1. Introduction
Evaporation of a volatile component in a magmatic system causes a change in the system's heat balance due to the work involved in gas expansion (PdV work) and latent heat of exsolution (evaporation). Additional minor effects include latent heat of crystallization or vitrification, heating of country rock, inter-melt radiation, etc. Navrotsky (1987) conducted a calorimetric study of water in albite melt at high pressure (100 MPa) and water content (about 4 wt. %) and found that at 1000 úC the heat of vaporization does not significantly effect the cooling of the system. The thermal effects of melt degassing have not drawn much attention from volcanologists because the temperature change of a magmatic system caused by degassing has been assumed to be small relative to the effects of thermal exchange with country rocks and the atmosphere. Furthermore, the cooling caused by degassing could be offset by latent heat of fusion, if it promotes crystallization. Nevertheless, degassing can make an important contribution to the total heat balance of a magmatic system (Marsh, 1989). There were a few early studies (Boyd, 1961; McBirney, 1963; Williams and McBirney, 1979) which crudely estimated the cooling effect of exsolution and expansion of gas, and found them to be significant in some specific cases. The present study is an attempt to refine our understanding and quantify the thermal effects of magmatic degassing.
Our analysis of the thermal effects of gas exsolution from magmatic melts is based on thermodynamic and experimental data for the albite-H2O system. We choose the albite-H2O system because it has been studied in much more detail than any other magmatic system, thus providing the most reliable thermodynamics data available. Major contributions to the study of albite-water system and its thermodynamics have been made by W. Burnham and others (Burnham, 1974; Burnham, 1994; Burnham and Davis, 1971; Burnham and Davis, 1974). These have mainly been directed toward water solubility and the prediction of phase relations. Additional studies have applied these experimental and theoretical results to interpretations of some magmatic systems (Burnham, 1983; Burnham, 1985), and have provided an important source of thermodynamic data for evaluation of the heat of water exsolution.
Oversaturation degassing is common for erupting volcanic systems near the vent where the rate of decompression to 0.1 MPa is so high that diffusive processes in the magma cannot keep pace with decompression. This can be exacerbated by a lack of bubble nucleation sites in particularly homogeneous melts. Highly silicic melt is most susceptible to oversaturation because of its low diffusivity, high viscosity, and high internal bubble pressure (Proussevitch et al., 1993). Oversaturation degassing may be responsible for explosive eruptions in which there is extreme expansion of the eruptive column at the vent.
First, it is important to note that adiabatic degassing in volcanic melts occurs at a very small spatial scale. The region of interest is an individual bubble surrounded by a limited volume of melt defined by the proximity of neighboring bubbles. A linear scale of such a 'bubble cell' is much smaller than the scale of an entire magma body. Therefore, local heat exchange between growing bubbles and melt at short time scales is not effected by heat exchange between magma and country rocks. Thus it is appropriate to consider adiabatic conditions for the thermodynamic evaluation of the melt degassing. Below we will consider the thermodynamics of reversible (equilibrium) and irreversible (oversaturation) degassing.
First, let us introduce molar quantities for a substance entropy as
Substitution of eqn. (19) to (16) yields the same equation (11) established for reversible water exsolution and ultimately also to subsequent eqs. (12)-(15). This demonstrates that regardless of the style of water exsolution (reversible or irreversible), the temperature change of the system is composed of two major terms (PdV work and heat of water exsolution). If the system (Fig. 2) is not in equilibrium and the melt is oversaturated with water, the concentration profile across the liquid phase must still come to saturation concentration at the liquid-vapor interface (Fig. 3).
Data and a method to calculate heat capacity of the melt is presented in Appendix 1. Along with the known solubility law (necessary for equilibrium degassing) it is possible to calculate the thermal effects (cooling) on albite melt of decompressive degassing.
An additional note is important regarding calculations of energy and cooling terms of gas expansion. The fact that gas actually expands in individual bubbles can make the illusion that bubble pressure should be used in eqs. (20b) and (21b), because it is different from local magma (ambient) pressure (Proussevitch et al., 1993; Sparks, 1978). The total pressure in a bubble consists of ambient pressure as well as surface tension pressure and dynamic pressure caused by viscous resistance to expansion such that
The dynamic pressure contribution to the work of expansion does not change the thermal balance of the system because PhdV cooling is completely compensated by dissipation heating caused by viscous resistance in the melt. We are thus left with only the ambient pressure of the system at the location of the bubble to be used in (21).
1). The heat of water exsolution from albite melt can be applied to other melts because all thermodynamic functions of dissolved water in the melt are proportional to the molecular weight of dry melt normalized to eight atoms of oxygen as demonstrated by Burnham (Burnham, 1974; Burnham, 1975; Burnham, 1994).
2). The pressure and temperature relations of the heat of exsolution are very complex. On a linear pressure scale (Fig. 5a) and for a wide range from atmospheric pressure (up to 1000 MPa) there is a steep decrease with increasing pressure to about 150 MPa. At that point, the heat of exsolution slowly decreases to a minimum at about 300-400 MPa. With further increase in pressure, it slowly and steadily increases until high pressures considered. At the minimum, some curves on Fig. 5a reach negative values. This is consistent with experimental results (Kadik et al., 1971) which demonstrate that at some high temperatures and pressures,
3). On logarithmic pressure scale (Fig. 5b) from low pressures the heat of exsolution increases sharply to a maximum at 1-3 MPa. This result can be examined by independent thermodynamic relations for dDeltaH/dP, as derived in Appendix 3. As an example, we apply (3.8) for the 1000 oC isotherm. It is convenient to use the results of Burnham (1994) for the change of kw with pressure. For low pressures,
An analogous exercise could be performed for the high pressure behavior of DeltaHev using the appropriate slightly more complex relations from Appendix 3.
