212
J. Phys. Chem. B 2009, 113, 212–223
State of Hydration Shells of Sodium Chloride in Aqueous Solutions in a Wide Concentration Range at 273.15-373.15 K Vladimir N. Afanasiev,*,† Alexandr N. Ustinov,† and Irina Yu. Vashurina‡ Institute of Solution Chemistry, Russian Academy of Sciences, 153045, Akademicheskaya Ul. 1, IVanoVo, Russia, and IVanoVo State UniVersity of Chemistry and Technology, 153007, Engelsa Prosp. 7, IVanoVo, Russia ReceiVed: December 7, 2007; ReVised Manuscript ReceiVed: October 3, 2008
On the basis of the measurements of the speed of ultrasound, density, and isobaric heat capacity, isentropic compressibilities of aqueous solutions of sodium chloride (from dilute up to highly concentrated) have been determined at 273.15-373.15 K. With the use of correct correlations, at electrolyte concentrations below complete hydration limit (21.53% wt), the following values have been calculated: solvation numbers (h), molar parameters of volume and compressibility of hydration complexes (Vh, βhVh), water in a hydration shell (V1h, β1hV1h), and a cavity containing a stoichiometric mixture of ions without a hydration shell (V2h, β2hV2h).The values of h and βhVh have been found to be independent of temperature in the indicated temperature interval, while the molar compressibilities of the hydration shell (β1hV1h) and of the stoichiometric mixture of ions (β2hV2h), of concentration. The electrostatic field of ions has been shown to influence the temperature dependence of the molar volume of water in the hydration shell much stronger than the mere change of pressure affects the temperature dependence of molar volume of water in the hydration shell of pure water. It is suggested that the reason for this effect is connected with the change in the dielectric permeability of water in direct proximity to ions. Below the complete solvation limit (CSL), the functions YK,S ) f(β1V /1), βhVh ) f(h), V1h ) f(T), and β1h ) f(T) are linear with a high correlation coefficient. Thus, at all studied concentrations of sodium chloride solutions in the temperature interval 273.15-323.15 K, the dependence YK,S ) f(β1 V /1) enables h and βhVh to be calculated using the obtained correlations; at higher temperatures (323.15-373.15 K), the dependence is reverse. Beyond the CSL, the dependence of the apparent molar volume φV′ upon the hydration number h′ remains linear and allows V2h′ and V1h′ to be estimated at various temperatures. 1. Introduction The determination of numerical parameters of solvation is one of the most topical issues of modern physical chemistry of solutions. The greatest attention is focused on the problems of calculating solvation parameters for aqueous electrolyte solutions including some difficulties connected with the existence of longrange electrostatic forces and the specific structure of water.1 To investigate the process of solvation, a great number of different theoretical and experimental methods have been used, such as X-ray and neutron diffraction, quantum chemical calculations, computer modeling, NMR relaxation, calorimetric and densimetric measurements, and different modifications of the Debye-Huckel method.2 The strategy of all these methods is to establish correlations between some property of a solution and the solvation processes. Publications devoted to molecular dynamics study of solvation numbers3 and those based on the results of IR spectroscopy4,5 or NMR relaxation6 can be cited as examples. Unfortunately, it is not quite right to compare the data obtained using the above-listed methods with those obtained by the method of adiabatic compressibility. First, the authors of the cited articles assume independence of hydration numbers on the electrolyte concentration. Second, the methods used permit the determination of hydration numbers only in the nearest vicinity of ions (mainly in the first solvation shell). Our * Corresponding author. Present address: Institute of Solution Chemistry, Russian Academy of Sciences, 153045, Akademicheskaya Ul. 1, Ivanovo, Russia. E-mail:
[email protected]. Phone: (0932)308528. Fax: (0932)336237. † Russian Academy of Sciences. ‡ Ivanovo State University of Chemistry and Technology.
