Enthalpies of Dilution of NaOH, KOH, and HCl and Thermodynamic

Aug 26, 2000 - The program uses the values of log K and ΔH° at zero ionic strength and the γ values to calculate the ... NaOH (T = 370 °C and P = ...
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3508

Ind. Eng. Chem. Res. 2000, 39, 3508-3515

Enthalpies of Dilution of NaOH, KOH, and HCl and Thermodynamic Quantities for the Formation of These Species from Their Constituent Ions in Aqueous Solution from 300 to 370 °C Saowarux Fuangswasdi, John L. Oscarson,* Li Zhou, and Reed M. Izatt Departments of Chemistry and Biochemistry and of Chemical Engineering, Brigham Young University, Provo, Utah 84602

Enthalpies of dilution measured using isothermal flow calorimetry are reported for aqueous solutions of NaOH, KOH, and HCl at 370 °C and 24.6 MPa. Previously collected data at 300, 325, and 350 °C were included with the new results when fitting the parameters for the Pitzer excess Gibbs energy ion-interaction equation. Equilibrium constants, enthalpy changes, entropy changes, and heat capacity changes for the association of Na+ and K+ with OH- ion and of H+ with Cl- ion were estimated from the heat data. Both enthalpy and entropy changes are positive for all of the systems and increase dramatically as the temperature increases. The resulting equilibrium constants show significant, but smaller, increases with temperature. 1. Introduction

Table 1. Summary of the Experimental Conditions

There is a need to obtain thermodynamic properties of aqueous electrolyte solutions and of reactions occurring in these solutions at high temperatures in order to understand various geological and industrial processes. While the chemistry of aqueous electrolyte solutions has been thoroughly studied near room temperature, few investigations have been reported for temperatures above 250 °C. The necessity for accurate thermodynamic data at elevated temperatures becomes apparent when one recognizes that the behavior of chemical species toward the solvent and toward each other differs markedly from that at room temperature. The current paper is a continuation of our investigation of ion-ion interactions in aqueous solutions at temperatures in the subcritical temperature region of water.1-5 The objective of the study is to provide equilibrium constants K, enthalpy changes ∆H°, entropy changes ∆S°, and heat capacity changes ∆C°p for the formation from their respective ions of NaOH, KOH, and HCl and activity coefficient γ parameters of the species in aqueous solutions from 300 to 370 °C:

T (°C)

P (MPa)

300 300 325 325 350 370

9.3 11.0 12.4 14.8 17.6 24.6

Na+ + OH- h NaOH(aq)

(1)

K+ + OH- h KOH(aq)

(2)

H+ + Cl- h HCl(aq)

(3)

These thermodynamic values have been estimated by analyzing calorimetric data for the heats of dilution of appropriate aqueous solutions at 300, 325, 350, and 370 °C at pressures above the saturation pressure of water at the respective temperatures. 2. Experimental Section 2.1. Materials. The solutions were prepared by dilution of concentrated NaOH (Fisher Scientific, certified ACS), KOH (Mallinckrodt AR), and HCl (Fisher * To whom correspondence should be addressed. Telephone: (801) 378-6243. Fax: (801) 378-7799. E-mail: [email protected].

NaOH

initial molality, mi KOH

HCl

0.5164, 1.9323 0.2324, 0.5191

0.5131

0.2471, 0.4955 0.2508, 0.5029 0.0786, 0.3493

0.2522, 0.5067 0.1007

0.5164, 1.9323 0.2207, 0.4411 0.0965, 0.3933

Scientific, certified concentrate) in distilled water, which had been deionized using a fixed-bed cation-exchange column and boiled for 30 min. The solutions were standardized after treatment with argon for 20 min to remove excess dissolved oxygen and CO2. The molar concentrations obtained were converted to molalities using the density data of NaOH and HCl solutions6,7 and the density of KOH solutions8 at 25 °C and 0.1 MPa with the assumption that the compressibility of the solution is the same as that of pure water. 2.2. Procedure. The calorimetric measurements were performed using two high-temperature flow calorimeters whose construction, calibration, and operation have been described.9 Heat of dilution data were collected at 370 °C and 24.6 MPa for NaOH, KOH, and HCl. Calorimetric data at lower temperatures were obtained from previous studies for NaOH,1,2 KOH,1,2 and HCl.3,4 A summary of experimental conditions is given in Table 1. Details of the experimental procedure and data reduction method are also available.10,11 The total flow rate was kept constant at 1 cm3‚min-1 at the pump conditions (25 °C and pressure of the system) in all experimental runs at 370 °C. The ratio of the flow of pure water to the flow of the solution was varied to change the final concentration of the solution. 2.3. Calculations. Models for γ are needed for the calculation of enthalpy changes valid at infinite dilution (I ) 0) from the heat data. A computer program12 was used to analyze the experimental heat data in order to find the best γ values based on the Pitzer ion-interaction model13 and to estimate values of log K and ∆H° values. The program uses the values of log K and ∆H° at zero ionic strength and the γ values to calculate the concentrations and heats at the experimental ionic strengths.