4). Temperature relations of the heat of exsolution are uniform for pressures less than 200 MPa. DeltaHev falls with temperature (Fig. 5) and the rate of fall decreases at high temperatures (negative second derivative). This can be explored through eq. (3.5) in Appendix 3 which shows that dDeltaHev/dT is approximately proportional to Deltacp. dDeltaHev/dT can be determined from the data used to construct Fig. 3 which shows that it is negative and decreases with increasing temperature. This agreement suggests that the behavior of the heat of exsolution heat at low pressures calculated by the entropy analysis is accurate.
At pressures above 200 MPa, the temperature relations of the heat of exsolution is more complex, and isotherms cross each other (Fig. 5). With temperature increase above 700 oC, DeltaHev initially decreases and then increases. A thermodynamic explanation of such behavior derives from (3.4b) in Appendix 3. Simple calculations reveal that the most important term in (3.4b) is Deltacp which results in dDeltaHev/dT is proportional to Deltacp . Heat capacities for gas and dissolved water can be found from published Steam Tables (Haar et al., 1983) and studies (Burnham and Davis, 1974). Heat capacity calculations show that at high pressures, lines for cp of steam and dissolved water cross each other at intermediate magmatic temperatures, with Dcp negative at low temperatures and positive at high temperatures. The change in the sign of Dcp explains the complex behavior of the heat of exsolution as a function of temperature at high pressures.
5). A qualitative relationship between the internal energy (U) of gaseous and dissolved water can be obtained on the basis of relationship between DeltaHev and PDeltaV (=RT) from the addition of gas volume into the bubble. According to the first law of thermodynamics
Equation (35) is not applicable to gas released from a volcanic foam after disruption into spray when the geometry inverts to melt fragments surrounded by gas. It is also not applicable to an eruption column in thermal contact with the atmosphere (atmospheric cooling of melt pyroclasts (Thomas and Sparks, 1992)). Consequently, eq. (35) is valid only until the point of disruption, where the final pressure P1 is ambient pressure at disruption. This corresponds to a gas volume fraction of about 80%, which is the limit of vesicularity in common scoria and pumice (Gardner et al., 1991; Houghton and Wilson, 1989; Mangan et al., 1993; Thomas et al., 1994). Given the appropriate solubility law, we can evaluate the disruption pressure (P1) from the initial pressure P0 from (see Appendix 4 for details)
The energy associated with degassing is normalized by the molecular weight of water (see (14)-(15) and (16)-(17)) and so do not depend on molecular weight of the melt. However, the corrections must be applied for water solubility and melt heat capacity if weight fractions are used, as described in Appendix 1. Thus, the results of this study are applicable to typical magmatic systems to the extent that their thermodynamics can be related to those of albite. As more experimental data become available for more complex systems, this analysis can be more readily to other magmatic compositions.
2. At high pressures, the heat of water exsolution (vaporization) can be indirectly determined from experimental water solubility data in conjunction with the Clausius-Clapeyron equation. The most reliable method for calculation of the heat of exsolution over a wide range of pressures is evaluation of the entropy of water in gas and in solution using empirical thermodynamic functions.)
3. Heat of vaporization is relatively high at low pressures and can reach 20 kJ/mole at 10-20 bars (1-2 MPa). It sharply decreases with increasing pressure to virtually nil at 200 MPa and even becomes negative at pressures of about 400 MPa. At higher pressures still, it slowly increases.)
4. The thermal effect of the work of bubble expansion is important only for equilibrium degassing, and then only if it occurs from relatively high starting pressures (over 100 MPa).)
5. During rapid magma rise at a volcanic vent during eruption, the thermal effects lead to cooling of bubbles more rapidly than can be equilibrated by thermal diffusivity. This can lead to extreme cooling near the bubble-melt interface with profound effects on melt rheology and volatile diffusivity. In the extreme case of vitrification, volatile diffusion is stopped, and the exsolution of the system becomes limited by thermal diffusivity rather than chemical diffusivity. In this case oversaturation can become extreme leading to explosive degassing and/or eruption of oversaturated melt. )
6. Cooled, chilled, or vitrified melt resulting from the thermal effects considered in this study may bear on fragmentation processes and the formation of fine ash during explosive eruptions.
Boyd, F.R., 1961. Welded tuffs and flows in the rhyolite plateau of Yellowstone Park, Wyoming. Geol. Soc. Amer. Bull., 72: 387-426.)
Burnham, C.W., 1974. NaAlSi3O8-H2O solutions: A thermodynamic model for hydrous magmas. Bull. Soc. Fr. Mineral. Crystallogr., 97: 223-230.)
Burnham, C.W., 1975. Water in magmas: a mixing model. Geochim. Cosmochim. Acta, 39: 1077-1084.)
Burnham, C.W., 1983. Deep submarine pyroclactic eruptions. Econ. Geol., Mono. 5: 142-148.)
Burnham, C.W., 1985. Energy release in subvolcanic environments: Implication for breccia formation. Econ. Geol., 80: 1515-1522.)
Burnham, C.W., 1994, Development of the Burnham model for prediction of H2O solubility in magmas. In: M.R. Carroll, and J.R. Holloway (M.R. Carroll, and J.R. Holloways), Volatiles in Magmas. Mineral. Soc. Amer., Washington, p. 123-129.)
