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Precision measurements of the colligative properties of solutions of strong electrolytes and of sea and brackish water: Theoretical calculation of the osmotic coefficient
PRECISION
MEASUREMENTS
OF
THE
COLLfCATfVE
PROPERTIES
OF SOLUTIONS OF STRONG ELECTROLYTES AND OF SEA AND BRACKISH WATER: THEORETfCAL CA*.CULATION OF THE OSMOTIC COEFFICIENT
The osmotic coefficients of sea and brackish water are determined by means of cryoscopic measurements. From these data the “best values” of the individual ionic radii and the dependence of the dielectricconstant on concentration are calcularcd using a two-parametric mjnjrnum-square-method of best fitting byapplyjng the Debye-HiickcI ~beo~a~ordjng to Bortino. with the elect of the”highcr terms*’ of Gronwaff and La Mer. The method is valid for recalculating experimental cryascopies with an error of a few thousandths of a degree over the whole concentraGon interval of sea and brackish wafer, which rcpresenrs its dilutions. These techniques may also be applied to the prediction of other thermodynamic prop&es of sea wafer, and above all. of brackish water, which today are assuming great importance in less expensive desalting methods such as reverse osmosis and clecfrodialysis. In order to make theoretica calcufations on the lhc~odynam~c properties of sea and brackish water, it is necessary fo refer to the theory of co~cenf~ted solutions of strong electrolytes. fn this field there have been a great number of studies, some quite recent, but it cannot be said that the problem has been definitely resolved. even fhongh the most modern theoretical formulations. based on Mayer’s theory of”ionicclusfers”(lf and on the “hyperreticulated chains*‘equation (21, may hopefufly lead to an interpretarion more closefy bound to the microscopic properties of the ions and of the solvent. fn any case. every theory regarding solutions of strong electrolytes amplifies the original formulation of Debye and Hiickel(3) and in fact is compared to it as a first test of validity for dilute solutions of salts of the NaCl or KC1 type. In this theory, an “ionic radius” averaged from atl ions present, is introduced for concentrations which are not grearty difuted, and at higher c~~~nt~tions it becomes DesuEnntion,IO( 1972) 263-272
C_ DESAK
264
et 01.
necessary to consider even the short range forces and introduce a relative parameter. The first treatment in this sense was carried out by Hiickei (4) on the basis
of dielectric saturation, congicbating such interactions toward the lower powers of l/r. There have also been attempts (5) to congiobate toward infinite powers, that is, toward rigid exclusion forces, but only :he cluster theory has met the prob!em satisfactorily. At any rate, even the original theory of Hiickei succeeds well in rendering exper~mentei results up to fairly high concentrations for solutions of simple electrolytes. so long as a more ample meaning than the initial one is given to the variation of the dielectric constant. Bonino (6) has interpreted this variation as the “efficacious dielectric constant”. finding an exponential law D = Do ems
(1)
with Do as the dieiectric constant of the solvent and [l proportional to an appropriate power of the concentration (94 in the appIications made): in this way absurd negative values of D are avoided and trends closer to the experimental ones are obtained for macroscopic dielectric constants (7). An attempt was also made to extend the theory to individual ionic radii, or averaged on the dimensions of the second ion only, which approaches the first. In fact, a definitive theory should contemplate the two-by-two minimum distances between all the ion couples for of Boninds these **maximum approach radii”. In any case, the introduction indi~duai ionic radii also aifows better separation of the characteristic behaviours of single ions in different solutions. than the mere average radius, and above all aids in the search for correlations with other physical quantities. In this way iinear
relations with the crystallographic-radius/polarizability cations (8) and for anions (9). from
This formulation has made experimemal data by means
ratio were found both for
possible the determination of these ion radii of hi-parametric optimization with electronic
computers_ This therefore requires the most precise and consistent experiments possible for the whole concentration field and on complete series of electrolytes. For this purpose, high precision cryoscopies represent the best experiments. but another series of apparatus for ebuiiioscopy (IO). vapour tension and osmotic pressure in order to obtain precise thermodynamic data comparable at different
temperatures are also being set up. From the cryoscopies
of aikali chlorides.
of D were found (II) for wide concent~tion
ionic radii and variation
constants
intervals (the smallest from 0.02 to
I -moifl) with average disparities between those calculated and those found of one thousandth of a degree, starting from the hypothesis of the equality of the ionic radii of the chlorine anion and the potassium cation, which was discussed in the paper mentioned. This work is being continued and extended to other halides. bivaient ions and different solvents. The theoretical extension to the cluster theory is also being planned.
