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The compressibility of ice to 2.0 kbar
Cold Regions Science and Technology 38 (2004) 211 – 218 www.elsevier.com/locate/coldregions
The compressibility of ice to 2.0 kbar G.M. Marion a,*, S.D. Jakubowski b,1 b
a Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512, USA Department of Oceanography, Florida State University, Tallahassee, FL 32306, USA
Received 18 November 2002; accepted 30 October 2003
Abstract To understand in situ properties of ice under pressure, we need an accurate model for ice compressibility. Many studies have estimated ice compressibility, but there still remains considerable controversy with respect to the magnitude and temperature dependence of ice compressibility. The specific objectives of this study were to (1) present a new model for estimating the compressibility of ice based on chemical thermodynamic principles, and (2) compare these model results with previous work. A thermodynamic equation that relates equilibrium constants to molal volumes and compressibilities of water and ice was the basic model used to estimate ice compressibilities. All terms in this equation (equilibrium constants, temperature, pressure, molal volumes of ice and water at 1 atm, and water compressibilities) are definable along the pressure ice/water melting curve except for ice compressibility, which we estimated from the equation. Our estimate of ice compressibility demonstrated a significant temperature dependence. In contrast, most other studies, especially those that rely on elastic parameters, only demonstrate a weak temperature dependence. All recent studies show a significantly lower ice compressibility than the classic Bridgman studies. The results of this work are in agreement with most other databases and models and indicate a general consensus on the compressibilities of water and ice up to 1200 bar of pressure. Between 1200 and 2000 bar, there are discrepancies that are unlikely to be resolved without further experimental work that directly estimate the compressibility of ice over a range of temperatures. A comparison of calculated ice core densities from Antarctica, with and without corrections for pressure, demonstrate the utility of this model for understanding in situ ice properties under pressure. D 2004 Elsevier B.V. All rights reserved. Keywords: Ice compressibility; Water compressibility; Pressure; Subzero temperatures; Chemical thermodynamic model
1. Introduction In order to understand in situ properties of ice under pressure, we need an accurate model for ice compress* Corresponding author. Tel.: +1-775-673-7349; fax: +1-775673-7485. E-mail addresses: [email protected] (G.M. Marion), [email protected] (S.D. Jakubowski). 1 Fax: +1-850-644-2581. 0165-232X/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2003.10.008
ibility. For example, Gow (1971) has shown that deep ice cores from Antarctica relax-out elastically as soon as the cores reach the surface, making it impossible to accurately determine the true densities at depth. There have been many studies of ice compressibility following the classic studies of Bridgman (1912). In the latter study, the ice compressibility was estimated indirectly from measurements of the total volume change (ice + water) along the pressure ice/water melting curve. Many subsequent studies have used the elastic proper-
212
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ties of ice in order to estimate ice compressibilities (e.g., Jona and Scherrer, 1952; Bass et al., 1957; Leadbetter, 1965; Dantl, 1969). A few studies have estimated ice compressibilities directly (e.g., Richards and Speyers, 1914; Gow and Williamson, 1972). All studies after Bridgman (1912) have reported substantially lower ice compressibilities than the Bridgman studies. The specific objectives of this study were to (1) present a new model for estimating the compressibility of ice based on chemical thermodynamic principles, and (2) compare these model results with previous work.
2. Materials and methods The baseline for pressure studies is 1 atm pressure (1.01325 bar). Compressibility studies report pressures either as total pressure ( P) or as gage pressure ( p) where P = p + 1.01325 (with pressure in bar). Both total pressure and gage pressure will be used in this work. Care will be taken to make clear which pressure is being used. Different units are used to report isothermal compressibilities:
1 bar
1 dV cm3 ¼ or V dP mol bar dV : ¼ dP
ð1Þ
Because the latter units are explicitly needed in our models, we will primarily rely upon them in this work; but the water and ice compressibility equations will be given with both sets of units. For a chemical reaction such as: H2 OðI; crÞ X H2 OðlÞ;
ð2Þ
where H2O(I,cr) is ice and H2O(l) is liquid water, the equilibrium constant (K) as a function of pressure can be estimated by: 0 KPT DV ðT ; P0 Þ ðP P0 Þ ¼ exp RT KpT0 DK 0 ðT ; P0 Þ 0 2 ðP P Þ þ 2RT
(Millero, 1983; Krumgalz et al., 1999), where 0 DV 0 ¼ Vl0 VI;cr
ð4Þ
and 0 DK 0 ¼ Kl0 KI;cr
ð5Þ
In these equations, KPT is the equilibrium constant at pressure P and temperature T, KP0T is the equilibrium constant at 1.0 atm (1.01 bar) and temperature T, R is the gas constant, T is temperature (K), V¯0 is the molal volume at 1.0 atm for either water (l) or ice (I,cr), and K¯0 is the isothermal compressibility for either water (l) or ice (I,cr). The molal volume for pure ice at 1 atm is given by 0 VI;cr ¼ 1:930447e1 7:988471e 4 T þ 7:563261e 6 T 2
ðR2 ¼ 0:9998Þ
ð6Þ
(Fig. 1). This equation is derived from crystal lattice parameters for ice Ih (Petrenko and Whitworth, 1999) and is valid over the temperature range from 160 to 273 K. Experimental measurements of the molal volumes of ice show significantly more variability than volumes based on crystal lattice parameters (Fig. 1) because of cracks and gas bubbles in experimental ice (Hobbs, 1974). Eq. (6) represents an idealized pure ice without cracks and gas bubbles. The molal volume for pure water at 1 atm is given by: Vl0 ¼ 9:479107e1 7:889356e 1 T þ 2:693176e 3 T 2 3:052859e 6 T 3 ðR2 ¼ 0:9949Þ
ð7Þ
The database for this relation (Fig. 1) was taken from the Smithsonian Meteorological Tables (Smithsonian Institute, 1951), Chen et al. (1977), and Hare and Sorensen (1986). The equilibrium constant for water and ice at 1 atm in Eq. (3) is given by: KPT 0 ¼ 1:833946e0 1:787269e 2 T þ 6:212940e 5 T 2 2:884752e 8 T 3 ðR2 ¼ 1:0000Þ
ð3Þ
ð8Þ
(Marion, 2002). The equilibrium temperature (T) at a pressure ( P) along the pressure ice/water melting curve
G.M. Marion, S.D. Jakubowski / Cold Regions Science and Technology 38 (2004) 211–218
213
Fig. 1. The molal volume of ice and liquid water as a function of temperature and pressure. Symbols represent experimental data. Lines are model estimates.
