This paper is the first of its kind which shows that the non ideal thermodynamic properties of strong electrolytes in aqueous solutions "at all concentrations" are due to partial dissociation and hydration.
(The author is grateful to Dr. D. Untereker of the Editorial Board of the ECS and to the referees for their excellent encouragement.)
Link to TABLE
Link to TABLE II
Physical Electrochemistry of Strong Electrolytes Based on Partial Dissociation and Hydration: Quantitative Interpretation of the Thermodynamic Properties of NaCl(aq) from "Zero to Saturation"
Academy of Sciences of the Czech Republic, J. Heyrovskı Institute of Physical Chemistry, 182 23 Prague 8, Czech Republic.
The author's earlier quantitative interpretations of the
non ideal properties of strong electrolytes in aqueous solutions have now
been successfully extended to cover the whole range of concentrations.
It is shown that i) non-ideality, expressed as osmotic and activity coefficients,
is due to the incomplete dissociation and hydration of the electrolyte,
ii) the hydration numbers at the vapor/solution interface and bulk regions
are different, iii) the degree of dissociation has a minimum value at the
molality at which the mean ionic activity is unity and iv) the molal volume
and the molal dissociation constant depend on the volumes of ions and ion
pairs. Simple quantitative relations, supporting graphs and Tables of data
for NaCl(aq) at 25oC are provided.
*Electrochemical Society Active Member
Aqueous solutions of strong electrolytes play an important role in electrochemical science and technology, physical chemistry, biochemistry, geochemistry, chemical engineering, atmospheric chemistry and environmental chemistry. However, a proper understanding of their non ideal properties has continued to exercise the minds of scientists throughout this century. The cause of this, as the author believes, is the assumption of complete dissociation of strong electrolytes at all concentrations based on the near success of the Debye-Hückel (DH) theory for very dilute solutions.
In short, interpretations based on the idea of complete dissociation (see, for example, Ref. 1) amounted to empirical extensions of the DH limiting equations for dilute solutions to fit the data for higher concentrations. The resulting complicated mathematical expressions for the deviations from ideality could not provide any unified interpretation of the solution properties over the whole concentration range. Further criticisms can be found for example, in the recent Ref. 2-5.
Therefore, the present author undertook a systematic research of the experimental data as such rather than of the deviations from ideal complete dissociation. The results, summarized in Ref. 2 show that partial dissociation and hydration are indeed the chief causes of nonideality as originally supposed6,7 and that the DH limiting equations are asymptotic laws for complete dissociation at infinite dilution. With the degrees of dissociation and hydration numbers evaluated from osmotic coefficients (or vapor pressures), many basic properties of solutions (like the electromotive force of concentration cells, diffusion coefficient, equivalent conductivity and solution density) are explained2 by simple equations, valid for large ranges of concentrations. Simple relations connecting the osmotic and activity coefficients with the degrees of dissociation and hydration numbers were also established.8a,b For a detailed introduction, references to literature and an account of the work, please see Ref. 2.
Turning now to NaCl(aq), a typical strong electrolyte, a recent review9 of the current interpretations of the thermodynamic properties, based on the assumption of complete dissociation,
NaCl(aq) ---> Na+ + Cl- 
(m-m) (m) + (m)
where m is the molality, illustrates the complexity of the extended DH equations tailored to fit the experimental data over a wide range of concentrations. For example, see Eq. 7, 8 and 19 in Ref. 9 for the osmotic coefficient, activity coefficient and molar volumes respectively. Such equations provide empirical representations of the data but do not provide explanations of the underlying molecular phenomena.
On the other hand, the earlier ideas6,7 of partial dissociation,
NaCl(aq) <===> Na+ + Cl- 
(m- am) (am) + (am)
where a < (or =) 1 is the degree of dissociation, m[(1- a) + 2a] = im is the total number of moles of solute and i is the van't Hoff factor, have been revived with much success over the last decade.2,8,10 Hydration numbers and degrees of dissociation at various molalities were calculated for NaCl(aq) and 99 other mono- and multivalent strong electrolytes over large ranges of concentrations. Recently, x-ray diffraction studies11 combined with molecular dynamic simulations12 of saturated alkali halide solutions have demonstrated the presence of about 30% ion pairs in NaCl(aq). These results provide strong support for the notion that the effects of partial dissociation must be considered.
