- Email: [email protected]
On the calculation of activity coefficients of electrolyte solutions from vapour pressure data up to high temperatures and concentrations
Chem., 180 (19843 57-65 Elsevter Sequoia S A., Lausanne - Prmted in The Netherlands
57
J. Eiectroanal
ON THE CALCULATION OF ACTIVITY COEFFICIENTS OF ELEmROLYTE SOLUTIONS FROM VAPOUR PRESSURE DATA UP TO HIGH TEMPERATURES
THE
SYSTEM
AND MgCl.
CONCENTRATIONS
+ H,O
H -H. EMONS,
W. VOIGT
Sektron Chemle,
Bergakademle
*
and W.-F. WOLLNY Frmberg.
DDR - 9200
Frezberg (G-D. R )
(Recetved 30th April 1984)
ABSTRACT Startmg from recent results of vapour Fressure meas-urements on hydrated melts of MgCI,. we have withm the temperature range of calculated the salt actrvlty coefficients for the system MgCI, &H,O SO-200°C and up to an rome strength of 35 mol/kg H,O. For the Gibbs-Duhem integration, the equattons of Pttzer and the modtfied B ET. equatton were used and their usefulness IS dtswssed
INTRODUCTION Recent developments for transforming and storing energy based on electrochemical and thermal processes often involve salt + water systems at high concentrations and temperatures as an integral part. For modelling and predicting ion association and
complex
formation
reactions
as well as solubilities,
knowledge
of the salt activity
is necessary_ At the present time, the direct determination of salt activity coefficients by means of EMF measurements is not possible for the halides of alkaline metals and alkaline earth metals at high concentrations and temperatures, because suitable galvanic cells which work reversibly are not available. Therefore the Gibbs-Duhem equation is used for the calculation of salt aciivities frcm water activity data for these binary systems. A presumption for these calculations is the complete knowledge of the water activities over the entire range of concentration, from dilute solutions up to interesting concentrations. La general, it is not possible to obtain experimental data of comparable accuracy over a large range of concentration and temperature with only ol~e given apparatus, for -instance a vapour pressure apparatus. From the experimental point of vie\v, four typical concentration/temperature ranges can be distinguished using experimental technology: (1) measurements from 25 to 100 o C and c,,,~ < saturation;
coefficients
* Dtdieated to the memory of Professor Dr. Dr. h c. Kurt Schwabe 002%0728/84/$03.00
0 1984 Elsevier Sequoia S.A.
58
(2) measuiements from 100 to 250 OC (3OOOC) and cmllSf saturation concentration at room temperature; (3) measurements from 100 to 250 OC (300 “C) and c,,,t >, saturation concentration at room temperature; (4) measurements above 300 OC. In particular the concentration range of the hydrated melts is comprised in the third experimental range. For investigations in ths range, an apparatus for vapour pressure measurements was developed by us and measurements were performed for hydrated melts of magnesium chloride over the temperature range 120 to 220 OC [1,2]. In this paper the salt activity coefficients are calculated for the system MgCl. + Hz0 up to nearly saturation concentrations and from 50 to 200 OC. For the first and second experimental range, the data have been taken from the literature_ For the Gibbs-Duhem integration, the equation of Prtzer [3] and a modified B.E.T. equation [4] are used and the applicability of these equations is discussed. WATER
ACTIVITY
DATA
A summary of the distribution of the experimental data for five selected temperatures from 50 to 200 OC is shown in Fig. 1. Apparently reliable values are available with the data published by Liu and Lindsay [5,6] and Mayrath and Wood [7] for relatively dilute solutions at high temperatures. The values determined by Holmes et al. [S] have not been taken into account, because they differ significantly from the data of refs. 5-7. For the hydrated melts, only our own data are available [2]. The values at low temperatures were essentially determined by Mayrath and Wood [7], Snipes et al. [9], Tammann [lo] and Huschenbett and Dahne [ll].