studies on the adiabatic compressibility of solutions,7,8 on the other hand, explicitly show that hydration numbers are a function of the solute concentration. Besides, the rather high values of the quoted hydration numbers denote that not only the first but also the subsequent shells are involved. In one of the first papers,9 it was shown that solvation numbers obtained using the adiabatic compressibility method can perfectly correlate with those based on the data of neutron and X-ray scattering provided concentrations are comparable. As solvation numbers depend on concentration, particular interest is drawn to the wide concentration range up to the CSL where one component, the solvent, is in great excess and the increase of solute concentration causes the decrease of its hydration degree due to the overlap of hydration shells of ions. In a series of publications,7,8,10,11 an attempt is made to conduct ultraacoustic studies at molecular level and, from the data on the speed of ultrasound, heat capacity, and density at different temperatures and concentrations, to obtain information on the volumetric compression (resulting from the ion-solvent interactions), concentration-dependent hydration numbers, the compressibility and density of water in hydration shells, etc. The most complete list of hydration parameters of different salts obtained by the adiabatic compressibility method is given in ref 11. Unfortunately, the available experimental data do not allow these parameters to be determined over a wider temperature range. It is due to this fact that the purpose of the present study was to apply the given method for description of volumetric-elastic properties of aqueous solutions of sodium chloride only, under the conditions of adiabatic compression in
10.1021/jp711542j CCC: $40.75 2009 American Chemical Society Published on Web 12/10/2008
State of Hydration Shells of Sodium Chloride
J. Phys. Chem. B, Vol. 113, No. 1, 2009 213
a wide range of concentrations and temperatures (273.15-373.15 K); no comparison with other salts was made. The adiabatic compressibility method was used to investigate many different systems, including nonorganic electrolytes, acids and bases, amino acids, dipeptides of glycil series, and nonelectrolytes (though at temperatures below 308.15 K), and in all cases, similar regularities were observed up to the CSL.
Let us demonstrate the simplest and most consistent way of deriving a correlation that links the adiabatic compressibility of a free solvent with that of a pure solvent. In this case, the derivative (∂ V 1//∂p)Sm will be the starting point. As all the included parameters are functions of pressure and temperature, the Jakobeans method can be used in order to derive these variables.
2. Thermodynamic Model The interpretation of properties of electrolyte solutions was made according to the approach used in the papers7,8 which is based on the classical thermodynamic interpretation of two states of water. Free water molecules that are not included in hydration shells correspond to the first state, and water molecules immobilized in hydration shells, to the second. The solution volume is supposed to obey the expression
Vm ) (x1 - hx2) · V /1 + x2Vh
(1)
where Vm is the molar volume of solution, V 1/ is the volume of a mole of free solvent that has not penetrated the hydration shells of the stoichiometric mixture of ions Na+ and Cl-, Vh is the volume of a mole of hydration complexes, x1 and x2 are the molar fractions of the solvent and the solute, respectively, and h is the hydration number. Here, the hydration number is the number of water molecules per molecule of solute whose density and compressibility substantially differ from those of a pure solvent. Differentiating eq 1 with respect to pressure p at constant entropy of solution Sm and supposing that in an acoustic experiment h is only slightly dependent on pressure, and the molar volume of pure solvent corresponds to the molar volume of free water in solution, we obtain an expression for the molar adiabatic compressibility of solution.
(∂V /1 /∂p)Sm
∂(V /1, Sm) ∂(V /1, Sm) ∂(p, T) ) ) ) ∂(P, Sm) ∂(p, Sm) ∂(p, T) / (∂V 1 /∂p)T (∂V /1 /∂T)p (∂Sm /∂p)T (∂Sm /∂T)p ) (∂p/∂p)T (∂p/∂T)p (∂Sm /∂p)T (∂Sm /∂T)p
|
|
(∂V /1 /∂p)T · (∂Sm /∂T)p
- (∂V /1 /∂T)p · (∂S/∂p)T (3) (∂p/∂p)T · (∂Sm /∂T)p - (∂p/∂T)p · (∂Sm /∂p)T
Expression 3 can be substantially simplified. Since the Gibbs free energy G is a state function and its differential dG ) -S dT + V dp is a full one, the correlation -(∂Sm/∂p)T ) (∂Vm/∂T)p must hold true. Besides, it is clear that (∂p/∂p)T is equal to unity and (∂p/∂T)p to zero. Bearing this in mind, one can express eq 3 as
(∂V /1 /∂p)Sm
)
(∂V /1 /∂p)T
(∂V /1 /∂T)p · (∂Vm /∂T)p + (4) (∂Sm /∂T)p
To connect the partial derivatives (∂ V 1//∂p)T and (∂ V 1//∂p)S /1, we again use the method of Jakobeans.