The author's earlier work2,8,10 on the solution properties of NaCl(aq) cover the concentration range from 0 to about 4 m. In this paper, the interpretations have been extended to cover concentrations up to saturation.13a The results are presented below.
Interpretation of Osmotic Coefficients (f) in Terms of Bulk and Surface Hydration Numbers and Degrees of Dissociation.
It has been shown13b-f that for large ranges of concentrations for many strong electrolytes, the following equation holds for the osmotic pressure,pos
pos = iRT/VAfb = iRTmdAfb/(1 - mnb/55.51) = nmfRTdAfb 
where VAfb is the volume of "free" solvent (water) in the bulk (subscript b) per mole of solute (NaCl), dAfb is the corresponding density, 55.51 is the number of moles of water in 1 kg., n (=2 for NaCl) is the number of moles of ions into which 1 mole of the electrolyte dissociates, and f, the osmotic coefficient, is the nonideality parameter. (Note that f = 1 and i = n for complete dissociation at m = 0.) From Eq. 3, one gets for the concentration dependence of f, the simple equation,
nf = 55.51(i/nAfb) 
where nAfb = 55.51 - mnb is the molality of free water and nb is the hydration number in the bulk. Equation 4 shows that f arises due to the partial dissociation and hydration of the electrolyte, as originally suggested.7
By definition,1 f is evaluated through the relation,
nf = -(55.51/m)ln (aA) 
where aA is the solvent activity, which is defined1 as the vapor pressure ratio (pA/pAo).
In Ref. 2, 7, 8 and 10, aA was shown to represent the mole fraction NAfs of free water,
aA = NAfs = nAfs/(nAfs + im) 
where nAfs = 55.51 - mns is the molality of free water and ns is the hydration number (subscript s: surface). Equation 6 is, in fact, Raoult's law, but modified for incomplete dissociation and hydration. It has been found2,8,10 that Eq. 6 is valid for over 100 strong electrolytes over large ranges of concentrations (e.g., 0 to 4 m NaCl).
An important observation8a was that Eq. 4 involving a bulk property and Eq. 6 for a surface property, gave almost the same values of i, but different (by about unity) hydration numbers. This means that nAfs differs from nAfb, probably because the water molecules at the vapor/solution interface are engaged in establishing equilibria with the bulk as well as the vapor phase, unlike the water molecules in the bulk.
On combining Eq. 4 and 6 and eliminating i, one obtains the ratio RAf,
RAf = -aAln aA/(1- aA) = nAfs/nAfb 
The value of RAf, which can be calculated by using the available aA or f (see Eq. 5 for the relation between f and aA) data in Eq. 7, represents the ratio of the molality of "free" water in the surface to that in the bulk. For NaCl(aq) at 25oC, RAf was calculated using the f data in Ref. 14 (see columns 2 and 4 in Table I). A computer linear best-fit plot of nAfs/RAf vs m from 0 to 6.144 m (satd. soln.), shown in Fig.1, gave ns = 3.348, nb= 2.457 (standard error (S.E.): 0.001) as the (-ve) slope and 55.51 (S.E.: 0.01) as the intercept.
For electrolytes like NaF, NaNO3, etc., with strong ion-water interactions, the hydration numbers are -ve, and ns is less negative than nb.8c In these cases, as nAfs and nAfb are both > 55.51, water is probably appreciably ionized. Conversely, in cases where ns and nb are positive, nAfs and nAfb are both < 55.51 and water can be considered to be associated.
The above values of nb and ns and the f data from Ref. 14 were used in Eq. 4 and 6, respectively, to calculate the two columns of a for NaCl(aq) in Table I. The near identity (±0.001) of the values of a obtained from the two equations, for the entire concentration range from zero to saturation, is noteworthy. Conversely, a (Eq. 4) and a (Eq. 6) perfectly reproduce (with an accuracy of ±0.001) the f values from Eq. 6 and Eq. 4, respectively, from zero to saturation. See for example, the column for f (Eq. 4) in Table I.
