(With permission from The Electrochemical Society, Inc., U.S.A., who has the copy right)

*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
I

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"**

**R. Heyrovská***

Academy of Sciences of the Czech Republic, J. Heyrovskı Institute of Physical Chemistry, 182 23 Prague 8, Czech Republic.

ABSTRACT

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 25^{o}C are provided.

*Electrochemical Society Active Member

**Introduction**

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 supposed^{6,7} 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 explained^{2} 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 review^{9} of the current interpretations of the thermodynamic
properties, based on the assumption of complete dissociation,

NaCl(aq) ---> Na^{+} + Cl^{-}
[1]

(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 ideas^{6,7} of partial
dissociation,

NaCl(aq) <===> Na^{+} + Cl^{-}
[2]

(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 studies^{11}
combined with molecular dynamic simulations^{12} 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 work^{2,8,10} 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 shown^{13b-f} that for large ranges
of concentrations for many strong electrolytes, the following equation
holds for the osmotic pressure,p_{os}

p_{os }= iRT/V_{Afb} = iRTmd_{Afb}/(1
- mn_{b}/55.51) = nmfRTd_{Afb}
[3]

where V_{Afb} is the volume of "free" solvent (water) in the
bulk (subscript b) per mole of solute (NaCl), d_{Afb} 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/n_{Afb})
[4]

where n_{Afb} = 55.51 - mn_{b} is the molality of free
water and n_{b} 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 (a_{A})
[5]

where a_{A} is the solvent activity, which is defined^{1}
as the vapor pressure ratio (p_{A}/p_{A}^{o}).

In Ref. 2, 7, 8 and 10, a_{A} was shown to represent
the mole fraction N_{Afs} of free water,

a_{A} = N_{Afs} = n_{Afs}/(n_{Afs }+
im)
[6]

where n_{Afs} = 55.51 - mn_{s} is the molality of free
water and n_{s} 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 found^{2,8,10} that Eq. 6 is valid for
over 100 strong electrolytes over large ranges of concentrations (*e.g*.,
0 to 4 m NaCl).

An important observation^{8a} 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 n_{Afs} differs from n_{Afb}, 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 R_{Af},

R_{Af} = -a_{A}ln a_{A}/(1- a_{A}) =
n_{Afs}/n_{Afb}
[7]

The value of R_{Af}, which can be calculated by using the available
a_{A} or f (see Eq. 5 for the relation
between f and a_{A}) 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 25^{o}C, R_{Af} was calculated
using the f data in Ref. 14 (see columns 2 and
4 in Table I).
A computer linear best-fit plot of n_{Afs}/R_{Af
}*vs*
m from 0 to 6.144 m (satd. soln.), shown in Fig.1, gave n_{s} =
3.348, n_{b}= 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, NaNO_{3}, etc., with
strong ion-water interactions, the hydration numbers are -*ve*, and
n_{s} is less negative than n_{b}.^{8c} In these
cases, as n_{Afs} and n_{Afb} are both > 55.51, water is
probably appreciably ionized. Conversely, in cases where n_{s}
and n_{b} are positive, n_{Afs} and n_{Afb} are
both < 55.51 and water can be considered to be associated.

The above values of n_{b} and n_{s} 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 K_{m} can be calculated
as,

K_{m} = a^{2}m/(1- a)
[8]

The concentration dependence of K_{m} (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/n _{Afs})**

The above values of a (see Table
I) have now enabled the extension up to saturation of the earlier interpretation^{2,8,10}
[from 0.0064m to 4m NaCl(aq)] of DE by the equation

DE = -d_{A}(2RT/F)ln
[(am/n_{Afs})/(am/n_{Afs})^{o}]
[9]

where DE was calculated from the g_{±}
(mean ionic activity coefficient) data^{14} using the relation^{1}

DE = (-2RT/F)ln mg_{±}
[10]

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/n_{Afs}) for 0.001 to 6.144m
NaCl(aq) is shown in Fig. 3. From the slope (= -0.04915V; S.E.= 0.0001),
d_{A}
(= 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/n_{Afs})^{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 a*vs* 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 n_{Afb}.)

