Validity Range of the Meissner Activity Coefficient Model
used in MULTEQ
Remote
Blockage
Project
Shirley Dickinson
, Martin BachetClearance
, Richard Eaker , Chuck
Marks ,
1
2
3
4
Peter Tremaine5 and Daniel M. Wells6
1National
Nuclear Laboratory (UK), 2Électricité de France (France), 3Richard W. Eaker LLC (USA),
4Dominion Engineering, Inc. (USA), University of Guelph (Canada), 6Electric Power Research Institute
(USA)
INTRODUCTION
MULTEQ TREATMENT OF ION ACTIVITY COEFFICIENTS
The MULTEQ computer model was developed by EPRI to calculate the composition, pH and electrochemical potential of
aqueous solutions at elevated temperature and pressure [1]. The model uses an extensive database of thermodynamic
equilibrium constants that determine the concentrations of aqueous species and the identities of precipitates formed in a
given system.
The treatment of activity of aqueous species in MULTEQ is based on that of Lindsay [2]. All solutes are assigned to
one of four classes, which include ionic species, dissolved permanent gases and other neutral species. Only the
former class is discussed here, and the contribution of other classes is excluded from the equations for clarity.
The activity coefficient for an ionic species, in the absence of an neutral species, is based on the work of Meissner [3],
who showed that the experimentally-observed dependence of the activity coefficient on the ionic strength for a variety
of electrolytes at 25°C could be represented by the function F(I), using a suitable value for the empirical parameter q.
The expression for q is based on experimental measurements for NaCl over the temperature range 100 – 350°C. The
natural log of the activity coefficient where zi is the ionic charge is shown in Equation 1 while that of F(I) and q are
defined in Equations 2 through 6.
An important element of MULTEQ is the activity coefficient model that allows reliable calculations to be performed at
temperatures and concentrations where the solutions cannot be treated as ideal. The model used in MULTEQ is an
extension of the Debye-Hückel equation based on empirical expressions that were fitted to the mean activity coefficients
of sodium chloride solutions at elevated temperatures.
It has been assumed that the MULTEQ model is valid over the temperature range 150 – 335°C. However, there has
been no assessment of the errors introduced by using the model outside this range, or by the application of Meissner’s
reduced activity coefficient for sodium chloride to other electrolytes, particularly at lower temperatures. In this paper, the
values calculated by MULTEQ for the ionic strength dependence of equilibrium quotients for several ionisation reactions,
including the self-ionisation of water, are compared with experimental measurements in the range 0 – 300°C and at ionic
strengths up to 5 mol kg-1. The activity coefficients calculated by MULTEQ are also compared with low-temperature
values for different electrolytes.
Equation 2
A  0.484582  0.00158173Tc  2.14065  105 Tc2
Equation 3
COMPARISON WITH EXPERIMENTAL DATA
The validity of the MULTEQ model was assessed by comparing the equilibrium quotient Q calculated for a number of
ionisation reactions over a range of I and T with the tabulated data in Reference 3, which were obtained from fits to
experimental data. The selected reactions, and the sources of experimental data, are listed in Table 1.
Table 1. Ionisation Reactions
Reaction
𝐻2 𝑂 ⇌ 𝐻 + + 𝑂𝐻 −
𝐶𝑂2 + 𝐻2 𝑂 ⇌ 𝐻 + + 𝐻𝐶𝑂3−
𝐻𝐶𝑂3− ⇌ 𝐻 + + 𝐶𝑂32−
𝐻𝑆𝑂4− ⇌ 𝐻 + + 𝑆𝑂42−
𝐻3 𝑃𝑂40 + 𝑂𝐻 − ⇌ 𝐻2 𝑃𝑂4− + 𝐻2 𝑂
𝐻2 𝑃𝑂4− + 𝑂𝐻 − ⇌ 𝐻𝑃𝑂42− + 𝐻2 𝑂
ln  i  zi2 F I 
1/ 2
 AI ln(10)
q

F (I ) 

ln
1

B
*
(
1

0
.