Desdinaxion. IO (I
972) 263-272
THEORETICAL
CALCULATIOK
THEORETICAL
FORMULATION
As already was worked
pointed
OF THE OSSlOTtC
out in previous
out for two parameters,
265
COEFFICYENT
works
(If),
an optimization
method
s and Ui: J, for the “efficacious”
dielectric
constant variation. and a, for one of the two “ionic radii”. while the other *.v+x obtained from previous analyses. starting from KCI where a, = o,,. This method was applied to alkali chloride s. and its application is now being studied for other halides. The constants found differed little from those found by Bonino er al. (8) and linearly correlated to other ionic parameters (9). However, it had not up to now been applied on such a vast scale to polyvalent electrolytes: but Dejalc (12) has already discussed this case and Dejak and hlauei (IS} have treated it and examined its theoretical basis. In order to calculate the thermodynamic properties of sea and brackish water, it \vrlt in fact be necessary to consider, besides Cl- and Na’ ions. a: least rb1g+ + and SO, - - ions as well, and then plurivalcnt and asymmetric electrolytes. It will also be indispensable to find the characteristic parameters of these latter two ;nnc whbh UVWP nnt nntimim-rl in the nrc=vinll< works Hnrvrver sinrr the infltlence of these ions is not preponderam in the mixture. it will be enough to use the relative parameters obtained directly from data in the literature and later on to determine the relative cryoscopies e~~rjmcntall~ in a more consistent way and on large concentration intervals. as well as a more adequate equalizing treatment. Caiculation of the activity coefhcient of the solvent or osnroiic cu~_ffiri~r will be obtained (IJ) with the expression In_/,
=
- I’(, c Zi v’i zi j, [Z,
where (15) the sum between electrostatic
refers energy
(xi)
+ 2 Ii A’, (.r,)]
(2)
to all the ion species present. j; is the relationship of an ion at distance u from a one atomic-unit charge
(2, s’jD oi) and the thermal agitation energy LT. while s, is the product the reciprocal thickness of the ionic atmosphere
of (I; by
where c is the electron charge in e.s.u ._ .zI“ the Avogadro number, k boltzmann’s constant. T absolute temperature. 2, and \ei the electrovalence and the number of ions of the ith species and finally c the concentration in moles per litre of solution and FO the volt;me in litres of a mole of solvent. both at temperature T. It is to be remembered that in (1)
with x0 calculated 0.001035 273,fli”K.
an-ording
to (3) but with D = 13, = ~8.23-0.4044
(T-To)’ for the solvent And finally it will be:
water,
referred
to its freezing
(T-
temperature
To)+
T,
=
Desulir~afion. 10 (1972) 263-272
266
C. DHAK
Xl (4 =
et Of.
j-&
(5) I
1t X,
.
(2)
dr =
0
In the analysis we referred to M&I, and Na$O, for the two bivvlent ions. since the opposite monovalent ion is prevalent in sea water. using the experimental cryoseopies for M&I, of Rivet1 (Id), Loomis (17) and Menzel (18). which were used in the authors’ previous works. while for Na,SO, besides those of Loomis (17), those of Roberts (19) and Harkins and Jones (3) are also used. From this optinli~tion of the parameters refative to the two salts. a very small (see Table I) ion radius for SO,- - appears, that is. such as to require a forI-ABLE
I
.-_._____-
_“.__
.-__-.-
_ .--.-
__-..___
(A)--+}
0,
-Solution
-_--___L?HB
LiCl NJCI KCl FEf
5.81 4.40 3.52 2.99 1.43
S-76 4.40 3.54 3.07 I.84
3.52 3.52 3.52 3.52
3.54 3.54 3.54 3.54
2.097 0.901 0.263 --0.092 0.233
2.162 0.883 0.216 -0.212 -0.200
0.8 0.7 0.7 0.3 I.1
0.7 0.R 0.5 0.7 1.3
NazSO4 M&k
4.40 5.20
4.G 4.80
1.76
3.52
2.13 3.54
--b 1.46
-0.307 1.246
7.6 15.2
---_ 6.0 15.2
S.W.
-
-
-
0.331
4.3
3.2
mulation truncated
Id)
--_-
Gi.
_
a_ (A)
s
__.-... .-.. _.-_ DHB GL
. -._. ._ _____ __ .___ DHB GL
Ah wr. TI0-J =cj ___
EkCfdj.k-
--
_--.- ----
-
---_-_-
-
. .._..- ____--
0.306
_
DIiB GL ._ .___ ____ _...
of the fundamental equation of the theory (Poisson-Bo~tzmann) not at the first term of the development in series of powers of Boltzmann’s
exponential (21). If we limit ourselves, as is commonly done (22), to the first three terms, we then obtain: Info = - 1’9 c zj Ii (Ziji [Z, (.A-&+ 3 6’ A’, (Xi)] (6) - =;Lif 92 CzZ (X3 + Q P x2 (Xi)] + + Gii
q3 I% 01) + 3 P X3 WI
+
-l- 5 if qf C& (x3 -l- 3 P % (Xi)] f + *..I Dedination.
10 (1972)
263-272
THEOREflCAL
CALCULATION
OF THE OSMOTIC COEFFiCIENT
267
where
X>(x) =
----
t tt* (3x) -- _-.- --.^._
x -
s
12(1 +5X)”
I
x + f
-
1
t1g (4x) ____-_.._
3
2
with
. (du/u) I = +? ._ ._.. J,
e-’
,t (x)
(C-“j.Y)-”
. I r”‘- ’ X, (f) df
1:
z,
(x,
= 3 X,, (x)
-
--:P
(9)
-0
while
In this way more reIiabIe vatucs are obtained for the parameters. &+&ally those of Na2SOS. which are reported in Table I in the cofumn headed GL (Gronwatl-La Mer et ol.) in comparison to those headed DHB (Debye-Hiickel-Bonino) based on (2). The values reported refer exclusively to the interval 0.02-I molil, and not to more ample intervals like those previously studied for aqueous solutions of pure electrolytes (II). In this cast the disparity between those calculated and those found does not substantiatly differ, excluding casts prcviousiy ci:ed, with the two different formulations. since for pure electrolytes the differences become notable only by taking into account concentrations higher than I mol/l: for mixtures of electrolytes the incidence already appears at lower concentrations, as will be seen in Fig_ I. Now, if one wishes to extend the theory to mixtures of different solutes, as DesaIiim!ion. 10 (I 972) 263-272
C. DUAK
268
t?f Of.
