PROCEEDINGS, Twenty-Eighth Workshop on Geothermal Reservoir Engineering Stanford University, Stanford, California, January 27-29, 2003 SGP-TR-173
A THERMODYNAMIC ASSESSMENT OF DISSOCIATION CONSTANT OF WATER Mahendra P. Verma Geotermia, Instituto de Investigaciones Electricas Av. Reforma 113, Col. Palmira, Apdo. 1-475 Cuernavaca, Morelos, 62000, México e-mail:
[email protected]
ABSTRACT A thermodynamically consistent formulation for the dissociation constant of water (KW) is presented. Based on experimental data along the vapor-water saturation, the following equation for KW is derived for temperature 0 to 380ºC and pressure 1 to 250 bar log KW = −10.02620 + 559 − 9581 − 9.678282 × 10 −3 P + T 5189145 2 + 0.176057 P + 1.805175 × 10 −3 P 2 T T
INTRODUCTION Knowledge of the dissociation constant of water (KW) at a specified temperature and pressure is fundamental in dealing the chemical modeling of aqueous systems in the laboratory or in nature. Working Group III of the International Association for Properties of Water-Steam (IAPWS) has compiled all the existing experimental values and established the representative values of KW for 01000ºC and 1-10,000 bar, which were published by Marshall and Franck (1981). The values are widely accepted and reproduction in handbooks on physicalchemical properties of water (e.g. Meyer et al., 1993; Lide and Frederikes, 1998). The present study scrutinizes the values of KW on the basis of basic laws of thermodynamics. Similarly, a formulation is presented to derive the consistent values of KW in the range approximately 0-380ºC and 1-250 bar. A STATE FUNCTION A thermodynamic variable (for example, pressure, temperature, volume, Gibbs free energy, internal energy, enthalpy or entropy represented by P, T, V, G, U, H or S, respectively), which does not depend on the past history of the substance or on the path it has followed in reaching a given state, is known as state function or point function (Smith and van Ness, 1975). A state function should be single valued and continuously differentiable unless there is a phase transition (Smith and van Ness, 1975; Chatterjee, 1991). On fixing the values of any two state functions (for example T and P), the values of all the other state
functions are uniquely defined. Similarly, the equilibrium constant (K) for a chemical reaction is also a state function and is expressed as log K = =
1 − ∆G FT , P 2.303 RT 1 − ∆H FT , P 2.303 RT
T ,P 1 − ∆S + 2.303 RT
…1 where R is gas constant, P is pressure and T is absolute temperature. F stands for formation. At low temperatures, the values of Gibbs free energy (∆GFT,P) or enthalpy (∆HFT,P) and entropy (∆ST,P) for the first order of approximation are constant and the variation of log K with 1/T is a straight line. There could be positive or negative deviation from the linear trend at higher temperatures, but the function should always be single valued (i.e. the trend of log K may be asymptotic at higher temperatures). Figure 1(a) shows a schematic diagram for the variation of log K with the inverse of T at constant P. Let us consider two other behaviors of log K as shown with curves I and II in Figure 1(b). In case I there are two values of log K at a given T, whereas in case II there are two values of T for a value of log K. Thus K(T) in case I and T(K) in case II are not single valued functions. Similarly, dT dK = ∞ in case I and dK dT = ∞ in case II. It means that T or K are not continuous differentiable. In other word either T or K is not thermodynamic state function in the respective cases. Figure 1(c) presents the behavior of Log K with 1/T at two pressures P1 and P2. The functions are crossing at temperature T1. Then at T=T1, dP dK = ∞ and
dP dT = ∞ . It means that P is not a state function. It is well known that T, P and K are state functions. Therefore, the behaviors of log K presented in Figure 1(b) and 1(c) are against the basic laws of thermodynamics. Thus the permissible behaviors of log K with 1/T are as given in Figure 1(a). This is valid for any property (for example, viscosity, thermal conductivity, etc.) of a substance or system, which does not depend on the past history or on the path to reach a given state (Verma, 2002a).
