• CN 62-1091/P
• ISSN 1001-8166
• 月刊 创刊于1986年

## 水热体系中Na2SO4/K2SO4溶解度的热力学计算

1. 贵州师范大学喀斯特生态文明研究中心，贵州 贵阳 550025

2. 中国科学院地球化学研究所地球内部物质高温高压院重点实验室，贵州 贵阳 550002

3. 中国科学院大学，北京 100039

## Thermodynamic Calculation of Solubility of Na2SO4/K2SO4 in Hydrothermal Fluids

Zhang Wei,1,2, Zhou Li,2, Tang Hongfeng2, Li Heping2, Wang Li2,3

1. Research Center of Karst Ecological Civilization, Guizhou Normal University, Guiyang 550025, China

2. Key Laboratory of High-Temperature and High-Pressure Study of the Earth’s Interior, Institute of Geochemistry, Chinese Academy of Sciences, Guiyang 550002, China

3. University of Chinese Academy of Sciences, Beijing 100039, China

 基金资助: 国家自然科学基金项目“独居石和磷钇矿在热液中溶解行为的实验研究：对U-Th-Pb年龄有效性以及稀土元素成矿的约束”.  编号：41773058贵州省科技厅—贵州师范大学联合基金项目“石阡县地热资源可持续利用研究”.  编号：黔科合LH字[2017]7339号

Corresponding authors: Zhou Li (1980-), male, Qianjiang City, Hubei Province, Associate professor. Research areas include experimental geochemistry at high-temperature and high pressure. E-mail：zhouli@vip.gyig.ac.cn

Received: 2018-11-08   Revised: 2019-03-19   Online: 2019-05-22

Abstract

Sulfate fluids are common fluids in nature, and their salinity studies can provide important information for the evolution of ore-forming fluids, migration and enrichment of ore-forming elements, and the classification of deposit types. Considerable research has been carried out to investigate the solubility of Na2SO4 and K2SO4 in hydrothermal fluids, however most of the literature reported experimental data were under saturated vapor pressure or the water supercritical region. A few data have been reported for the low temperature hydrothermal mineralization region. Thermodynamic model is a useful method to study the properties of hydrothermal geofluids, especially for mineral solubility. Pitzer interaction model is one of the most widely used model to calculate the thermodynamic properties of hydrothermal fluids, but few work have ever been carried out to calculate the solubility of sulfate at high temperature and pressure. With Pitzer specific interaction model, using the literature reported density data of Na2SO4 and K2SO4 solutions at high temperature and pressure, the pressure effect on Pitzer activity coefficient of sulfate and the standard partial molar volume change during sulfate dissolution process were evaluated and related parameters were obtained. The standard partial molar volumes of Na2SO4 and K2SO4 calculated with these parameters agreed well with those reported in the literature. Combined with the relevant parameters in the literature under saturated vapor pressure, a thermodynamic model for Na2SO4 and K2SO4 solubility calculation with temperature up to 250 ℃ and pressure up to 40 MPa was developed. The model gave very good agreement with the experimental solubility data. With this model, Na2SO4 and K2SO4 solubility was calculated at high temperature and pressure. The calculation results showed that pressure had a positive effect on both the average activity coefficient and solubility product of Na2SO4 and K2SO4, but the solubility of Na2SO4 and K2SO4 decreased with pressure due to the larger change of the average activity coefficient with pressure. And as the temperature increased, the degree of such reduction became larger. The results herein can provide instructions for the compositional analysis of sulfate fluid inclusions.

Keywords： Sulfate fluids inclusion ; Solubility ; Thermodynamic model ; Pitzer model.

Zhang Wei. Thermodynamic Calculation of Solubility of Na2SO4/K2SO4 in Hydrothermal Fluids. Advances in Earth Science[J], 2019, 34(4): 414-423 doi:10.11867/j.issn.1001-8166.2019.04.0414

## 1 引 言

### 2 理论背景及计算方法

$Na2SO4(s)↔2Na++SO42-$
$K2SO4(s)↔2K++SO42-$

$KNa2SO4=aNa+2∙aSO42-=mNaγNa2∙(mSO42-γSO42-)$
$KK2SO4=aK+2∙aSO42-=mKγK2∙(mSO42-γSO42-)$