4 m.8 -xi
I
..
* :+-or. .
0pl.
..
. C.IC.
v
opt
GLG
. L.~c.
. s&L
300
100
t
- 2.0
-1.5
-1 .o
-0.5
Log
m+
0
Fig. I.
is necessary for applications to sea and brackish water. there are very few changes in the above-mentioned formulas. Only in q* must the sums be intended as not only extended to the numbers of r ions in which the only dissolved sait is disassociated. but in Eq. (IO) at effective (c v)~ concentrations of every single ion. deriving from the various salts which make up the dissolved misture: in place of (c v),. (nr I.)~may also be used, given the adimensionality of the qn’s_ For the parameters, every sin& ion radius optimized for the corresponding concentration intervat is to be used. while for the constant s a weighted average of relative values for solulions of singk =Its may be attempted. In this casr. in place of (4), we shall have: 1, Do ~0 ?li2 =
where the summation
Zj
7,
Sj
[Zi
Yi
a: Jr
VII
,wiIl be extended
to the singIe saltsj which are disassociated. between the motar fraction of the single salts and their total. Another alternative will be that of directly og timizing only the’ parameter S, defining it in this case by means of
each one into i ions and where yj is the relationship
@0&J=
= S xj -;;r&j 5.iu&
(12)
On the other hand. the simptest treatment wouid be that of utilizing only the optimized parameters for the predominant NaCI solutes, while for the secondary ions the theoretical constants, that is to say. those obtained from the linear correlation of Bonino (8) and Rolla (9) of the ai’s with the crystallographic-radius/ poIarizabiIity ratio and s with a weighted average of crystalline compressibility could be used, Desalination,
10 (1972) 263-272
TNEORETICAL
CALCULATBXU OF THE OSMOTK- COEFFKIENT
249
EXPERIMENTAL
The cryoscopy apparatus has previously been described (fl). Temperatures were measured by a platinum resistance thermometer coupled with a Miiller bridge. Precision thus obtainable is within a few ten thousandths of a degree and constant throughout the temperature interval. Equilibrium is assured by a pneumatic stirrer made of high quality glass. The concentration must bc known so as to assure regularity oferror throughoui the interval with a constant precision which for alkali halides should be on the order of a few hundred thousandths of molality. This is carried cuit with a “HilgerRayleigh” interferometer. This method yields really precise measurements at high concentrations, where more objective data arc required. while for the lower concentrations they have the advantage of being consistent with the rest, with no pretense of matching the precision of those obtained with other instruments. However, a tesssr relative precision, that is, an acxidentat error higher in percentage at lower concentrations has negligible influence on the parameters obtained, while uncertainty in accuracy. that is systematic errors in the measurements of other workers in more concentrated solutions have often made every determination of objective ionic parameters quite problematic. For sea water we used so-called “reconstructed sea water” (23) with the following concentrations: NaCI 0.337, MgCIL 0.0665 and Na,SO,0.029 for a total of 0.5325 mole per kg H,O (including crystallization water of the latter two salts). The salts used were: Merck type “Suprapur”cat. no, 6406for NaCI, Merck type”P.A.” cat. no. 5833 for M&X1-6H20 and Merck “Supnpui’ cat. no. 6647 for Na,SO,.
Magnesium chloride was determined with EDTA, whife the usual caution was used for NatSO in order to avoid errors owing to its well-known hygroscopicity. Brackish water was obtained by proportionally reducing these concentrations practically in geometric progression in the dilution interval till about 0.02 mo!/kg. Experimental data are shown in Table 11. Here m stands for the total mofality reported above, but if aFtdated as the sum of the ion mofslities, it must be divided by a factor C, the weighted average of the number of ions into which every single salt is disassociated. In this way a A T/m is obtained which tends towards infinite dilution at i; times the extrapolated value for non-electrotytes. At this point Info (24) and ionic force are calculated, always keeping in mind the detinltion of m given above- The Info/nx data were equalize!d and interpolated with the maximum possible objectivity at intervals of 0.1 in log tn. For the higher dilutions a trend parallel to the limit law of Debye-Hiickel, as already discussed, (IQ, 25) was sought. The equalized data thus obtained are also reported in Table II and may represent an experimental basis for calculating other thermodynamic quantities of sea and brackish water. DesuIinafiun, 10 (1972) 263-272
270 TABLE
I1
0.023085
3.8033
0.033954
3.7462
0.071785
3.6685
0.093905
3.7187
0.189279
3.5651
0.309489
3.5465
0.424423
3.5507
0.497147
3.540)
Ih?ERPRITATiON
1.7 1.6 1.5 1.4 1.3 3.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3
220
247 275 303 331 358 383 406 427 446 459 465 465 461 451
OF THE RESULTS
In order to interpret these experimental results we first tried to utilize Eq. (2) with optimized ionic radii for the predominant NaCI salt and the theoretical values for the other ionic radii with an average value of 5 according to (I I). This curve is shown in Fig. I indicated by T-T. (a theoretical-s theoretical). The trend is quite unsatisfactory. It was therefore necessary to resort to the optimization of new parameters for MgClr and Na,SO,, first with (2) then with (9 the results are shown$n Table I. With these parameters, still using (I I) a& (2), we calculated a new trend indicated by CC in Fig. 1 (a calculated-s calculated) and improved it by the introduction of the higher terms of Gronwah-La Mer according to (6) and reported with the same letters but in bold type. These trends curye too much at high concentrations, even though the second already more nearly approaches theexperimenta! values.