Burnham, C.W., and Davis, N.F., 1971. The role of H2O in silicate melts. I. P-V-T relations in the system NaAlSi2O8-H2O to 10 kilobars and 1000 C. Am. J. Sci., 270: 54-79.)
Burnham, C.W., and Davis, N.F., 1974. The role of H2O in silicate melts: II. Thermodynamic and phase relations in the system NaAlSi3O8-H2O to 10 kilobars, 700 to 1100 C. Am. J. Sci., 274: 902-940.)
Dunbar, C., and Kyle, P., 1992. Volatile contents of obsidian clasts in tephra from the Taupo volcanic zone, New Zealand: Implications to eruptive processes. J. Volc. Geotherm. Res., 49: 127-145.)
Eichelberger, J.C., and Westrich, H.R., 1981. Magmatic volatiles in explosive rhyolitic eruptions. Geophys. Res. Lett., 8: 757-760.)
Gardner, J.E., Sigurdsson, H., and Carey, S.N., 1991. Eruption dynamics and magma withdrawal during the plinian phase of the Bishop Tuff eruption, Long Valley Caldera. J. Geophys. Res., 96: 8097-8111.)
Haar, L., Gallagher, J.S., and Kell, G.S., 1983. NBS/NRC Steam Tables: Thermodynamic and Transport Properties and Computer Programs for Vapor and Liquid States of Water in SI Units. Hemisphere Publishing Corporation, Washington, 312 pp.)
Houghton, B., and Wilson, C., 1989. A vesiculartiy index for pyroclastic deposits. Bull. Volcanol., 51: 451-462.)
Hurwitz, S., and Navon, O., 1994. Bubble nucleation in rhyolitic melts: Experiments at high pressure, temperature, and water content. Earth Planet. Sci. Lett., 122: 267-280.)
Kadik, A.A., and Frenkel, M.Y., 1980. Thermodynamics of decompression in the albite-water system and the role of pressure decrease in magma production. Geochem. Int., 17: 1-21.)
Kadik, A.A., Lebedev, E.B., and Khitarov, N.I., 1971. Water in magmatic melts. Nauka, Moscow, 268 pp.)
Lupis, C.H.P., 1983. Chemical thermidynamics of materials. Elsevier Science Publishing Co., NY-Amsterdam-Oxford, )
Mangan, M.T., Cashman, K.V., and Newman, S., 1993. Vesiculation of basaltic magma during eruption. Geology, 21: 157-160.)
Marsh, B.D., 1989. Magma chambers. Ann. Rev. Earth Planet. Sci., 17: 439-474.)
McBirney, A.R., 1963. Factors governing the nature of submarine volcanism. Bull. Volc., 26: 455-469.)
Melson, W.G., Allan, J.F., Jerez, D.R., Nelen, J., and etc., 1990. Water contents, temperatures and diversity of the magmas of the catastrophic eruption of Nevado-Del-Ruiz, Columbia, November 13, 1985. J. Volcan Geotherm. Res., 41: 97-126.)
Navrotsky, A., 1987, Calorimetric studies of melts, crystals, and glasses, especially in hydrous systems. In: B.O. Mysen (B.O. Mysens), Magmatic processes: Physicichemical principles. Geochemical Society, Washington, p. 411-422.)
Newman, S., Stolper, E., and Epstein, S., 1986. Measurement of water in rhyolitic glasses: Calibration of an infrared spectroscopic technique. Am. Mineral., 71: 1527-1541.)
Proussevitch, A.A., Sahagian, D.L., and Anderson, A.T., 1993. Dynamics of diffusive bubble growth in magmas: Isothermal case. J. Geophys. Res., 98: 22283-22308.)
Richet, P., and Bottinga, Y., 1984. Glass transition and thermodynamic properties of NaAlSinO2n-z and KAlSi3O8. Geochim. Cosmochim. Acta, 48: 453-470.)
Richet, P., and Bottinga, Y., 1985. Heat capacity of aluminium-free liquid silicates. Geochim. Cosmochim. Acta, 49: 471-486.)
Sparks, R.S.J., 1978. The dynamics of bubble formation and growth in magmas: A Review and analysis. J. Volcanol. Geotherm Res., 3: 1-38.)
Thomas, N., Jaupart, C., and Vergniolle, S., 1994. On the vesicularity of pumice. J. Geophys. Res., 99: 15,633-15,644.)
Thomas, R.M.E., and Sparks, R.S.J., 1992. Cooling of tephra during fallout from eruption columns. Bull. Volcan., 54: 542-553.)
Williams, H., and McBirney, A.R., 1979. Volcanology. Freeman, Cooper & Co., San Francisco, 397 pp.
2. Degassing Styles
When melt reaches the solubility curve of a dissolved volatile during decompression, exsolution commences unless a metastable condition is allowed to persist. We define two end-member degassing styles, according to the direction of approach to the solubility curve during decompression (Fig. 1). If a magma is initially undersaturated, moves to saturation, and then proceeds along the solubility curve, we call it equilibrium degassing. If, on the other hand, a magma is somehow initially oversaturated, it degasses as it proceeds toward the solubility curve. We call this oversaturation degassing. Natural systems undoubtedly involve both end-members and intermediate situations. We will restrict our present discussion to the simple case of the end-members because natural systems depend completely on their individual decompression rate history. More realistic system evolution will be left for a subsequent study.
Fig. 1. Two styles of degassing for which thermal effects are considered in this study. The curve represents the solubility curve of water in albite melt (Burnham, 1974). Equilibrium degassing occurs along the solubility curve. Oversaturation degassing occurs at constant pressure as volatiles enter bubbles from the melt causing reduction in oversaturation.