Pitzer[28]模型被广泛应用于水热体系中物质活度系数的计算[29]，被PHREEQC[30]，GEM-Selektor[31]和ScaleSoftPitzer[32]等世界知名的地球化学计算软件所采用。Greenberg等[26]利用Pitzer模型建立了0~250 oC，饱和蒸气压下Na2SO4和K2SO4的溶解度计算模型。但是，Shi等[33]发现不考虑压力情况下重晶石的溶解度要比实验结果低27$%$，为了准确描述矿物在高温高压下的溶解度就必须考虑压力对活度系数和溶解平衡常数的影响[34]

### 2.1　压力对溶解平衡常数和活度系数的影响

$lnKspT,P=lnKspT,P0-∆rV0RTP-P0$

$∆rV0=V0Na2SO4,aq-V0(Na2SO4,s)$

$lnγP=lnγPsat-AϕP-AϕPsat×I1+bI+2bln(1+bI)+2mβ0v+β1v(2/α2I)1-1+αI×exp-αIP-Psat+3m2Cv(P-Psat)$ ,

### 2.2　高温高压下溶解平衡常数和活度系数的计算

Pitzer模型表示的是物种的吉布斯自由能，经过适当的微分可以得到焓、热容和体积（密度）等性质。相反也可以利用热容、焓和密度等数据来评价温度、压力对模型参数的影响[41]。电解质溶液的密度可以用来计算压力对电解质的偏摩尔体积和Pitzer模型表示的溶质的活度系数的影响[20,40]

$V∅=1m(1000+mMsρsol-1000ρ0)$

$式中:V∅$表示电解质的表观摩尔体积（apparent molar volume，cm3/mol）；m是电解质的质量摩尔浓度(mol/kg H2O)；Ms是电解质的摩尔质量(g/mol)；$ρsol$$ρ0$分别为目标温度压力下溶液和纯水的密度(g/cm3)；其中纯水的密度依据IAPWS97方程来计算[42]

$V∅=V∅0+v|z+z-|Avln1+bI/2b+2v+v-mRT(Bv+v+z+mCv)$
$Bv=β0v+β1v(2/α2I)1-1+αI×exp-αI$

$V∅,mr=V∅0+v|z+z-|Avln1+bImr/2b+2v+v-mrRT(β0v+v+z+mrCv)$

$V∅,m=V∅,mr+v|z+z-|Avln1+bIm/2b-ln1+bImr/2b+2v+v-RTβ0vm-mr+z+v+Cvm2-mr2$

$1000+mMsmρsol-1000mρ0$
$=Vmrmr-1000mrρ0+v|z+z-|Avln1+bIm/2b$
$-ln1+bImr/2b+2v+v-RTβ0vm-mr$
$+z+v+Cvm2-mr2$

$V∅0=Vmrmr-1000mrρ0-v|z+z-|Avln1+bImr/2b-2v+v-mrRT(β0v+v+z+mrCv)$

Pitzer等[40]利用水热体系中NaCl溶液的热容和焓等热力学数据确定了0~300 oC，0.1~100.0 MPa，0~6 m范围内NaCl活度系数。Møller[47]确定了0~250 oC，饱和蒸气压下NaCl的活度系数参数；Mao等[25]利用Pitzer模型确定了水热体系中NaCl标准偏摩尔体积和压力对Pitzer参数影响的参数，将二者结合利用公式（7），我们计算了水热体系中NaCl的平均活度系数（图1）。从图1可看出，结合饱和蒸汽压下参数与压力积分项计算出来的不同浓度下NaCl的平均活度系数和Pitzer等[40]完整模型参数计算出来的结果吻合得很好，证明了该方法和计算过程的可行性。从图1还可看出，在NaCl溶液体系中，压力对活度系数有较大的影响，活度系数随着压力的升高而增大。

### 图1

Fig. 1   The model calculated NaCl mean activity coefficient at 25 oC as a function of concentration under different pressure

## 3 高温高压条件下Na2SO4和K2SO4溶解度的计算

Table 1  Summary of density data of Na2SO4-H2O system

25.0~300.09.0~30.0[48]
25.0~200.01.1~68.6[49]
5.0~60.00.1[50]
20.0~200.02.0~10.0[51]
0~50.00.1~80.0[52]
50.0~200.02.0[53]
25.0~100.00.6[54]
25.0~250.00.1~40[55]
30.0~300.02.4~39.8[56]
5.0~100.00.1[57]

Table 2  Summary of density data of K2SO4-H2O system

25.0~300.09.0~30.0[48]
50.0~200.02.0[53]
25.0~100.00.6[54]
0~90.00.1[57]
20.0~300.05.0~40.0[58]