We then attempted
to optimize the parameter f with (12) and obtained. with of Brsnino et al. for tbheMg+ l and SO,-- ion radii, a trend which better interpolates the experimental results (the curve T-0. that is, u’s theoretical-s’s optimized). A further improvement of the agreement is reached with the a’s calculated as mentioned above (indicated by CO.) but only the use of (6) in this case succeeds in giving the proper curve to the trend (CO. in bold type) the theoretical
values
in order to render the experimental one we& Desalinorion.10 (1972)263-272
THEORETliCAL
CALCULATION
OF THE OSMOTlC
COEFFICIENT
271
CONCLUSIONS It can
agreement
thus
be
between
seen that with this formulation it is possible to reach a good theory and practice, arrivieg not only at a root-mean square
deviation of a few thousandths of “C throughout the interval but niso a trend which substantially reproduces the one found experimentally. In Fig. 2 it can be seen how the osmotic cocficicnt varies from salt to salt in an extremely sensitive way and how the data far sea and brackish water fit between these.
.
ihe experimental
obtained and the theoretical fli‘rmulations
confirmed by them. any other thermodynamic application is possible. and thus a basis for predictions at other quantities is formed. in any case, it will be interesting to see later on the dependence on temperature of the trends studied as a function of concentration when the apparata for ebullioscopy, vapour tension and osmotic pressure are definitely ready. Osmotic pressure, even though measurable with much less precision than vapour tension, as is amply shown in the literature, is however of great interesi in that osmotic equilibrium serves as a starting point for all calculations of reverse
osmosis. This,
in fact,
today
represents one of the most economical desalting Desalinarion, IO ( 1972) 263-272
mtihods, above all for small and medium plants and for brackish water and a more theoretical study of it will certainly iead to useful ideas. At any rate, under the aspect of the theory of the solutions of mixtures of strong electrolytes it must be pointed out that the agreement reached is noteworthy, considering that one of the most common objections to the DebyeHiickel theory is that generally it does not succeed in rendering well the experimental trends for mixtures. A study, which has already been started (26). on Mayer’s clusters theory and on its relative applications (27) will Perhaps further improve on the agreement between theory and experimental data. The authors wish to than& Prof. F. Momicchioli and Dr. G. Grandi of the Institute of Physical Chemistry of the University of Modena and Dr. G. Paschina of Istituto Chimico Policattedra of the University of Cagliari for their collaboration as well as the Consiglio Nazionale delle Ricerche for its financial contribution. For the calculations we used the IBM 1130 of the Computing Centre of the University of Cagliari and the IBM 7090 of the “CNUCE” of Pisa by teleprocessing. REFERh’cEs 1. 2.
J. E. MA=
J. Chem. Phys., 18 (1950) 1426. A&D 3. C. R.wshtt. J. Chem. Phys., 48 (1968) 2742; J. Whys. Chem., 72 ( 1968)
H. L F-MAN 3352.
P. Drew AND E. Hiiclru. Physik 2.. 24 (1923) 165. E HitPhys& 2.. 26 (192.5) 93. M. Durr~ A!SVS. N. BAG, indian J. Physics, 24 (1950) 61; M. EIGLV ASD E. WI-E, 2. H&r0&m., ss (19s)) s34. 6. G. B. BohrnO. Atti &de Acznd. ?taf.., 4 (I 933) 41 S. 7. c. DEBA& Ann. Chim., 47 (1957) 1039. 8. G. B. E.b&wo Abm G. f3Ehntu, Atii &air Aecad- Itufia, 4 f 1933) 445. 9. M. Rou, Atti X corigr. Naz. CAim.. 2 (1938) 459. 10. G. cocc0, C. Duiur phi 0. DEVOTE. Chem. Phys. Lcrrtrs, If (1971) 198. 11. F. Mowx?uou, 0. bo~0, G. GRAKDI AND G. Cocco, Ber. Bunsenges_ Physik. Cht-m., 74 3. 4. 5.
(1970) 59; P. Cwowou, f2 13.
14. 1s. Id. 17.
18. 19. 20.
21. 22
23. 24.
25.
26. 27.