3.897e 5 = 17.23 cm3/mol).It can be shown that the equilibrium constant for water and pure ice is given by:
is calculated from the equations of Wagner et al. (1994). The compressibility of water (K¯ 0l ) is based on the experimental database of Ter Minassian et al. (1981). In the latter study, water compressibilities are reported from 40 to 120 jC and from 0 to 5000 bar. We used a subset of this database that covered the temperature range from 30 to 30 jC and the pressure range from 0 to 2000 bar (Table 1). In our work, water compressibilities are reported in units of cm3/(bar mol), while in Ter Minassian et al. (1981), compressibilities are reported in units of 1/bar. Dividing an entry in Table 1 by the corresponding datum in Table 3 of Ter Minassian et al. (1981) gives the molal volume (e.g., at 0 jC and 1000 bar of pressure, 6.715e 4/
K ¼ PðH2 O;iceÞ =PðH2 O;waterÞ ¼ aH2 O ;
ð9Þ
where P(H2O,ice) and P(H2O,water) are partial pressures of water vapor above pure ice and water, respectively, and aH2O is the activity of water (Marion, 2002). To find the temperature where pure ice and pure water are in equilibrium at a specified pressure, it suffices to find the temperature where P(H2O,ice) = P(H2O,water) and therefore aH2O = 1.0 (Eq. (9)). For example, at 1 atm pressure, P(H2O,ice) = P(H2O,water) at 273.15 K (Lide, 1994), which is the freezing point of pure water at 1 atm pressure.
Table 1 The isothermal compressibility of water [cm3/(mol bar)] as a function of temperature and pressure (gage) Pressure Temperature (jC) (bar) 30 25 0 200 400 600 800 1000 1200 1400 1600 1800 2000
1.579e 3 1.348e 3 1.175e 3 1.042e 3 9.353e 4 8.483e 4 7.761e 4 7.154e 4 6.636e 4 6.190e 4 5.804e 4
1.331e 3 1.186e 3 1.064e 3 9.623e 4 8.768e 4 8.044e 4 7.423e 4 6.890e 4 6.427e 4 6.021e 4 5.666e 4
20
15
10
5
0
5
10
20
30
1.179e 3 1.075e 3 9.822e 4 9.006e 4 8.294e 4 7.673e 4 7.132e 4 6.655e 4 6.237e 4 5.867e 4 5.539e 4
1.077e 3 9.954e 4 9.202e 4 8.520e 4 7.909e 4 7.365e 4 6.881e 4 6.451e 4 6.070e 4 5.728e 4 5.422e 4
1.004e 3 9.364e 4 8.724e 4 8.131e 4 7.593e 4 7.107e 4 6.670e 4 6.277e 4 5.924e 4 5.607e 4 5.319e 4
9.513e 4 8.916e 4 8.349e 4 7.821e 4 7.335e 4 6.892e 4 6.490e 4 6.127e 4 5.798e 4 5.500e 4 5.230e 4
9.114e 4 8.572e 4 8.055e 4 7.573e 4 7.126e 4 6.715e 4 6.341e 4 6.002e 4 5.690e 4 5.408e 4 5.153e 4
8.813e 4 8.305e 4 7.825e 4 7.372e 4 6.953e 4 6.570e 4 6.216e 4 5.895e 4 5.600e 4 5.332e 4 5.086e 4
8.584e 4 8.098e 4 7.639e 4 7.211e 4 6.816e 4 6.450e 4 6.114e 4 5.806e 4 5.526e 4 5.267e 4 5.031e 4
8.277e 4 7.813e 4 7.383e 4 6.986e 4 6.616e 4 6.278e 4 5.965e 4 5.677e 4 5.413e 4 5.170e 4 4.947e 4
8.102e 4 7.651e 4 7.234e 4 6.851e 4 6.496e 4 6.172e 4 5.872e 4 5.596e 4 5.342e 4 5.108e 4 4.893e 4
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G.M. Marion, S.D. Jakubowski / Cold Regions Science and Technology 38 (2004) 211–218
Bearing in mind that KPT = aH2O = 1.0 at equilibrium for pure ice/water systems means that all terms in Eq. (3) are definable along the pressure ice/water melting curve see above except for K¯I,cr0. In practice, we substituted values for K¯I,cr0 into Eq. (3) until the relationship predicted aH2O = 1.00. This is the chemical thermodynamic basis for calculating the isothermal compressibility of ice. We incorporated the above relationships into the FREZCHEM model (Marion and Farren, 1999; Marion, 2001, 2002) to facilitate the calculations. In what follows, the molal volumes of ice and water as functions of temperature and pressure are calculated by: VPT ¼ VPT0 K T DP;
ð10Þ
where V¯PT is the molal volume at pressure P and temperature T, V¯P0T is the molal volume at 1 atm and temperature T, K¯T is the compressibility at temperature T, and DP is the change in pressure. We used a secondorder interpolating polynomial algorithm (Press et al., 1992; Subroutines: Locate, Polint, Polin2) in Table 1 for calculating the isothermal compressibility of water as a function of temperature and pressure. An underlying assumption in the application of Eqs. (3) and (10) is that the compressibilities of water and ice can be approximated by a constant at a given temperature. Clearly, there is a strong pressure dependence for the isothermal compressibility of water (Table 1). In what follows, we calculated an average water compressibility for a given temperature by dividing the pressure range of interest (e.g., 1000 bar) into 20 equally spaced intervals (e.g., 50 bar), estimated the compressibility at the mid-point of each interval, and used the average of the 20 values in Eqs. (3) and (10). A more accurate alternative to using Eq. (10) is to directly integrate: dV ¼ K T ðPÞ: ð11Þ dP T For example, we fitted a quadratic equation to the water compressibility data at 0 jC (Table 1) and integrated Eq. (11) over the pressure range from 1.01 to 2001.01 bar (DP = 2000 bar). The change in the molal volume over this pressure range (DV) calculated with Eq. (11) is 1.3717 cm3/mol, compared to 1.3719 cm3/mol cal-