(∂Vm /∂p)Sm ) (x1 - hx2) · (∂V /1 /∂p)Sm + x2 · (∂Vh /∂p)Sm (2) In ref 9, it was assumed that the derivatives of the molar volume of pure solvent with respect to pressure at constant solvent entropy S 1/ and solution entropy Sm are equal: (∂ V 1// ∂p)Sm ) (∂ V 1//∂p)S /1. However, it is evident that the adiabatic compressibilities of the solvent are determined under different conditions: for (∂ V 1//∂p)Sm at constant solution entropy, Sm ) const, while, for (∂ V 1//∂p)S 1/ at constant solvent entropy, S 1/ ) const. Since the equilibrium state of a system can be described by a set of independent variables (T, p, N1, N2), the solution entropy being a function of these variables, it is obvious that to each composition there is a specific entropy value. The foregoing makes the possibility to replace the compressibility of the solvent not included in the hydration shells by the compressibility of pure water doubtful, which, in turn, casts doubt upon the validity of the results obtained by the adiabatic compressibility method.9 The invalidity of the correlation (∂ V 1//∂p)Sm ) (∂ V 1//∂p)S /1 was pointed out by Blandamer who studied the question of the apparent adiabatic compressibility of a solute φK,S.12 From this, the problem arises of how to determine (∂ V 1//∂p)Sm from experimental data. In our opinion, the best solution is to develop an expression that would relate the derivative of the molar volume of pure solvent with respect to pressure at constant solution entropy Sm to the derivative of the molar volume of pure solvent at constant solvent entropy S 1/.
|
|
(∂V /1 /∂p)S/1 )
∂(V /1, S/1) ∂(p, T) ) ) ∂(p, S/1) ∂(p, T) / (∂V 1 /∂p)T (∂V /1 /∂T)p
∂(V /1, S/1) ∂(p, S/1)
|
(∂S/1 /∂p)T (∂p/∂p)T
|
(∂S/1 /∂T)p (∂p/∂T)p
(∂S/1 /∂p)T (∂S/1 /∂T)p
(∂V /1 /∂p)T · (∂S/1 /∂T)p (∂p/∂p)T · (∂S/1 /∂T)p
-
|
|
)
(∂V /1 /∂T)p · (∂S/1 /∂p)T (∂p/∂T)p · (∂S/1 /∂p)T
(5)
This expression can also be simplified, provided we use the condition of a full differential and the fact that some partial derivatives are equal to zero or unity:
(∂V /1 /∂p)T
)
(∂V /1 /∂p)S/1
-
(∂V /1 /∂T)2p (∂S/1 /∂T)p
(6)
The correlation between the molar adiabatic compressibility of a free solvent -(∂ V 1//∂p)Sm and the compressibility of pure water -(∂ V 1//∂p)S /1 is obtained by substituting eq 6 into eq 4
214 J. Phys. Chem. B, Vol. 113, No. 1, 2009
-(∂V /1 /∂p)Sm ) -(∂V /1 /∂p)S1/ + (∂V /1 /∂T)p ·
[
(∂V /1 /∂T)p (∂S/1 /∂T)p
-
Afanasiev et al.