F. Mo~lcanou Ah?) G. GRAND), Bof. Sri. Fat. Chim. id Bulogna, 24 (1966) 133. C. DUAIE, Ga==.C&m. Ztaf., 83 (I9S3) 3. C. DESADZAND 1. Mm, Ann. Chim., SO (1950) 956. G. B. B~‘N;IFIO Ah= M. ROLL*, Atri I&& Accad. Italia, 4 (1933) 465. C. I)UAR, d&z?& chim., 49 (1959) 1016. A. C. D. Rfwrr. Z- Phys&. Chem.. 80 (1912) S46. E H. Loom Wipdm. Ann.. 50 (18%) 503. H. MENU, 2. Ebkrrockm., 33 (1927) 68. W. D. Hnmr tiw W. A. Rosurn, 1. Am. Cfrun. Sac.. 38 (1915) 2679. H. C. 3o?cls,H.&ties in Aguemu Sofutiom, Camtgk Inst. Washington,1907,p. 36. T. H. GRONWAU, V. K. Ijr MARmz K. SA~W’ED, Phys& 2.. 29 (1928) 358. V. K. LA AQ%,T. H. GRON~~ Ah-D L S. GRUFF, J. Phys. Chci*., 2 t?;H) 35. D. F. MuR?~. Marinr Chcmislry,Vol. I, p. 59, M. Dekkcr Inc., XJw York, N.Y., 1959. c. DEJA& Ann. chim., 43 (19S3)90. C.DEaAxAND1.MAzzEl.Ann. c&7., 47 (1957) 1019. C -A*; Gnu. C&m_ fraf.., 95 (l%S) loJ!L C ikr*JE. G. Lrannr AND M. Mlsucm, Gazz. Chim. id., 95 (1963 1156. Desalination,10 (1972) 263-272
MEASUREMENTS
OF
THE
COLLfCATfVE
PROPERTIES
OF SOLUTIONS OF STRONG ELECTROLYTES AND OF SEA AND BRACKISH WATER: THEORETfCAL CA*.CULATION OF THE OSMOTIC COEFFICIENT
The osmotic coefficients of sea and brackish water are determined by means of cryoscopic measurements. From these data the “best values” of the individual ionic radii and the dependence of the dielectricconstant on concentration are calcularcd using a two-parametric mjnjrnum-square-method of best fitting byapplyjng the Debye-HiickcI ~beo~a~ordjng to Bortino. with the elect of the”highcr terms*’ of Gronwaff and La Mer. The method is valid for recalculating experimental cryascopies with an error of a few thousandths of a degree over the whole concentraGon interval of sea and brackish wafer, which rcpresenrs its dilutions. These techniques may also be applied to the prediction of other thermodynamic prop&es of sea wafer, and above all. of brackish water, which today are assuming great importance in less expensive desalting methods such as reverse osmosis and clecfrodialysis. In order to make theoretica calcufations on the lhc~odynam~c properties of sea and brackish water, it is necessary fo refer to the theory of co~cenf~ted solutions of strong electrolytes. fn this field there have been a great number of studies, some quite recent, but it cannot be said that the problem has been definitely resolved. even fhongh the most modern theoretical formulations. based on Mayer’s theory of”ionicclusfers”(lf and on the “hyperreticulated chains*‘equation (21, may hopefufly lead to an interpretarion more closefy bound to the microscopic properties of the ions and of the solvent. fn any case. every theory regarding solutions of strong electrolytes amplifies the original formulation of Debye and Hiickel(3) and in fact is compared to it as a first test of validity for dilute solutions of salts of the NaCl or KC1 type. In this theory, an “ionic radius” averaged from atl ions present, is introduced for concentrations which are not grearty difuted, and at higher c~~~nt~tions it becomes DesuEnntion,IO( 1972) 263-272
C_ DESAK
264
et 01.
necessary to consider even the short range forces and introduce a relative parameter. The first treatment in this sense was carried out by Hiickei (4) on the basis
of dielectric saturation, congicbating such interactions toward the lower powers of l/r. There have also been attempts (5) to congiobate toward infinite powers, that is, toward rigid exclusion forces, but only :he cluster theory has met the prob!em satisfactorily. At any rate, even the original theory of Hiickei succeeds well in rendering exper~mentei results up to fairly high concentrations for solutions of simple electrolytes. so long as a more ample meaning than the initial one is given to the variation of the dielectric constant. Bonino (6) has interpreted this variation as the “efficacious dielectric constant”. finding an exponential law D = Do ems
(1)
with Do as the dieiectric constant of the solvent and [l proportional to an appropriate power of the concentration (94 in the appIications made): in this way absurd negative values of D are avoided and trends closer to the experimental ones are obtained for macroscopic dielectric constants (7). An attempt was also made to extend the theory to individual ionic radii, or averaged on the dimensions of the second ion only, which approaches the first. In fact, a definitive theory should contemplate the two-by-two minimum distances between all the ion couples for of Boninds these **maximum approach radii”. In any case, the introduction indi~duai ionic radii also aifows better separation of the characteristic behaviours of single ions in different solutions. than the mere average radius, and above all aids in the search for correlations with other physical quantities. In this way iinear
relations with the crystallographic-radius/polarizability cations (8) and for anions (9). from
This formulation has made experimemal data by means
ratio were found both for
possible the determination of these ion radii of hi-parametric optimization with electronic
computers_ This therefore requires the most precise and consistent experiments possible for the whole concentration field and on complete series of electrolytes. For this purpose, high precision cryoscopies represent the best experiments. but another series of apparatus for ebuiiioscopy (IO). vapour tension and osmotic pressure in order to obtain precise thermodynamic data comparable at different
temperatures are also being set up. From the cryoscopies
of aikali chlorides.
of D were found (II) for wide concent~tion
ionic radii and variation
constants
intervals (the smallest from 0.02 to
I -moifl) with average disparities between those calculated and those found of one thousandth of a degree, starting from the hypothesis of the equality of the ionic radii of the chlorine anion and the potassium cation, which was discussed in the paper mentioned. This work is being continued and extended to other halides. bivaient ions and different solvents. The theoretical extension to the cluster theory is also being planned.