culated with Eq. (10). An error of this magnitude is insignificant. The isothermal compressibility of ice calculated by solving Eq. (3) is assumed to represent an average over the pressure range of interest. The change in compressibility of ice with pressure is not large (Richards and Speyers, 1914; Nagornov and Chizhov, 1990). Estimates of the molal volumes of water at 500, 1000, 1500, and 2000 bar of pressure based on Eq. (10) and the associated assumptions are given in Fig. 1. For comparative purposes, we also included the experimental measurements of Chen et al. (1977) at 500 and 1000 bar between 273 and 298 K. At 298 K (25 jC), which constitutes an interpolation for our model (Table 1), the estimates of the molal volumes at 500 bar are 17.689 and 17.683 cm3/mol, and at 1000 bar, the molal volumes are 17.358 and 17.351 cm3/mol for the Chen et al. (1977) and our model, respectively. This close agreement is partial validation of our approach for estimating the molal volume of water based on Eq. (10).
3. Results Our estimates of ice isothermal compressibility showed a significant temperature dependence (Fig. 2). A quadratic equation fit to our model-derived estimates gives: K 0 ðcm3 =ðmol barÞÞ ¼ 2:790102e 2 2:235440e 4 T þ 4:497731e 7 T 2 ðR2 ¼ 0:9989Þ
ð12Þ 0 K ð1=barÞ ¼ 1:398427e 3 1:121305e 5 T þ 2:258122e 8 T 2 ðR2 ¼ 0:9988Þ
ð13Þ
The earlier Bridgman (1912) study also demonstrated a significant temperature dependence. There have been several studies that have estimated ice compressibilities based on elastic parameters of ice; these studies are designated in Fig. 2 with solid (filled) symbols. These elastic-parameterization studies show only a slight temperature dependence. The data points for the Dantl (1969) study (Fig. 2) are estimated from their derived equation. The two studies that have estimated ice compressibility di-
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215
Fig. 2. The isothermal compressibility of ice as a function of temperature.
rectly (Richards and Speyers, 1914; Gow and Williamson, 1972) fall in the midrange of temperatures and are compatible with both our chemical thermodynamic model and the elastic-parameter models. All works subsequent to Bridgman (1912), including our study, indicate significantly lower ice compressibilities than the Bridgman studies (Fig. 2). This is an important point because, despite the age of the Bridgman (1912) study, it is still widely cited in the literature (e.g., Hobbs, 1974; Petrenko and Whitworth, 1999), which it should be for historical purposes but not for ice compressibilities. In Fig. 3, we compare the molal volumes of water and ice along the pressure melting curve for our study, the Bridgman (1912) study, and a recent modeling study of Nagornov and Chizhov (1990). In the latter modeling study and in our study, the Ter Minassian et al. (1981) database was used to estimate water compressibilities, but the specific algorithms of the two models are slightly different which leads to a small discrepancy between models (Fig. 3). For example, at 2000 bar of pressure, the extrapolated molal volumes
for water are 16.65, 16.57, and 16.53 cm3/mol for the Bridgman (1912) study, the Nagornov and Chizhov (1990) model, and our model, respectively. The discrepancies for ice volumes are much larger than for water volumes (Fig. 3). Our model estimates of the molal volume of ice flatten out between 1200 and 2000 bar of pressure because ice compressibilities decrease to low values at low temperatures (Fig. 2). The Bridgman (1912) study, on the other hand, which reports high ice compressibilities (Fig. 2), has, as a consequence, low ice volumes (Fig. 3). The Nagornov and Chizhov (1990) model uses an ice compressibility algorithm based on temperature-insensitive elastic parameters (Fig. 2); their calculated ice molal volumes fall inbetween our study and the Bridgman (1912) study at high pressures. Note, however, that our model and the Nagornov and Chizhov (1990) model are in reasonably good agreement up to 1200 bar of pressure (Fig. 3). The pressure ice/water melting database of Wagner et al. (1994) was a major source of data used to estimate ice compressibilities in our study. So
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Fig. 3. The molal volumes of ice and liquid water along the pressure ice/water melting curve.
naturally, our derived model fits this database very well (Fig. 4). We examined the consequences of using a different equation for ice compressiblity on the resulting temperature – pressure relationship. For
the latter, we used our previously described model except we substituted the Dantl (1969) equation (Fig. 2) for our equation (Eq. (12)) for ice compressibility. At 2000 bar of pressure, the Dantl
Fig. 4. The freezing point depression of an ice/water system as a function of pressure.