59
The calculation of the water activity a,” from vapour pressure data taken from literature was accomplished in the same manner as for our data using the quotient of the fugacity q,,,: a, = qL, (solution) r/G, (water) T
(1)
The fugacity was calculated from the equatron SL = PW exPKJhJv4
(2)
where B is the second virial coefficient of water vapour. The PVT data of pure water [12] were used for the calculation of B. A five-parameter equation was needed to represent the temperature dependence of B: B
[m3/kg] = - 2.98407 x 1O-2 + 27.1085 T-’ -2.1080
x lo6 T-3 + 1.41176 x 10’ T-4
The pressure dependence regarded as negligible. GIBBS-DUHEM
- 4.81991 x lo3 T-’
of the activity
coefficient
within
the liquid
(3) phase was
INTEGRATION
Pitzer equation
From Fig. 1 it can be recognized that the values are very irregmarly distributed and that the number of data points IS very limited. In this case, the usual description of the concentration dependence of the osmotic coefficient + using an exponential function with more than four adJustable parameters cannot be recommended. In the last few years, the equations proposed by Pitzer [13] have been applied to an increasing extent. Normally, for these equations only three adJustable parameters are needed. The validity of these equations was confirmed for MgCl, for concentrations up to 4.5 mol/kg H,O, that means ionic strengths up to I = 13.5 at 25 o C [3]. All available thermodynamic data for NaCl, KC1 and CsCl in aqueous solutions up to 250 “C have been described within the limits of the experimental errors as well
P41.
The equations for the osmotic coefficient Y+(,,,) of a salt M,, X,_ possess the form: +-l=(z,zx
f”=
)f@ + m (2%?J
-A*[Y/(l
B~=fl(*)+p”)
lny
*(In)
=
and the mean salt activity
Y ) B+ 9 m2 [2(
P&px)3’2/Y]
coefficient
c*
+bY’)]
(4)
exp(-&‘/2) (Z&x)f’t-
m(2vM~x/zJ)B~
f 7 = -A, [I"'/(1 + bl”‘) j17 = 2/3(O)+ (2@“/a21)[l Cy = (3/2) Co
+ (2/b)
+ “~[2(v,vx)3’2/v]
cy
ln(1 + bI’/‘)]
- (1 + cyJ’/” - 1/2&I)
(3 exp( -11/2)]
60
where A, represents the Debye-Htickel coefficient, whose values have been taken from ref. 3 for the different temperatures. Following Pitzr, we have set a and b equal to 2.0 and 1.2 kg’j2 mol-“2, respectively_ /3(O), p(‘) and Ca are parameters, whose values are determined from a least-squares fit of pqn_ (4) to all the experjmental data mentioned above. Thereby, all the data were weighted equally in the fitting procedure. The deviations of the experimental + values from those recalculated by means of eqn. (4) are illustrated in Fig. 2 at 50, 80 and 100°C and in Fig. 3 at 150 and 200 o C. It should be noted that the experimental data up to 100°C (Fig. 2) are scattered irregularly around the calculated ones. For concentrations m > 6 mol kg;’ (Fig. 3), the deviations exceed the experimental error calculated by us, of a maximum of 2.3% the water actrvity. At the highest concentration, the value A+ = 0.104 corresponds Aa, = 7%.
-0.10
e
1
3 Molality/mol
5 kg-’
FIN 2 Dewatlon of the expenmental 80 (A) and 100 o C (0)
Fsg (0)
3. Dewations and 200 o C (0)
of the experimental
7
Q values from the values recalculated
by means of eqn. (4) at 50 (x),
+ values from the values recalculated
by means of eqn
(4) at 150
61
Figure 4 demonstrates that no essential the data points at very high concentrations points at low concentrations.
improvement of the fit is attainabie for using diminished weights for the data
Fig 4. Dewat~ons of the expenmental Q valuesfrom the values recalculated usmg dmuc.shed weights for the data pomts at low concentrations. (s+) All data points, 20; (0) 13 data points, equal number m the ddute and concentrated range, (A) 9 data points, m > 1.5 mol kg-‘.
