during degassing of C02. A quantitative evaluation and application to the Kushikino gold mining area in Japan. Y.-E Zheng. Geochemisches Institut, Universit/it ...
Mineral. Deposita 25, 246-250 (1990)
MINERALIUM DEPOSITA 9 Springer-Verlag 1990
Carbon-oxygen isotopic covariation in hydrothermal calcite during degassing of C02 A quantitative evaluation and application to the Kushikino gold mining area in Japan Y.-E Zheng Geochemisches Institut, Universit/itG6ttingen, GoldsehmidtstraBe1, D-3400 G6ttingen, Federal Republic of Germany
Abstract. The effect of the outgassing of CO 2 from a hydrothermal fluid on the C- and O-isotopic compositions of calcite, which is precipitated from this fluid, is quantitatively modelled in terms of batch and Rayleigh distillation equations. Both CO2 degassing and calcite precipitation are considered to be the removal mechanisms for dissolved carbon species from the fluid. Combined degassing-precipitation models are then developed by taking H2CO 3 and H C O ] , respectively, as the dominant dissolved carbon species. A positive correlation array between 613C and 8130 values of calcite can be yielded by the precipitation of calcite from a H2CO 3dominant fluid, accompanied by a progressive decrease in temperature during CO 2 degassing, whereas calcite precipitated from a HCO3-dominant fluid under the same conditions tends to display much smaller variation in 613C values than in 61so values. The combined processes of CO 2 degassing and calcite precipitation result in lowering the 613C value of calcites with respect to those precipitated in a closed system simply due to temperature effect. Carbon and oxygen isotopic data for calcite from the Kushikino gold-mining area in Japan illustrate the application of quantitative modelling, and degassing of CO2 is suggested as a more likely cause for the precipitation of the calcite and quartz in this mining area.
The solubility of calcite in a hydrothermal fluid increases with decreasing temperature, and decreases with decreasing pressure (Holland and Malinin 1979). Therefore, calcite cannot be deposited from a hydrothermal fluid by simple cooling under the conditions of a closed system. Instead, degassing of CO2 from the fluid can be an effective process responsible for the precipitation of calcite. Other processes (such as boiling and change in pH) are also important for calcite precipitation from a hydrothermal fluid. The purpose of this communication is to describe calculations that model both carbon and oxygen isotopic variations in hydrothermal calcite as a
function of both progressively decreasing temperature and a given fraction of carbon in degassed CO2. Such a model has been frequently applied to the variation of carbon isotope composition during decarbonation (Nabelek et al. 1984; Valley 1986). However, it has not previously been used to describe the covariation in both carbon and oxygen isotopic compositions of hydrothermal calcite. The simultaneous determination of carbon and oxygen isotopic compositions for calcite using the conventional experimental techniques (McCrea 1950) enables the examination of geochemical processes by which both carbon and oxygen isotopes have been fractionated. As an example of the calculations, carbon and oxygen isotopic data of calcites for the Kushikino gold mining area (Matsuhisa et al. 1985) are modelled to illustrate the application of quantitative evaluation.
Model for CO2 degassing Both batch and Rayleigh distillation models can be used to describe the covariation in carbon and oxygen isotopic compositions of calcite during both CO2 degassing and calcite precipitation from an aqueous fluid. It is assumed that an instantaneous isotopic equilibrium fractionation occurred among degassed CO 2 , remaining dissolved carbon species and precipitated calcite. Following Valley (1986), the effect of batch degassing on the isotopic composition of a fluid can be expressed as: (~fluid
= 6 :il , i d _ ( 1
_ F1 ) .
103 m,
~ :co~ ,.ia
(1)
where F1 is the mol fraction of the element of interest that remains in the fluid after degassing, CZfluid -co2 is the isotopic fractionation factor between CO2 and fluid, and t~i and 6: are the initial and final isotopic values in standard permil notation. Similarly, the effect of the batch crystallization of calcite on the isotopic composition of the fluid can be described by: f' = Jfluia--( -r (~fluid 1 F2)" 10 3 In o{flui dcalcite (2)
247
where F2 is the mol fraction of the element of interest that remains in the fluid after calcite crystallization, Ctfl,i ac"tc~e is the isotopic fractionation factor betwen calcite and fluid, and superscripts i' and f ' refer to the initial and final with respect to the calcite crystallization. Because the precipitation of calcite from the fluid is caused by CO2 degassing, the following approximation can be made: 6ifluid
(3)
f (~ fluid
o A
-2
~ -6
"
10 3
In ,,calcite t~ - F O " ~ fluid - - \ ~
10 3
z~
O.~o.
f'y.~
.'f-o ''~-
~f
~, ~ - - ~ "
fh.~"
~ : 2~ _
--
~ , ~ z~ .~
~
.m
~o02
/ / ~
9
/ ~
~-~'~
o.~
~
....
