 0.9492
0.9769 0.9776 0.9792 0.9807 0.9821 0.9834 0.9846 0.9857 0.9867 0.9877 0.9886
The solubility of methane in water is quite small, and consequently, in the absence of dissolved salts, the partial molar volume of water can be taken to be the same as its molar volume. The volumetric properties of water as a function of temperature and pressure have been given by Kell.lz The molar volume of water, both at 273.15 and 278.15 K, is 18.02 cm3 mol-] and decreases by 2.5% for a pressure change of 500 bar. Leyendekker~'~ has analyzed literature data to derive an expression for the partial molar volume of water in seawater as a function of temperature, pressure, and salinity. This expression was used to obtain VI,again ignoring the effect of dissolved gas on VI. Within the temperature and pressure ranges of interest here, the volumetric properties of seawater of salinity 3.5% thus calculated are the same as those of pure water. Partial molar volumes of methane V2 in water were calculated from the expression, based on a fit of the experimental data, given by Rettich et al.I4 The values are 34.5 cm3 mol-] at 273.15 K and 34.7 cm3 mol-' at 278.15 K. O'Sullivan and SmithIS have reported V2values in 0.0, 1 .O, and 4.0 M aqueous NaCl solutions at 324.7 K, where M stands for molar, derived from the pressure dependence of solubility using eq 8. Their results show a negligibly small effect of electrolyte and pressure on V2. Tiepel and GubbinsI6have reported dilatometric partial molar volumes of methane in 0.0, 2.0, and 4.0 M aqueous KC1 solutions at 298 K. The results indicate a decrease of about 1 .O cm3 mol-I M-] in V2. Solubilities of methane in salt solutions and seawater at 298 K and pressures up to 70 bar have been reported by Duffy et a1.I7 and Stoessell and Byrne.I* However, these studies are not precise enough to allow a meaningful evaluation of V2. Because of a small dependence of V2 on salt concentration and of the fact that 3.5% saline seawater is at most 0.6 M with respect to NaC1, we can take partial molar volumes of methane in seawater at 273.15 and 278.15 K to be the same as those in pure water. High-precision solubilities, at about 1 bar and at a number of temperatures, of methane in water14 and in seawater of different salinitiesI9have been reported in the literature. These results were used to calculate solubilities of methane in water x& and in seawater xi" as a function of pressure with use of eq 8 and the fugacities of methane calculated from the IUPAC equation of state.20 For these calculations, it was assumed that no hydrate forms, and thus, represents the saturation solubility. The results at 273.15 and 278.15 K are given in Tables I and 11. ~
(12) Kell, G.S . J . Chem. Eng. Data 1967, 12, 66. (13) Leyendekkers, J. V. Thermodynamics of Seawater; Dekker: New York. 1976: Part 1. D 237. (14) Rettich, T. R.;Handa, Y. P.; Battino, R.; Wilhelm, E. J. Phys. Chem. 1981, 85, 3230. (15) O'Sullivan, T. D.; Smith, N. 0. J . Phys. Chem. 1970, 74, 1460. (16) Tiepel, E. W.; Gubbins, K. E. J . Phys. Chem. 1972. 76, 3044. (17) Duffy, J. R.; Smith, N. 0.;Nagy, B. Geochim. Cosmochim. Acta 1961, 24, 23. (18) Stoessell, R. K.; Byrne, P. A. Geochim. Cosmochim. Acta 1982, 46, 11?7
(19) Yamamoto, S.; Alcauskas, J. B.; Crozier, T. E. J . Chem. Eng. Dara 19'16, 21. 78. (20) Angus, S.; Armstrong, B.; de Reuck, K. M. International Thermo-
( I O ) Bertie, J . E.; Jacobs, S. M . J . Chem. Phys. 1982, 77, 3230. (1 1 ) Tse, J. S.: McKinnon, W. R.; Marchi, M . J . Phys. Chem. 1987, 91,
4188.
dynamic Tables ofthe Fluid State-5. Methane. International Union of Pure and Applied Chemistry, Chemical Data Series No. 16; Pergamon Press: New
York, 1978.