1
I
)
 B *
1/ 2
1  CI
Equation 1
Quotient Reference
Qw
[5]
QC1
[6]
QC2
[7]
QS
[8]
QP1
[9]
QP2
[9]
 2.56199  107 Tc3  1.05332  109 Tc4  1.57603  1012 Tc5
B*  0.75  0.065q
Equation 4
 0.023 I 3
Equation 5
C  1  0.055 qe
Equation 6
q  2.95869  3.21502  103 Tc  1.7233  105 Tc2
The two main assumptions in the MULTEQ approach, which are tested in the current work, are: 1) that the q
parameter for NaCl can be applied to other electrolytes without introducing excessive errors, and 2) that the
expression for q is valid only in the range 100-350°C. It should be noted that the calculations performed by MULTEQ
are, generally, also dependent on the presence of neutral species, which can affect the activity of ionic species. This
work did not explicitly evaluate the other activity models used in MULTEQ. However, these models were included in
the MULTEQ calculations used to assess the validity of the Meissner model for ionic species.
The results for the carbonic acid dissociation reactions are shown below; similar plots for the other reactions in Table 1 are presented in the full paper. The plots show the difference between the log K and log Q(I) values as a function of temperature,
with the reference data on the left and the MULTEQ data on the right. This shows the effect of ionic strength on the measured or calculated reaction quotient. The tables beneath the plots give the following quantities for each value of Q(T,I):
Delta ( I )  log10 Q(T , I ) ref  log10 Q(T , I ) MULTEQ
Delta ( I  0)  (log10 Q(T , I ) ref  log10 Q(T , I ) MULTEQ )  (log10 K (T ) ref  log10 K (T ) MULTEQ )
This form was used rather than the inverse (MULTEQ – Ref) because it gives positive Delta values in most cases. In the second of these, any differences arising from the equilibrium constant expression used in MULTEQ (K(T)MULTEQ) are subtracted
out so the remaining differences arise only from the activity coefficient model.
Comparison of log Q calculated by MULTEQ with reference values for carbonic acid dissociation reactions
𝐶𝑂2 + 𝐻2 𝑂 ⇌ 𝐻𝐶𝑂3−
2.9
2.4
1.9
2.4
I = 3.0
I = 5.0
Q(I) - Q(I=0)
Q(I) - Q(I=0)
I = 1.0
Reference
I = 0.5
1.4
0.9
0.4
-0.1
I = 0.1
I = 0.5
I = 1.0
I = 3.0
1.4
0.9
0.4
0
50
100
150
200
250
300
-0.1
I=0
0.012
0.027
0.032
0.027
0.021
0.008
-0.022
-0.032
0.065
I = 0.1
0.025
0.041
0.047
0.044
0.036
0.022
-0.009
-0.014
0.090
I = 0.1
I = 0.5
I = 1.0
I = 3.0
I=3
0.240
0.218
I=5
0.350
0.302
0.072
0.103
0.194
0.255
0.047
0.014
0.010
0.105
0.078
0.046
0.042
0.130
0.170
0.153
0.179
0.314
0.238
0.253
0.342
0.595
2.4
I = 5.0
1.9
1.4
0.9
0
50
100
150
200
250
-0.1
300
I = 0.1
0.013
0.014
0.014
0.017
0.014
0.014
0.012
0.018
0.025
I = 0.1
I = 0.5
I = 1.0
I = 3.0
I = 5.0
1.9
1.4
0.9
0
50
100
150
200
250
-0.1
0
50
Temperature °C
Delta(I-0)
I = 0.5
I=1
0.046
0.087
0.042
0.078
I=3
0.228
0.191
I=5
0.338
0.275
0.044
0.075
0.167
0.228
0.040
0.036
0.043
0.040
0.071
0.067
0.074
0.064
0.163
0.175
0.212
0.249
0.231
0.275
0.375
0.530
T
25
50
75
100
125
150
200
250
MULTEQ
0.4
Temperature °C
Delta(I)
I = 0.5
I=1
0.058
0.099
0.069
0.104
Reference
0.4
Temperature °C
T
25
50
75
100
125
150
200
250
300
2.4
I = 5.0
1.9
2.9
MULTEQ
Q(I) - Q(I=0)
I = 0.1
2.9
Q(I) - Q(I=0)
2.9
𝐻𝐶𝑂3− ⇌ 𝐻 + + 𝐶𝑂32−
I=0
-0.031
-0.014
-0.004
0.007
0.040
0.112
0.220
0.436
I = 0.1
0.002
0.023
0.036
0.054
0.081
0.121
0.255
0.481
Delta
10^(Delta)
150
200
250
Temperature °C
Delta(I)
I = 0.5
I=1
0.078
0.167
0.105
0.200
I=3
0.460
0.506
I=5
0.692
0.748
0.143
0.242
0.553
0.796
0.204
0.325
0.545
0.297
0.405
0.611
0.583
0.799
0.646
0.820
0.798
0.924
Colour Scale
-0.2
0.63
0
1.00
-0.4
0.40
100
0.2
1.58
I = 0.1
0.033
0.037
0.039
0.047
0.041
0.009
0.035
0.044
0.4
2.51
I = 0.5
0.109
0.119
Delta(I-0)
I=1
0.198
0.214
I=3
0.492
0.520
I=5
0.723
0.762
0.136
0.235
0.546
0.789
0.092
0.106
0.109
0.185
0.185
0.175
0.471
0.426
0.361
0.687
0.600
0.488
0.6
3.98
0.8
6.31
1
10.00
DISCUSSION
In general, these comparisons confirm the expectations that the Meissner model, using generalized NaCl fitting
parameters, is a good predictor of actual activity coefficients for a diverse group of ions when the temperature is high and
the concentrations are low. This work has, for the first time, provided a quantitative evaluation of the extent to which
deviations from this model and reality may be expected as temperature is decreased and/or concentration is increased.