. 2-o7--. . . . . . ~
o0.3
% -10
The isotopic composition of calcite deposited from the fluid can then be expressed as: (~calcite = ( ~ if l u i d + F 2
~ . ~ o , I
H2CO 3 as the dominant carbon species
-12 12
In ,,cos ~calcite
I
I
I
I
I
14
16
18
20
22
~L ~ 9 ~ 24
26
0 I,
28
30
32
fiis OsMowof Catcite (%o1
(4) In the case of Rayleigh degassing, the isotopic composition of a fluid can be quantified by the relationship (Rayleigh 1902):
-
if. %
~
-
t,,co
Fx ~ l . i d
_ 1)
-6
....
020
6~l,id
R f
6~l,,id+ In G " 103 In O[flui _co~d
(6)
Likewise, the effect of Rayleigh crystallization on the isotopic composition of the fluid can be described by: (~;iuid :
i' 3 fl.id + ln F 2 9 1 0 3 I n O~flui dcalcite
(7)
Thus, the isotopic composition of calcite deposited from the fluid due to the Rayleigh degassing-precipitation is represented by: 6c,aci, r = J~l~ia + l n
+(1 + i n
F 1 9 103 In ~.co~ fluid F2)"
10 3 In (~ fluid calcite
(8)
For oxygen, a hydrothermal fluid is generally dominated by HEO and the influence of both CO2 degassing and calcite precipitation on the oxygen isotope composition of the fluid is not significant. But for carbon, such an influence is considerable. The formation of calcite demands that a fluid must contain oxidized carbon species such as CO2 (aqueous), HzCO3, HCO~- and CO ] - . Apparently, both CO2 degassing and calcite precipitation from the fluid can significantly affect the carbon isotope composition of the fluid. Assuming that chemically the mol fraction of lost carbon from the fluid through the C O / degassing is identical to that through the calcite crystallization, i.e. F~ = 1 - Xco ~ F2 = G - X c o ~ = 1 - 2 X c o ~
(9) (10)
where Xco~ represents the mol fraction of carbon in the degassed CO2. Then F~ and Fz in Eqs. 4 and 8 can be
~
0.2o.~ ~ o
(5)
and R i are the final and initial 13C/12C or 1ao/~ 60 ratio of the fluid respectively; other terms are as same as defined for Eq. 1. Following the approximation by Zheng (1990) for the Rayleigh distillation equation, Eq. 5 becomes: where
HCO~as the dominant carbon species ~
0.1
oJ
Ry Ri
B
~
~~
-'-.Lo___.__ ~ ~ -- ~:-~'~ o
~
~
~-8
% 0.3
-10 -12
I
12
lt,
16
i
I
I
I
I
I
i
18
20
22
24
26
28
30
32
6~e Osnow of Calcite (%.)
Fig.
1 a, b. Effects o f C O 2 d e g a s s i n g f r o m a h y d r o t h e r m a l fluid o n
the isotopic compositions of both carbon and oxygen in precipitated calcite during batch (dashed curve) and Rayleigh (solid curve) degassing-precipitation at progressively changing temperatures and mol fractions of carbon in the degassed CO 2 (indicated by figures near the curves) for (a) H2CO3 as the dominant dissolved carbon species and (b) HCO~ as the dominant dissolved carbon species. The precipitation of calcite under conditions of a closed system is shown in the dot-and-dash curve. The diagrams quantitatively illustrate that the significant variations in 613C and 6180 values of hydrothermal calcite can be caused by simple CO2 degassing along with the change in temperature. The 613C and 6tso values of the fluid are taken as - 7.0%0(relative to PDB) and + 8.0%o(relative to SMOW) for both cases
replaced by Xco~ to obtain: i 6caZcite= 6/luid + Xco~ ' 103 In fX- fluid co2 calcite + (1 - 2 Xco2) " 103 In Ctflui d
(11)
for the batch degassing-precipitation model, and: r
-~-
6IZ.id + i
I n (1 - - X c o 2 )
9 10
3 In
co~ ~~ ,fluid
+ [1 + ln(1 - 2 Xco)] 9 10 3 1. ~vcalcite AA~ ~ f l u i d
(12)
for the Rayleigh degassing-precipitation model. The aqueous carbon species which may become important in carbon isotope fractionation for the present calculations are taken as CO2 (aqueous), H2CO ~ . Following the approximation by O h m o t o (1972) for the carbon isotope fractionation between H2CO3 (apparent) and CO2 (gas): 6 1 3 C n 2 c o 3 ,~ 6 1 3 C c o 2
(13)
248 two extreme cases can then be taken into account in order to perform specific calculation as detailed below.