Figure 2 shows the i vs m curve, which passes through a minimum. (Note: the f vs m and g±vs m curves also pass through minima.) It was verified that this is true for many electrolytes.8c This observation confirms the phenomena of "dissociation minima" suggested by Davies.15 He discusses the conditions for a-minimum in terms of the ionic strength and activity coefficient. It is shown below that DE = 0 (i.e., the mean ionic activity is unity) at the a-minimum.
With the present knowledge of a at any value of m (see the last two columns in Table I and column 4 in Table II), the molal dissociation constant Km can be calculated as,
Km = a2m/(1- a) 
The concentration dependence of Km (see the last column in Table II) is treated below after dealing with the molal volumes.
The Linear Dependence of the EMF, DE, of Concentration Cells on ln(am/nAfs)
The above values of a (see Table I) have now enabled the extension up to saturation of the earlier interpretation2,8,10 [from 0.0064m to 4m NaCl(aq)] of DE by the equation
DE = -dA(2RT/F)ln [(am/nAfs)/(am/nAfs)o] 
where DE was calculated from the g± (mean ionic activity coefficient) data14 using the relation1
DE = (-2RT/F)ln mg± 
The product mg± is termed the mean ionic activity.16 (Note that the difference, [DE + (2RT/F) ln m], from which g± is evaluated, is the free energy deficit or excess over that expected for complete dissociation. g± was, in fact, originally considered as the thermodynamic degree of dissociation.16)
The linear dependence of DE on ln(am/nAfs) for 0.001 to 6.144m NaCl(aq) is shown in Fig. 3. From the slope (= -0.04915V; S.E.= 0.0001), dA (= 0.957; S.E.= 0.002), a constant (which is thought to characterize the solvent-solute polarization) is obtained. From the intercept (-0.1845V; S.E.= 0.0014), the value of (am/nAfs)o = 0.0276 at DE = 0 (i.e., at mg± = 1) is obtained. This value corresponds to a solution of 1.5m for which a is at its minimum. DE is of opposite sign on either side of the avs m curve. Equation 9 reproduces the DE values to better than ± 2mV over the entire concentration range up to saturation, except at 0.001m where it is 3 mV off. (Note: for NaF and a few other electrolytes,8c Eq. 9 involves nAfb.)
Thus, Eq. 9 based on the idea of partial dissociation and hydration affords a simple and unified interpretation of DE over the entire concentration range. Therefore, the elusive "single ionic activity"16,17no longer is necessary.
Dependence of the Solution Densities/Molal Volumes on am, Ion and Ion Pair Volumes and Electrostriction
The interpretation of the solution density (d) of a solution of molality m amounts to that of the molal volume V, since V = (1000 + 58.443m)/d, where 58.443 is the molecular weight of NaCl. The density data tabulated in Ref. 18 and reviewed in Ref. 9 and the corresponding molal volumes are given Table II.
Molal volumes V(1) for molalities to the left of the a-minimum
It was shown earlier2,8,10 that the contraction in volume attending dissolution of NaCl in water is due to partial dissociation and electrostriction.19-22 The following equations were shown to quantitatively account for V from 0 to about 1.4m
V = VA + mVB [< (VAo+ mVcr)] 
VB = (1- a)Vcr+ a(V+ + V-± dVel) = Vcr- a[Vcr- (V+ + V- ± dVel) 
where VA is the volume of 1 kg of water in the solution and VAo is that of pure water, VB is the volume occupied by one mole of solute (B) (NaCl) in the dissolved state, Vcr = 26.8 cm3/mole is the volume of one mole of the crystalline salt,20 (V+ + V-) (<Vcr) is the sum of the volumes per mole of the ions and ±adVel is the volume change due to electrostriction.19-22 Equation12 is similar to those proposed for many strong19 as well as weak electrolytes.23 See also the review.21 The validity of Eq.11 and 12 for the data from 0 to 1m in Table II is demonstrated by the linear dependence of V- mVcr on am in Fig. 4. From the slope, [Vcr- (V+ + V- ± dVel)] = 10.55 (±0.13)cm3, (V+ + V- ± dVel) was found to be 16.25 cm3. This value is in the range (±0.5 cm3) of the reported21,23 apparent molal volumes at infinite dilution, as pointed out earlier.2,8a On using the values of the crystal ionic radii Rico in Ref. 24, one gets (V+ + V-) = 18.0 cm3, and hence dVel = -1.75 cm3. The intercept of the straight line in Fig. 4 gives VA = 1002.86 (±0.09) cm3, which is close to VAo (=1002.92 at 25oC). The molal volumes Vcal calculated back from Eq. 11 and 12, using the above slope and intercept are given in Table II. It can be seen as before8a that V - Vcal < (or =) 0.1 cm3. The corresponding differences between the actual and calculated densities, d - dcal presented in the next column are < (or =) 0.00012 g/cm3.