(∂Vm /∂Vm)p (∂Sm /∂T)p
]
β1V /1 ) β/1V /1 + R/1V /1T ·
[
R/1 σ/1
Rm σm
-
]
[
-
]
Rm + x2 · βhVh (8) σm
[
Rm ) -h β/1V /1 + R/1V /1T · / σm σ1
])
It can be easily seen that the only difference between eqs 11 and 12 is that in the right-hand part of eq 11 there is the derivative -(∂V /1/∂p)Sm, whereas in the right-hand part of eq 12, the derivative -(∂V 1//∂p)S /1. At the same time, as is seen from eq 7a, these derivatives differ by the term R1/V 1/T · [(R1//σ 1/) (Rm/σm)]. Hence, the derivative of the apparent molar volume of the solute is connected with its apparent adiabatic compressibility through
[
x1 / / Rm R/1 - R1V 1T · / x2 σm σ1
]
(13)
Comparison of eqs 13 and 9 leads to the conclusion that YK,S ) -(∂φV/∂p)Sm and thus
-
where β1V /1 ) -(∂V /1/∂p)Sm is the molar adiabatic compressibility of the free solvent, βhVh ) -(∂Vh/∂p)Sm is the molar adiabatic / compressibility of hydration complexes, and σ 1/ ) C 1,P /V 1/, σm ) Cm,p/Vm. Substituting eq 8 into the expression φK,S )[βSVm - x1 · β/1V /1]/ x2 and introducing a new variable YK,S, we obtain an equation that makes it possible to calculate the hydration number of electrolyte ions (h) and the molar adiabatic compressibility of hydration complexes (βhVh)
R/1
x1 1 -(∂φV /∂p)Sm ) - (∂Vm /∂p)Sm + (∂V /1 /∂p)Sm (12) x2 x2
-(∂φV /∂p)Sm ) φK,S
βSVm ) (x1 - hx2) · β/1V /1 + (x1 - hx2) · R/1V /1T · R/1 σ/1
The expression for the apparent adiabatic compressibility expressed in partial derivatives has the form of
(7a)
In view of the above, eq 2 can be transformed into
(
x1 1 -(∂φV /∂p)Sm ) - (∂Vm /∂p)Sm + (∂V /1 /∂p)Sm (11) x2 x2
(7)
Equation 7 can be written in the form of eq 7a if the partial derivatives in eq 7 are interpreted as experimentally determined parameters: R1/ ) 1/ V 1/(∂V 1//∂T)p, the heat expansion rate of the solvent; Rm ) 1/Vm(∂Vm/∂T)p, the heat expansion rate of the solution determined from the temperature dependencies of both / ) T · (∂S 1//∂T)p, the isobaric heat the solvent and solution; C 1,P capacity of the solvent; Cm,p ) T · (∂Sm/∂T)p, the isobaric heat capacity of solution measured calorimetrically; and β1/ ) -(1/ V 1/)(∂V 1//∂P)S /1, the adiabatic compressibility coefficient of the pure solvent calculated by the Newton-Laplace equation, βS ) (1/FU2), where U is the speed of propagation of ultrasound and F is the density.