∆G=-∞
+ve
K2
a
-ve deviation
log K
0
-ve
con s
∆G=0
1/T ∆G
ve =+
K3
tan t
I T2
0 con
II
nt sta
T3
T1
1/T c P2
log K
∆G =
May be asymptotic at high T
b
K1
log K
+ve deviation
-ve ∆G=+∞
P1
0
T1
1/T
Figure. 1. A schematic diagram for the variation of log K with inverse of absolute temperature (after Verma, 2002a).
DISSOCIATION CONSTANT OF WATER The most accepted values of dissociation constant of water (KW) from Marshall and Franck (1981) are plotted in Figure 2. Figure 2(a) shows the temperature behavior of KW. It can be observed that KW increases with T and P. It is well known that T and P produce, in general, adverse effect on the physical and chemical properties of a substance. The increase in KW with T and P does not seem reasonable. Accordingly, there will be an increase in the concentration of ionized water as the water is buried deeper into the Earth. The chemical reactivity of water should increase with P. It looks quite unreasonable. Additionally, KW has irregular behavior with T. For example, there is a drastic drop around 400ºC for the curve corresponding to 250 bar. Similarly, there are two temperatures for a given values of KW. It means that Kw or T is not a state function. Figure 2(b) shows the pressure dependence of KW. The curves corresponding to different temperatures are crossing each other. This is again against the basic laws of thermodynamics. Thus the values of KW reported by Marshall and Franck (1981) are violating fundamental laws of thermodynamics and are unreliable. Clever (1968) reviewed the earlier experimental data and techniques for the determination of KW. The mostly used techniques are electrical conductivity, emf of cells without transference and thermal measurements. In all the methods, a salt solution like
of KCl is used and the measured properties are extrapolated to the zero salt effect, as it is difficult to measure the properties of pure water. Sweeton et al. (1974) performed a comprehensive study for the determination of KW along the water-vapor saturation and their values have been used for geochemical modeling at along the water-vapor saturation up to 300ºC. Since an aqueous solution in lab or in nature is not always along the saturation curve (Verma, 2002), a numerical formulation is carried out to derive the pressure and temperature dependence of K W. The equation for log KW as a function of temperature and pressure is written as
log KW = a + b / T + c P + d / T 2 + e P / T + f P 2 …2 where the values for the constants are a=-10.02620, b=559.9581, c=-9.678282 10-3, d=518914.5 and f=1.805175 10-5. Figure 3 shows the values of log KW calculated by eq. (2) together with the experimental values (Sweeton et al., 1974). The calculated and experimental values of KW are in good agreement. Mathematically, a quadratic function (for example Y= a+bX+CX2) must always have one value for two values of independent variable (X). Thus Y cannot be single valued for all the range of X. Since KW is a state function, the extrapolation of KW is only permissible up to approximately 380ºC and 250 bar. The calculated values of KW by eq. (2) are tabulated in Table 1. At P=1 bar, the feasible temperature range is 0 to 100ºC. If T is higher than 100ºC, the water
-5
-5
Increasing with T & P
-10
-15
log KW
log KW
-10
250 bar 500 750 1000 1500 2000 3000 5000 10000
-20
-15 T=
Crossing -20
Anomalous
0 ºC
T=
25
T= 100
T= 300
T= 500
T= 800
T=1000
-25
-25
0
500
0
1000
2000
4000
T (ºC)
6000
8000
10000
P (bar)
Figure. 2. Temperature and pressure dependence of log KW for the values reported by Marshall and Franck (1981).
-11
log K
-12
-13
Experimental
-14
C alculated
-15 1.5
2
2.5
3
3.5
1000/T (in K) Figure. 3. A comparison between the experimental (sweeton et al, 1974) and theoretical values of log KW along the water-vapor saturation curve.