G.M. Marion, S.D. Jakubowski / Cold Regions Science and Technology 38 (2004) 211–218
217
Fig. 5. Calculated densities of an Antarctic ice core as a function of pressure and temperature.
equation underpredicts the freezing point depression by 1.02 jC. However, at least down to 1200 bar of pressure, the two models are in reasonably good agreement (Fig. 4).
4. Discussion The chemical thermodynamic equation used to estimate the ice compressibility (Eq. (3)) depends fundamentally on four components that must be known accurately: the equilibrium constants, the molal volumes of ice and water at 1 atm (Eqs. (6) and (7)), the isothermal compressibility of water (Table 1), and the pressure and temperature along the ice/water melting curve. The latter relationship appears well established (Wagner et al., 1994). It would also seem that the molal volumes for water and ice at 1 atm are well established (Fig. 1) as are the equilibrium constants. We showed that the molal volumes for water at 1 atm (Eq. (7)) used in conjunction with the water compressibility data (Table 1) leads to accurate estimates of the molal volume for water at higher pressures over the temperature range from 273 to 298 K, based on a comparison with the independent Chen et al. (1977) database (Fig. 1). Our model for water at subzero temperatures is also
consistent with the Nagornov and Chizhov (1990) model (Fig. 3); but this is not a true validation as both models use the water molal compressibility database of Ter Minassian et al. (1981). An independent validation of the Ter Minassian et al. (1981) water compressibility database for work at subzero temperatures is desirable. We terminated our use of this database at 30 jC (Table 1) because water compressibility data at temperatures of 35 and 40 jC given by Ter Minassian et al. (1981) led to a precipitous drop in calculated water molal volumes at all pressures (calculations not presented), in sharp contrast to the trends in the data at higher temperatures (Fig. 1). The results of this work are in agreement with most other databases and models and indicate a general consensus on the compressibilities of water and ice up to 1200 bar of pressure (Figs. 1– 4). Between 1200 and 2000 bar, there are discrepancies, which are unlikely to be resolved without further experimental work that directly estimate the compressibility of ice (Richards and Speyers, 1914; Gow and Williamson, 1972) over a range of temperatures. Earlier we alluded to the importance of an accurate ice compressibility model in understanding the physical properties of deep ice cores where both pressure and temperature are changing simultaneous-
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G.M. Marion, S.D. Jakubowski / Cold Regions Science and Technology 38 (2004) 211–218
ly. Gow (1971) has shown that the density of deep ice cores under pressure relaxes elastically as soon as the cores are extracted. In Fig. 5, we used our model to calculate how the density of an ice core from Antarctica (Gow et al., 1968; Gow, 1971) would vary with core temperature at 1 atm, which is what is measured at the surface with corrections for temperature, to the same core under both temperature and pressure constraints. At 1 atm pressure, the core density changes linearly with temperature (Fig. 5), in agreement with our model (Fig. 1) and the Gow (1971) results (see his Table 1). In contrast, the density of the ice core subjected to both temperature and pressure constraints is always higher and becomes progressively more separated from the 1 atm curve as pressure increases (Fig. 5). At the base of the core, the predicted densities are 0.9171 and 0.9203 g/cm3 for 1 and 192 bar, respectively. Between approximately 1730 and 2150 m, the calculated ice density only fluctuates between 0.9201 and 0.9203 g/cm3. Over this range, the increasing pressure raises the density and the increasing temperature lowers the density; the two forces of temperature and pressure are in relative balance over this range. Gow et al. (1968) estimated that the pressure melting point of the ice at the base of this core at 197 bar of pressure is 1.6 jC. Our model predicts 1.5 jC at 197 bar for a pure ice/water system. The presence of solutes at the ice/water junction would depress our estimate to lower temperatures.
Acknowledgements We thank Dr. A.J. Gow for providing guidance on the ice compressibility literature. This work was funded by a NASA PG&G Grant on An Aqueous Geochemical Model for Cold Planets, a NASA EPSCoR project, and a NASA Space Grant award.
References Bass, R., Rossberg, D., Ziegler, G., 1957. Die elastischen Konstanten des Eises. Z. Phys. 149, 199 – 203. Bridgman, P.W., 1912. Water, in the liquid and five solid forms, under pressure. Proc. Am. Acad. Arts Sci. 47, 441 – 558. Chen, C.-T., Fine, R.A., Millero, F.J., 1977. The equation of state of
pure water determined from sound speeds. J. Chem. Phys. 66, 2142 – 2144. Dantl, G., 1969. Elastic moduli of ice. In: Riehl, N., Bullemer, H., Engelhardt, H. (Eds.), Physics of Ice. Plenum, New York, pp. 223 – 230. Gow, A.J., 1971. Relaxation of ice in deep drill cores from Antarctica. J. Geophys. Res. 76, 2533 – 2541. Gow, A.J., Williamson, T.C., 1972. Linear compressibility of ice. J. Geophys. Res. 77, 6348 – 6352. Gow, A.J., Ueda, H.T., Garfield, D.E., 1968. Antarctic ice sheet: preliminary results of first core hole to bedrock. Science 161, 1011 – 1013. Hare, D.E., Sorensen, C.M., 1986. Densities of supercooled H2O and D2O in 25 A glass capillaries. J. Chem. Phys. 84, 5085 – 5089. Hobbs, P.V., 1974. Ice Physics Clarendon Press, Oxford. Jona, F., Scherrer, P., 1952. Die elastichen Konstanten von Eis-einkristallen. Helv. Phys. Acta 25, 35 – 54. Krumgalz, B.S., Starinsky, A., Pitzer, K.S., 1999. Ion-interaction approach: pressure effect on the solubility of some minerals in