The Journal ofPhysical Chemistry, Vol. 94, No. 6, 1990 2655
Hydrostatic Pressure and Salinity Effects on Gas Hydrates
TABLE 11: Solubilities and Cage Occupancies in Methane Hydrate-Water and Methane Hydrate-Seawater Systems at 278.15 K Plbar 1 42.73 49.54 100 1so 200 250 300 350 400 450
500
103x;.
f,lbar 0.9979 38.92 44.45 80.62 110.3 137.3 163.8 191.4 220.6 252.0 285.9 322.8
103~:~
0:
0:
I 03s"-
I
owh
fiW
0:"
0.031 33
0.040 22 1.473
1.473
0.8991
0.9816
2.801 3.555 4.105 4.544 4.926 5.267 5.582 5.875 6.153
1.445 1.422 1.400 1.379 1.360 1.341 1.323 1.305 1.289
0.9067 0.9130 0.9188 0.9243 0.9295 0.9343 0.9388 0.9430 0.9470
0.9831 0.9843 0.9855 0.9865 0.9875 0.9884 0.9892 0.9900 0.9907
For the methane hydrate-water-gas system the equilibrium temperature-pressure data21-24in the range 273.7-301 K can be represented by In ( P e / P o ) = -1205.907 + 44097.00/(T/K) 186.7594 In (T/K) (12)
1.297 2.182 2.769 3.198 3.539 3.836 4.102 4.347 4.576 4.793
1.297 1.276 1.256 1.236 1.218 1.201 1.184 1.168 1.153 1.139
0.9106 0.9 166 0.9221 0.9274 0.9322 0.9368 0.941 1 0.9452 0.9489 0.9524
0.9838 0.9850 0.9860 0.9870 0.9880 0.9888 0.9896 0.9904 0.991 1 0.9917
2.0 I
1
+
1 1 1 1
where Po = 1.013 25 bar. Equation 12 gives the equilibrium pressures to be 26.29 bar at 273.15 K and 42.73 bar at 278.15 K. There is no phase-equilibrium study reported in the literature on the formation of methane hydrate from seawater. Recently, de Roo et al.25 have reported phase-equilibrium data for the methane hydrate-aqueous NaC1-gas system in the temperature range 268 to 278 K and at NaCl concentrations in the range 0-3 1.7% by mass. Their results give, for a 3.5% by-mass NaCl solution, the equilibrium pressures of 28.95 bar at 273.15 K and 49.54 bar at 278.15 K. Hand et a1.26have reported, in a graphical form, the conditions of formation of methane hydrate from a 3.5% by-mass aqueous solution of NaCl extrapolated from the measurements for 10% and 20% by-mass NaCl solutions. As read off the graph, values of 28.5 bar and 52.9 bar for the equilibrium pressures at 273.15 K and 278.15 K, respectively, are obtained. These are in reasonably good agreement with the results of de Roo et al. We shall assume that the equilibrium pressures for the methane hydrate-seawater-gas system are the same as those for the methane hydrate-aqueous NaC1-gas system reported by de Roo et al. For the hydrate-water-gas equilibrium at T and P, the right-hand side of eq 4, Ap, can be written as -ACL = - - (pi -
RT
RT"
JpT(Hi -
zy
pv, TT+ -
AHr d T
AVr d P
+
+ In (1 - x&)
(13)
where y, has been assumed to be unity, TO = 273.15 K, AHris the enthalpy of melting of ice (6010 J mol-'), AVf is the volume change on the melting of ice (-1.7 cm3 mol-'), and is the molar volume of ice at 273.15 K (19.7 cm3 mol-'). The chemical potential difference between ice and the empty lattice ( p i- pe) is2' -1297 J mol-] and the enthalpy difference is2*-931 J mol-' at 273.15 K and 1 bar. The solubility of methane in water x;,~ at 273.1 5 K and 26.29 bar is 0.001 11 3 and at 278.15 K and 42.73 bar is 0.001 473. Integrating and substituting the various values in eq 13 gives for Ap -1 3 10.9 J mol-' at 273.1 5 K and 26.29 bar and -1435.9 J mol-' at 278.15 K and 42.73 bar. For the methane hydrate prepared at 260 K, the ratio of the = 0.916 as determined by I3C NMR.29 cage occupancies (21) Deaton, W. M.; Frost, E. M. Monogr.-US., Bur. Mines. 1946, No.