10.1021/ie0001016 CCC: $19.00 © 2000 American Chemical Society Published on Web 08/26/2000

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3509 Table 2. Measured (meas) and Calculated (calc) Calorimetric Data for the Dilution of NaOH, KOH, and HCl Solutions mi (mol‚kg-1)

mf (mol‚kg-1)

meas ∆H (kJ‚mol-1)

0.0965 0.0965 0.0965 0.0965 0.0965 0.0965 0.0965 0.0965 0.0965

0.0096 0.0193 0.0290 0.0386 0.0483 0.0579 0.0676 0.0772 0.0868

-70.6 ( 0.5 -50.2 ( 0.4 -37.62 ( 0.05 -28.6 ( 0.1 -21.5 ( 0.2 -16.0 ( 0.3 -11.18 ( 0.08 -7.09 ( 0.02 -3.3 ( 0.2

NaOH (T ) 370 °C and P ) 24.6 MPa) -76.42 0.3933 -52.81 0.3933 -39.44 0.3933 -30.14 0.3933 -22.99 0.3933 -17.12 0.3933 -12.10 0.3933 -7.68 0.3933 -3.68

0.0394 0.0788 0.1182 0.1575 0.1969 0.2755 0.3148 0.3541

-67.5 ( 0.3 -51.0 ( 0.2 -39.0 ( 0.2 -30.2 ( 0.1 -22.7 ( 0.3 -11.7 ( 0.4 -7.5 ( 0.1 -3.6 ( 0.2

0.2207 0.2207 0.2207 0.2207 0.2207 0.2207 0.2207 0.2207 0.2207

0.0221 0.0442 0.0663 0.0884 0.1104 0.1325 0.1546 0.1766 0.1987

-56 ( 6 -43.2 ( 0.1 -32 ( 2 -24.92 ( 0.08 -19.1 ( 0.6 -13.6 ( 0.2 -10.1 ( 0.2 -5.4 ( 0.3 -3.0 ( 0.1

NaOH (T ) 350 °C and P ) 17.6 MPab) -52.38 0.4411 -36.06 0.4411 -26.82 0.4411 -20.42 0.4411 -15.51 0.4411 -11.51 0.4411 -8.11 0.4411 -5.12 0.4411 -2.44

0.0443 0.0885 0.1769 0.2210 0.2651 0.3092 0.3532 0.3972

-69.92 ( 0.03 -46.14 ( 0.02 -27.06 ( 0.01 -20.54 ( 0.07 -15.34 ( 0.02 -10.70 ( 0.06 -6.81 ( 0.01 -3.13 ( 0.06

-53.94 -38.32 -23.01 -17.89 -13.53 -9.67 -6.18 -2.97

0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164

0.0259 0.0518 0.1034 0.1551 0.2068 0.2585 0.3101 0.3616 0.4132 0.4648 0.4905

NaOH (T ) 325 °C and P ) 12.4 MPac) -43.8 ( 0.6d -46.04 1.9323 -35.9 ( 0.5d -36.33 1.9323 -26.13 1.9323 -26.6 ( 0.4d -20.6 ( 0.3d -20.02 1.9323 -15.9 ( 0.2d -15.57 1.9323 d -12.3 ( 0.2 -12.01 1.9323 -9.5 ( 0.1d -9.01 1.9323 -6.8 ( 0.1d -6.40 1.9323 -4.19 ( 0.06d -4.06 1.9323 -1.94 1.9323 -2.08 ( 0.06d d -1.08 ( 0.06 -0.96 1.9323

0.0967 0.1935 0.3867 0.5800 0.7733 0.9665 1.1596 1.3526 1.5456 1.7380 1.8349

-55.1 ( 0.8d -44.1 ( 0.7d -32.8 ( 0.5d -25.4 ( 0.4d -19.9 ( 0.3d -15.2 ( 0.2d -11.3 ( 0.2d -8.0 ( 0.1d -4.98 ( 0.08d -2.57 ( 0.07d -1.31 ( 0.07d

-53.58 -43.01 -31.66 -24.24 -18.58 -14.02 -10.26 -7.08 -4.38 -2.04 -0.99

0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164 0.5164

0.0259 0.0518 0.1034 0.1551 0.2068 0.2585 0.3101 0.3616 0.4132 0.4648 0.4905

-27.8 ( 0.4d -23.9 ( 0.4d -18.3 ( 0.3d -14.3 ( 0.2d -11.5 ( 0.2d -8.8 ( 0.1d -6.6 ( 0.1d -4.68 ( 0.07d -3.08 ( 0.05d -1.46 ( 0.04d -0.71 ( 0.04d

NaOH (T ) 300 °C and P ) 9.3 MPac) -30.45 1.9323 -24.88 1.9323 -18.41 1.9323 -14.27 1.9323 -11.16 1.9323 -8.64 1.9323 -6.49 1.9323 -4.61 1.9323 -2.93 1.9323 -1.40 1.9323 -0.69 1.9323

0.0967 0.1935 0.3867 0.5800 0.7733 0.9665 1.1596 1.3526 1.5456 1.7380 1.8349

-36.7 ( 0.6d -29.9 ( 0.4d -22.6 ( 0.3d -17.8 ( 0.3d -14.1 ( 0.2d -10.7 ( 0.2d -8.0 ( 0.1d -5.63 ( 0.08d -3.53 ( 0.05d -1.79 ( 0.04d -0.88 ( 0.04d