2 Fig. 5 Dewations for the experimental (0) - rdr-. m XI mol/kg H20. - wm’fl
(Q
and calculated (e) vdues of + at 200 o C in the
COD~I~MCC
62
TABLE
1
Parameters
of P~tzer;s equation
Temperature/
for the system MgCl,
p (0)
oC
50 80 100 150 200 200’ 200 h
T Hz0
C+
0 3151 0 2879 0 3120 0.3122 0 3167 0.3218 0.3308
2.2601 2.6610 1 9774 2 3640 2 5344 2 3870 -1 3905
0.0071 0 0067 0.0013 - 0.0023 -00065 -00069 - 0 0075
_ max. concentratlon/mol (No_ of data pomts) 6.2 6 8 7.6 10.1 12.1 12.1 12 1
(kg H,O)-’
(27) (26) (34) (18) (20 (13) (9)
It Only 50 per cent of the number of the data pomts are used between 0.1 and 1 5 moi/kg h Only values nz 21 5 mol kg-’
H,O
In order to estimate the effect of these deviations on the calculation of the salt activity coefficient, the experimental and calculated values of 9 at 200°C were plotted within the coordmates (+ - 1)/m”‘. - m”” (Fig. 5). The curves show that the Pitzer equation does not give a maximum which occurs in the concentration range of the hydrated melts. From the area below the curves, the salt activity coefficient 1s obtained m the usual way from the appropriate general form of the Gibbs-Duhem equation:
Jn Y -e(nt) = @ - 1 t 2/0”‘( Q - l)/m”‘dm”’ In relation concentrations
to the total area, the dtfference between both curves is small at high and accordmgly the error of the y* values is small. Therefore the salt
8h-
0-l
-0
Fig
2
16 Molahty/mol
6 Concentratron
9b
12
'kg-'
dependence
of the mean molal actwty
coeffictent
Y+(,,,,.
63
activity coefficient can be calculated from eqn. (5) with the parameters presented in Table 1. Figure 6 shows the concentration dependence of y-Frm)for the five chosen temperatures. The y*(,,,) values decrease with increasing temperature smnlar to other alkaline^metd chlorides [15-171 and alkaline earth metal chlorides [17,18]. The start of the steep slope is shifted to higher concentrations at higher temperatures_ BE
T. equatzon
Quantitatively, the water activity data of hydrated melts can be described successfully by a suitably modified B.E.T. adsorption equatron [19-221 proposed by Robinson:
aw(l--x,)
x,(1-a_,)=&
1 +
(c-1) Cr
a
w
This equation contains only the two adjustable parameters c and r. The applicability of the adapted B.E.T. equation for hydrated melts of MgClz has already been shown elsewhere [l]. It must be emphasized that this equation is not suitable for more dilute solutions with water activities a, 2 0 5. Recently, Abraham [23] has also derived the analogous notation for the salt activity a, by means of a simple statistical treatment:
X(1 - XJ
x,(1 -hj
r+r(c--ljX
=C
C
where X = az*/r) and x, = the stoichiometric mole fraction of the salt. The parameters I’ and c are the same as those in eqn. (7). Unfortunately up to now, eqn. (8) has been not tested regardmg this applicability, not even by Abraham. One reason can be seen in the unusual standard state for the salt activity which refers to the pure molten salt. At present, no suitable thermodynamic values for MgCl, are available to perform the conversion from the standard state of the pure molten salt into the infinite dilution standard state. If eqn. (8) describes the concentration dependence of the salt activity quite well for high concentrations, and also the values Y+(,,~) calculated from the Pitzer equation are correct, that of loganthmic values of-the activity coefficients has to remain constant at a given temperature according to the general relationship:
h3r -kdI+ RT
ycko, -=lny(Pitzer)-lny(B.E.T.) Yo.,
This fact was checked for our data at 2OC“C. Equation (7) was used for the calculation of the constants c and r from a fit of the water activities and values of 27.41 and 5.23 were obtamed. The salt activities calculated with these constants using eqn. (8) are listed in Table 2. The activity coefficients in the last three columns refer to the mole fraction scale
64
TABLE
2
Comparison
of the activity coefficients
Molahty MgCIJmol
calculated
by eqns. (5) and (8)
7;t(rn> Pltzer
(kg H,O)-’
In7 ft.=-) B ET.