8. (22) Kobayashi, R.; Katz, D. L. Petr. Trans. AIME 1949, 6 6 . (23) McLeod, H. 0.; Campbell, J. M. J . Pet. Technol. 1961, 222, 590. (24) Marshall, D. R.; Saito, S.; Kobayashi, R. AIChE J . 1964, IO, 202. (25) de ROO,J. L.; Peters, C. J.; Lichtenthaler, R. N.; Diepen, G. A. M. AIChE J . 1983, 29, 651. (26) Hand, J . H.; Katz, D. L.; Verma, V. K. In Natural Gases in Marine Sediments; Kaplan, I. R., Ed.; Plenum: New York, 1974; p 186. (27) Davidson, D. W.; Handa, Y. P.; Ripmeester, J. A. J . Phys. Chem. 1986, 90, 6549. (28) Handa, Y . P.; Tse, J . S. J . Phys. Chem. 1986, 90, 5917
1.2
x2x103
1
------
r------------
0.8
0.4
0
100
200
300
400
500
Plbar Figure 1. Solubilities of methane in water (w) and seawater (sw) plotted against hydrostatic pressure. The curves at 278.15 K have been shifted upward by 0.0003. Solubilities below the point of inflection are for the two-phase liquid-gas system, a t the point of inflection are for the three-phase hydrate-liquid-gas system, and above the point of inflection are for the two-phase hydrate-liquid system.
Assuming the ratio 0,/02 to be the same at higher temperatures, eq 4 can be solved with values of 01/02 and A p calculated above to give Oy = 0.8929 and 0; = 0.9748 at 273.15 K and 26.29 bar; 0; = 0.8991 and 0; = 0.9816 at 278.15 K and 42.73 bar. The Langmuir constants can be obtained from the relation
with use of the cage occupancies calculated above. The cage occupancies of the hydrates formed from seawater can then be obtained from eq 7. The values are 0;" = 0.9013 and Oiw = 0.9769 at 273.15 K and 28.95 bar; 0s" = 0.9106 and @iw = 0.9838 at 278.15 K and 49.54 bar. The cage occupancies in the case of seawater are slightly higher than in the case of water because of slightly higher gas pressures of formation. The values of the various parameters discussed above were substituted in eqs 10 and 11 to obtain the equilibrium solubilities and the cage occupancies. The calculations were performed at 273.15 and 278.15 K and at hydrostatic pressures up to 500 bar. For seawater, the calculations were limited to a salinity of 3.5%. Calculations were performed at intervals of 10 bar; however, some representative results only are given in Tables I and 11, in which the superscripts w and sw refer to water and seawater, respectively, x ~to, the ~ solubility of methane in the liquid in equilibrium with hydrate and in the absence of a gas phase, and Oi to the corresponding cage occupancies. It should be noted that a free gas (29) Ripmeester, J. A.; Ratcliffe, C. I. J . Phys. Chem. 1988, 92, 337.
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The Journal of Physical Chemistry, Vol. 94, No. 6,1990
0'15
I------
Handa 75
60
I
1
45
0.09
/bar
1-8 I
30
0.06
c 0.03
1
15 1-62 W
0
100
200
300
400
0
500
Plbar Figure 2. Fraction of unoccupied cages at 278.15 K in the two-phase hydrate-liquid system plotted against hydrostatic pressure: w, water; sw, seawater.