-39.50 -32.38 -24.23 -18.87 -14.73 -11.32 -8.42 -5.91 -3.70 -1.75 -0.86

0.0786 0.0786 0.0786 0.0786 0.0786 0.0786 0.0786 0.0786 0.0786

0.0079 0.0157 0.0236 0.0314 0.0393 0.0472 0.0550 0.0629 0.0707

-88 ( 1 -64.2 ( 0.3 -49.6 ( 0.4 -38.1 ( 0.2 -29.2 ( 0.1 -21.83 ( 0.07 -15.6 ( 0.1 -9.8 ( 0.3 -4.5 ( 0.1

KOH (T ) 370 °C and P ) 24.6 MPa) -88.09 0.3493 -64.35 0.3493 -49.43 0.3493 -38.41 0.3493 -29.59 0.3493 -22.16 0.3493 -15.71 0.3493 -9.97 0.3493 -4.77

0.0349 0.1046 0.1395 0.1744 0.2094 0.2443 0.2793 0.3143

-70 ( 1 -40.9 ( 0.1 -31.89 ( 0.08 -24.0 ( 0.1 -17.5 ( 0.2 -12.2 ( 0.1 -7.46 ( 0.08 -3.3 ( 0.1

a a a a a a a a

0.2508 0.2508 0.2508 0.2508 0.2508 0.2508 0.2508 0.2508 0.2508

0.0250 0.0501 0.0752 0.1002 0.1253 0.1504 0.1755 0.2006 0.2257

-59.01 ( 0.01 -43.8 ( 0.06 -33.66 ( 0.05 -26.00 ( 0.01 -19.88 ( 0.06 -14.77 ( 0.02 -10.41 ( 0.02 -6.51 ( 0.01 -3.05 ( 0.03

KOH (T ) 350 °C and P ) 17.6 MPab) -57.58 0.5029 -41.83 0.5029 -32.01 0.5029 -24.78 0.5029 -19.01 0.5029 -14.18 0.5029 -10.01 0.5029 -6.32 0.5029 -3.01 0.5029

0.0501 0.1003 0.1505 0.2007 0.2510 0.3013 0.3516 0.4020 0.4525

-61.0 ( 0.9 -45 ( 4 -35.92 ( 0.01 -27 ( 2 -21.14 ( 0.01 -15.8 ( 0.3 -10.88 ( 0.01 -6.9 ( 0.1 -2.96 ( 0.02

-62.86 -45.76 -35.16 -27.30 -20.98 -15.65 -11.04 -6.96 -3.31

0.2471 0.2471 0.2471 0.2471 0.2471 0.2471 0.2471 0.2471 0.2471

0.0247 0.0494 0.0741 0.0988 0.1235 0.1482 0.1729 0.1976 0.2224

-28.55 ( 0.04 -21.66 ( 0.09 -16.80 ( 0.07 -13.17 ( 0.04 -10.12 ( 0.01 -7.59 ( 0.06 -5.34 ( 0.03 -3.39 ( 0.02 -1.62 ( 0.03

KOH (T ) 325 °C and P ) 14.8 MPab) -29.80 0.4955 -22.16 0.4955 -17.10 0.4955 -13.28 0.4955 -10.20 0.4955 -7.60 0.4955 -5.36 0.4955 -3.38 0.4955 -1.60 0.4955

0.0494 0.0988 0.1482 0.1950 0.2473 0.2968 0.3464 0.3961 0.4458

-32.61 ( 0.5 -24.50 ( 0.09 -19.11 ( 0.05 -15.18 ( 0.06 -11.46 ( 0.03 -8.50 ( 0.02 -6.04 ( 0.01 -3.78 ( 0.02 -1.76 ( 0.03

-33.17 -24.29 -18.61 -14.60 -11.01 -8.18 -5.75 -3.62 -1.71

calc ∆H (kJ‚mol-1)

mi (mol‚kg-1)

mf (mol‚kg-1)

meas ∆H (kJ‚mol-1)

calc ∆H (kJ‚mol-1) a a a a a a a a

3510

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000

Table 2 (Continued) mi (mol‚kg-1)

mf (mol‚kg-1)

meas ∆H (kJ‚mol-1)

0.2324 0.2324 0.2324 0.2324 0.2324 0.2324 0.2324 0.2324 0.2324

0.0233 0.0464 0.0696 0.0928 0.1161 0.1393 0.1626 0.1858 0.2091

-14.7 ( 0.2d -11.6 ( 0.2d -9.7 ( 0.1d -7.6 ( 0.1d -5.88 ( 0.09d -4.74 ( 0.07d -3.20 ( 0.05d -2.26 ( 0.03d -0.59 ( 0.01d

KOH (T ) 300 °C and P ) 11.0 MPab) -16.47 0.5191 -12.50 0.5191 -9.76 0.5191 -7.64 0.5191 -5.90 0.5191 -4.41 0.5191 -3.12 0.5191 -1.97 0.5191 -0.94 0.5191

0.1007 0.1007 0.1007 0.1007 0.1007 0.1007 0.1007 0.1007 0.1007

0.0100 0.0201 0.0302 0.0402 0.0503 0.0604 0.0704 0.0805 0.0906

-80 ( 1 -50.7 ( 0.4 -35.21 ( 0.06 -25.4 ( 0.3 -18.4 ( 0.4 -13.0 ( 0.2d -8.6 ( 0.1d -5.17 ( 0.08d -2.31 ( 0.04d

HCl (T ) 370 °C and P ) 24.6 MPa) -73.75 -45.88 -31.98 -23.14 -16.83 -12.02 -8.17 -5.00 -2.31