y*<=> In y*(x)
6.939
2 209
5 3890
- 2 8410
8.2300
7.929
3 420
5 8418
- 2 4282
8 2700
9 243 10100
5.907 8 238
6.4088 6.7546
- 1.8828 -15467
8 2916 8 3013
- 1.1922 -0.9088
8.3277 8.3778
11.111
11875
7 1355
12069
16341
7.4690
for total dissoclatron of the salt. The equations
as= uMgC12
=
m:Y:(,,)
=
00)
XX(r,
and Y+(t) = Y *(>>*)41’3(n-z, + 55.5087)
(11)
are used for the conversions. In fact an appropriate constant value of 8.30 f 0.5 was obtained for the difference [In y,(Pitzer) - In y,(B.E.T.)]. CONCLUSIONS
A necessary requirement for the calculatron of salt actrvity coeffrcrents from water activity data is the quantitative description of the concentration dependence over the entire range of concentration, from infinite dilution to the given concentration for these data. Pitzer’s equation can be used for the description of the osmotic coefficients up to ionic strengths of 35 mol/kg HzO, if the demands on the accuracy of the return are lower, which 1s mostly the case under conditions of high concentrations and temperatures_ Because this equation involves only three adjustable parameters, it can be used for treatment of incomplete data series. In the present paper the salt activity coeffrcrents were calculated for the system rvigClz + Hz0 up to I = 35 mol/kg H,O and temperatures up to 200 o C._The simple two-parameter B.E.T. equation should be suitable for the description of changes of the salt activity coefficients within the concentratron range of the hydrated melts. REFERENCES H -H Emons, W. Voqg and W.-F_ Wollny Molten Salts, San Francmx, 1983 The 377-391. H.-H
Emons.
KS
Pitzer m R
Boca Raton.
W. Volgt
and W -F. WoI!ny,
m Proceedings of the Fourth Intematlonal Symposium Electrochemical Society, Princeton, PV 84-2, 1984, Z. Phys
Chem
(Leipzig),
on pp_
in press.
M. wtkowlcz (Ed ). ActtvrtyCoefficientsin ElectrolyteSoluhons.Vol. 1, CRC Press,
Florida,
1979,
p 157
65 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
R.H. Stokes and RA Robmson, J. Am. Chem. Sot., 70 (1948) 1870 Chia-Tsun Lm and WT. Lmdsay. OSW Report, RDPR-71-722, Washington, 1971. Chia-Tsun Lnt and WT. Lmdsay, OSW Report, RDPR 68-347, Washington 1968. J.E. Mayrath and R.H. Wood, J. Chem. Eng. Data, 28 (1983) 56. H-F. Holmes, CF. Baes, Jr_ and RE Mesmer, J. Chem. Therrnodyn., 10 (1978) 953. H.P. Snipes, C Manly and D D. Ensor, J. Chem Eng Data. 20 (1975) 287. G. Tarnmann, Gmelins Handbuch der Anorganischen Chernie, Bd Magnesmm System No 27 B Verlag Chemie, Berlin, 1939, 8th ed. R Huschenbett and C Dahne, Fretb Forsch.-H. A, 384 (1966) 31. Landoldt-Bornstem, Zahlenwerte u. Funktionen IV. Bd. Technik Teil 3, Springer Veriag. Berhn. Gottmgen, Hetdelberg, 1957. KS. Pttzer, J. Phys Chem. 77 (1973) 268. H F. Holmes and R E Mesmer, J Phys. Chem , 87 (1983) 1242 Chia-Tsun Liu and WT. Lmdsay, J. Solution Chem.. 1 (1972) 45. H F. Holmes, C F. Baes, Jr. and RE. Mesmer, J. Chem. Thermodyn., 10 (1978) 983 H F. Holmes and R.E. Mesmer, J. Chem. Thermodyn.. 13 (1981) 1035. H F. Holmes and R.E Mesmer, J Chem Thermodyn ,13 (1981) 1025 J. Sangster, M.-Ch Abraham and M. Abraham, J. Chem. Thermodyn., 14 (1982) 599. J. Sangster, M -Ch. Abraham and M. Abraham, J. Chem Therrnodyn , 11 (1979) 619. M.-Ch. Trudelle, M Abraham and J Sangster, Can. J. Chem.. 55 (1977) 1713. M.-Ch. Trudelle, M. Abraham and J Sangster J. Chem., Eng. Data. 25 (1980) 331. M. Abraham, Rev. Roum Chtm ,26 (1981) 829. J.H. Derby and Z. Ingve, J. Am. Chem Sot., 38 (1916) 1439. F Pohle., Mitt. Kah-Forsch.-Anst., 4 (1927) 5, 26. N W. Kondyrev and G W. Beresowski. Zh Obshch Khim , 5 (1935) 1249.