Desdinaxion. IO (I
972) 263-272
THEORETICAL
CALCULATIOK
THEORETICAL
FORMULATION
As already was worked
pointed
OF THE OSSlOTtC
out in previous
out for two parameters,
265
COEFFICYENT
works
(If),
an optimization
method
s and Ui: J, for the “efficacious”
dielectric
constant variation. and a, for one of the two “ionic radii”. while the other *.v+x obtained from previous analyses. starting from KCI where a, = o,,. This method was applied to alkali chloride s. and its application is now being studied for other halides. The constants found differed little from those found by Bonino er al. (8) and linearly correlated to other ionic parameters (9). However, it had not up to now been applied on such a vast scale to polyvalent electrolytes: but Dejalc (12) has already discussed this case and Dejak and hlauei (IS} have treated it and examined its theoretical basis. In order to calculate the thermodynamic properties of sea and brackish water, it \vrlt in fact be necessary to consider, besides Cl- and Na’ ions. a: least rb1g+ + and SO, - - ions as well, and then plurivalcnt and asymmetric electrolytes. It will also be indispensable to find the characteristic parameters of these latter two ;nnc whbh UVWP nnt nntimim-rl in the nrc=vinll< works Hnrvrver sinrr the infltlence of these ions is not preponderam in the mixture. it will be enough to use the relative parameters obtained directly from data in the literature and later on to determine the relative cryoscopies e~~rjmcntall~ in a more consistent way and on large concentration intervals. as well as a more adequate equalizing treatment. Caiculation of the activity coefhcient of the solvent or osnroiic cu~_ffiri~r will be obtained (IJ) with the expression In_/,
=
- I’(, c Zi v’i zi j, [Z,
where (15) the sum between electrostatic
refers energy
(xi)
+ 2 Ii A’, (.r,)]
(2)
to all the ion species present. j; is the relationship of an ion at distance u from a one atomic-unit charge
(2, s’jD oi) and the thermal agitation energy LT. while s, is the product the reciprocal thickness of the ionic atmosphere
of (I; by
where c is the electron charge in e.s.u ._ .zI“ the Avogadro number, k boltzmann’s constant. T absolute temperature. 2, and \ei the electrovalence and the number of ions of the ith species and finally c the concentration in moles per litre of solution and FO the volt;me in litres of a mole of solvent. both at temperature T. It is to be remembered that in (1)
with x0 calculated 0.001035 273,fli”K.
an-ording
to (3) but with D = 13, = ~8.23-0.4044
(T-To)’ for the solvent And finally it will be:
water,
referred
to its freezing
(T-
temperature
To)+
T,
=
Desulir~afion. 10 (1972) 263-272
266
C. DHAK
Xl (4 =
et Of.
j-&
(5) I
1t X,
.
(2)
dr =
0
In the analysis we referred to M&I, and Na$O, for the two bivvlent ions. since the opposite monovalent ion is prevalent in sea water. using the experimental cryoseopies for M&I, of Rivet1 (Id), Loomis (17) and Menzel (18). which were used in the authors’ previous works. while for Na,SO, besides those of Loomis (17), those of Roberts (19) and Harkins and Jones (3) are also used. From this optinli~tion of the parameters refative to the two salts. a very small (see Table I) ion radius for SO,- - appears, that is. such as to require a forI-ABLE
I
.-_._____-
_“.__
.-__-.-
_ .--.-
__-..___
(A)--+}
0,
-Solution
-_--___L?HB
LiCl NJCI KCl FEf
5.81 4.40 3.52 2.99 1.43
S-76 4.40 3.54 3.07 I.84
3.52 3.52 3.52 3.52
3.54 3.54 3.54 3.54
2.097 0.901 0.263 --0.092 0.233
2.162 0.883 0.216 -0.212 -0.200
0.8 0.7 0.7 0.3 I.1
0.7 0.R 0.5 0.7 1.3
NazSO4 M&k
4.40 5.20
4.G 4.80
1.76
3.52
2.13 3.54
--b 1.46
-0.307 1.246
7.6 15.2
---_ 6.0 15.2
S.W.
-
-
-
0.331
4.3
3.2
mulation truncated
Id)
--_-
Gi.
_
a_ (A)
s
__.-... .-.. _.-_ DHB GL
. -._. ._ _____ __ .___ DHB GL
Ah wr. TI0-J =cj ___
EkCfdj.k-
--
_--.- ----
-
---_-_-
-
. .._..- ____--
0.306
_
DIiB GL ._ .___ ____ _...
of the fundamental equation of the theory (Poisson-Bo~tzmann) not at the first term of the development in series of powers of Boltzmann’s
exponential (21). If we limit ourselves, as is commonly done (22), to the first three terms, we then obtain: Info = - 1’9 c zj Ii (Ziji [Z, (.A-&+ 3 6’ A’, (Xi)] (6) - =;Lif 92 CzZ (X3 + Q P x2 (Xi)] + + Gii
q3 I% 01) + 3 P X3 WI
+
-l- 5 if qf C& (x3 -l- 3 P % (Xi)] f + *..I Dedination.
10 (1972)
263-272
THEOREflCAL
CALCULATION
OF THE OSMOTIC COEFFiCIENT
267
where
X>(x) =
----
t tt* (3x) -- _-.- --.^._
x -
s
12(1 +5X)”
I
x + f
-
1
t1g (4x) ____-_.._
3
2
with
. (du/u) I = +? ._ ._.. J,
e-’
,t (x)
(C-“j.Y)-”
. I r”‘- ’ X, (f) df
1:
z,
(x,
= 3 X,, (x)
-
--:P
(9)
-0
while
In this way more reIiabIe vatucs are obtained for the parameters. &+&ally those of Na2SOS. which are reported in Table I in the cofumn headed GL (Gronwatl-La Mer et ol.) in comparison to those headed DHB (Debye-Hiickel-Bonino) based on (2). The values reported refer exclusively to the interval 0.02-I molil, and not to more ample intervals like those previously studied for aqueous solutions of pure electrolytes (II). In this cast the disparity between those calculated and those found does not substantiatly differ, excluding casts prcviousiy ci:ed, with the two different formulations. since for pure electrolytes the differences become notable only by taking into account concentrations higher than I mol/l: for mixtures of electrolytes the incidence already appears at lower concentrations, as will be seen in Fig_ I. Now, if one wishes to extend the theory to mixtures of different solutes, as DesaIiim!ion. 10 (I 972) 263-272
C. DUAK
268
t?f Of.
