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A Thermodynamic Assessment of Dissociation Constant of Water

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A Thermodynamic Assessment of Dissociation Constant of Water 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] functions are uniquely defined. Similarly, the ABSTRACT equilibrium constant (K) for a chemical reaction is also a state function and is expressed as A thermodynamically consistent formulation for the − ∆G T ,P = 1 F dissociation constant of water (KW) is presented. log K Based on experimental data along the vapor-water 2.303 RT T ,P − ∆ T ,P saturation, the following equation for KW is derived 1 − ∆H 1 S = F + for temperature 0 to 380ºC and pressure 1 to 250 bar 2.303 RT 2.303 RT …1 = − + 559 − 9581 − × −3 + log KW 10.02620 9.678282 10 P where R is gas constant, P is pressure and T is T − absolute temperature. F stands for formation. At low 5189145 + 0.176057 P +1.805175×10 3 P2 T 2 T temperatures, the values of Gibbs free energy T,P T,P T,P (∆GF ) or enthalpy (∆HF ) and entropy (∆S ) for INTRODUCTION the first order of approximation are constant and the variation of log K with 1/T is a straight line. There Knowledge of the dissociation constant of water (KW) at a specified temperature and pressure is could be positive or negative deviation from the fundamental in dealing the chemical modeling of linear trend at higher temperatures, but the function aqueous systems in the laboratory or in nature. should always be single valued (i.e. the trend of log Working Group III of the International Association K may be asymptotic at higher temperatures). Figure for Properties of Water-Steam (IAPWS) has 1(a) shows a schematic diagram for the variation of compiled all the existing experimental values and log K with the inverse of T at constant P. Let us consider two other behaviors of log K as established the representative values of KW for 0- 1000ºC and 1-10,000 bar, which were published by shown with curves I and II in Figure 1(b). In case I Marshall and Franck (1981). The values are widely there are two values of log K at a given T, whereas in accepted and reproduction in handbooks on physical- case II there are two values of T for a value of log K. chemical properties of water (e.g. Meyer et al., 1993; Thus K(T) in case I and T(K) in case II are not single Lide and Frederikes, 1998). The present study valued functions. Similarly, dT dK = ∞ in case I scrutinizes the values of KW on the basis of basic and dK dT = ∞ in case II. It means that T or K are laws of thermodynamics. Similarly, a formulation is not continuous differentiable. In other word either T presented to derive the consistent values of K in the W or K is not thermodynamic state function in the range approximately 0-380ºC and 1-250 bar. respective cases. Figure 1(c) presents the behavior of Log K with 1/T A STATE FUNCTION at two pressures P1 and P2. The functions are crossing A thermodynamic variable (for example, pressure, at temperature T1. Then at T=T1, dP dK = ∞ and temperature, volume, Gibbs free energy, internal dP dT = ∞ . It means that P is not a state function. energy, enthalpy or entropy represented by P, T, V, G, U, H or S, respectively), which does not depend It is well known that T, P and K are state functions. on the past history of the substance or on the path it Therefore, the behaviors of log K presented in Figure has followed in reaching a given state, is known as 1(b) and 1(c) are against the basic laws of state function or point function (Smith and van Ness, thermodynamics. Thus the permissible behaviors of 1975). A state function should be single valued and log K with 1/T are as given in Figure 1(a). This is continuously differentiable unless there is a phase valid for any property (for example, viscosity, transition (Smith and van Ness, 1975; Chatterjee, thermal conductivity, etc.) of a substance or system, 1991). On fixing the values of any two state functions which does not depend on the past history or on the (for example T and P), the values of all the other state path to reach a given state (Verma, 2002a). ∆G=-∞ +ve K2 +ve deviation a b II K1 -ve deviation K ∆ g May be G K 3 I = o asymptotic - l ve at high T con sta nt T2 TT3 1 K ∆G=0 0 g 0 1/T o t l 1/T n sta on e c c +v ∆G= P2 K g o l P1 -ve T 0 1 ∆G=+∞ 1/T Figure. 