57
J. Eiectroanal
ON THE CALCULATION OF ACTIVITY COEFFICIENTS OF ELEmROLYTE SOLUTIONS FROM VAPOUR PRESSURE DATA UP TO HIGH TEMPERATURES
THE
SYSTEM
AND MgCl.
CONCENTRATIONS
+ H,O
H -H. EMONS,
W. VOIGT
Sektron Chemle,
Bergakademle
*
and W.-F. WOLLNY Frmberg.
DDR - 9200
Frezberg (G-D. R )
(Recetved 30th April 1984)
ABSTRACT Startmg from recent results of vapour Fressure meas-urements on hydrated melts of MgCI,. we have withm the temperature range of calculated the salt actrvlty coefficients for the system MgCI, &H,O SO-200°C and up to an rome strength of 35 mol/kg H,O. For the Gibbs-Duhem integration, the equattons of Pttzer and the modtfied B ET. equatton were used and their usefulness IS dtswssed
INTRODUCTION Recent developments for transforming and storing energy based on electrochemical and thermal processes often involve salt + water systems at high concentrations and temperatures as an integral part. For modelling and predicting ion association and
complex
formation
reactions
as well as solubilities,
knowledge
of the salt activity
is necessary_ At the present time, the direct determination of salt activity coefficients by means of EMF measurements is not possible for the halides of alkaline metals and alkaline earth metals at high concentrations and temperatures, because suitable galvanic cells which work reversibly are not available. Therefore the Gibbs-Duhem equation is used for the calculation of salt aciivities frcm water activity data for these binary systems. A presumption for these calculations is the complete knowledge of the water activities over the entire range of concentration, from dilute solutions up to interesting concentrations. La general, it is not possible to obtain experimental data of comparable accuracy over a large range of concentration and temperature with only ol~e given apparatus, for -instance a vapour pressure apparatus. From the experimental point of vie\v, four typical concentration/temperature ranges can be distinguished using experimental technology: (1) measurements from 25 to 100 o C and c,,,~ < saturation;
coefficients
* Dtdieated to the memory of Professor Dr. Dr. h c. Kurt Schwabe 002%0728/84/$03.00
0 1984 Elsevier Sequoia S.A.
58
(2) measuiements from 100 to 250 OC (3OOOC) and cmllSf saturation concentration at room temperature; (3) measurements from 100 to 250 OC (300 “C) and c,,,t >, saturation concentration at room temperature; (4) measurements above 300 OC. In particular the concentration range of the hydrated melts is comprised in the third experimental range. For investigations in ths range, an apparatus for vapour pressure measurements was developed by us and measurements were performed for hydrated melts of magnesium chloride over the temperature range 120 to 220 OC [1,2]. In this paper the salt activity coefficients are calculated for the system MgCl. + Hz0 up to nearly saturation concentrations and from 50 to 200 OC. For the first and second experimental range, the data have been taken from the literature_ For the Gibbs-Duhem integration, the equation of Prtzer [3] and a modified B.E.T. equation [4] are used and the applicability of these equations is discussed. WATER
ACTIVITY
DATA
A summary of the distribution of the experimental data for five selected temperatures from 50 to 200 OC is shown in Fig. 1. Apparently reliable values are available with the data published by Liu and Lindsay [5,6] and Mayrath and Wood [7] for relatively dilute solutions at high temperatures. The values determined by Holmes et al. [S] have not been taken into account, because they differ significantly from the data of refs. 5-7. For the hydrated melts, only our own data are available [2]. The values at low temperatures were essentially determined by Mayrath and Wood [7], Snipes et al. [9], Tammann [lo] and Huschenbett and Dahne [ll].