YK,S
we can write -(∂φV/∂p)Sm as
( ) ∂φV ∂p
Sm
) -hβ1V /1 + βhVh
(14)
In the model employed, βhVh is the sum of the molar adiabatic compressibility of the stoichiometric mixture of electrolyte ions without hydration shells (β2hV2h) and water in hydration shells (β1hV1h)
βhVh ) β2hV2h + h · β1hV1h
(15)
As the apparent molar volume is expressed by
+ βhVh
(9)
φV ) (Vm - x1V /1)/x2
(16)
where YK,S ) φK,S - (x1/x2)R1/V 1/T · ((R1//σ 1/) - (Rm/σm)) or as a linear dependence
substitution of eq 1 into eq 16 gives the equation for the molar volume of hydration complexes:
YK,S ) -h · β1V /1 + βhVh
φV ) -hV /1 + Vh
(10)
where β1V 1/ ) β1/V 1/ + R1/V 1/T · [(R1//σ 1/) - (Rm/σm)]. Equation 10 is fundamental for determining hydration numbers and the molar adiabatic compressibility of hydration complexes. The variable β1V 1/ in eq 10 is the molar adiabatic compressibility of a free solvent at constant solution entropy, and YK,S performs the function similar to the apparent molar compressibility corrected for (x1/x2) · R/1V /1T · [(R/1/σ /1) - (Rm/σm)]. It is possible to demonstrate that YK,S is a derivative of the apparent molar volume of solute with respect to pressure at constant solution entropy. Differentiating with respect to pressure of the known expression for the apparent volume of solute,
(17)
As the molar volume of a hydration complex can be divided into the molar volume of the stoichiometric mixture of electrolyte ions free of hydration shell V2h and the molar volume of water in hydration shell V1h:
Vh ) V2h + hV1h
(18)
substitution of eq 18 into eq 17 gives the equation for molar volume of a stoichiometric mixture of ions without hydration shells and the molar volume of hydration water:
State of Hydration Shells of Sodium Chloride
φV ) V2h - h(V /1 - V1h)
J. Phys. Chem. B, Vol. 113, No. 1, 2009 215
(19)
where (V 1/ - V1h) characterizes the volumetric compression of water in a hydration shell. Thus, the use of eq 15 together with eq 19 enables one to calculate the coefficient of adiabatic compressibility of the stoichiometric mixture of ions (β2h) and, what is especially important, the coefficient of adiabatic compressibility and molar volume of the solvent in hydration shells (β1h, V1h). The change of the molar adiabatic compressibility of hydration complex βhVh with the increase of temperature is negligible compared to the change of the molar adiabatic compressibility of pure water β1V 1/. Thus, it is reasonable to study the dependence of the apparent molar compressibility upon temperature at fixed solute concentrations rather than the concentration dependence at fixed temperatures. As in the present paper, the function YK,S is similar to the apparent molar compressibility and the variable β1V 1/ is the molar adiabatic compressibility of a free solvent in solution; the dependence YK,S ) f(β1V 1/) was investigated at constant solute concentrations in order to determine h.7,8,10,11 Application of the given approach in a wider range of conditions and the assessment of structural parameters of the formed solute complexes require precise experimental measurements. In the present study, the data on the speed of sound (U), density (F),13,14 and isobaric capacity15 in a wide concentration and temperature range have been used. 3. Results and Discussion 3.1. Influence of Solvent. The molar coefficients of adiabatic compression (βSVm) were calculated on the basis of the data obtained which are referred to as rational parameters in ref 16. It has been shown that the molar adiabatic compressibility (βSVm) of NaCl aqueous solution decreases with the rise of electrolyte temperature and concentration. The influence of these factors on solution compressibility can be expressed by the vibration and configuration components as in the case of the temperature dependence of water compressibility:17,18
βSVm ) (βSVm)vib + (βSVm)conf
Figure 1. Concentration dependence of molar adiabatic compressibility of NaCl aqueous solutions at different temperatures (1, 278.15 K; 2, 288.15 K; 3, 298.15 K; 4, 308.15 K).
(20)
where (βSVm)vib is the vibration component due to the intermolecular thermal vibrations and (βSVm)conf is the configuration component. According to Hall‘s two-structure model, the latter is attributed to the displaced equilibrium between the spatial structure of water (characterized by the tetrahedral coordination) and the closely packed structure. With the rise of temperature, the vibration component in βSVm augments due to the heat expansion which causes the increase of compressibility, whereas the configuration component decreases due to the equilibrium shift toward a denser structure; thus, compressibility also diminishes. At low temperatures, the second process prevails. Consequently, below the CSL, the lowering of the molar compressibility in the studied temperature range (273.15-323.15) results from the predominant role of the configuration component.19 Figure 1 shows that for NaCl aqueous solutions the CSL is reached at concentration x2 ) 0.08, where βSVm ) f(x2) curves corresponding to different temperatures intersect in one point. It is to be noted that water compressibility is characterized by several peculiarities connected with structural transformations of the net of hydrogen bonds that affect the compressibility of ionic hydration shells. One of these peculiarities is the existence of a minimum on the temperature dependence of water
Figure 2. Temperature dependence of pair correlation functions of water at 1000 atm.20
compressibility at 338.15 K, and another is the minimum of the molar adiabatic compressibility at 329.15 K. The latter becomes apparent from the temperature dependence of the radial distribution function at high pressure (Figure 2) when the
216 J. Phys. Chem. B, Vol. 113, No. 1, 2009
Afanasiev et al.