submarine brines and seawater. J. Solution Chem. 28, 667 – 692. Leadbetter, A.J., 1965. The thermodynamic and vibrational properties of H2O ice and D2O ice. Proc. R. Soc. Lond., A 287, 403 – 425. Lide, D.R. (Ed.), 1994. CRC Handbook of Chemistry and Physics. CRC Press, Boca Raton, pp. 14 – 15. Marion, G.M., 2001. Carbonate mineral solubility at low temperatures in the Na – K – Mg – Ca – H – Cl – SO4 – OH – HCO3 – CO3 – CO2 – H 2 O system. Geochim. Cosmochim. Acta 65, 1883 – 1896. Marion, G.M., 2002. A molal-based model for strong acid chemistry at low temperatures ( < 200 to 298 K). Geochim. Cosmochim. Acta 66, 2499 – 2516. Marion, G.M., Farren, R.E., 1999. Mineral solubilities in the Na – K – Mg – Ca – Cl – SO4 – H2O system: a re-evaluation of the sulfate chemistry in the Spencer – Møller – Weare model. Geochim. Cosmochim. Acta 63, 1305 – 1318. Millero, F.J., 1983. Influence of pressure on chemical processes in the sea. In: Riley, J.P., Chester, R. (Eds.), Chem. Oceanogr, vol. 8. Academic Press, London, pp. 1 – 88. Nagornov, O.V., Chizhov, V.E., 1990. Thermodynamic properties of ice, water, and their mixtures at high pressures (in Russian). Prikl. Mekh. Tekh. Fiz. 31, 41 – 48. Petrenko, V.F., Whitworth, R.W., 1999. Physics of Ice. Oxford Univ. Press, Oxford. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge Univ. Press, Cambridge. Richards, T.W., Speyers, C.L., 1914. The compressibility of ice. J. Am. Chem. Soc. 36, 491 – 494. Smithsonian Institute, 1951. Smithsonian Meteorological Tables. Smithsonian Institute, Washington, DC. Ter Minassian, L., Pruzan, P., Soulard, A., 1981. Thermodynamic properties of water under pressure up to 5 kbar and between 28 and 120 jC. Estimates in the supercooled region down to 40 jC. J. Chem. Phys. 75, 3064 – 3072. Wagner, W., Saul, A., PruB, A., 1994. International equations for the pressure along the melting and along the sublimation curve of ordinary water substance. J. Phys. Chem. Ref. Data 23, 515 – 525.
The compressibility of ice to 2.0 kbar G.M. Marion a,*, S.D. Jakubowski b,1 b
a Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512, USA Department of Oceanography, Florida State University, Tallahassee, FL 32306, USA
Received 18 November 2002; accepted 30 October 2003
Abstract To understand in situ properties of ice under pressure, we need an accurate model for ice compressibility. Many studies have estimated ice compressibility, but there still remains considerable controversy with respect to the magnitude and temperature dependence of ice compressibility. The specific objectives of this study were to (1) present a new model for estimating the compressibility of ice based on chemical thermodynamic principles, and (2) compare these model results with previous work. A thermodynamic equation that relates equilibrium constants to molal volumes and compressibilities of water and ice was the basic model used to estimate ice compressibilities. All terms in this equation (equilibrium constants, temperature, pressure, molal volumes of ice and water at 1 atm, and water compressibilities) are definable along the pressure ice/water melting curve except for ice compressibility, which we estimated from the equation. Our estimate of ice compressibility demonstrated a significant temperature dependence. In contrast, most other studies, especially those that rely on elastic parameters, only demonstrate a weak temperature dependence. All recent studies show a significantly lower ice compressibility than the classic Bridgman studies. The results of this work are in agreement with most other databases and models and indicate a general consensus on the compressibilities of water and ice up to 1200 bar of pressure. Between 1200 and 2000 bar, there are discrepancies that are unlikely to be resolved without further experimental work that directly estimate the compressibility of ice over a range of temperatures. A comparison of calculated ice core densities from Antarctica, with and without corrections for pressure, demonstrate the utility of this model for understanding in situ ice properties under pressure. D 2004 Elsevier B.V. All rights reserved. Keywords: Ice compressibility; Water compressibility; Pressure; Subzero temperatures; Chemical thermodynamic model
1. Introduction In order to understand in situ properties of ice under pressure, we need an accurate model for ice compress* Corresponding author. Tel.: +1-775-673-7349; fax: +1-775673-7485. E-mail addresses: [email protected] (G.M. Marion), [email protected] (S.D. Jakubowski). 1 Fax: +1-850-644-2581. 0165-232X/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2003.10.008
ibility. For example, Gow (1971) has shown that deep ice cores from Antarctica relax-out elastically as soon as the cores reach the surface, making it impossible to accurately determine the true densities at depth. There have been many studies of ice compressibility following the classic studies of Bridgman (1912). In the latter study, the ice compressibility was estimated indirectly from measurements of the total volume change (ice + water) along the pressure ice/water melting curve. Many subsequent studies have used the elastic proper-
212
G.M. Marion, S.D. Jakubowski / Cold Regions Science and Technology 38 (2004) 211–218
ties of ice in order to estimate ice compressibilities (e.g., Jona and Scherrer, 1952; Bass et al., 1957; Leadbetter, 1965; Dantl, 1969). A few studies have estimated ice compressibilities directly (e.g., Richards and Speyers, 1914; Gow and Williamson, 1972). All studies after Bridgman (1912) have reported substantially lower ice compressibilities than the Bridgman studies. The specific objectives of this study were to (1) present a new model for estimating the compressibility of ice based on chemical thermodynamic principles, and (2) compare these model results with previous work.