phase can exist at hydrostatic pressures up to Pe, so that the values , ~x * , are ~ the same at P,. of x ~and Figure 1 shows the solubilities of methane in the liquid phases at 273.1 5 and 278.15 K plotted against pressure. For the sake of clarity, the results at 278.15 K have been shifted upward by 0.0003. As seen in Figure 1, the concentration of methane in water increases with hydrostatic pressure up to the point where hydrostatic pressure becomes equal to the equilibrium hydrate formation pressure Pe, Le., the point of inflection. Up to this point, we have a two-phase liquid-gas equilibrium. At P, we have a three-phase hydrate-liquid-gas equilibrium, the concentration of methane in the liquid phase becomes constant, and the hydrate formation process continues until all the free gas has been used up. At pressures higher than P,,we are left with a two-phase hydrateliquid system only in which the concentration of dissolved methane decreases slightly with pressure. Figure 2 shows the fraction of unoccupied small and large cages plotted against pressure. For the sake of clarity, the results at 278.15 K only are shown. At hydrostatic pressures higher than P,,the number of unoccupied cages decreases with pressure even though the concentration of methane in the coexisting liquid is decreased. The decrease in the number of unoccupied small cages is greater than for the large cages primarily because the large cages already are mostly occupied. Thus, from Figure I and 2, the overall effect of the hydrostatic pressure is to squeeze the gas out of the solution and transfer it to the hydrate phase. The effect is slightly more pronounced at the higher temperature. It should be noted that because of an increase in cage occupancy the liquid phase fugacity of methanefk increases with pressure (Figure 3), in accordance with eq 7, but its solubility in the liquid phase decreases, which is contrary to what one would expect in accordance with eq 8. Since the solubility of methane in seawater is smaller than in pure water, methane concentrations required to stabilize the hydrates at the Ocean bottom are even smaller than those required in the case of pure water (Figure 1). The results presented here show that the free gas does not have to be present for the hydrates to be stable. Moreover, contrary to earlier rep o r t ~no , ~ abrupt and large change in the solubility of the dissolved gas is observed. Figure 4 shows the ratio between the concentration of dissolved methane necessary to form hydrate to that in equilibrium with free gas (in the absence of hydrate) at 273.15 and 278.15 K plotted against pressure. At pressures greater than 100 bar, conditions under which hydrates are normally encountered in nature, the methane content of water in equilibrium with hydrate is much smaller than the saturation solubility. This partial saturation,
100
200
300
400
500
Plbar Figure 3. Fugacity of methane in the liquid phase plotted against hydrostatic pressure at 273.15 K. Fugacities below the point of inflection are for the two-phase liquid-gas system, at the point of inflection are for the three-phase hydrate-liquid-gas system, and above the point of inflection are for the two-phase hydrate-liquid system: w, water; sw, seawater. 1.0
0.8
0.6
'2,h -
x2,g 0.4
0.2
-
0
278.15 K
100
200
300
400
500
P Ibar Figure 4. Ratio of mole fraction of methane in the liquid in equilibrium with hydrate (in the absence of a gas phase) to its mole fraction in the liquid in equilibrium with a gas phase (in the absence of a hydrate phase) plotted against hydrostatic pressure: w, water; sw, seawater. required to stabilize the hydrate, increases slightly with temperature and is slightly higher for seawater than for pure water. But in all cases, contrary to commonly held opinion,'*3z the present analysis shows that water does not have to be saturated with gas for the hydrates to form or be stable. In the above analysis, an assumption has been made that the overall composition of the system is unaffected by the hydrostatic pressure, Le., the gas hydrateaqueous solution system is physically isolated from the pressure-transmitting medium (and not, for (30) Kvenvolden, K. A,; Barnard, L. A,; Brooks, J. M.; Wiesenberg, D. A. Adu. Org. Geochem. 1981, 422.
(31) Brooks, J. M.; Kennicutt, M. C.; Fay, R. R.; McDonald, T. J. Science 1984, 225, 409. (32) Brooks, J. M.; Cox, H. B.; Bryant, W. R.; Kennicutt, M. C . ; Mann, R. G.;McDonald, T.J . Org. Geochem. 1986, 10, 221.
J . Phys. Chem. 1990, 94, 2651-2665 example, in contact with the seawater above). This trapping of gas is an obvious requirement for the local accumulation of gas. The assumption can be justified by the following argument. Let us assume that equilibrium exists between the methane hydrate-aqueous solution system and the seawater and that the dissolved methane is in equilibrium at all depths d . Then, the solubility of methane at depth d , x 5 , is related to the solubility at the surface XCby33
where M is the molar mass of the gas, p the density of seawater, and g the acceleration due to gravity. The surface solubility in eq 15 refers to methane partial pressure of 1 bar. The solubilities of methane at 273.15 K calculated at various depths are given in Table I. In these calculations, 1 m of a seawater column has been taken to exert a hydrostatic pressure of 0.1 bar, and it has been assumed that the temperature is the same at all depths. As seen in Table I, x s values are much smaller than those required xi; for the hydrates to form and to be stable. Thus, methane hydrate is always stable at any reasonable pressure above the three-phase equilibrium pressure, and in fact the amount of hydrate will generally increase at the expense of liquid water as pressure increases. In the case of seawater, the behavior is somewhat more complex. The thermodynamic parameters not only depend on the concentration of salt initially present, but it is obvious that the more water is converted into hydrate, the more concentrated the salt solution becomes. As seen in Figure 3, the effect of dissolved salts is to raise the value of If the hydrostatic pressure is not greatly in excess of the three-phase equilibrium pressure, hydrate formation at a fixed pressure will proceed until the concentration of salt in the liquid phase has reached a critical value. At this critical value,f$ will be raised to the value of the fugacity of free gas at that pressure and we have again a hydrate-salt-solution-gas equilibrium. Further addition of methane will simply increase its relative amounts in the condensed phases and hardly change the amount of hydrate present. If the hydrostatic pressure is relatively large, increased amounts of methane in the system may lead to so much hydrate formation
fi.