Figure 5. Molar adiabatic compressibility of hydration complexes as a function of hydration number for aqueous NaCl solutions.
Figure 3. Temperature dependence of molar volume of hydration shells for sodium, potassium, and rubidium chlorides.
Figure 6. Apparent volume of NaCl as a function of hydration number (1, 273.15 K; 2, 278.15 K; 3, 283.15 K; 4, 288.15 K; 5, 293.15 K; 6, 298.15 K; 7, 303.15 K; 8, 313.15 K; 9, 323.15 K).
Figure 4. Correlations between the pressure derivative of apparent molar volume at constant solution entropy YK,S ) -(∂φV/∂p)Sm and the compressibility of free solvent β1V /1 for aqueous solutions of sodium chloride (1, 0.001 m; 2, 0.2 m; 3, 0.6 m; 4, 1.0 m; 5, 1.6 m; 6, 2.4 m; 7, 3.0 m; 8, 3.6 m; 9, 4.4 m; 10, 5.0 m).
tendency of rupture of the tetrahedral water structure with temperature (lowering of the 4.5 Å peak) is replaced by the tendency of its stabilizing (growth of the T < 573.15 K peak).20 We believe that this effect can manifest itself not only under supercritical conditions, but it can occur in hydration shells of ions close to normal temperatures as well. Electrostriction that
provides high pressure around ions reduces the probability of formation of tetrahedral water structure, and considerable increase of the kinetic energy of water molecules is needed to recreate it. This may be accompanied by an additional growth of the molar volume of water in hydration shells with temperature. It can be exemplified by V1h ) f(T) dependencies for aqueous solutions of alkali metals including sodium chloride (Figure 3). 3.2. Complete Solvation Limit. In the presence of ions, water acquires more compact structures which results in the reduction of Vm of the solution due to electrostriction of water in hydration shells. Below the CSL, constriction of free water decreases, the configuration contribution to compressibility becomes smaller, and having passed through the minimum value, βSVm gradually increases with salt content. As evidenced by experimental research,10,11 below the CSL, the βSVm ) f(x2) function is drop-down, although the dependence of the speed of ultrasound for some electrolytes (e.g., KI) is extreme. The higher the salt concentration, the closer are the indicated dependencies for different temperatures, and in the point of their intersection, βSVm becomes independent of temperature (Figure 1). The correlation x1 ) hx2 is the complete hydration limit: x1 - hx2 ) 0, Vh ) Vm/x2, βSVm ) x2βhVh f βS ) βh. Thus, the interval of admissible concentrations turns out
State of Hydration Shells of Sodium Chloride
J. Phys. Chem. B, Vol. 113, No. 1, 2009 217
to be rather broad and stretches from dilute solutions up to rather high molalities (NaCl, 5.1; KI, 4.56; NaNO3, 4.46). It substantially exceeds those calculated from thermodynamic correlations obtained by the Debye-Huckel method even with semiempirical assumptions. It is important to stress that, in the point of intersection, βS coincides with the coefficient of adiabatic compressibility of hydration complex βh of solute which enables the compressibility of the hydration complex to be measured. 3.3. Research at Concentrations below the Complete Solvation Limit. Thorough analysis of literary data13-15 on density, heat capacity, and the speed of ultrasound in aqueous solutions of sodium chloride revealed that the function YK,s ) f(β1V /1) is linear at temperatures up to 323.15 K with correlation coefficient Rcorr g -0.9998 (Figure 4). This is indicative of the fact that the hydration numbers h and the molar adiabatic compressibility βhVh are independent of temperature.21-23 Hence, hydration numbers can be defined as
h)
d(∂φV /∂p)Sm d(β1V /1)
(21)