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"^{16,17}no 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 earlier^{2,8,10} 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 = V_{A} + mV_{B} [< (V_{A}^{o}+
mV_{cr})]
[11]

V_{B} = (1- a)V_{cr}+ a(V_{+
}+
V_{-}± dV_{el}) = V_{cr}-
a[V_{cr}-
(V_{+ }+ V_{- }±
dV_{el})
[12]

where V_{A} is the volume of 1 kg of water in the solution and
V_{A}^{o} is that of pure water, V_{B} is the volume
occupied by one mole of solute (B) (NaCl) in the dissolved state, V_{cr}
= 26.8 cm^{3}/mole is the volume of one mole of the crystalline
salt,^{20} (V_{+ }+ V_{-}) (<V_{cr})
is the sum of the volumes per mole of the ions and ±adV_{el}
is the volume change due to electrostriction.^{19-22} Equation12
is similar to those proposed for many strong^{19} 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- mV_{cr} on
am in Fig. 4. From the slope, [V_{cr}-
(V_{+ }+ V_{- }±
dV_{el})]
= 10.55 (±0.13)cm^{3}, (V_{+ }+ V_{- }± dV_{el})
was found to be 16.25 cm^{3}. This value is in the range (±0.5
cm^{3}) of the reported^{21,23} apparent molal volumes
at infinite dilution, as pointed out earlier.^{2,8a} On using the
values of the crystal ionic radii R_{ic}^{o} in Ref. 24,
one gets (V_{+ }+ V_{-}) = 18.0 cm^{3}, and hence
dV_{el}
= -1.75 cm^{3}. The intercept of the straight line in Fig. 4 gives
V_{A} = 1002.86 (±0.09) cm^{3}, which is close to V_{A}^{o}
(=1002.92 at 25^{o}C). The molal volumes V_{cal} calculated
back from Eq. 11 and 12, using the above slope and intercept are given
in Table II.
It can be seen as before^{8a} that V - V_{cal} < (or
=) 0.1 cm^{3}. The corresponding differences between the actual
and calculated densities, d - d_{cal} presented in the next column
are < (or =) 0.00012 g/cm^{3}.

Therefore, V(1) is given by

V(1) = 1002.86 + m(26.8 - 10.55a) [13]

* 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) cm^{3}
and 1002.38 (±0.28)cm^{3} respectively. The equation for V(2) is
therefore

V(2) = 1002.38 + 24.74 am [14]

where V_{A} = 1002.38 (= V_{A}^{o}- 0.54)cm^{3}
and V_{B} = 24.74 a. The agreement between
V_{cal} and V and the smallness of d - d_{cal} 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 - V_{cal}
on this basis is higher (±0.3 cm^{3}) than that (±0.1 cm^{3})
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]
+ amdV_{el}
[15]

V(2) = 1002.38 + am(18.31 + dV_{d})
+ amdV_{el}
[16]

where 18.31 cm^{3} is the sum (V_{+ }+ V_{-})
(which is close to the sum of the crystal ionic volumes, 18.0 cm^{3}
of Ref. 24), dV_{d} = V_{cr}-
(V_{+ }+ V_{-}) = 8.49 cm^{3} is the volume of
the void that accompanies the dissociation of the crystal (or association
of the ions), and -dV_{el} = 2.06 cm^{3}
= 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 cm^{3} per mole as ion pairs.

**Interpretation of the Molal Dissociation Constant, K _{m}**

K_{m} is defined by Eq. 8 and its variation with
m can be seen in Table
II.

* K _{m}(1) in the range of molalities to the left
of the a-minimum*

In this range, it follows from Eq. 13 and 15 that

K_{m}(1) = K^{o} [V_{B}(1) - 26.8(1- a)
m]^{2}/[V_{B}(1) - 18.31am]
[17]

where K^{o} = 26.8/(18.31)^{2} = 0.080 cm^{-3}.mol
and V_{B}(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 K_{v}(1).

* K _{m}(2) in the range of molalities to the right
of the a-minimum:*

It follows from Eq. 14 and 16 that,

K_{m}(2) = K^{o} [V_{B}(2) - 8.49 am]^{2}/[-V_{B}(2)
+ 26.8m]
[18]

where K^{o} is the same as in Eq. 17 and V_{B}(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 K_{v}(2).

The correlation of K_{m} as defined by Eq. 8 with
the K_{m}(cal.) values calculated using Eq. 17 and 18 for the respective
concentration ranges is shown graphically in Fig. 6, where K_{m}
is shown to be directly proportional to K_{v}(1) and K_{v}(2).
The intercept and slope (K^{o}) of this line are 0.02 [standard
error (S.E.) 0.3] and 0.0799 (S.E.: 0.0007) cm^{-3}.mol, respectively.

**Acknowledgment**

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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