4 m.8 -xi
I
..
* :+-or. .
0pl.
..
. C.IC.
v
opt
GLG
. L.~c.
. s&L
300
100
t
- 2.0
-1.5
-1 .o
-0.5
Log
m+
0
Fig. I.
is necessary for applications to sea and brackish water. there are very few changes in the above-mentioned formulas. Only in q* must the sums be intended as not only extended to the numbers of r ions in which the only dissolved sait is disassociated. but in Eq. (IO) at effective (c v)~ concentrations of every single ion. deriving from the various salts which make up the dissolved misture: in place of (c v),. (nr I.)~may also be used, given the adimensionality of the qn’s_ For the parameters, every sin& ion radius optimized for the corresponding concentration intervat is to be used. while for the constant s a weighted average of relative values for solulions of singk =Its may be attempted. In this casr. in place of (4), we shall have: 1, Do ~0 ?li2 =
where the summation
Zj
7,
Sj
[Zi
Yi
a: Jr
VII
,wiIl be extended
to the singIe saltsj which are disassociated. between the motar fraction of the single salts and their total. Another alternative will be that of directly og timizing only the’ parameter S, defining it in this case by means of
each one into i ions and where yj is the relationship
@0&J=
= S xj -;;r&j 5.iu&
(12)
On the other hand. the simptest treatment wouid be that of utilizing only the optimized parameters for the predominant NaCI solutes, while for the secondary ions the theoretical constants, that is to say. those obtained from the linear correlation of Bonino (8) and Rolla (9) of the ai’s with the crystallographic-radius/ poIarizabiIity ratio and s with a weighted average of crystalline compressibility could be used, Desalination,
10 (1972) 263-272
TNEORETICAL
CALCULATBXU OF THE OSMOTK- COEFFKIENT
249
EXPERIMENTAL
The cryoscopy apparatus has previously been described (fl). Temperatures were measured by a platinum resistance thermometer coupled with a Miiller bridge. Precision thus obtainable is within a few ten thousandths of a degree and constant throughout the temperature interval. Equilibrium is assured by a pneumatic stirrer made of high quality glass. The concentration must bc known so as to assure regularity oferror throughoui the interval with a constant precision which for alkali halides should be on the order of a few hundred thousandths of molality. This is carried cuit with a “HilgerRayleigh” interferometer. This method yields really precise measurements at high concentrations, where more objective data arc required. while for the lower concentrations they have the advantage of being consistent with the rest, with no pretense of matching the precision of those obtained with other instruments. However, a tesssr relative precision, that is, an acxidentat error higher in percentage at lower concentrations has negligible influence on the parameters obtained, while uncertainty in accuracy. that is systematic errors in the measurements of other workers in more concentrated solutions have often made every determination of objective ionic parameters quite problematic. For sea water we used so-called “reconstructed sea water” (23) with the following concentrations: NaCI 0.337, MgCIL 0.0665 and Na,SO,0.029 for a total of 0.5325 mole per kg H,O (including crystallization water of the latter two salts). The salts used were: Merck type “Suprapur”cat. no, 6406for NaCI, Merck type”P.A.” cat. no. 5833 for M&X1-6H20 and Merck “Supnpui’ cat. no. 6647 for Na,SO,.
Magnesium chloride was determined with EDTA, whife the usual caution was used for NatSO in order to avoid errors owing to its well-known hygroscopicity. Brackish water was obtained by proportionally reducing these concentrations practically in geometric progression in the dilution interval till about 0.02 mo!/kg. Experimental data are shown in Table 11. Here m stands for the total mofality reported above, but if aFtdated as the sum of the ion mofslities, it must be divided by a factor C, the weighted average of the number of ions into which every single salt is disassociated. In this way a A T/m is obtained which tends towards infinite dilution at i; times the extrapolated value for non-electrotytes. At this point Info (24) and ionic force are calculated, always keeping in mind the detinltion of m given above- The Info/nx data were equalize!d and interpolated with the maximum possible objectivity at intervals of 0.1 in log tn. For the higher dilutions a trend parallel to the limit law of Debye-Hiickel, as already discussed, (IQ, 25) was sought. The equalized data thus obtained are also reported in Table II and may represent an experimental basis for calculating other thermodynamic quantities of sea and brackish water. DesuIinafiun, 10 (1972) 263-272
270 TABLE
I1
0.023085
3.8033
0.033954
3.7462
0.071785
3.6685
0.093905
3.7187
0.189279
3.5651
0.309489
3.5465
0.424423
3.5507
0.497147
3.540)
Ih?ERPRITATiON
1.7 1.6 1.5 1.4 1.3 3.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3
220
247 275 303 331 358 383 406 427 446 459 465 465 461 451
OF THE RESULTS
In order to interpret these experimental results we first tried to utilize Eq. (2) with optimized ionic radii for the predominant NaCI salt and the theoretical values for the other ionic radii with an average value of 5 according to (I I). This curve is shown in Fig. I indicated by T-T. (a theoretical-s theoretical). The trend is quite unsatisfactory. It was therefore necessary to resort to the optimization of new parameters for MgClr and Na,SO,, first with (2) then with (9 the results are shown$n Table I. With these parameters, still using (I I) a& (2), we calculated a new trend indicated by CC in Fig. 1 (a calculated-s calculated) and improved it by the introduction of the higher terms of Gronwah-La Mer according to (6) and reported with the same letters but in bold type. These trends curye too much at high concentrations, even though the second already more nearly approaches theexperimenta! values.