0.2522 0.2522 0.2522 0.2522 0.2522 0.2522 0.2522 0.2522 0.2522

0.0251 0.0503 0.0754 0.1006 0.1258 0.1510 0.1763 0.2016 0.2269

-70.6 ( 0.5 -45.2 ( 0.4 -32.0 ( 0.1 -23.4 ( 0.2 -16.9 ( 0.1 -12.10 ( 0.09 -8.2 ( 0.2 -4.93 ( 0.07 -2.11 ( 0.08

HCl (T ) 350 °C and P ) 17.6 MPae) -70.95 0.5067 -45.90 0.5067 -32.67 0.5067 -23.96 0.5067 -17.60 0.5067 -12.67 0.5067 -8.67 0.5067 -5.33 0.5067 -2.47 0.5067

0.5131 0.5131 0.5131 0.5131 0.5131 0.5131 0.5131 0.5131 0.5131

0.0510 0.1020 0.1531 0.2043 0.2556 0.3070 0.3584 0.4099 0.4614

-30.5 ( 0.4d -22.0 ( 0.3d -16.4 ( 0.2d -12.8 ( 0.2d -9.8 ( 0.1d -7.4 ( 0.1d -5.28 ( 0.08d -3.52 ( 0.05d -1.74 ( 0.03d

HCl (T ) 300 °C and P ) 11.0 MPaf) -31.07 -23.14 -17.80 -13.75 -10.49 -7.76 -5.42 -3.39 -1.60

calc ∆H (kJ‚mol-1)

mi (mol‚kg-1)

mf (mol‚kg-1)

meas ∆H (kJ‚mol-1)

calc ∆H (kJ‚mol-1)

0.0518 0.1037 0.1555 0.2074 0.2593 0.3112 0.3632 0.4151 0.4671

-19.8 ( 0.3d -14.9 ( 0.2d -11.7 ( 0.2d -9.2 ( 0.1d -7.1 ( 0.1d -5.44 ( 0.08d -3.92 ( 0.06d -2.54 ( 0.04d -1.22 ( 0.02d

-19.28 -14.28 -10.99 -8.51 -6.51 -4.83 -3.39 -2.13 -1.01

0.0502 0.1006 0.1510 0.2015 0.2522 0.2669 0.3537 0.4046 0.4556

-58.90 ( 0.03 -37.57 ( 0.08 -26.48 ( 0.01 -19.46 ( 0.01 -14.19 ( 0.01 -13.68 ( 0.01 -6.99 ( 0.02 -4.33 ( 0.01 -2.00 ( 0.01

-60.65 -38.67 -27.37 -20.03 -14.70 -13.40 -7.23 -4.45 -2.07

a No calc ∆H values obtained as meas ∆H values could not be fitted with the Pitzer model. b Reference 1. c Reference 2. of value is (1.5% or 0.0005 J‚s-1, whichever is greater. e Reference 3. f Reference 4.

The procedure used to calculate ion-interaction parameters in the Pitzer equation, as well as the log K and standard enthalpy change at zero ionic strength ∆H°, has been described.14,15 Only interaction parameters of the two most concentrated species in the solution make a significant contribution in the Pitzer model. Numerical values of parameters of other ion interactions are assumed to be zero. The effect of H+ is assumed to be negligible in NaOH and KOH solutions because of its low concentration. For the same reason, the effect of OH- is assumed to be negligible in HCl solution. When interaction parameters of neutral species are included, only an insignificant improvement of the results is obtained. It is thus assumed that the interactions of neutral species are negligible at the ionic strength of the solutions studied, despite the notable existence of these species. The error caused by these assumptions is expected to be less than 5%. The reaction for the formation of H2O from its constituent ions was included in the parameter calculation using the equation to determine log K and ∆H° derived earlier.16 Only heat effects due to the change in ionic strength and ion association to form water and NaOH, KOH, or HCl were considered as major contributors to the overall measured enthalpy change. The Pitzer parameters β(0) and β(1) have been expressed as a function of absolute temperature T and the

d

Uncertainty

density of water Fw in the form

β(0) or β(1) ) q0 + q1(T - Tref) + q2 ln(Fw/Fref) + q3(Fw - Fref) + q4 exp(x) (4) The value of Fw at the temperature and pressure of interest was calculated using the equation of state of Haar et al.,17 and the properties of water used to calculate the Debye-Hu¨ckel limiting slope were those of Haar et al.17 and Uematsu and Franck.18 The values of q0, q1, q2, q3, and q4 are fitting parameters, reference temperature Tref is 298.15 K, reference density Fref is 0.99707 g/mL, and x in the exponential term is defined by

x ) (T - 473.15)/20

(5)

The parameters C(φ) and λ from the Pitzer ion-interaction model13 were not used for it was felt that they were not necessary in the molality range studied. Regression of the heat data to find the optimal γ parameters, log K, and ∆H° was accomplished in an iterative manner. First, approximate γ values were calculated by fitting values generated by a simplified form of the activity coefficient equation. Second, these γ values were used in the data reduction process to find the best log K and ∆H° values at each temperature.