2.1. Equilibrium Degassing
Previous studies on degassing (Boyd, 1961; McBirney, 1963) have been restricted to equilibrium degassing. There is little doubt that this style is common for all basaltic melts, as well as for silicic intrusions where the rate of decompression is small, and the melt may have some solid phase (crystals) which serve as bubble nucleation sites (Hurwitz and Navon, 1994). However, for erupting volcanic systems having extreme magma rise and decompression rates, oversaturation degassing is more relevant, particularly for highly silicic magmas. The equilibrium conditions involve a reversible process where there are no kinetic factors or increase in system entropy.
2.2. Oversaturation Degassing
If a melt finds itself substantially oversaturated with dissolved volatiles, it can begin to degas at constant ambient pressure. We refer to this process as oversaturation degassing (Fig. 1). This style represents an extreme case of irreversible volatile exsolution from a melt. In natural systems, decrease in the level of oversaturation caused by degassing usually proceeds at variable (decreasing) pressures until the magma reaches the surface. This is very complex problem not to be assessed here.
3. Thermal Effects of Degassing
Heat of vaporization and the work of bubble expansion (PdV work) are the two major thermal effects of bubble growth during adiabatic decompression of volcanic melts. There are a number of minor cooling terms which affect magma temperature such as heat of radiation, heat flux to country rocks, heat of crystallization, etc., but these are not directly related to degassing processes prior to volcanic eruption and thus are not considered here.
Table 1
Fig. 2. Idealized system of albite-water for thermodynamic evaluation of volcanic degassing processes.
3.1. Adiabatic reversible water exsolution from magmatic melts
Internal energy conservation (first law of thermodynamics) of the whole system (Fig. 2) can be written as (for notations see Table 1):
(1)
For all reversible or irreversible adiabatic processes, the change of internal energy occurs only due to external PdV work so that
The change of internal energy for the system can be expanded term by term for each component (albite melt, water in solution, water in bubble). Water can be exsolved or dissolved, and the solution is always in equilibrium with the gas. These conditions refer to reversible processes and are applicable if pressure changes very slowly to maintain saturation of H20 in the melt. Equation (1) thus becomes (2)
At equilibrium, (3)
And mass conservation requires (4)
(5)
Substitution of (4)-(6) into (3) yields (6)
If (7) is combined with (2), then (7)
TdS=0 (8)
and an important condition for reversible processes arises (dS=0). This result also demonstrates that equilibrium degassing is a reversible process. . For water number of moles (N) is variable, then it follows
... . (9)
Substitution of (9) into (8) with (5)-(6) yields
. (10)
Because entropy is a function of pressure, temperature and concentration, we can introduce terms with partial derivatives for "TdS" parts in (10) as follows:
In (10a) we used well known relations (10a)
, and
. Concentration of water in gas is always equal to unity, therefore
. For an ideal gas,
. In eq. (10) we can also substitute
. Taking these relations and eq. (5) with eq. (10) and transforming from number of moles (N) to molar concentration (X), we can write
where (11)
is the mixing enthalpy of the albite-water solution. Thus eq. (11) is the general energy conservation equation for the reversible process of gas exsolution from magmatic melts. For the albite-water system, numerous partial thermodynamic functions and their derivatives are known (Burnham and Davis, 1974), and the temperature change due to exsolution can be found from this equation. However, the variation in magnitude between terms in eqn. (11) exceeds an order of magnitude, so that some terms can be neglected. These terms are those which include the small coefficients of thermal expansion of liquids, and mixing enthalpy (Burnham and Davis, 1974). All these terms together contribute less than 1 % of dU. We thus have a practical energy conservation equation in the form
From this equation follows that there are two important cooling terms- 1) heat of water exsolution and 2) effect of external PdV work. We can consider and treat each term separately to highlight the contribution of each to the total change of magma temperature. For practical calculations it is also useful to approximate the coefficient of dT in (12) as total heat capacity of the melt . (12)
and change from mole fractions (X) to mass, which leads to
For many practical cases and application of eq. (14) to magmatic systems, the mass fraction of gas is usually much smaller than that for melt. In this case (13) reduces to. (13)
In terms of concentration, and applying ideal gas law, we can rewrite (14) as . (14)
(15)
Although the last equation is much more useful for practical calculations and integration, eq. (14) explicitly accounts for the two cooling terms (exsolution and expansion) during bubble growth in a melt.
3.2. Adiabatic irreversible water exsolution from magmatic melts
Adiabatic irreversible processes are characterized by increasing entropy DeltaS>0. Equations (1)-(3) are also applicable for adiabatic irreversible processes, and can be used for evaluation of the temperature change of the system. But unlike reversible processes, condition (4) is not maintained because water vapor is not in equilibrium dissolved water and the terms in eqn. (3) with chemical potential (mi) must remain. Substitution of eqn. (10) into (3) combined with (2) results in
Equation (16) differs from (11) only by the terms which contain chemical potentials. These terms must be considered in detail. With the introduction of water fugacities and activities for chemical potentials, e.g. (Burnham, 1994), we can write (16)
In addition, . (17)
Thus, differentiating (17) we can obtain the TS terms in (16). With the solubility law as (18)
(Burnham, 1994), it is possible to write the dX terms as
which is the heat of exsolution at equilibrium. (19)
Fig. 3. Scheme of water exsolution from oversaturated melt (irreversible process)
Thus, despite oversaturation throughout the bulk of the melt, the heat of exsolution is always that for equilibrium so that it would not be appropriate to use where entropy of water in the melt is integrated over the melt volume. So, the form of the "cooling" factors (PdV and DeltaH) is independent of kinetics (reversible vs. irreversible) for integration and evaluation of the final temperature of the system, but their values are different because entropy increases for irreversible processes (2nd law of thermodynamics). Thus, the final change of temperature is expected to be much smaller for the irreversible processes than for reversible "equilibrium" processes, if the same amount of water is exsolved.