The values of hydration numbers, adiabatic compressibility of hydration complexes, and linear correlation coefficients calculated using eq 10 are given in Table 1. As is seen, the values of h and βhVh gradually decrease with the increase of electrolyte concentration which is probably connected with the overlap of hydration shells of ions at increased salt concentrations. The form of the hydration number dependence on solute concentration is close to exponential:
h ) h0 exp(-kx2)
(22)
where h0 is the hydration number at infinite dilution and k is a constant that defines the dependence of hydration number upon electrolyte concentration. For sodium chloride solutions, the following values were obtained: h0 ) 24.43, k ) 9.99. Experimental curves of the hydration number logarithm via the molar fraction of solute indicate that the choice of the exponential function for the description of concentration dependence of hydration numbers was quite justified: Rcorr ) -0.999. It is to be noted that the choice of the approximating function for eq 22 was based on the best coincidence between the experimental data and the describing function. Thus, expression 22 is purely empirical and does not rest upon deep theory. The molar adiabatic compressibility of the stoichiometric mixture of ions without hydration shells β2hV2h and the adiabatic compressibility of water in hydration shells β1hV1h were determined by the correlation 15. Further experimental data treatment reveals that β2hV2h and β1hV1h do not depend on
TABLE 1: Concentration Dependence of Molar Adiabatic Compressibility of Hydration Complexes βhVh, Hydration Numbers h, and Coefficients of Linear Correlation Rcorr for Aqueous Solutions of NaCl at 273.15-323.15 K (below the CSL) m, mol kg-1
βhVh · 1014, Pa-1 · m3 · mol-1
h
Rcorr
0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.5 1.6 1.8 2.0 2.2 2.4 2.5 2.6 2.8 3.0 3.2 3.4 3.5 3.6 3.8 4.0 4.2 4.4 4.5 4.6 4.8 5.0
15.718 15.702 15.667 15.621 15.547 15.369 15.119 14.686 14.297 13.936 13.598 13.277 12.972 12.680 12.402 12.134 11.630 11.162 10.940 10.726 10.320 9.939 9.582 9.246 9.086 8.930 8.632 8.351 8.084 7.833 7.712 7.594 7.367 7.152 6.948 6.753 6.659 6.567 6.391 6.222
25.279 25.256 25.206 25.141 25.036 24.781 24.424 23.803 23.243 22.723 22.233 21.768 21.325 20.900 20.492 20.100 19.359 18.668 18.339 18.021 17.415 16.845 16.307 15.800 15.556 15.320 14.865 14.434 14.025 13.635 13.448 13.265 12.912 12.575 12.253 11.946 11.797 11.652 11.370 11.101
-0.99961 -0.99962 -0.99963 -0.99964 -0.99966 -0.99970 -0.99975 -0.99980 -0.99984 -0.99986 -0.99987 -0.99988 -0.99988 -0.99987 -0.99986 -0.99984 -0.99980 -0.99975 -0.99972 -0.99969 -0.99963 -0.99956 -0.99948 -0.99941 -0.99937 -0.99933 -0.99926 -0.99918 -0.99910 -0.99903 -0.99899 -0.99895 -0.99888 -0.99881 -0.99874 -0.99868 -0.99865 -0.99861 -0.99855 -0.99850
concentration, and all the changes of adiabatic compressibility of hydration complexes with the increase of solute concentration result from the concentration dependence of hydration numbers. Figure 5 shows a linear correlation between βhVh and hydration number h (linear correlation coefficients for sodium chloride Rcorr g 0.9997), which confirms our assumption about β2hV2h and β1hV1h being independent of concentration. The isotherms of the apparent molar volume φV ) f(h) shown in Figure 6 correspond to eq 19 where (V 1/ - V1h) is the change
TABLE 2: Volumetric Compression (V 1/ - V1h), Volume V1h, and Compressibility β1h of Water in Hydration Shells in Aqueous Solutions of NaCl, Volume V2h and Compressibility β2h of a Stoichiometric Mixture of Ions at Concentrations below the CSL T, K
(V /1 - V1h) · 107, m3 · mol-1
V1h · 105, m3 · mol-1
β1h · 1010, Pa-1
V2h · 105, m3 · mol-1
β2h · 1010, Pa-1
273.15 278.15 283.15 288.15 293.15 298.15 303.15 313.15 323.15
4.345 3.923 3.562 3.254 3.000 2.781 2.607 2.366 2.242
1.758 1.762 1.766 1.771 1.775 1.779 1.783 1.792 1.801
3.821 3.812 3.803 3.794 3.785 3.776 3.767 3.749 3.730
2.408 2.407 2.406 2.405 2.404 2.403 2.403 2.407 2.414
5.495 5.496 5.499 5.501 5.504 5.505 5.505 5.498 5.482
218 J. Phys. Chem. B, Vol. 113, No. 1, 2009
Afanasiev et al.