We then attempted
to optimize the parameter f with (12) and obtained. with of Brsnino et al. for tbheMg+ l and SO,-- ion radii, a trend which better interpolates the experimental results (the curve T-0. that is, u’s theoretical-s’s optimized). A further improvement of the agreement is reached with the a’s calculated as mentioned above (indicated by CO.) but only the use of (6) in this case succeeds in giving the proper curve to the trend (CO. in bold type) the theoretical
values
in order to render the experimental one we& Desalinorion.10 (1972)263-272
THEORETliCAL
CALCULATION
OF THE OSMOTlC
COEFFICIENT
271
CONCLUSIONS It can
agreement
thus
be
between
seen that with this formulation it is possible to reach a good theory and practice, arrivieg not only at a root-mean square
deviation of a few thousandths of “C throughout the interval but niso a trend which substantially reproduces the one found experimentally. In Fig. 2 it can be seen how the osmotic cocficicnt varies from salt to salt in an extremely sensitive way and how the data far sea and brackish water fit between these.
.
ihe experimental
obtained and the theoretical fli‘rmulations
confirmed by them. any other thermodynamic application is possible. and thus a basis for predictions at other quantities is formed. in any case, it will be interesting to see later on the dependence on temperature of the trends studied as a function of concentration when the apparata for ebullioscopy, vapour tension and osmotic pressure are definitely ready. Osmotic pressure, even though measurable with much less precision than vapour tension, as is amply shown in the literature, is however of great interesi in that osmotic equilibrium serves as a starting point for all calculations of reverse
osmosis. This,
in fact,
today
represents one of the most economical desalting Desalinarion, IO ( 1972) 263-272
mtihods, above all for small and medium plants and for brackish water and a more theoretical study of it will certainly iead to useful ideas. At any rate, under the aspect of the theory of the solutions of mixtures of strong electrolytes it must be pointed out that the agreement reached is noteworthy, considering that one of the most common objections to the DebyeHiickel theory is that generally it does not succeed in rendering well the experimental trends for mixtures. A study, which has already been started (26). on Mayer’s clusters theory and on its relative applications (27) will Perhaps further improve on the agreement between theory and experimental data. The authors wish to than& Prof. F. Momicchioli and Dr. G. Grandi of the Institute of Physical Chemistry of the University of Modena and Dr. G. Paschina of Istituto Chimico Policattedra of the University of Cagliari for their collaboration as well as the Consiglio Nazionale delle Ricerche for its financial contribution. For the calculations we used the IBM 1130 of the Computing Centre of the University of Cagliari and the IBM 7090 of the “CNUCE” of Pisa by teleprocessing. REFERh’cEs 1. 2.
J. E. MA=
J. Chem. Phys., 18 (1950) 1426. A&D 3. C. R.wshtt. J. Chem. Phys., 48 (1968) 2742; J. Whys. Chem., 72 ( 1968)
H. L F-MAN 3352.
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F. Mo~lcanou Ah?) G. GRAND), Bof. Sri. Fat. Chim. id Bulogna, 24 (1966) 133. C. DUAIE, Ga==.C&m. Ztaf., 83 (I9S3) 3. C. DESADZAND 1. Mm, Ann. Chim., SO (1950) 956. G. B. B~‘N;IFIO Ah= M. ROLL*, Atri I&& Accad. Italia, 4 (1933) 465. C. I)UAR, d&z?& chim., 49 (1959) 1016. A. C. D. Rfwrr. Z- Phys&. Chem.. 80 (1912) S46. E H. Loom Wipdm. Ann.. 50 (18%) 503. H. MENU, 2. Ebkrrockm., 33 (1927) 68. W. D. Hnmr tiw W. A. Rosurn, 1. Am. Cfrun. Sac.. 38 (1915) 2679. H. C. 3o?cls,H.&ties in Aguemu Sofutiom, Camtgk Inst. Washington,1907,p. 36. T. H. GRONWAU, V. K. Ijr MARmz K. SA~W’ED, Phys& 2.. 29 (1928) 358. V. K. LA AQ%,T. H. GRON~~ Ah-D L S. GRUFF, J. Phys. Chci*., 2 t?;H) 35. D. F. MuR?~. Marinr Chcmislry,Vol. I, p. 59, M. Dekkcr Inc., XJw York, N.Y., 1959. c. DEJA& Ann. chim., 43 (19S3)90. C.DEaAxAND1.MAzzEl.Ann. c&7., 47 (1957) 1019. C -A*; Gnu. C&m_ fraf.., 95 (l%S) loJ!L C ikr*JE. G. Lrannr AND M. Mlsucm, Gazz. Chim. id., 95 (1963 1156. Desalination,10 (1972) 263-272