3.3. Individual cooling terms of degassing: work of gas expansion and heat of exsolution
Equations (13)-(15) are applicable for both reversible and irreversible exsolution processes. However, there is a significant difference in the relations of pressure to water concentration. For reversible (equilibrium) processes the concentration is a function of pressure, and cooling can be determined from pressure change (Fig 1). For irreversible processes, concentration and pressure changes are independent, and according to the second law of thermodynamics, the total cooling must be less than that for reversible processes.
Equations (13)-(15) have important implications. Simple integration yields additive terms of cooling heat (E) and, therefore, cooling temperatures (DeltaT)
.
Applying (5) we obtain
(20a)
After integration of the first term in (15) we can obtain temperature change from the energy terms, so that. (20b)
where
(21a)
Equation (21) is an approximate solution of (15) in that it is based on the assumption that DeltaHev , c'm and xwm are independent of temperature, which valid for small DeltaT. From these relations (21) it follows that we can treat energy and cooling terms of water exsolution (related to DeltaHev) and gas expansion (related to PdV) independently. Expressions (21) are particularly useful because pressure is the only variable of integration for reversible (equilibrium) degassing since concentration is a function of pressure. For irreversible processes the integral in (21b) is always smaller than for reversible processes because the pressure range of integration is smaller for the same amount of exsolved water. This results in smaller total cooling of the magmatic system. (21b)
According to (22), PdV term for bubbles breaks into three PidV parts.. (22)
Considering them individually, it could be noted that PsdV work is small for grown bubbles (>> nuclear size) and is also small for bubbles near nuclear size. This is because surface tension pressure is very low for large bubbles, and the change of volume DeltaV is very small for small bubbles for which surface tension pressure is significant. Thus, we can neglect PsdV work.
4. Evaluation of heat of water exsolution from albite melt
The temperature change associated with adiabatic bubble growth in a volcanic melt results from volatile (water) exsolution and external PdV work. There are no direct measurements of the molar heat of vaporization of water from albite melt, so it must be calculated from indirect data using appropriate thermodynamic methods. In this section, we discuss a few ways to evaluate heat of water exsolution.
4.1. dT/dP Method.
An early attempt to calculate the heat of water exsolution (Kadik et al., 1971) involved the Clausius-Clapeyron equation (Lupis, 1983) in the form of
where DeltaV could be taken as Vwg at low pressures. (See Appendix 2 for the details of deviation.) In order to determine the heat of vaporization (DeltaHev) it is necessary to know (dT/dP)x which can be visualized as the slope of the line of constant concentration on the phase diagram of the volatile-melt system. The slope of this line for the system albite-water has been investigated and DeltaHev estimated by Kadik and others (Kadik and Frenkel, 1980; Kadik et al., 1971). The results for the 1000 oC isotherm and for a range of pressures from 10 to 1000 MPa are presented in the Table 2. The accuracy of original experimental data and the ability to track constant concentration on isotherms make the results of the Table 2 of questionable utility, particularly at low and high pressures where extreme values of heat of exsolution were found. This, and a lack of data for pressures under 10 MPa compels us to seek other methods to find DeltaHev.. (23)
Table 2
Reproduction of (dP/dT)x and heat of water exsolution calculations for 1000 oC isotherm. Based on Kadik et al. (1971).
4.2. Water Activity Method.
A very explicit and potentially useful way to evaluate the heat of water exsolution from albite melt is through a function for water activity in the melt (awm) and Henry's law analog constant (kwma) (Burnham, 1994) which is derived from interpolation of experimental data (Burnham, 1994; Burnham and Davis, 1971; Burnham and Davis, 1974). On this basis,
where activity depends on concentration range such that (24)
(25a)
Assuming that (25b)
, and
, and
, which is required for equilibrium, we can differentiate (24) and (25) to find the enthalpy of water exsolution from the Henry's law analog constant
(26a)
An analysis of these results shows that they are invalid for a wide range of temperatures and pressures, and in many cases are misleading. For instance, at low pressures (see eq. (3.5) in Appendix 3), the temperature derivative of DeltaHev should be close to the difference between heat capacity (Delta cp) of water in gas and melt, and at low temperatures (< 1200 oC) should decrease with temperature (Fig. 4). (26b)
Fig. 4. Plot of heat capacity of water vapor (Steam Tables, 1984) and water in solution in albite melt at 0.1 MPa. Partial molar heat capacity of water in the melt is derived from Burnham (1974 - eq. 35).
The second derivative should be negative (Fig. 4 and eq. (3.5)). However, a calculation of DeltaHev based on kw (Burnham, 1994) produces a positive second derivative, and the heat capacity of water vapor, calculated as , becomes negative at relatively high temperatures which is thermodynamically impossible. Thus, kw is useful for water solubility evaluation, but can not be used for calculation of the heat of exsolution by eq. (26). The reason for this is that eq. 7 of Burnham (1994) was obtained from multiple nested interpolations, which prevents it from being used for differentiation because uncertainty would be magnified by a factor of (RT^2), i.e. about 10^7 times.