2. Materials and methods The baseline for pressure studies is 1 atm pressure (1.01325 bar). Compressibility studies report pressures either as total pressure ( P) or as gage pressure ( p) where P = p + 1.01325 (with pressure in bar). Both total pressure and gage pressure will be used in this work. Care will be taken to make clear which pressure is being used. Different units are used to report isothermal compressibilities:
1 bar
1 dV cm3 ¼ or V dP mol bar dV : ¼ dP
ð1Þ
Because the latter units are explicitly needed in our models, we will primarily rely upon them in this work; but the water and ice compressibility equations will be given with both sets of units. For a chemical reaction such as: H2 OðI; crÞ X H2 OðlÞ;
ð2Þ
where H2O(I,cr) is ice and H2O(l) is liquid water, the equilibrium constant (K) as a function of pressure can be estimated by: 0 KPT DV ðT ; P0 Þ ðP P0 Þ ¼ exp RT KpT0 DK 0 ðT ; P0 Þ 0 2 ðP P Þ þ 2RT
(Millero, 1983; Krumgalz et al., 1999), where 0 DV 0 ¼ Vl0 VI;cr
ð4Þ
and 0 DK 0 ¼ Kl0 KI;cr
ð5Þ
In these equations, KPT is the equilibrium constant at pressure P and temperature T, KP0T is the equilibrium constant at 1.0 atm (1.01 bar) and temperature T, R is the gas constant, T is temperature (K), V¯0 is the molal volume at 1.0 atm for either water (l) or ice (I,cr), and K¯0 is the isothermal compressibility for either water (l) or ice (I,cr). The molal volume for pure ice at 1 atm is given by 0 VI;cr ¼ 1:930447e1 7:988471e 4 T þ 7:563261e 6 T 2
ðR2 ¼ 0:9998Þ
ð6Þ
(Fig. 1). This equation is derived from crystal lattice parameters for ice Ih (Petrenko and Whitworth, 1999) and is valid over the temperature range from 160 to 273 K. Experimental measurements of the molal volumes of ice show significantly more variability than volumes based on crystal lattice parameters (Fig. 1) because of cracks and gas bubbles in experimental ice (Hobbs, 1974). Eq. (6) represents an idealized pure ice without cracks and gas bubbles. The molal volume for pure water at 1 atm is given by: Vl0 ¼ 9:479107e1 7:889356e 1 T þ 2:693176e 3 T 2 3:052859e 6 T 3 ðR2 ¼ 0:9949Þ
ð7Þ
The database for this relation (Fig. 1) was taken from the Smithsonian Meteorological Tables (Smithsonian Institute, 1951), Chen et al. (1977), and Hare and Sorensen (1986). The equilibrium constant for water and ice at 1 atm in Eq. (3) is given by: KPT 0 ¼ 1:833946e0 1:787269e 2 T þ 6:212940e 5 T 2 2:884752e 8 T 3 ðR2 ¼ 1:0000Þ
ð3Þ
ð8Þ
(Marion, 2002). The equilibrium temperature (T) at a pressure ( P) along the pressure ice/water melting curve
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213
Fig. 1. The molal volume of ice and liquid water as a function of temperature and pressure. Symbols represent experimental data. Lines are model estimates.
3.897e 5 = 17.23 cm3/mol).It can be shown that the equilibrium constant for water and pure ice is given by:
is calculated from the equations of Wagner et al. (1994). The compressibility of water (K¯ 0l ) is based on the experimental database of Ter Minassian et al. (1981). In the latter study, water compressibilities are reported from 40 to 120 jC and from 0 to 5000 bar. We used a subset of this database that covered the temperature range from 30 to 30 jC and the pressure range from 0 to 2000 bar (Table 1). In our work, water compressibilities are reported in units of cm3/(bar mol), while in Ter Minassian et al. (1981), compressibilities are reported in units of 1/bar. Dividing an entry in Table 1 by the corresponding datum in Table 3 of Ter Minassian et al. (1981) gives the molal volume (e.g., at 0 jC and 1000 bar of pressure, 6.715e 4/
K ¼ PðH2 O;iceÞ =PðH2 O;waterÞ ¼ aH2 O ;
ð9Þ
where P(H2O,ice) and P(H2O,water) are partial pressures of water vapor above pure ice and water, respectively, and aH2O is the activity of water (Marion, 2002). To find the temperature where pure ice and pure water are in equilibrium at a specified pressure, it suffices to find the temperature where P(H2O,ice) = P(H2O,water) and therefore aH2O = 1.0 (Eq. (9)). For example, at 1 atm pressure, P(H2O,ice) = P(H2O,water) at 273.15 K (Lide, 1994), which is the freezing point of pure water at 1 atm pressure.