H 2 C O 3 as the d o m i n a n t dissolved carbon ~pecies
If HCO 3 is present only in trace amount in the fluid, the dissolved carbon species in a low-pH fluid is represented by H2CO3. Then in the case of the batch model, the carbon and oxygen isotopic composition of calcite can be described by: t~ 13 Ccalcite = (~13ciftuid~-(1
- 2 X c o ) 9 103 In o~calciteCO 2
(14a)
(~18Ocalcite = t~lSOifluid-~-( l - - 2 X c o 2 ) " 103 i n ~calci~eH20
-(1 - X'co2)9 103 In ~co2 .~o
(14b)
where X'COz represents the tool fraction of oxygen in the degassed CO 2 . In the case of the Rayleigh model, we have: 0"13 Ccatcit e = ~13CZftuid+ [] +ln(1 - 2Xco)] " l 0 3 In ~co2Calcite
(15a) (~ 18 Oc~tcit e ---- i)lSo)t~idd-[1
+ l n ( l - 2Xco~) ] ' 103 In %~oCalcite
+ ( 1 - X'co) 9 1031n~co2 ,2o
(15 b)
H C O 3 as the d o m i n a n t dissolved carbon species
If the dominant carbon species in a hydrothermal fluid is HCO3, which could be due to the increase in pH of the fluid with decreasing temperature and partial pressure of CO2 (Helgeson 1969), the carbon isotopic variation in precipitated calcite due to CO 2 degassing is quite different from the case for H2CO3 as the dominant carbon species, because the carbon isotopic fractionation between calcite and HCO3 is significantly different from that between calcite and CO 2 (Robinson 1975; Ohmoto and Rye 1979). Then for the batch model we have: (~ 13 Ccah, i t e : ~ 13C~.luid-]-(l -- 2Xco2) 9 103 In %cog calcite
- X c o 2 9 103 In ~CO~n2o
(16a)
t~ is Ocatcit e = 6180~.id +(1 - 2 X ~ o ) 9 103 In %~o calcite
--Xco 2 9 103 In ~co:.2o
(16b)
Similarly for the Rayleigh model, it follows: ~13Ccalcit e = 3 1 3 c i f l u i d + l n ( l
- CO2 - - X c 0 2 ) 9 103 In ~.co~
+[1 +ln(1 - 2 X c o ) ] " 103 in (xcalcite.co~
(17a)
~5180 ,.,m ., = 0180ift,id + ln(1 -- Xc02) " 103 In ~CO2H20
+[1 +ln(l - 2 Xco~)] 9 103 In %~o~alcite
(17b)
By using carbon isotopic fractionation factors among calcite, CO2 and HCO3 (Ohmoto and Rye 1979) and oxygen isotope fractionation factors between calcite and H20 (O'Neil et al. 1969) and between CO2 and H2O (Truesdell 1974), and by assuming the initial carbon and oxygen isotopic compositions of the fluid and the mol fractions of carbon and oxygen in the degassed CO2, the
equations listed above enable calculation of the changes in both carbon and oxygen isotopic compositions of hydrothermal calcite as a function of progressively changing temperature. This is illustrated in Fig. 1 A for HECO 3 as the dominant carbon species and in Fig. 1 B for HCO 3 as the dominant carbon species respectively. The diagrams indicate that the batch and Rayleigh degassingprecipitation models give identical results when the mol fraction of carbon in the degassed CO z is less than about 0.2. When Xco2 is significantly greater than 0.2, the Rayleigh model yields a larger variation in 613C values of calcite than the batch model.