4.3. Entropy Method
The most accurate way to evaluate the heat of water exsolution is based on evaluation of thermodynamic functions of the albite-water system (Burnham and Davis, 1974). Burnham and Davis (1974) found an explicit way to calculate for a range of temperatures at 2 kbar (at this pressure
). It follows that
Knowing the reference state of water vapor it is possible to find (27)
. The reference state can be obtained from the data of Haar et al. (Haar et al., 1983) which encompass the temperature and pressure ranges relevant to magmatic conditions. Integration of entropy over pressure and composition leads to
where Po=200 MPa, T' is the temperature used for DeltaS at Po from (22), Xwom is the equilibrium water concentration (saturation) at Po and T', and Xwm is the equilibrium water concentration (saturation) at P and T'. An important note is necessary about P-T-X relations. All values of entropy in (27), (28) and below refer to conditions of water-melt equilibrium. Thus Xwm is a function of P and T, and therefore, "equilibrium" entropy is actually a function of two variables, i.e. P-T, and not of three. The first integral in (28) can be written as (Burnham and Davis, 1974)
(28)
where P is in MPa and T is in K. The second integral in (28) can be found from (Burnham and Davis, 1974) (29)
(30a)
It is important to observe that eqs. (30) are not consistent with each other and lead to a singularity at Xwm=0.5 as a result of a kink in the DeltaH function. To avoid this problem, we revised (30b) so that the two would be consistent throughout the range of DH by setting (30b) to -4R. Thus, sequential application of eqs. (27)-(30) with the reference state of water (steam) allows evaluation of the partial molar entropy of water in albite melt at equilibrium, as well as the molar heat of water exsolution ( (30b)
). The resulting heat of exsolution is illustrated in Fig. 5 and in Table 3.
Table 3
Calculated heat of exsolution of water from albite melt
Fig. 5. Heat of water exsolution from albite melt calculated from thermodynamic functions (Burnham and Davis, 1974). Figures (a) and (b) have different pressure scales.
In addition, a polynomial interpolation equation for the heat of vaporization can be written as
where coefficients kj,i are presented in Table 4; DeltaHev is in J/mole; P is in MPa; T is in K. (31)
Table 4
Coefficients for eq. (31)
4.4. Discussion of DeltaHev
It is important to explore a few significant aspects of the results concerning DeltaHev before subsequent discussion. becomes negative. Thus, the heat of exsolution should be negative according to the Clausius-Clapeyron equation
(Burnham and Davis, 1974). At higher pressures still, we find that it becomes positive again (Fig. 5a).
is about 8/7 (P in CGS units) (Burnham, 1994). After some simple substitutions, and using Dcp and the mean value at low pressure for DHev from Figs. 3 and 4 respectively, (3.8) reduces to
where DeltaHev is in cal/mole, and P is in bars. From this analysis, it follows that at low pressures, DeltaHev should increase with pressure until it reaches a maximum at about 16 bars, and then decrease. This estimate agrees with Fig. 5b and Table 3, where the maximum is observed at 20 bars. (32)
where the last term approaches RT at low pressures. Using this relation and data from Table 3 we observe that in the pressure range 0.1 to 1000 MPa, the internal energy of gaseous and dissolved water are approximately equal. Another result of the first law of thermodynamics is that
which demonstrates that the PdV work involved in gas exsolution is included in the enthalpy term of exsolution (vaporization). That demonstrates that if oversaturation degassing occurs at constant pressure, then only one heat of exsolution cooling term is applicable.
5. Quantification of Thermal Effects of Equilibrium and Oversaturation Degassing
5.1. Equilibrium Degassing
Equilibrium degassing involves volatile exsolution along the solubility curve such that the gas is in equilibrium with the melt at all times during decompression. We will consider the thermal effects of the work of gas expansion and heat of exsolution.
5.1.1. Work of Gas Expansion
In this section we will assess the cooling of magma by expansion of a gas bubble due to decompressive and diffusive growth as it rises in a magmatic conduit or vent. PdV work is considered here as a reversible process of gas expansion with complete heat exchange between bubbles and host magma. The dissolved mass fraction is determined by the solubility curve of the volatile in the melt. No kinetic effects are included, though we realize that these effects could be very important, especially at the top of volcanic column where ambient conditions for a given parcel of magma can evolve rapidly. Decompressive magma cooling due to PdV work can be obtained from
(see constraints of eq. (21b), wherein thermodynamic functions are temperature-independent). For the case of equilibrium degassing, dissolved water concentration is a function of pressure, so both terms on the right-hand side of (21b) can be integrated over pressure. This requires application of Henry's solubility law with an isothermal approximation which can be written for an albite-water isotherm at 1273 K as (33)
x = b P1/2 = 4.24 10-6 P1/2 (34)
Using the solubility law, integration of (33) results in
Application of (35) to volcanic systems presumes that temperature is equilibrated in the bubble-melt system as a result of thermal conductivity through the bubble walls. This is not an instantaneous process, and for rapid bubble growth at or near the vent during eruption, the system can be driven far from thermal and solubility equilibrium leading to irreversible water exsolution. As a result, the actual melt temperature could be higher, and the gas temperature could be lower than indicated in eq. (35), and the total cooling could be less, as discussed in thermodynamics terms earlier (irreversible degassing). (35)
The solution of this equation is presented graphically in Figure 6. . (36)
Fig. 6. Plot of disruption pressure of magmatic foam versus starting pressure of water exsolution from (albite) melt. Disruption pressure is based on 80 % volume fraction of bubbles.
The disruption pressure can be used in (35), and Figure 7 illustrates DeltaT as such.