59
The calculation of the water activity a,” from vapour pressure data taken from literature was accomplished in the same manner as for our data using the quotient of the fugacity q,,,: a, = qL, (solution) r/G, (water) T
(1)
The fugacity was calculated from the equatron SL = PW exPKJhJv4
(2)
where B is the second virial coefficient of water vapour. The PVT data of pure water [12] were used for the calculation of B. A five-parameter equation was needed to represent the temperature dependence of B: B
[m3/kg] = - 2.98407 x 1O-2 + 27.1085 T-’ -2.1080
x lo6 T-3 + 1.41176 x 10’ T-4
The pressure dependence regarded as negligible. GIBBS-DUHEM
- 4.81991 x lo3 T-’
of the activity
coefficient
within
the liquid
(3) phase was
INTEGRATION
Pitzer equation
From Fig. 1 it can be recognized that the values are very irregmarly distributed and that the number of data points IS very limited. In this case, the usual description of the concentration dependence of the osmotic coefficient + using an exponential function with more than four adJustable parameters cannot be recommended. In the last few years, the equations proposed by Pitzer [13] have been applied to an increasing extent. Normally, for these equations only three adJustable parameters are needed. The validity of these equations was confirmed for MgCl, for concentrations up to 4.5 mol/kg H,O, that means ionic strengths up to I = 13.5 at 25 o C [3]. All available thermodynamic data for NaCl, KC1 and CsCl in aqueous solutions up to 250 “C have been described within the limits of the experimental errors as well
P41.
The equations for the osmotic coefficient Y+(,,,) of a salt M,, X,_ possess the form: +-l=(z,zx
f”=
)f@ + m (2%?J
-A*[Y/(l
B~=fl(*)+p”)
lny
*(In)
=
and the mean salt activity
Y ) B+ 9 m2 [2(
P&px)3’2/Y]
coefficient
c*
+bY’)]
(4)
exp(-&‘/2) (Z&x)f’t-
m(2vM~x/zJ)B~
f 7 = -A, [I"'/(1 + bl”‘) j17 = 2/3(O)+ (2@“/a21)[l Cy = (3/2) Co
+ (2/b)
+ “~[2(v,vx)3’2/v]
cy
ln(1 + bI’/‘)]
- (1 + cyJ’/” - 1/2&I)
(3 exp( -11/2)]
60
where A, represents the Debye-Htickel coefficient, whose values have been taken from ref. 3 for the different temperatures. Following Pitzr, we have set a and b equal to 2.0 and 1.2 kg’j2 mol-“2, respectively_ /3(O), p(‘) and Ca are parameters, whose values are determined from a least-squares fit of pqn_ (4) to all the experjmental data mentioned above. Thereby, all the data were weighted equally in the fitting procedure. The deviations of the experimental + values from those recalculated by means of eqn. (4) are illustrated in Fig. 2 at 50, 80 and 100°C and in Fig. 3 at 150 and 200 o C. It should be noted that the experimental data up to 100°C (Fig. 2) are scattered irregularly around the calculated ones. For concentrations m > 6 mol kg;’ (Fig. 3), the deviations exceed the experimental error calculated by us, of a maximum of 2.3% the water actrvity. At the highest concentration, the value A+ = 0.104 corresponds Aa, = 7%.
-0.10
e
1
3 Molality/mol
5 kg-’
FIN 2 Dewatlon of the expenmental 80 (A) and 100 o C (0)
Fsg (0)
3. Dewations and 200 o C (0)
of the experimental
7
Q values from the values recalculated
by means of eqn. (4) at 50 (x),
+ values from the values recalculated
by means of eqn
(4) at 150
61
Figure 4 demonstrates that no essential the data points at very high concentrations points at low concentrations.
improvement of the fit is attainabie for using diminished weights for the data
Fig 4. Dewat~ons of the expenmental Q valuesfrom the values recalculated usmg dmuc.shed weights for the data pomts at low concentrations. (s+) All data points, 20; (0) 13 data points, equal number m the ddute and concentrated range, (A) 9 data points, m > 1.5 mol kg-‘.