Application to the Kushikino gold-mining area
The Kushikino Mine is located in the Tertiary goldmining area of southern Kyushu, Japan. The ore deposits occur in andesitic volcanics in the form of fissure-filling epithermal veins, consisting of gold- and silver-bearing quartz and calcite with minor amounts of adularia, sericite, and sulfides. Quartz and calcite are dominant throughout the mineralization. Calcite crystallized along with milky quartz to form an intricately intergrown finegrained quartz-calcite assemblage interlain with quartz. The fine-grained quartz-calcite assemblage is closely associated with the gold-silver minerals such as electrum, argentite, naumannite, polybasite group minerals and pyrargyrite. The gold-silver minerals and the relatively Au-poor quartz-calcite assemblage precipitated repeatedly, forming a finely banded texture. The earlier precipitation was occassionally brecciated or fractured and was followed by later open-space fillings. Carbon and oxygen isotopic compositions were determined by Matsuhisa et al. (1985) for calcite and quartz from the Kushikino 1 and the Arakawa 4 veins in the mining area. The results are reproduced in Figs. 2 and 3. T h e 3 1 3 C - t ~ 1 8 0 trend of calcite in Fig. 2 was interpreted by the authors to show a decrease in temperature and a change in the dissolved carbon species of the hydrotherreal fluid. In this regard, a crystallization of calcite from a fluid under conditions of a closed system has been used by the authors to model the variations in both 513C and 6180 values. On the other hand, however, mixing of two fluids, both of which were saturated with calcite but which only had different temperatures, was assumed to cause supersaturation and thus precipitation of the calcite along with a decrease in temperature. Such a mixing process is hardly tenable, because the two fluids would have to have the same isotopic compositions of both carbon and oxygen. This is incompatible with the assumption that the rapid cooling of the fluid was due to mixing of a deep hydrothermal fluid and a low-temperature meteoric ground water. Furthermore, if the calcite could have crystallized due to mixing of the two fluids, the coexisting quartz would have to be respectively saturated in the two fluids as well. This seems unlikely. Instead of a mixing mechanism, the degassing of CO2 from a fluid is a plausible process to cause precipitation of calcite in the gold-mining area. The loss of CO 2 from the fluid leads to changes in the saturation state of the fluid
249 -6
o
HCOj dominant
18o0~ 1'0 14`0 ~ o --~-..o_t._!~i../_!20 9 0~/
9
~
o 10o c
. 160 ~
200 >qlJ~mlm~1B 0'm to
24,0 o ,
-12
0
HzC0 3 d o m i n a n t l
l
i
2
t+
6
5 '18 OSMOWof
l
i
8
10
12
Catcite (%,)
Fig. 2. A plot of bl~ versus 61sO for calcite from the Kushikino 1 (solid circle) and the Arakawa 4 (triangle) veins (data from Matsuhisa et al. 1985). Theoretical curves are calculated using the Rayleigh degassing-precipitation model and by assuming H2CO 3 and HCO~ as the dominant dissolved carbon species in the fluid, respectively; the temperature information is obtained from the quartz-calcite oxygen isotope geothermometer (for details see text). The 6180 value of the fluid is taken as -6.0%0, and the 6 1 3 C a s - 1 0 . 5 % for CO: degassing from the HzCO3-dominant fluid and as - 10.0%o for the case of HCO ~ as the dominant carbon species; the mol fraction of carbon in degassed CO 2 is taken as 0.05 and that of oxygen as 0.005 7
~ 5 t../
o
3
2
/
i
i
i
i
6
7
B
9
6 la 0SM0W o f O, uartz
10
(%,)
Fig. 3. A plot of 6180 values of calcite versus 61~O valuesof coexisting quartz for the Kushikino 1 (circle) and the Arakawa 4 (triangle) veins (data from Matsuhisa et al. 1985). Temperature scale is calculated by applying a combined fractionation curve of quartzwater (Clayton et al. 1972) and calcite-water (O'Neil et al. 1969)
with respect to calcite. In general, CO2 is lost, pH increases, and the fluid become more supersaturated with respect to calcite. As a result, the system responds by precipitating calcite. The CO 2 degassing could occur simultaneously with boiling of the fluid, but the fluid inclusion data do not indicate boiling (Izawa et al. 1981). However, the absence of evidence of hydrothermal boiling does not exclude a slow loss of CO 2 from the fluid, because threephase inclusions are expected to be completely absent when the CO2 content in epithermal fluids is less than 1.0 mol-% (Bodnar et al. 1985). Moreover, oxygen isotope compositions of the coexisting quartz and calcite pairs display a good positive correlation array with slope 1 on an 6180 vs. 6180 diagram (Fig. 3). This indicates