Fig. 7. Temperature drop of albite melt due to adiabatic expansion of water bubbles during equilibrium degassing. Horizontal axis represents pressure at which degassing commences. Dashed line indicates 'formal' magma expansion cooling if it ends at 0.1 MPa. At this final pressure gas fraction is usually close to unity, the melt is disrupted to spray, and thermal equilibrium with gas is unlikely. The solid line shows the case for melt cooling only until disruption of magmatic foam at 80 % gas volume fraction. Disruption pressure is indicated on Fig. 6.
For example, magma cooling due to PdV work from exsolution of reasonable water contents during eruption (up to 4 wt. % which is saturated at about 100 MPa) is about 20 K at the disruption pressure and about 100 K at atmospheric pressure. Figure 7 indicates that cooling due to bubble expansion is considerable for low to medium starting pressures (< 100-200 MPa) and should not be neglected in physical models of volcanic eruption because the temperature change effects viscosity, diffusivity and other properties which control transport of magmatic melts. Expansion cooling is especially substantial for high starting pressures and can reach tens of K (Fig. 7). These are considerably lower than those of previous studies (Boyd, 1961; McBirney, 1963), because of our new interpretation stemming from (18) and (23). If the final pressure of bubble expansion is taken as atmospheric in (35), then the resulting cooling would increase by a factor of 5 for medium starting pressures (up to few 100 MPa) and a factor of 3 for higher pressures, but as described above (P1 in (36)) is normally greater than atmospheric due to foam disruption.
5.1.2. Heat of Exsolution
In order to estimate the cooling effect of water vaporization during equilibrium exsolution (along the solubility curve), it is necessary to integrate equation (21a) within the relevant pressure range after replacement of solubility terms by eq. (34). Thus, from (21a) we have
where DeltaHev is assumed to be temperature-independent (valid for small temperature variations). An appropriate numerical integration of (37) with pressure-dependent DeltaHev, as taken from Fig. 5 or Table 2, results in the exsolution (evaporation) cooling terms of equilibrium magma degassing plotted in Fig. 8. The results indicate that equilibrium exsolution (vaporization) from a starting pressure of 100 MPa decompressing to atmospheric pressure causes a temperature decrease of the magmatic system of about 20 K. At low pressures this cooling factor increases rapidly, but at higher pressures the exsolution cooling changes less dramatically because the heat of exsolution is small at high pressures. (37)
Fig. 8. Temperature drop of albite melt due to heat of vaporization for equilibrium degassing of water.
The contributions of heat of exsolution and gas expansion are displayed in Figure 9. Exsolution cooling dominates the total unless the starting pressure greater than 100 MPa. Total cooling is significant throughout the pressure range. The total amount of cooling exceeds 40 K for starting pressures over 120 MPa (saturated by about 4 wt. % of water, common for volcanic eruptions (Anderson et al., 1989; Dunbar and Kyle, 1992; Eichelberger and Westrich, 1981; Melson et al., 1990; Newman et al., 1986)). This cooling could lead to major changes in the system including crystallization and even glass formation, which would serve to at least partially compensate the thermal effect of exsolution cooling.
Fig. 9. Total temperature drop of albite-water melt for equilibrium degassing.
5.2. Oversaturation Degassing
Oversaturation degassing is an extreme case of an irreversible thermodynamic process and proceeds at constant ambient and bubble pressure (Fig.1). This style of degassing is strongly dependent on kinetic factors. While pressure is constant, the amount of volatile exsolution can vary considerably from 1 to 4 or more wt. %. The thermal effects of oversaturation degassing are simply additive, and as follows from integration of (15b) with the reasonable approximations discussed above, can be stated as
As is evident from (38), only the exsolution (vaporization) cooling term is effective for oversaturation degassing at constant pressure, and its value is easily calculated. Values of the heat of water exsolution (Table 3 and Fig. 5) indicate that this cooling effect can vary widely. For high pressures (> 200 MPa) the heat of exsolution is very small and there is no significant cooling with oversaturation degassing. However, at pressures less than 100 MPa, the heat of exsolution cooling term is significant and can reach 8 K per wt. % of exsolved water at pressures of 1-2 MPa. At low pressures for a given amount of exsolved water, this effect increases exponentially with decreasing pressure (Fig. 5b). Thus, the cooling is important if oversaturation degassing occurs at low pressures corresponding to magma depths less than 0.5 km. This is the region where the most rapid magma rise would be expected, producing most likely conditions for oversaturation degassing and consequent thermal effects. (38)
6. Application of albite-water system for other melt compositions
The methodology of calculating the thermal effects of water exsolution from albite melt can be applied to other magmatic melts. The issue of similarity of the thermodynamic properties of albite melt with dissolved water with other melt compositions has been addressed and it has been shown that the thermodynamics of albite melt - water system is applicable to a wide range of magmatic melt compositions (Burnham, 1975; Burnham, 1994). Water solubility in magmatic melts ranging from rhyolites to basanites is the same in terms of molecular fractions. With appropriate weight fractions the same formulation derived for albite can be used for other melts (Burnham, 1975).