300
-11
200
100
0
b
a
-12
-13
-12
Log KW
log KW
-11
-14
P=
-15
1 bar
P= 10
P= 50
P=100
P=150
P=200
-13
-14 T= 25ºC 100 200 300
-15
P=250
-16
50 150 250 350
-16
1.5
2
2.5
1000/T (in K)
3
3.5
0
50
100
150
200
P (bar)
Figure. 4. Temperature and pressure dependence of log KW for the values derived in the present study.
250
Table 1: Logarithm (base 10) of the dissociation constant of water (KW). T (ºC) 0 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 380
1 -14.940 -13.995 -13.272 -12.708 -12.262
10 -15.020 -14.075 -13.352 -12.789 -12.343 -11.984 -11.692 -11.452
50 -15.338 -14.395 -13.674 -13.112 -12.668 -12.310 -12.019 -11.780 -11.581 -11.414 -11.274
will be in vapor phase at P=1 bar. Similarly, at T=125ºC, the minimum pressure for the existence of water is 4.76 bar. Figure 4 shows the temperature and pressure dependence of KW for all the possible values from the vapor-water saturation to the maximum values of T and P. The behavior is consistent with the basic laws of thermodynamics. CONCLUSIONS The values of the dissociation constant of water for temperature 0 to 380ºC and pressure 1 to 250 bar, presented here are thermodynamically consistent and are in good agreement with the experimental values reported in the literature7. There are practically very little efforts to improve the analytical quality of experimental values of KW since 1974. Therefore the values of KW at higher temperatures and pressures need an evaluation for analytical uncertainty and its propagation in different formulation to study the temperature and pressure dependence of KW. REFERENCES Chatterjee, N.D. (1991). “Applied mineralogical thermodynamics”, Springer-Verlag, New York, 320pp. Clever, H.L. (1968). “The Ion Product Constant of Water”, Journal of Chemical Education, 45, pp. 231235. Lide, D.R. and Frederikse, H.P.R. (eds.) (1998). “CRC Handbook of Chemistry and Physics”, CRC Press, London, pp. 8.56-8.57.
P (bar) 100 -15.654 -14.714 -13.995 -13.436 -12.992 -12.636 -12.347 -12.108 -11.911 -11.745 -11.606 -11.487 -11.385
150 -15.880 -14.943 -14.226 -13.669 -13.227 -12.872 -12.584 -12.347 -12.150 -11.986 -11.847 -11.729 -11.628 -11.542
200 -16.016 -15.081 -14.367 -13.811 -13.372 -13.018 -12.731 -12.495 -12.300 -12.136 -11.998 -11.881 -11.781 -11.695 -11.621
250 -16.061 -15.129 -14.418 -13.864 -13.426 -13.074 -12.788 -12.554 -12.359 -12.196 -12.059 -11.943 -11.843 -11.758 -11.685 -11.621 -11.609
Marshall, W.L. and Franck, E.U. 1981. “Ion Product of Water Substance, 0-1000ºC, 1-10,000 Bars New International Formulation and Its Background”, Journal of Physical Chemical Reference Data, 10, pp. 295-304. Meyer, C.A., McClintock, R.B., Silvestri, G.J. and Spencer, R.C. (1983). “Steam Tables: Thermodynamic and Transport Properties of Steam “, ASME Press, New York, pp. 147-152. Smith, J.M. and van Ness, H.C. (1974). “Introduction to chemical thermodynamics” McGraw-Hill Kogakusha, ltd., Tokyo, 632pp. Sweeton, F.H., Mesmer, R.E. and Baes, C.F. (1974). “Acidity Measurements at Elevated Temperatures. VII. Dissociation of Water”, Journal of Solution Chemistry, 3, pp. 191-214. Verma, M.P. (2002a). “Geochemical techniques in geothermal development”, In: D. Chandrasekharam and J. Bundschuh (eds.) Geothermal Energy Resources for Developing Countries, Swets & Zeitlinger Publishers, The Netherlands, pp. 225-251. Verma, M.P. (2002b). “A numerical simulation of H2O-CO2 heating in a geothermal reservoir”. Proceedings XXVII Workshop Geothermal Reservoir Engineering, Stanford University, Stanford, CA, pp.217-222.