2 Fig. 5 Dewations for the experimental (0) - rdr-. m XI mol/kg H20. - wm’fl
(Q
and calculated (e) vdues of + at 200 o C in the
COD~I~MCC
62
TABLE
1
Parameters
of P~tzer;s equation
Temperature/
for the system MgCl,
p (0)
oC
50 80 100 150 200 200’ 200 h
T Hz0
C+
0 3151 0 2879 0 3120 0.3122 0 3167 0.3218 0.3308
2.2601 2.6610 1 9774 2 3640 2 5344 2 3870 -1 3905
0.0071 0 0067 0.0013 - 0.0023 -00065 -00069 - 0 0075
_ max. concentratlon/mol (No_ of data pomts) 6.2 6 8 7.6 10.1 12.1 12.1 12 1
(kg H,O)-’
(27) (26) (34) (18) (20 (13) (9)
It Only 50 per cent of the number of the data pomts are used between 0.1 and 1 5 moi/kg h Only values nz 21 5 mol kg-’
H,O
In order to estimate the effect of these deviations on the calculation of the salt activity coefficient, the experimental and calculated values of 9 at 200°C were plotted within the coordmates (+ - 1)/m”‘. - m”” (Fig. 5). The curves show that the Pitzer equation does not give a maximum which occurs in the concentration range of the hydrated melts. From the area below the curves, the salt activity coefficient 1s obtained m the usual way from the appropriate general form of the Gibbs-Duhem equation:
Jn Y -e(nt) = @ - 1 t 2/0”‘( Q - l)/m”‘dm”’ In relation concentrations
to the total area, the dtfference between both curves is small at high and accordmgly the error of the y* values is small. Therefore the salt
8h-
0-l
-0
Fig
2
16 Molahty/mol
6 Concentratron
9b
12
'kg-'
dependence
of the mean molal actwty
coeffictent
Y+(,,,,.
63
activity coefficient can be calculated from eqn. (5) with the parameters presented in Table 1. Figure 6 shows the concentration dependence of y-Frm)for the five chosen temperatures. The y*(,,,) values decrease with increasing temperature smnlar to other alkaline^metd chlorides [15-171 and alkaline earth metal chlorides [17,18]. The start of the steep slope is shifted to higher concentrations at higher temperatures_ BE
T. equatzon
Quantitatively, the water activity data of hydrated melts can be described successfully by a suitably modified B.E.T. adsorption equatron [19-221 proposed by Robinson:
aw(l--x,)
x,(1-a_,)=&
1 +
(c-1) Cr
a
w
This equation contains only the two adjustable parameters c and r. The applicability of the adapted B.E.T. equation for hydrated melts of MgClz has already been shown elsewhere [l]. It must be emphasized that this equation is not suitable for more dilute solutions with water activities a, 2 0 5. Recently, Abraham [23] has also derived the analogous notation for the salt activity a, by means of a simple statistical treatment:
X(1 - XJ
x,(1 -hj
r+r(c--ljX
=C
C
where X = az*/r) and x, = the stoichiometric mole fraction of the salt. The parameters I’ and c are the same as those in eqn. (7). Unfortunately up to now, eqn. (8) has been not tested regardmg this applicability, not even by Abraham. One reason can be seen in the unusual standard state for the salt activity which refers to the pure molten salt. At present, no suitable thermodynamic values for MgCl, are available to perform the conversion from the standard state of the pure molten salt into the infinite dilution standard state. If eqn. (8) describes the concentration dependence of the salt activity quite well for high concentrations, and also the values Y+(,,~) calculated from the Pitzer equation are correct, that of loganthmic values of-the activity coefficients has to remain constant at a given temperature according to the general relationship:
h3r -kdI+ RT
ycko, -=lny(Pitzer)-lny(B.E.T.) Yo.,
This fact was checked for our data at 2OC“C. Equation (7) was used for the calculation of the constants c and r from a fit of the water activities and values of 27.41 and 5.23 were obtamed. The salt activities calculated with these constants using eqn. (8) are listed in Table 2. The activity coefficients in the last three columns refer to the mole fraction scale
64
TABLE
2
Comparison
of the activity coefficients
Molahty MgCIJmol
calculated
by eqns. (5) and (8)
7;t(rn> Pltzer
(kg H,O)-’
In7 ft.=-) B ET.