7. Thermal Disequilibrium
In this analysis, we have assumed complete thermal equilibrium between bubbles and melt. However, oversaturation degassing at low pressures proceeds very rapidly and can lead to greater cooling of gas than melt. We will leave the solution of this dynamic problem to a subsequent study, but it is possible even now to anticipate some of its implications. Depending on the rate of magma rise (which ultimately controls the extent of volatile exsolution and gas cooling), a significant thermal gradient can develop across the half-wall of melt between bubbles. The cooling of melt near the bubble interface can cause a reduced rate of volatile diffusion toward the bubble because diffusivity depends exponentially on temperature. An extreme case of this could even cause vitrification of a thin film of melt at the bubble interface, stopping diffusion altogether (Fig. 10). In the latter case, bubble growth is limited to decompressive expansion during magma rise. In effect, diffusive bubble growth would be limited by thermal diffusion and reduction of the temperature gradient across bubble walls, rather than by its normal limitation of chemical diffusion (of volatiles).
Fig. 10. Dynamic effects of gas cooling on volatile diffusion into a bubble. If water exsolution from albite melt occurs at high pressures where thermal effects are small (see Table 3), then dynamic cooling of the melt can be neglected. At low pressures the thermal effects of exsolution are considerable and can lead to substantial cooling of gas (and consequently of melt) and to reduction or even cessation of diffusion due to the formation of glass at the bubble interface.
Chilling of the bubble-melt interface can delay degassing until the melt reaches lower or even atmospheric pressure, resulting in extreme oversaturation. This can result in either explosive degassing and bubble growth in the eruption column, or, depending on heat exchange with the atmosphere, to chilling of oversaturated melt. The latter can be tested by additional studies on water contents of glassy eruption materials from highly explosive eruptions.
8. Conclusions
1. The thermal effect of water exsolution (from albite melt) can be divided into two terms including the work associated with bubble expansion, and heat of exsolution. Together, these can lead to substantial cooling of a magmatic system (by more than 100 K under some volcanic conditions).)
Acknowledgments
The authors are grateful to A.T. Anderson for helpful discussions and comments, and to C. Burnham and another friendly reviewer for insightful reviews. This study was supported by NSF grant EAR9317314.
Appendix 1
For estimation of the temperature decrease of the system it is necessary to know the heat capacity of the melt. Here we use a simple empirical equation (Richet and Bottinga, 1985)
where xi and (1.1)
(cp)i = ai + bi T + ci T-2 (1.2)
are the moles and additive temperature dependent coefficients of heat capacity respectively (see Table 1.1 for ai, bi, and ci). For albite melt at 1273 K, calculated heat capacity is (cp)m = 28.31 J g atom-1 K-1 or (cp)m = 1.40 J g-1 K-1. Published experimental measurements of the heat capacity of albite melt at 1096 K are (cp)m = 26.72 J g-atom-1 K-1 (Richet and Bottinga, 1984) which is very close to the calculated value.
Table 1.1
Coefficients for calculations of heat capacity of magmatic melts (in J mole-1 K-1) (Richet and Bottinga, 1985)
Appendix 2
The classical Clausius-Clapeyron equation, which is commonly used for dT/dP relations of phase transition for pure substances in both phases, can not be used for albite-water system because thermodynamic functions of water in the melt depend on concentration as well as on temperature and pressure. Consequently, we must derive an analog of the Clausius-Clapeyron equation for the albite-water system.
For the gas melt equilibrium we have
Therefore, it follows that (2.1)
The partial derivative in the last term of (2.2) can be derived from Burnham (1994 - Eqs. 5 and 6) and yields (2.2)
(2.3a)
Differentiation of (2.3) and algebraic rearrangement yields an equivalent to the Clausius-Clapeyron relation for the albite-water system (2.3b)
(2.4a)
In (2.4b), (2.4b)
always, making it possible to neglect the second term in square brackets of (2.4b). Thus,
The same approach can be used for derivation of dT/dP, where (2.4c)
(2.5a)
Thus, equations (2.4) and (2.5) should be used for albite-water system instead of the classical Clausius-Clapeyron, which is relevant only to pure substances. (2.5b)
Appendix 3
It is important to establish expressions for temperature and pressure differentials of the heat of exsolution. Knowing that the full differential of any thermodynamic function (A) for the albite-water system is pressure, temperature, and composition dependent, we can write
(3.1)
We can apply (3.1) to find the change in the heat of water exsolution from albite melt with change in temperature. Using (3.2)
, from (3.1), it follows that
Using . (3.3)
,
, and (25) for
, equation (3.3) becomes
(3.4a)
(3.4b)For the range of low pressures, (3.4a) can be simplified assuming ideal gas behavior for water vapor and the relations
and
. Thus, for low pressures we have
The last term in (3.5) must not be neglected because it is comparable with Dcp at temperatures greater than 1000 oC. (3.5)
Using the same method, it is possible to find a differential equation for the heat of exsolution change with pressure from eq. (3.2). The result is
(3.6a)
This can be modified using eq. (20), taking into account that at water-melt equilibrium the water activity in the melt is equal to unity, and assuming that the temperature derivative of ln(kw) is small compared to the pressure derivative (Burnham, 1994). After simple transformations we obtain (3.6b)
(3.7a)
Eq. (3.7) is much more useful because the term (3.7b)
can be obtained simply (Burnham, 1994 - Fig. 3 and eq. 7).
Using the same method as for derivation of eq. (3.5), we can obtain an approximate form of (3.7a) as
(3.8)
Appendix 4
The pressure at which foam disrupts can be evaluated. We assume that the foam disrupts at gas volume fraction of 80 %. It follows that
After applying the ideal gas law and Henry's solubility law into (4.1), we obtain the disruption pressure P1 as (See (36)
(4.1)
(4.2)
References
Anderson, A.T.J., Skirius, C.M., Lu, F., and Davis, A.M., 1989. Preeruption gas content of Bishop Plinian rhyolitic magma. Geol. Soc. Am. Abst. w. Prog., 21: A270.