The generalized Meissner model used in MULTEQ provides reasonable predictions of the activity coefficients of ions when
the ionic strength is below about 2 molal and the temperature is above about 100°C.
Alternatives to the MULTEQ approach have been developed that use activity models that are more species-specific. Such
models may provide higher fidelity predictions when they are used within the range of the data from which they were
developed. The general Meissner model used in MULTEQ, however, would be expected to be more reliable when
extrapolating outside the temperature range for which data are available or when new ionic species, for which no relevant
activity coefficient measurements exist, are to be modelled.
CONCLUSIONS
As formulated by Lindsay [2] in MULTEQ, the Meissner expression provides an accurate method for estimating activity
coefficients of electrolyte solutions over the range 100 to 325 °C, with accurate limiting law behaviour and a precision of
about ±30% at concentrations up to ~ 0.1 m. At higher concentrations the uncertainties increase sharply, in part because of
ion pairing effects. The value of this approach is that the ionic strength dependence of equilibrium quotients at elevated
temperatures can be estimated directly from measured or estimated equilibrium constants, without concentration-dependent
experimental data. The results are sufficiently accurate to calculate bulk water chemistry in the primary and secondary
coolant circuits of nuclear power reactors, which generally operate at low ionic strengths within tight specifications. The
model is useful for calculating under-deposit chemistry responsible for "hide-out" reactions and corrosion at higher
concentrations, because these processes require data sufficiently accurate to model the precipitation of major deposits and
precipitates that cause large swings in pH, but otherwise do not require high precision.
REFERENCES
1. MULTEQ: Equilibrum of an Electrolytic Solution with Vapor-Liquid Partitioning and Precipitation, Version 7.0. EPRI, Palo Alto, CA: 2012. 1025010.
2. W. T. Lindsay Jr., The ASME Handbook on Water Technology for Thermal Power Systems, edited by Paul Cohen, 1989, Chapter 7.
3. H. P. Meissner, “Prediction of Activity Coefficients of Strong Electrolytes in Aqueous Systems”, In Thermodynamics of Aqueous Systems with Industrial Applications; Newman, S., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.
4. P. Tremaine, K Zhang, P Bénézeth and C Xiao, “Ionic equilibria of acids and bases under hydrothermal conditions”, in Aqueous Systems at Elevated Temperatures and Pressures: Physical Chemistry in Water, Steam and Hydrothermal Solutions, Elsevier Ltd., 2004.
5. F. H. Sweeton, R. E. Mesmer and C. F. Baes, C F, "Acidity Measurements at Elevated Temperatures, VII. Dissociation of Water", J. Solution Chem. 3, 1974, pp. 191-214.
6. C. S. Patterson, G. H. Slocum, R. H. Busey and R. E. Mesmer, “Carbonate equilibria in hydrothermal systems: first ionization of carbonic acid in NaCl media to 300°C”, Geochimica et Cosmochimica Acta 46, 1982, pp. 1653-1663.
7. C. S. Patterson, R. H. Busey and R. E. Mesmer, “Second ionization of carbonic acid in NaCl media to 250°C”, J. Solution Chem. 13 (9), 1984, pp. 647-661.
8. A. G. Dickson, D. J. Wesolowski, D. A. Palmer and R. E. Mesmer, “Dissociation Constant of Bisulfate Ion in Aqueous Sodium Chloride Solutions to 250 °C”, J. Phys. Chem. 94, 1990, pp. 7918-7985.
9. R. E. Mesmer and C. F. Baes, “Phosphoric acid dissociation equilibrium in aqueous solutions to 300°C”, J. Solution Chem. 3, 1974, pp. 307-322.