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3511

Figure 2. Plot of ionic strength against final molality mf at 370 °C. Table 3. Ion-interaction qi Values for Eq 4 NaOH KOH HCl

β(0) β(1) β(0) β(1) β(0) β(1)

q013

q1

0.0864 0.235 0.1298 0.32 0.1775 0.2945

1.316 × 10-3 -4.782 × 10-3 5.228 × 10-4 -9.30 × 10-4 3.158 × 10-3

where Lφ is the apparent relative molar enthalpy of the electrolyte solution being considered at the initial and final molalities mi and mf, respectively. Experimental values of ∆dilH were calculated from the observed change in frequency of the pulses (corresponding to a known energy increment) from the heater required to maintain the reaction vessel at isothermal conditions. The final molality values were calculated using the mass flow rates of the solution and the water. These mass flow rates were determined using volumetric flow rates and fluid densities at the reservoir conditions. The measured and calculated heats of dilution are given in Table 2. The molar flow of solute in moles per second can be calculated using the following equation:

molsol ) 1.667 × 10-5mf

Figure 1. Experimental and calculated enthalpies of dilution in aqueous solutions of (a) NaOH, (b) KOH, and (c) HCl at 300, 325, 350, and 370 °C. The maximum error is approximately the size of the symbols. Experimental values are represented by symbols (9, 0.1 m; [, 0.25 m; 2, 0.5 m) and calculated values by solid line.

Then, the values of the parameters a, b, c, and d in eqs 6 and 7 were calculated as described previously:14

log K ) a + b/T + c ln T + d ln Fw

(6)

∆H° ) 2.303R{-b + cT + dT2 [∂ ln Fw/∂T]} (7) Third, log K and ∆H° values were used to calculate more consistent temperature derivatives of the γ parameters β(0) and β(1) at each temperature. These derivatives together with q0 values found using γ values valid at 25 °C13 were used to find the values of q1 through q4 in eq 4. Steps 2 and 3 were repeated until the difference between calculated and measured heats was minimized. 3. Results and Discussion The dilution enthalpy change ∆dilH value is defined as

∆dilH ) Lφ(mf) - Lφ(mi)

(8)

(9)

where molsol is the molar flow of solute in moles per second. The measured values correspond to the arithmetic means of at least two experiments with standard deviation σn-1 of n experiments on the mean, except as stated otherwise. At 370 °C, experimental heat of dilution data for NaOH and KOH solutions at higher molality (g0.3493 m) cannot be fitted with the Pitzer model used. This may be due to the fact that the parameters C(φ) and λ were ignored. When available, calculated values generated by the computer program agree well with measured values, except in the very dilute regions at 370 °C, where the difference between these values might be as high as 6 kJ‚mol-1. This result is, nevertheless, acceptable considering that measured heats in the dilute region were very small and thus result in larger percentage errors. It appears that there is a systematic error in the data that causes up to 10% error when the heats measured are small (at high dilution) but causes only an error of 1-2% when the heats measured are large. In Figure 1, the apparent ∆dilH relative to that of 0.5 m solution is plotted against the square root of the final molality. For NaOH and KOH, Figure 1 shows, as expected, that ∆dilH becomes more negative as the temperature increases. However, the same trend is not observed in the case of HCl as ∆dilH at 370 °C is more positive than that at 350 °C. This can be explained by the higher degree of association at higher temperature,

3512

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000

Table 4. Thermodynamic Data Valid at Zero Ionic Strength from 300 to 370 ˚C for the Formation in Aqueous Solution of NaOH, KOH, and HCl from Their Constituent Ionsa T (°C)

P (MPa)

log K

∆S° (J‚mol-1‚K-1)

∆H° (kJ‚mol-1) Na+

300

325

350

370

300

325

350

370

300

320

325

350

370

11.0 11.0 11.0 9.3 12.4 11.0 14.8 14.8 14.8 12.4 14.8 17.6 17.6 17.6 17.6 24.6 24.6 24.6

0.82 0.55 1.32 0.25 0.19 1.41 1.24 0.97 1.50 0.68 1.95 1.76 1.47 1.78 2.77 2.20 2.00 3.33

97 95 43 121 113 120 131 130 64 145 185 187 180 125 367 226 171 523

11.0 11.0 11.0 11.0 14.8 14.8 14.8 14.8 17.6 17.6 17.6 17.6 24.6 24.6 24.6

0.68 0.58 1.26 1.45 0.96 0.89 1.42 1.80 1.40 1.33 1.68 2.28 1.81 1.88 2.58

55 60 38 80 101 105 58 115 160 165 117 192 204 163 250

11.0 11.0 11.0 11.0 11.0 11.0 10.3 10.3 sat. sat.

0.62 0.50 0.63 0.65 1.75 1.16 0.51 0.87 1.0 1.2 1.51 1.24 1.00 1.04 1.37 2.08 1.55 1.31 1.40 1.61 2.11 1.4 1.5 2.42 1.70 2.12 2.01 2.44 2.40 2.43 2.4 2.4 3.14 2.55 2.83 3.15 3.06 2.60 3.0 3.0 4.00 4.11 3.18

152 154 163 192

11.0 12.8 12.8 12.8 13.2 14.8 14.8 14.8 sat. sat. 14.8 17.6 17.6 17.6 17.6 17.6 sat. sat. 17.6 24.6 24.6 24.6 24.6 sat. sat. 24.6

+

∆C°p (J‚mol-1‚K-1)

methoda

ref

1304 1300 670 1200 1200 2301 1676 1784 1389 1387 5078 4523 4157 6533 18580 6765 11262 32614

cal cal con cal cal mod cal cal con cal mod cal cal con mod cal con mod

this study 1 19b 2 2 21 this study 1 19b 2 21 this study 1 19b 21 this study 19b 21