1. A schematic diagram for the variation of log K with inverse of absolute temperature (after Verma, 2002a). of KCl is used and the measured properties are DISSOCIATION CONSTANT OF WATER extrapolated to the zero salt effect, as it is difficult to measure the properties of pure water. Sweeton et al. The most accepted values of dissociation constant of (1974) performed a comprehensive study for the water (KW) from Marshall and Franck (1981) are determination of KW along the water-vapor saturation plotted in Figure 2. Figure 2(a) shows the and their values have been used for geochemical temperature behavior of KW. It can be observed that modeling at along the water-vapor saturation up to KW increases with T and P. It is well known that T 300ºC. Since an aqueous solution in lab or in nature and P produce, in general, adverse effect on the is not always along the saturation curve (Verma, physical and chemical properties of a substance. The 2002), a numerical formulation is carried out to increase in KW with T and P does not seem derive the pressure and temperature dependence of reasonable. Accordingly, there will be an increase in K . the concentration of ionized water as the water is W The equation for log KW as a function of temperature buried deeper into the Earth. The chemical reactivity and pressure is written as of water should increase with P. It looks quite unreasonable. = + + + 2 + + 2 log KW a b /T c P d /T e P /T f P …2 Additionally, KW has irregular behavior with T. For example, there is a drastic drop around 400ºC for the where the values for the constants are a=-10.02620, curve corresponding to 250 bar. Similarly, there are b=559.9581, c=-9.678282 10-3, d=518914.5 and -5. two temperatures for a given values of KW. It means f=1.805175 10 that Kw or T is not a state function. Figure 3 shows the values of log KW calculated by eq. Figure 2(b) shows the pressure dependence of KW. (2) together with the experimental values (Sweeton et The curves corresponding to different temperatures al., 1974). The calculated and experimental values of are crossing each other. This is again against the KW are in good agreement. basic laws of thermodynamics. Thus the values of KW Mathematically, a quadratic function (for example reported by Marshall and Franck (1981) are violating Y= a+bX+CX2) must always have one value for two fundamental laws of thermodynamics and are values of independent variable (X). Thus Y cannot be unreliable. single valued for all the range of X. Since KW is a Clever (1968) reviewed the earlier experimental data state function, the extrapolation of KW is only and techniques for the determination of KW. The permissible up to approximately 380ºC and 250 bar. mostly used techniques are electrical conductivity, The calculated values of KW by eq. (2) are tabulated emf of cells without transference and thermal in Table 1. At P=1 bar, the feasible temperature range measurements. In all the methods, a salt solution like is 0 to 100ºC. If T is higher than 100ºC, the water -5 -5 Increasing with T & P -10 -10 W W -15 250 bar -15 500 T= 0 ºC T= 25 log K log K 750 Crossing T= 100 T= 300 1000 1500 T= 500 T= 800 -20 2000 -20 3000 Anomalous T=1000 5000 10000 -25 -25 0 500 1000 0 2000 4000 6000 8000 10000 T (ºC) P (bar) Figure. 2. Temperature and pressure dependence of log KW for the values reported by Marshall and Franck (1981). -11 -12 -13 log K log -14 Experimental Calculated -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. 300200 100 0 -11 -11 a b -12 -12 -13 W -13 W log K log -14 Log K -14 P= 1 bar P= 10 P= 50 P=100 T= 25ºC 50 -15 P=150 P=200 -15 100 150 P=250 200 250 300 350 -16 -16 1.5 2 2.5 3 3.5 0 50 100 150 200 250 1000/T (in K) P (bar) Figure. 4. Temperature and pressure dependence of log KW for the values derived in the present study. Table 1: Logarithm (base 10) of the dissociation constant of water (KW). P (bar) T (ºC) 1 10 50 100 150 200 250 0 -14.940 -15.020 -15.338 -15.654 -15.880 -16.016 -16.061 25 -13.995 -14.075 -14.395 -14.714 -14.943 -15.081 -15.129 50 -13.272 -13.352 -13.674 -13.995 -14.226 -14.367 -14.418 75 -12.708 -12.789 -13.112 -13.436 -13.669 -13.811 -13.864 100 -12.262 -12.343 -12.668 -12.992 -13.227 -13.372 -13.426 125 -11.984 -12.310 -12.636 -12.872 -13.018 -13.074 150 -11.692 -12.019 -12.347 -12.584 -12.731 -12.788 175 -11.452 -11.780 -12.108 -12.347 -12.495 -12.554 200 -11.581 -11.911 -12.150 -12.300 -12.359 225 -11.414 -11.745 -11.986 -12.136 -12.196 250 -11.274 -11.606 -11.847 -11.998 -12.059 275 -11.487 -11.729 -11.881 -11.943 300 -11.385 -11.628 -11.781 -11.843 325 -11.542 -11.695 -11.758 350 -11.621 -11.685 375 -11.621 380 -11.609 will be in vapor phase at P=1 bar.
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