$V∅=V∅0+6Avln1+bI/2b+4mRT(β0v+2mCv)$
$V∅0=V(mr)mr-1000mrρ0-6Avln1+bImr/2b-4RT(mrβ0v+2Cvmr2)$

$V(mr)=a1+a2T+a3T2+a4T3+(a5+a6T+a7T2)P$
$β0v=a8+a9T-227+a10T$
$Cv=a11+a12T-227+a13T$

### 图2

Fig.2   Plot of density deviations between model calculation and literature data of Na2SO4 + H2O (a)and K2SO4 + H2O (b) system

Table 3  Parameters for the volumetric properties calculation of Na2SO4 and K2SO4

mr1.51.0
a19.69382688×1021.10686112×103
a22.83105828×10-1-5.98247606×10-1
a3-9.00393580×10-45.96259241×10-4
a42.57624184×10-62.39804594×10-6
a5-1.43199512×100-2.71085716×100
a67.37402186×10-31.41493666×10-2
a7-1.17539062×10-5-2.08587301×10-5
a8-4.86834961×10-45.43596750×10-3
a98.87142143×10-24.61245819×10-3
a102.75966178×10-7-1.21299746×10-5
a112.13350752×10-4-5.55380818×10-3
a12-1.36823713×10-26.41685688×10-2
a13-2.95936943×10-71.44544684×10-5

### 图3

Fig.3   The standard partial molar volumes of Na2SO4(a) and K2SO4(b)against temperature and pressure

Greenberg等[26]确定了温度0~250 oC，饱和蒸汽压下Na2SO4和K2SO4活度系数随温度变化的参数。依据公式（7）和以上获得的参数，我们可以评价压力对Na2SO4和K2SO4活度系数的影响（图4）。从图4中可以看出，压力的增大会导致Na2SO4和K2SO4的平均活度系数的增大，并且随着溶液浓度的增大和温度的增高，这种增加的趋势更明显。对于Na2SO4体系，250 oC和30 MPa，1.5 m时Na2SO4的平均活度系数比相同温度浓度下，饱和蒸汽压时增加了0.02（图4a）；K2SO4体系中，250 oC和30 MPa， 0.5 m时的平均活度系数比饱和蒸汽压时增加了0.03（图4b）。

### 图4

Fig. 4   The model calculated Na2SO4(a) and K2SO4(b) mean activity coefficient as a function of concentration at different temperature and pressure

Table 4  Comparison between model calculated Na2SO4 and K2SO4 with experimental data

Na2SO4

P=Psat

60.03.153.34[16,27]
100.02.993.04
120.02.952.94
140.72.962.89
200.73.153.16

K2SO4

P=Psat

50.000.950.94[15]
80.001.231.20
100.01.401.33
150.01.691.57
190.01.971.71

Na2SO4

P= 25 MPa

50.03.533.19[60]
100.02.342.74
150.02.332.53
180.02.392.48
200.02.412.46
220.02.372.40

5和图6分别表示水热体系模型计算的Na2SO4和K2SO4的溶度积的自然对数和溶解度。从图5a中可以看出，压力能够促进电解质溶解平衡常数的增加，在150 oC，压力为饱和蒸气压，10 MPa和30 MPa时，Na2SO4溶解平衡常数的自然对数分别为-2.71，-2.65和-2.55；相同条件下K2SO4溶度积的自然对数分别为-3.50，-3.45和-3.36（图6a）。但是压力对电解质溶解度的影响，却恰好与之相反。从图5b中可以看出，随着压力的增大，电解质的溶解度却迅速降低。150 oC时，随着压力的增加，Na2SO4的溶解度从饱和蒸气压的2.89 mol/kg降低到10 MPa的2.73 mol/kg，最后降低到30 MPa的2.48 mol/kg。在相同温压条件下，K2SO4的溶解度分别为1.57，1.26和1.00 mol/kg。随着温度的升高，溶解度的减少量更大（图6b）。这是因为压力对Na2SO4和K2SO4的溶度积和平均活度系数都有正向的促进作用，但是压力对Na2SO4和K2SO4平均活度系数的影响更明显，结果就导致了Na2SO4和K2SO4溶解度的降低。

### 图5

Fig.5   The natural logarithm of Na2SO4 solubility product (a) and solubility (b) at different temperature and pressure

### 图6

Fig.6   The natural logarithm of K2SO4 solubility product (a) and solubility (b) at different temperature and pressure

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