1790 1761 655 1325 2026 2024 1357 2680 3832 3996 6379 6512 5254 11002 10370

cal cal con mod cal cal con mod cal cal con mod cal con mod

this study 1 20c 22 this study 1 20c 22 this study 1 20c 22 this study 20c 22

2165 2164 1989 2166

cal cal cal con cal cal cal cal cal cal sol con mod cal cal sol mod cal con cal cal cal cal sol mod cal cal con cal cal cal cal sol mod cal con cal cal cal cal sol con mod

this study 3 4 23d 25e 25g 3 5 25f 25g 26 27 28 3 5 26 28 4 23d 25g 25e 25f 25g 26 28 this study 3 23d 25g 25e 25f 25g 26 28 this study 23d 25g 25e 25f 25g 26 27 28

OH-

h NaOH(aq) 185 176 101 215 200 236 243 236 135 255 246 333 317 234 642 393 305 876 K+ + OH- h KOH(aq) 109 117 90 166 187 196 124 226 284 290 220 352 352 290 438 H+ + Cl- h HCl(aq) 278 278 296 348

104 154 141 220 120 164

203 279 263 403 232 315

2187 2534 1844

161 199 184 210 206 233 256 173

300 356 336 394 376 415 456 319

1798 2564 2431 2530 3481 5220 4819 4895

300 220 223 223 291 293 475 397

528 397 416 405 508 508 809 683

2640 4361 6187 6186 25127 24652

370 305 297 380 351 610 582

637 533 536 658 600 1009 963

3246 18090 8987 41115 43988

470 445 367

788 749 647

3774

530

885

32890

2104

cal ) calorimetry, con ) conductance, sol ) solubility, mod ) modeling with the HKF equation. b Calculated using eq 10 from ref 19. c Calculated using eq 9 from ref 20. d Calculated using eq 19 from ref 23. e Calculated using eq 35 from ref 25 and β(2) from ref 24. f Calculated using the ion association-interaction model. g Calculated using the chemical equilibrium model. a

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3513

Figure 4. Plot of log K as a function of temperature.

Figure 3. Plot of ∆G°, ∆H°, and -T∆S° as a function of temperature for the formation from their constituent ions of (a) NaOH, (b) KOH, and (c) HCl.

thus leading to fewer free ions. This is consistent with results predicted by the model. Figure 2 shows a plot of ionic strength against final molality at 370 °C. This plot demonstrates that the ionic strength of HCl is much lower than that of NaOH and KOH, as a result of the larger value of the HCl association constant. The ion-interaction qi values in eq 4 found by fitting the ∆dilH values are given in Table 3. It was found that only q0 and q1 values were needed to fit the data over the temperature and pressure range of this study. Insignificant improvement in the fits of the data was realized when additional terms were used in eq 4. If data had been collected at more than one pressure at each temperature, the density-dependent terms would have been necessary. However, use of nonzero q0 and q1 values made a significant difference in the log K and ∆H° values and the goodness of the fit of the data. The log K values calculated using interaction coefficients equal to zero were about 2 log K units higher for NaOH and KOH association and 1.4 log K units higher for HCl association. The ∆H° values calculated using no interaction parameters were 12, 13, and 9 kJ more negative for the association of NaOH, KOH, and HCl, respectively. The average sum of squared differences was roughly 3 times as large for all systems when using interaction parameters of 0 as opposed to using the values found. The log K, ∆H°, ∆S°, and ∆C°p values for the formation of NaOH, KOH, and HCl from their respective constituent ions are given in Table 4 together with literature values. In Figure 3, ∆G°, ∆H°, and -T∆S° are plotted against temperature for the interaction of OH- with Na+ and K+ and of Cl- with H+. For all systems, both enthalpy changes and entropy changes are positive and the magnitude of these quantities increases dramatically as the temperature increases.

Hence, the ion association at high temperature is entropy driven. The compensating effect of enthalpy and entropy leads to only a small variation of Gibbs free energy changes. However, the resulting ∆G° values do show significant but small increases with temperature. The log K values increase with increasing temperature for all of the reactions (Figure 4), with the greatest increase seen in the case of HCl. The log K value for the association of Na+ or K+ with OH- to form the ion pair from 375 to 600 °C and up to 300 MPa was reported by Ho and Palmer.19,20 These log K values are larger than those determined earlier by us1,2 from 300 to 350 °C. The data collected at 300 and 350 °C reported earlier1,2 together with the data collected at 370 °C for this study were used in the regression to find the best log K and ∆H° values. The ∆H° values are roughly the same, but the log K values are significantly higher. This is due to the fact that, as the temperature increases, the log K values also increase. When log K values are small, the predicted heats of dilution are approximately the same for a fairly wide range of log K values. As log K values increase, the heat of dilution is a stronger function of log K. The log K values found in the present study are again lower than those calculated from the equations of Ho and Palmer,19,20 with the exception of the NaOH results at 370 °C. However, they are in better agreement with log K values obtained from the studies of Ho and Palmer than those obtained from our previous study,1 especially at higher temperatures. This is likely due to the inclusion of heats of dilution at 370 °C in the present study, which makes the experimental conditions closer to those used by Ho and Palmer. The values for the thermodynamic quantities, log K, ∆H°, ∆S°, and ∆C°p, can be calculated for NaOH and KOH association using the revised Helgeson-Kirkham-Flowers (HKF) equation of state. Those values using the parameters found for NaOH by Pokrovskii and Helgeson for NaOH21 and for KOH22 are shown in Table 4. The values found using the HKF equation are considerably larger than the values found in this study. The log K values for NaOH formation are slightly greater than those for KOH, as has been observed in previous studies.1,19-22 This result is consistent with the increase in the ionic charge density for M+ with decreasing ionic radius. The enthalpy and entropy changes are consistent with this observation. The determination of association constants for HCl from its constituent ions from 400 to 700 °C and up to 400 MPa was reported by Frantz and Marshall from conductance studies.23 Heats of dilution of HCl to 375 °C and 40 MPa over the molality range of 0.01-15.6 m were reported by Holmes et al.24 These data together with literature results were correlated by them using the Pitzer ion-interaction model.13 The β(2) term was included in that study to describe the effect of ion