(33) Klotz, I . M. Limnol. Oceonog. 1963, 8, 149.
2657
that the aqueous solution becomes saturated with salt, which then tends to precipitate out. We then have a hydratesalt-solutionsalt equilibrium that determines the amount of hydrate present. In general, then, under hydrostatic pressure hydrate can coexist with a salt solution of methane alone, with a salt solution and free gas, or with a salt solution and solid salt, depending on the hydrostatic pressure and the relative amounts of methane and water in the system.
Methane HydrateIce Equilibrium Here, we need only to consider the effect of hydrostatic pressure on the cage occupancy for the two-phase methane hydrate-ice system. For this case, eq 5 becomes
which gives for T = 273.15 K = -1.251 X
bar-'
(17)
Thus, the effect of hydrostatic pressure is to decrease the number of unoccupied cages. Unlike the case with water where methane gas is available in the liquid phase, in the present case the increase in cage occupancy can only be accomplished by the conversion of some hydrate to ice, the methane released being absorbed by the remaining hydrate. At temperatures slightly below 273.1 5 K, the effect will be similar because the volume difference (Vi - V,) remains essentially the same. The thermal expansivity of the hydrates is about twice that of so that at low temperatures V, - V, decreases slightly with temperature. However, this decrease is small and the effect of pressure is about the same as at higher temperatures. For the temperature and pressure ranges to be encountered on earth in regions where hydrates are found under the permafrost, only about 1% of the water in the hydrate will be converted to ice. However, at large hydrostatic pressures, of the order of 12-14 kbar, which may be encountered in the interiors of the massive icy planets, it has been shown2 that methane hydrate becomes unstable and should dissociate into ice and solid or fluid methane, depending on the temperature. Registry No. Methane hydrate, 14476-19-8. (34) Tse,
J. S. J . Phys. 1987, CI,543.
Phenomenological Models for Diffusion of Vibrational Energy in Polyatomic Gases John R.Ferron Department of Chemical Engineering, University of Rochester, Rochester, New York 14627 (Received: July 25, 1989; I n Final Form: September 29, 1989)
Transfer during collisional processes of the internal energy carried by small, polyatomic molecules may be characterized by rotational and vibrational diffusion coefficients. These are evaluated here by use of simplified models for vibrational relaxation together with literature data on excitation-relaxation experiments. The results are used for prediction of thermal conductivities and/or of various internal diffusivities. Pure nitrogen and carbon dioxide, and binary mixtures of each with rare gases and with each other, are studied, all for a pressure of 1 atm. Accurate predictions are obtained, and the method suggests explicit means for identification and numerical evaluation of the various elastic and inelastic contributions to thermal conductivity.
Introduction When a gas containing polyatomic species encounters an environment that causes a temperature or concentration gradient, the status of internal energy modes changes from point to point. Movement of the gas from a relatively warm location to a cooler one, for example, is accompanied by relaxation of the more highly excited states to those representative of the new condition. In 0022-3654/90/2094-2657$02.50/0
particular, considering now only moderately high temperatures and neglecting electronic changes, we recognize that vibrational and rotational states rapidly exchange energy among themselves and with the translational mode as the gas flows. Some exchanges (not a major part of what we consider here) produce radiation, which may be lost from the system directly or by self-absorption and reradiation. Other interactions are 0 1990 American Chemical Society