that the quartz and calcite were coprecipitated from a hydrothermal fluid in isotopic equilibrium with each other (Gregory 1986). Using a combination of experimentally determined calibration curves for the quartzwater system (Clayton et al. 1972) and for the calcitewater system (O'Neil et al. 1969), isotopic equilibrium temperatures are obtained between 140 ~ and 220 ~ For both samples, almost identical 6180 values between - 5%o and - 7.5%o are calculated as the isotopic compositions of the hydrothermal fluid which could have equilibrated with both quartz and calcite. Apparently, the quartz-calcite deposition did not take place as a result of a rapid cooling of the fluid due to mixing of the deep hydrothermal fluid and the low-temperature meteoric ground water. In case CO2 degassing is taken to be responsible for the precipitation of calcite in the Kushikino I and Arakawa 4 veins and the crystallization of quartz is ascribed to a decrease in fluid temperature during CO2 degassing, either a batch or a Rayleigh model can be used to simulate the covariation in both carbon and oxygen isotopic compositions of the calcite. The theoretical curves are calculated for the isotopic covariation as a function of temperature (Fig. 2). Both CO2 degassing and calcite precipitation are acting as removal mechanisms for dissolved carbon species from the fluid. It is assumed that the calcite crystallized in instantaneous equilibrium with the fluid, in which H 2 0 and H2CO 3 were oxygen and carbon reservoirs respectively, resulting in the positive correlation array between 613C and 6180 values of the calcite. If a constant 6180 value of -6.0%o (relative to SMOW) is taken for the hydrothermal fluid and the temperature range from 220 ~ to 140~ for the calcite precipitation, a 613C value of - 10.5%o (relative to PDB) is obtained for H 2 C O 3 in the fluid. With a loss of 5% carbon as degassed CO2 from the fluid, the data points with 6180 values lower than +6%0 can be well modelled by the combined degassing-precipitation equations, corresponding to a temperature change from 220 ~ to 160 ~ The data points with 61 sO values greater than + 6%~ show a much larger variation in di180 values than in 613C values. The nearly horizontal negative correlation array can be modelled by changing the dominant dissolved carbon species in the fluid from H2CO 3 to HCO3 at temperatures of about 160-140 ~ The curve fitting these data points is calculated by assuming that the isotopic composition of oxygen in the fluid remained unchanged but the isotopic composition of carbon was changed from the initial - 10.5%o to - 10.0%o due to the loss of CO2 which leads to a concomitant increase in pH of the fluid and a change in dissolved carbon species from H 2 C O 3 to H C O 3 9 The data points near 140~ to 160~ are scattered between the two theoretical curves, indicating crystallization in a transitional condition of mixedcarbon species (Matsuhisa et al. 1985). The present calculations show that the calcite in the gold-bearing hydrothermal veins of the Kushikino mining area crystallized in equilibrium with a fluid in which H2CO 3 was dominant at temperatures higher than 160~176 whereas HCO~- was dominant at lower temperatures due to a change in pH of the fluid during
250
CO2 degassing. Although the present results are not significantly different from the previous treatments by Matsuhisa et al. (1985) using the closed-system model, the introduction of the degassing model throws new light on the genesis of the calcite as well as the coexisting quartz in this mine.
Conclusions
Quantitative modelling has demonstrated how batch and Rayleigh distillation during CO2 degassing can yield significant variations in carbon and oxygen isotopic compositions of calcite precipitated from a hydrothermal fluid. A positive correlation array between ~13C and 6;80 values of the calcite can be best explained by the precipitation of calcite from a H2CO3-dominant fluid accompanied by a progressive decrease in temperature during CO2 degassing. Only when a strong fluid boiling occurs, can calcites precipitated from a HCO]--dominant fluid show the positive correlation between 613C and 6180 values. Such a strong fluid boiling is nevertheless more compatible with the case where the H z C O 3 is dominant in the fluid. Then a negative correlation could be yielded between the 6~3C and 61sO values of the calcites through the strong fluid boiling, as illustrated in Fig. 1 A by the curve with )(co 2= 0.4. However, loss of 0.4 mol of carbon through CO2 degassing seems unlikely to take place in the true epithermal system. When the dissolved carbon species in the fluid is dominated by H C O ~ , the precipitated calcite displays a smaller variation in 6~3C values with respect to 6~sO values under conditions of progressively decreasing temperature and CO2 degassing. The carbon and oxygen isotopic variations of calcite in the Kushikino gold-mining area can be modelled by a combined degassing-precipitation model as a function of progressively decreasing temperature. The modelling can be performed by taking H z C O 3 as the initially dominant dissolved carbon species in a hydrothermal fluid and then by taking H C O ; as the dominant carbon species for the later stage calcite. According to the above calculations, the coexisting calcite and quartz may have formed from the same fluid at temperatures 220 ~ 140~ the later stage calcite could crystallize at a temperature as low as I10~ Degassing of CO2 can be regarded as a more likely process responsible for genesis of calcite and quartz in this mining area, as compared to the mixing mechanism previously suggested by Matsuhisa et al. (1985).