3514

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000

Table 5. Parameters for Eqs 6 and 7 NaOH KOH HCl

a

b (K)

c

d

-376.32 -609.05 -673.65

25752.19 45503.79 46591.58

52.26 83.47 93.31

-0.85 -0.54 -1.06

association above 250 °C. Because data were collected at a wide range of temperatures and concentrations, Holmes et al.24 included additional parameters not considered in the present study in their activity coefficient equations. Later, Simonson et al.25 published log K and ∆H° values for HCl association from its constituent ions based on two different models. Values corresponding to the ion association-interaction model of Simonson et al.25 are omitted in Table 4 because this model used extrapolated K values from Frantz and Marshall’s study.23 It was shown in our previous study that values from the literature over the 300-350 °C range3 were generally higher than our values. Pokrovskii28 reports the parameters for the HFK equation of state for HCl. The values for log K, ∆H°, ∆S°, and ∆C°p found using their values are shown in Table 4. The values found using the HFK are somewhat larger but in qualitative agreement with those found in this study. The values obtained in the present study are higher than those obtained earlier by us3 but show better agreement with values found in the literature. The parameters for eqs 6 and 7, which can be used to calculate log K and ∆H° at temperatures from 300 to 370 °C, are given in Table 5. These two equations have proven to be useful in fitting data over wide temperature and pressure ranges. However, because our data were fitted at only three or four temperatures, a large covariance among the parameters can occur. Therefore, unreasonable values might be obtained if these equations are used outside of the specified temperature range (300-370 °C). More precise values for these thermodynamic quantities can be obtained by mixing the two streams containing the reacting ions. It is intended to do this soon. Because heats of dilution depend on the log K, ∆H°, and γ values, the effects of each of these quantities cannot be independently determined using only ∆dilH values. Hence, the log K, ∆H°, and γ values reported in this study are dependent on each other. Acknowledgment This material is based upon work supported by the U.S. Army Research Office under Grant DAAG55-971-01129. Literature Cited (1) Gillespie, S. E.; Chen, X.; Oscarson, J. L.; Izatt, R. M. Enthalpies of Dilution of Aqueous Solutions of NaOH, KOH, and CsOH at 300, 325, and 350 °C. J. Solution Chem. 1998, 27, 183. (2) Chen, X.; Gillespie, S. E.; Oscarson, J. L.; Izatt, R. M. Enthalpy of Dissociation of Water at 325 °C and Log K, ∆H, ∆S, and ∆Cp Values for the Formation of NaOH(aq) from 250 to 325 °C. J. Solution Chem. 1992, 21, 803. (3) Gillespie, S. E.; Chen, X.; Oscarson, J. L.; Schuck, P. C.; Izatt, R. M. Enthalpies of Dilution of Aqueous Solutions of HCl, MgCl2, CaCl2, and BaCl2 at 300, 325, and 350 °C. J. Solution Chem. 2000, in press. (4) Gillespie, S. E.; Oscarson, J. L.; Chen, X.; Izatt, R. M.; Pando, C. Thermodynamic Quantities for the Interaction of Clwith Mg2+, Ca2+ and H+ in Aqueous Solution from 250 to 325 °C. J. Solution Chem. 1992, 21, 761.