Table 1 The isothermal compressibility of water [cm3/(mol bar)] as a function of temperature and pressure (gage) Pressure Temperature (jC) (bar) 30 25 0 200 400 600 800 1000 1200 1400 1600 1800 2000
1.579e 3 1.348e 3 1.175e 3 1.042e 3 9.353e 4 8.483e 4 7.761e 4 7.154e 4 6.636e 4 6.190e 4 5.804e 4
1.331e 3 1.186e 3 1.064e 3 9.623e 4 8.768e 4 8.044e 4 7.423e 4 6.890e 4 6.427e 4 6.021e 4 5.666e 4
20
15
10
5
0
5
10
20
30
1.179e 3 1.075e 3 9.822e 4 9.006e 4 8.294e 4 7.673e 4 7.132e 4 6.655e 4 6.237e 4 5.867e 4 5.539e 4
1.077e 3 9.954e 4 9.202e 4 8.520e 4 7.909e 4 7.365e 4 6.881e 4 6.451e 4 6.070e 4 5.728e 4 5.422e 4
1.004e 3 9.364e 4 8.724e 4 8.131e 4 7.593e 4 7.107e 4 6.670e 4 6.277e 4 5.924e 4 5.607e 4 5.319e 4
9.513e 4 8.916e 4 8.349e 4 7.821e 4 7.335e 4 6.892e 4 6.490e 4 6.127e 4 5.798e 4 5.500e 4 5.230e 4
9.114e 4 8.572e 4 8.055e 4 7.573e 4 7.126e 4 6.715e 4 6.341e 4 6.002e 4 5.690e 4 5.408e 4 5.153e 4
8.813e 4 8.305e 4 7.825e 4 7.372e 4 6.953e 4 6.570e 4 6.216e 4 5.895e 4 5.600e 4 5.332e 4 5.086e 4
8.584e 4 8.098e 4 7.639e 4 7.211e 4 6.816e 4 6.450e 4 6.114e 4 5.806e 4 5.526e 4 5.267e 4 5.031e 4
8.277e 4 7.813e 4 7.383e 4 6.986e 4 6.616e 4 6.278e 4 5.965e 4 5.677e 4 5.413e 4 5.170e 4 4.947e 4
8.102e 4 7.651e 4 7.234e 4 6.851e 4 6.496e 4 6.172e 4 5.872e 4 5.596e 4 5.342e 4 5.108e 4 4.893e 4
214
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Bearing in mind that KPT = aH2O = 1.0 at equilibrium for pure ice/water systems means that all terms in Eq. (3) are definable along the pressure ice/water melting curve see above except for K¯I,cr0. In practice, we substituted values for K¯I,cr0 into Eq. (3) until the relationship predicted aH2O = 1.00. This is the chemical thermodynamic basis for calculating the isothermal compressibility of ice. We incorporated the above relationships into the FREZCHEM model (Marion and Farren, 1999; Marion, 2001, 2002) to facilitate the calculations. In what follows, the molal volumes of ice and water as functions of temperature and pressure are calculated by: VPT ¼ VPT0 K T DP;
ð10Þ
where V¯PT is the molal volume at pressure P and temperature T, V¯P0T is the molal volume at 1 atm and temperature T, K¯T is the compressibility at temperature T, and DP is the change in pressure. We used a secondorder interpolating polynomial algorithm (Press et al., 1992; Subroutines: Locate, Polint, Polin2) in Table 1 for calculating the isothermal compressibility of water as a function of temperature and pressure. An underlying assumption in the application of Eqs. (3) and (10) is that the compressibilities of water and ice can be approximated by a constant at a given temperature. Clearly, there is a strong pressure dependence for the isothermal compressibility of water (Table 1). In what follows, we calculated an average water compressibility for a given temperature by dividing the pressure range of interest (e.g., 1000 bar) into 20 equally spaced intervals (e.g., 50 bar), estimated the compressibility at the mid-point of each interval, and used the average of the 20 values in Eqs. (3) and (10). A more accurate alternative to using Eq. (10) is to directly integrate: dV ¼ K T ðPÞ: ð11Þ dP T For example, we fitted a quadratic equation to the water compressibility data at 0 jC (Table 1) and integrated Eq. (11) over the pressure range from 1.01 to 2001.01 bar (DP = 2000 bar). The change in the molal volume over this pressure range (DV) calculated with Eq. (11) is 1.3717 cm3/mol, compared to 1.3719 cm3/mol cal-
culated with Eq. (10). An error of this magnitude is insignificant. The isothermal compressibility of ice calculated by solving Eq. (3) is assumed to represent an average over the pressure range of interest. The change in compressibility of ice with pressure is not large (Richards and Speyers, 1914; Nagornov and Chizhov, 1990). Estimates of the molal volumes of water at 500, 1000, 1500, and 2000 bar of pressure based on Eq. (10) and the associated assumptions are given in Fig. 1. For comparative purposes, we also included the experimental measurements of Chen et al. (1977) at 500 and 1000 bar between 273 and 298 K. At 298 K (25 jC), which constitutes an interpolation for our model (Table 1), the estimates of the molal volumes at 500 bar are 17.689 and 17.683 cm3/mol, and at 1000 bar, the molal volumes are 17.358 and 17.351 cm3/mol for the Chen et al. (1977) and our model, respectively. This close agreement is partial validation of our approach for estimating the molal volume of water based on Eq. (10).
3. Results Our estimates of ice isothermal compressibility showed a significant temperature dependence (Fig. 2). A quadratic equation fit to our model-derived estimates gives: K 0 ðcm3 =ðmol barÞÞ ¼ 2:790102e 2 2:235440e 4 T þ 4:497731e 7 T 2 ðR2 ¼ 0:9989Þ
ð12Þ 0 K ð1=barÞ ¼ 1:398427e 3 1:121305e 5 T þ 2:258122e 8 T 2 ðR2 ¼ 0:9988Þ
ð13Þ
The earlier Bridgman (1912) study also demonstrated a significant temperature dependence. There have been several studies that have estimated ice compressibilities based on elastic parameters of ice; these studies are designated in Fig. 2 with solid (filled) symbols. These elastic-parameterization studies show only a slight temperature dependence. The data points for the Dantl (1969) study (Fig. 2) are estimated from their derived equation. The two studies that have estimated ice compressibility di-
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215
Fig. 2. The isothermal compressibility of ice as a function of temperature.