Therefore, V(1) is given by
V(1) = 1002.86 + m(26.8 - 10.55a) 
Molal volumes V(2) for molalities to the right of the a-minimum
The molal volumes from 2 to 6.144m (saturation) (see Table II), where a rises with concentration, are directly proportional to am as can be seen from Fig. 5. The slope and intercept of this line are 24.74 (±0.10) cm3 and 1002.38 (±0.28)cm3 respectively. The equation for V(2) is therefore
V(2) = 1002.38 + 24.74 am 
where VA = 1002.38 (= VAo- 0.54)cm3 and VB = 24.74 a. The agreement between Vcal and V and the smallness of d - dcal can be seen from Table II. The last row gives the values for the saturated solution calculated on the basis of Eq. 14.
[In fact, the V vs am linearity nearly holds for the entire region from 0 to saturation, as can be seen from Fig. 5. However, for solutions < 1.4 m, V - Vcal on this basis is higher (±0.3 cm3) than that (±0.1 cm3) given by Eq. 13.]
The results obtained in these preceding sections can be summarised thus
V(1) = 1002.86 + m[26.8(1- a)+18.31a] + amdVel 
V(2) = 1002.38 + am(18.31 + dVd) + amdVel 
where 18.31 cm3 is the sum (V+ + V-) (which is close to the sum of the crystal ionic volumes, 18.0 cm3 of Ref. 24), dVd = Vcr- (V+ + V-) = 8.49 cm3 is the volume of the void that accompanies the dissociation of the crystal (or association of the ions), and -dVel = 2.06 cm3 = 18.31 - 16.25 = 26.8 - 24.74. Therefore, at molalities to the left of the a-minimum, the ions and ion pairs are separate entities with their respective volumes, whereas at higher molalities to the right of the minimum, the ions occupy a volume of (18.31 + 8.49) = 26.8 cm3 per mole as ion pairs.
Interpretation of the Molal Dissociation Constant, Km
Km is defined by Eq. 8 and its variation with m can be seen in Table II.
Km(1) in the range of molalities to the left of the a-minimum
In this range, it follows from Eq. 13 and 15 that
Km(1) = Ko [VB(1) - 26.8(1- a) m]2/[VB(1) - 18.31am] 
where Ko = 26.8/(18.31)2 = 0.080 cm-3.mol and VB(1) = V(1) - 1002.86 + 2.06 am is the total volume of solute. For short, the term in the square brackets can be denoted as Kv(1).
Km(2) in the range of molalities to the right of the a-minimum:
It follows from Eq. 14 and 16 that,
Km(2) = Ko [VB(2) - 8.49 am]2/[-VB(2) + 26.8m] 
where Ko is the same as in Eq. 17 and VB(2) = V(2) - 1002.38 + 2.06 am is the total volume of the solute. As in Eq. 17, the term in the square brackets, can be written as Kv(2).
The correlation of Km as defined by Eq. 8 with the Km(cal.) values calculated using Eq. 17 and 18 for the respective concentration ranges is shown graphically in Fig. 6, where Km is shown to be directly proportional to Kv(1) and Kv(2). The intercept and slope (Ko) of this line are 0.02 [standard error (S.E.) 0.3] and 0.0799 (S.E.: 0.0007) cm-3.mol, respectively.
The author is grateful to the Academy of Sciences of the Czech Republic for the award of a research grant (Grant GA AV CR 440401).
Manuscript submitted July 25, 1995; revised manuscript
received Jan.19, 1996. This was Paper 662 presented at the Reno, NV, Meeting
of the Society, May 21-26, 1995.
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