(5) Oscarson, J. L.; Gillespie, S. E.; Christensen, J. J.; Izatt, R. M.; Brown, P. R. Thermodynamic Quantities for the Interaction of H+ and Na+ with C2H3O2- and Cl- in Aqueous Solution from 275 to 320 °C. J. Solution Chem. 1988, 17, 865. (6) Simonson, J. M.; Ryther, R. J. Volumetric Properties of Aqueous Sodium Hydroxide from 273.15 to 348.15 K. J. Chem. Eng. Data 1989, 34, 57. (7) Hershey, J. P.; Damesceno, R.; Millero, F. J. Densities and Compressibilities of Aqueoous HCl and NaOH from 0 to 45 °C. The Effect of Pressure on the Ionization of Water. J. Solution Chem. 1984, 13, 825. (8) Lazarev, M. A.; Sorochenko, V. R. (translator). In Properties of Aqueous Solutions of Electrolytes; Zaytsev, I. D., Aseyev, G. G., Eds.; CRC Press: Boca Raton, FL, 1992. (9) Zhou, L. Design and Construction of a New High-Temperature Calorimeter and Calorimetry Study of Aqueous NaCl, KCl, NaOH, KOH and HCl solutions at 370 and 380 °C. M.S. Thesis, Brigham Young University, Provo, UT, 1999. (10) Chen, X. Calorimetric Determination of Thermodynamic Quantities for Chemical Reactions in Aqueous Solutions at High Temperatures. Ph.D. Dissertation, Brigham Young University, Provo, UT, 1991. (11) Izatt, R. M.; Oscarson, J. L.; Chen, X.; Gillespie, S. E. Determination of Thermodynamic Data for Modeling Corrosion. CO2-NaOH-H2O System; EPRI Report NP-5708; Electric Power Research Institute: Palo Alto, CA, 1992; Vol. 3. (12) Gillespie, S. E.; Oscarson, J. L.; Izatt, R. M.; Wang, P.; Renuncio, J. A. R.; Pando, C. Thermodynamic Quantities for the Protonation of Amino Acid Amino Groups from 323.15 to 398.15 K. J. Solution Chem. 1995, 24, 1219. (13) Pitzer, K. S. In Activity Coefficients in Electrolyte Solutions, 2nd ed.; Pitzer, K. S., Ed.; CRC Press: Boca Raton, FL, 1991. (14) Gillespie, S. E.; Chen, X.; Oscarson, J. L.; Izatt, R. M. Enthalpies of Dilution of Aqueous Solutions of LiCl, KCl, and CsCl at 300, 325 and 350 °C. J. Solution Chem. 1997, 26, 47. (15) Izatt, R. M.; Gillespie, S. E.; Oscarson, J. L.; Wang, P.; Renuncio, J. A. R.; Pando, C. The Effect of Temperature and Pressure on the Protonation of o-Phosphate Ions at 348.15 and 398.15 K and at 1.52 and 12.50 MPa. J. Solution Chem. 1994, 23, 449. (16) Chen, X.; Oscarson, J. L.; Gillespie, S. E.; Cao, H.; Izatt, R. M. Determination of Enthalpy of Ionization of Water from 250 to 350 °C. J. Solution Chem. 1994, 23, 747. (17) Haar, L.; Gallagher, J. S.; Kell, G. S. NBS/NRC Steam Tables: Thermodynamic and Transport Properties and Computer Programs for Vapor and Liquid States of Water in SI Units; Hemisphere: Bristol, PA, 1984. (18) Uematsu, M.; Franck, E. U. Static Dielectric Constant of Water and Steam. J. Phys. Chem. Ref. Data 1980, 9, 1291. (19) Ho, P. C.; Palmer, D. A. Ion Association of Dilute Aqueous Sodium Hydroxide Solution to 600 °C and 300 MPa by Conductance Measurements. J. Solution Chem. 1996, 25, 711. (20) Ho, P. C.; Palmer, D. A. Ion Association of Dilute Aqueous Potassium Chloride and Potassium Hydroxide Solutions to 600 °C and 300 MPa Determined by Electrical Conductance Measurements. Geochim. Cosmochim. Acta 1997, 61, 3027. (21) Pokrovskii, V. A.; Helgelson, H. C. Thermodynamic Properties of Aqueous Species and the Solubilities of Minerals at High Pressures and Temperatures: the System Al2O3-H2O-NaCl. Am. J. Sci. 1995, 295, 1255. (22) Pokrovskii, V. A.; Helgelson, H. C. Thermodynamic Properties of Aqueous Species and the Solubilities of Minerals at High Pressures and Temperatures: the System Al2O3-H2O-KOH. Chem. Geol. 1997, 137, 221. (23) Frantz, J. D.; Marshall, W. L. Electrical Conductances and Ionization Constants of Salts, Acids, and Bases in Supercritical Aqueous Fluids: I. Hydrochloric Acid from 100° to 700 °C and at Pressures to 4000 Bars. Am. J. Sci. 1984, 284, 651. (24) Holmes, H. F.; Busey, R. H.; Simonson, J. M.; Mesmer, R. E.; Archer, D. G.; Wood, R. H. The Enthalpy of Diltuion of HCl(aq) to 648 K and 40 MPa. Thermodynamic Properties. J. Chem. Thermodyn. 1987, 19, 863. (25) Simonson, J. M.; Holmes, H. F.; Busey, R. H.; Mesmer, R. E.; Archer, D. G.; Wood, R. H. Modelling of the Thermodynamics of Electrolyte Solutions to High Temperatures Including Ion Association. Application to Hydrochloric Acid. J. Phys. Chem. 1990, 94, 7675.

Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 3515 (26) Ruaya, J. R.; Seward, T. M. The Ion-pair Constant and Other Thermodynamic Properties of HCl up to 350 °C. Geochim. Cosmochim. Acta 1987, 51, 121. (27) Pearson, D.; Copeland, C. S.; Benson, S. W. The Electrical Conductance of Aqueous Hydrochloric Acid in the Range 300 to 383°. J. Am. Chem. Soc. 1963, 85, 1047. (28) Pokrovskii, V. A. Calculation of the Standard Partial Molal Thermodynamic Properties and Dissociation Constants of Aqueous

HCl0 and HBr0 at Temperatures to 1000 °C and Pressures to 5 kbar. Geochim. Cosmochim. Acta 1999, 63, 1107.

Received for review January 13, 2000 Revised manuscript received June 26, 2000 Accepted June 26, 2000 IE0001016