Acknowledgements. I am grateful to an anonymous reviewer of the Mineralium Deposita for his constructive comments which led to appropriate improvements in this paper.
References Bodnar, R. J., Reynolds, T. J., Kuehn, C. A. : Fluid inclusion systematics in epithermal systems. In: Berger, B. R., Bethke, P. M. (eds.) Geology and Geochemistry of Epithermal Systems, pp.73-97. Reviews in Economic Geology, Vol. 2, Texas: Soc. Econ. Geol. (1985) Clayton, R. N., O'Neil, J. R., Mayeda, T. K. : Oxygen isotope exchange between quartz and water. J. Geophys. Res. 77:30573067 (1972) Gregory, R. T.: Oxygen isotopic systematics of quartz-magnetite pairs from Precambrian iron formations: Evidence for fluidrock interaction during diagenesis and metamorphism. In: Walter, J. V,, Wood, B. J. (eds.) Fluid-Rock Interaction during Metamorphism, pp. 132-153. Berlin Heidelberg New York: Springer 1986 Helgeson, H. C.: Thermodynamics of hydrothermal systems at elevated temperatures and pressure. Am. J. Sci. 267:729-804 (1969) Holland, H. D., Malinin, S. D. : The solubility and occurrence of non-ore minerals. In: Barnes, H. L. (ed.) Geochemistry of Hydrothermal Ore Deposits, 2nd Edition, pp. 461-508. New York: John Wiley and Sons (1979) Izawa, E., Yoshida, T., Sakai, T.: Fluid inclusion studies on the gold-sliver quartz veins at Kushikino, Kagoshima, Japan. Soc, Mining Geologists Japan Spec. Issue 10:25-34 (1981) Matsuhisa, Y., Morishita, Y., Sato, T.: Oxygen and carbon isotope variations in gold-bearing hydrothermal veins in the Kushikino mining area, southern Kyushu, Japan. Econ. Geol. 80:283-293 (1985) McCrea, J. M.: On the isotopic chemistry of carbonates and a paleo-temperature scale. J. Chem. Phys. 18:849-857 (1950) Nabelek, P. I., Labotka, T. C., O'Neil, J. R., Papike, J. J. : Contrasting fluid/rock interaction between the Notch Peak granitic intrusion and argillites and limestones in western Utah: evidence from stable isotope and phase assemblages. Contrib. Mineral. Petrol. 86:25-34 (1984) Ohmoto, H.: Systematics of sulfur and carbon isotopes in hydrothermal ore deposits. Econ. Geol. 67:551-578 (1972) Ohmoto, H., Rye, R. O. : Isotopes of sulfur and carbon. In : Barnes, H. L. (ed.) Geochemistry of Hydrothermal Ore Deposits, 2nd Edition, pp. 509-567. New York: John Wiley and Sons (1979) O'Neil, J. R., Clayton, R. N., Mayeda, T. K. : Oxygen isotope fractionation in divalent metal carbonates. J. Chem. Phys. 51 : 55475558 (1969) Rayleigh, L. : On the distillation of binary mixtures. Phil. Mug. Sci. 6.4:521-537 (1902) Robinson, B. W.: Carbon and oxygen isotopic equilibria in hydrothermal calcites. Geochem. J. 9:43-46 (1975) Truesdell, A. H.: Oxygen isotope activities and concentrations in aqueous salt solutions at elevated temperatures - Consequences for isotope geochemistry. Earth Planet. Sci. Len. 23:387-396 (1974) Valley, J. W. : Stable isotope geochemistry of metamorphic rocks. In: Valley, J. W, Taylor, H. P., Jr., O'Neil, J. R. O. (eds.) Stable Isotopes in High Temperature Geological Processes, pp. 445489. Reviews in Mineralogy Vol. 16, Washington: Miner. Soc. America (1986) Zheng, Y.-F. : The effect of Rayleigh degassing of magma on sulphur isotope composition: A quantitative evaluation. Terra Nova 2, 74-78 (1990) Received: January 20, 1990 Accepted: April 12, 1990