y*<=> In y*(x)
6.939
2 209
5 3890
- 2 8410
8.2300
7.929
3 420
5 8418
- 2 4282
8 2700
9 243 10100
5.907 8 238
6.4088 6.7546
- 1.8828 -15467
8 2916 8 3013
- 1.1922 -0.9088
8.3277 8.3778
11.111
11875
7 1355
12069
16341
7.4690
for total dissoclatron of the salt. The equations
as= uMgC12
=
m:Y:(,,)
=
00)
XX(r,
and Y+(t) = Y *(>>*)41’3(n-z, + 55.5087)
(11)
are used for the conversions. In fact an appropriate constant value of 8.30 f 0.5 was obtained for the difference [In y,(Pitzer) - In y,(B.E.T.)]. CONCLUSIONS
A necessary requirement for the calculatron of salt actrvity coeffrcrents from water activity data is the quantitative description of the concentration dependence over the entire range of concentration, from infinite dilution to the given concentration for these data. Pitzer’s equation can be used for the description of the osmotic coefficients up to ionic strengths of 35 mol/kg HzO, if the demands on the accuracy of the return are lower, which 1s mostly the case under conditions of high concentrations and temperatures_ Because this equation involves only three adjustable parameters, it can be used for treatment of incomplete data series. In the present paper the salt activity coeffrcrents were calculated for the system rvigClz + Hz0 up to I = 35 mol/kg H,O and temperatures up to 200 o C._The simple two-parameter B.E.T. equation should be suitable for the description of changes of the salt activity coefficients within the concentratron range of the hydrated melts. REFERENCES H -H Emons, W. Voqg and W.-F_ Wollny Molten Salts, San Francmx, 1983 The 377-391. H.-H
Emons.
KS
Pitzer m R
Boca Raton.
W. Volgt
and W -F. WoI!ny,
m Proceedings of the Fourth Intematlonal Symposium Electrochemical Society, Princeton, PV 84-2, 1984, Z. Phys
Chem
(Leipzig),
on pp_
in press.
M. wtkowlcz (Ed ). ActtvrtyCoefficientsin ElectrolyteSoluhons.Vol. 1, CRC Press,
Florida,
1979,
p 157
65 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
R.H. Stokes and RA Robmson, J. Am. Chem. Sot., 70 (1948) 1870 Chia-Tsun Lm and WT. Lmdsay. OSW Report, RDPR-71-722, Washington, 1971. Chia-Tsun Lnt and WT. Lmdsay, OSW Report, RDPR 68-347, Washington 1968. J.E. Mayrath and R.H. Wood, J. Chem. Eng. Data, 28 (1983) 56. H-F. Holmes, CF. Baes, Jr_ and RE Mesmer, J. Chem. Therrnodyn., 10 (1978) 953. H.P. Snipes, C Manly and D D. Ensor, J. Chem Eng Data. 20 (1975) 287. G. Tarnmann, Gmelins Handbuch der Anorganischen Chernie, Bd Magnesmm System No 27 B Verlag Chemie, Berlin, 1939, 8th ed. R Huschenbett and C Dahne, Fretb Forsch.-H. A, 384 (1966) 31. Landoldt-Bornstem, Zahlenwerte u. Funktionen IV. Bd. Technik Teil 3, Springer Veriag. Berhn. Gottmgen, Hetdelberg, 1957. KS. Pttzer, J. Phys Chem. 77 (1973) 268. H F. Holmes and R E Mesmer, J Phys. Chem , 87 (1983) 1242 Chia-Tsun Liu and WT. Lmdsay, J. Solution Chem.. 1 (1972) 45. H F. Holmes, C F. Baes, Jr. and RE. Mesmer, J. Chem. Thermodyn., 10 (1978) 983 H F. Holmes and R.E. Mesmer, J. Chem. Thermodyn.. 13 (1981) 1035. H F. Holmes and R.E Mesmer, J Chem Thermodyn ,13 (1981) 1025 J. Sangster, M.-Ch Abraham and M. Abraham, J. Chem. Thermodyn., 14 (1982) 599. J. Sangster, M -Ch. Abraham and M. Abraham, J. Chem Therrnodyn , 11 (1979) 619. M.-Ch. Trudelle, M Abraham and J Sangster, Can. J. Chem.. 55 (1977) 1713. M.-Ch. Trudelle, M. Abraham and J Sangster J. Chem., Eng. Data. 25 (1980) 331. M. Abraham, Rev. Roum Chtm ,26 (1981) 829. J.H. Derby and Z. Ingve, J. Am. Chem Sot., 38 (1916) 1439. F Pohle., Mitt. Kah-Forsch.-Anst., 4 (1927) 5, 26. N W. Kondyrev and G W. Beresowski. Zh Obshch Khim , 5 (1935) 1249.