rectly (Richards and Speyers, 1914; Gow and Williamson, 1972) fall in the midrange of temperatures and are compatible with both our chemical thermodynamic model and the elastic-parameter models. All works subsequent to Bridgman (1912), including our study, indicate significantly lower ice compressibilities than the Bridgman studies (Fig. 2). This is an important point because, despite the age of the Bridgman (1912) study, it is still widely cited in the literature (e.g., Hobbs, 1974; Petrenko and Whitworth, 1999), which it should be for historical purposes but not for ice compressibilities. In Fig. 3, we compare the molal volumes of water and ice along the pressure melting curve for our study, the Bridgman (1912) study, and a recent modeling study of Nagornov and Chizhov (1990). In the latter modeling study and in our study, the Ter Minassian et al. (1981) database was used to estimate water compressibilities, but the specific algorithms of the two models are slightly different which leads to a small discrepancy between models (Fig. 3). For example, at 2000 bar of pressure, the extrapolated molal volumes
for water are 16.65, 16.57, and 16.53 cm3/mol for the Bridgman (1912) study, the Nagornov and Chizhov (1990) model, and our model, respectively. The discrepancies for ice volumes are much larger than for water volumes (Fig. 3). Our model estimates of the molal volume of ice flatten out between 1200 and 2000 bar of pressure because ice compressibilities decrease to low values at low temperatures (Fig. 2). The Bridgman (1912) study, on the other hand, which reports high ice compressibilities (Fig. 2), has, as a consequence, low ice volumes (Fig. 3). The Nagornov and Chizhov (1990) model uses an ice compressibility algorithm based on temperature-insensitive elastic parameters (Fig. 2); their calculated ice molal volumes fall inbetween our study and the Bridgman (1912) study at high pressures. Note, however, that our model and the Nagornov and Chizhov (1990) model are in reasonably good agreement up to 1200 bar of pressure (Fig. 3). The pressure ice/water melting database of Wagner et al. (1994) was a major source of data used to estimate ice compressibilities in our study. So
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Fig. 3. The molal volumes of ice and liquid water along the pressure ice/water melting curve.
naturally, our derived model fits this database very well (Fig. 4). We examined the consequences of using a different equation for ice compressiblity on the resulting temperature – pressure relationship. For
the latter, we used our previously described model except we substituted the Dantl (1969) equation (Fig. 2) for our equation (Eq. (12)) for ice compressibility. At 2000 bar of pressure, the Dantl
Fig. 4. The freezing point depression of an ice/water system as a function of pressure.
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217
Fig. 5. Calculated densities of an Antarctic ice core as a function of pressure and temperature.
equation underpredicts the freezing point depression by 1.02 jC. However, at least down to 1200 bar of pressure, the two models are in reasonably good agreement (Fig. 4).
4. Discussion The chemical thermodynamic equation used to estimate the ice compressibility (Eq. (3)) depends fundamentally on four components that must be known accurately: the equilibrium constants, the molal volumes of ice and water at 1 atm (Eqs. (6) and (7)), the isothermal compressibility of water (Table 1), and the pressure and temperature along the ice/water melting curve. The latter relationship appears well established (Wagner et al., 1994). It would also seem that the molal volumes for water and ice at 1 atm are well established (Fig. 1) as are the equilibrium constants. We showed that the molal volumes for water at 1 atm (Eq. (7)) used in conjunction with the water compressibility data (Table 1) leads to accurate estimates of the molal volume for water at higher pressures over the temperature range from 273 to 298 K, based on a comparison with the independent Chen et al. (1977) database (Fig. 1). Our model for water at subzero temperatures is also
consistent with the Nagornov and Chizhov (1990) model (Fig. 3); but this is not a true validation as both models use the water molal compressibility database of Ter Minassian et al. (1981). An independent validation of the Ter Minassian et al. (1981) water compressibility database for work at subzero temperatures is desirable. We terminated our use of this database at 30 jC (Table 1) because water compressibility data at temperatures of 35 and 40 jC given by Ter Minassian et al. (1981) led to a precipitous drop in calculated water molal volumes at all pressures (calculations not presented), in sharp contrast to the trends in the data at higher temperatures (Fig. 1). The results of this work are in agreement with most other databases and models and indicate a general consensus on the compressibilities of water and ice up to 1200 bar of pressure (Figs. 1– 4). Between 1200 and 2000 bar, there are discrepancies, which are unlikely to be resolved without further experimental work that directly estimate the compressibility of ice (Richards and Speyers, 1914; Gow and Williamson, 1972) over a range of temperatures. Earlier we alluded to the importance of an accurate ice compressibility model in understanding the physical properties of deep ice cores where both pressure and temperature are changing simultaneous-
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ly. Gow (1971) has shown that the density of deep ice cores under pressure relaxes elastically as soon as the cores are extracted. In Fig. 5, we used our model to calculate how the density of an ice core from Antarctica (Gow et al., 1968; Gow, 1971) would vary with core temperature at 1 atm, which is what is measured at the surface with corrections for temperature, to the same core under both temperature and pressure constraints. At 1 atm pressure, the core density changes linearly with temperature (Fig. 5), in agreement with our model (Fig. 1) and the Gow (1971) results (see his Table 1). In contrast, the density of the ice core subjected to both temperature and pressure constraints is always higher and becomes progressively more separated from the 1 atm curve as pressure increases (Fig. 5). At the base of the core, the predicted densities are 0.9171 and 0.9203 g/cm3 for 1 and 192 bar, respectively. Between approximately 1730 and 2150 m, the calculated ice density only fluctuates between 0.9201 and 0.9203 g/cm3. Over this range, the increasing pressure raises the density and the increasing temperature lowers the density; the two forces of temperature and pressure are in relative balance over this range. Gow et al. (1968) estimated that the pressure melting point of the ice at the base of this core at 197 bar of pressure is 1.6 jC. Our model predicts 1.5 jC at 197 bar for a pure ice/water system. The presence of solutes at the ice/water junction would depress our estimate to lower temperatures.
Acknowledgements We thank Dr. A.J. Gow for providing guidance on the ice compressibility literature. This work was funded by a NASA PG&G Grant on An Aqueous Geochemical Model for Cold Planets, a NASA EPSCoR project, and a NASA Space Grant award.
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