Figure 7. (A) Temperature dependence of molar volume V1h and compressibility β1h of water in hydration shells of sodium chloride ions at concentrations below the CSL. (B) Temperature dependence of molar volume V2h and compressibility β2h of a stoichiometric mixture of ions of sodium chloride at concentrations below the CSL.
Figure 9. Correlation between Y true K,S and the molar adiabatic compressibility of free solvent for aqueous solutions of sodium chloride (1, 0.001 m; 2, 0.2 m; 3, 0.6 m; 4, 1.0 m; 5, 1.6 m; 6, 2.4 m; 7, 3.0 m; 8, 3.6 m; 9, 4.4 m; 10, 5.0 m).
Figure 8. Temperature dependence of molar volume of neat water and of water in hydration shells of sodium chloride (1) at different pressures.
of volumetric compression of water in hydration shell, V2h is the molar volume of the stoichiometric mixture of ions without hydration environment, and V1h corresponds to the molar volume of water in hydration shell. From Figure 6, it is seen that the apparent molar volume is lineally dependent on h (Rcorr g 0.999). This fact suggests that the values of V2h and V1h do not depend on the electrolyte concentration and that the concentration dependence of the apparent molar volume results from the
change of hydration number with salt concentration. With temperature rise, increase of the molar volume of hydration water V1h is observed. It is explained by a stronger temperature dependence of V1h compared with that of the molar volume of free solvent V /1. The use of both eqs 15 and 19 makes it possible to calculate such numeric solvation parameters as the coefficient of adiabatic compressibility of the stoichiometric mixture of ions β2h, and, more importantly, the coefficient of adiabatic compressibility β1h and the molar volume of the solvent in hydration shells V1h. Temperature dependencies of V1h, (V2h) and β1h, (β2h) are presented in Figure 7, while their numerical descriptions (data on basic volumetric properties) are brought together in Table 2. It is noteworthy that the molar volume V1h and the coefficient of adiabatic compressibility β1h of water in the liquid phase is linearly dependent on temperature in the interval 273.15-323.15 K while the molar volume of the stoichiometric mixture without hydration environment V2h and its coefficient
State of Hydration Shells of Sodium Chloride
J. Phys. Chem. B, Vol. 113, No. 1, 2009 219
Figure 10. Correlation between verified activity coefficient upon hydration number for aqueous solutions of sodium chloride. Figure 11. Molar adiabatic compressibility as a function of hydration number at concentrations beyond the CSL.
of adiabatic compressibility β2h in this temperature range pass through extremes (Figure 7). This points to the fact that, below 300 K, the molar volume decreases and the compressibility of the stoichiometric mixture of ions without hydration shell increases which may be due to dehydration of ions and increase of their compressibility. The dehydration effect is stronger at lower temperatures where solvation is of great importance and becomes predominant for the molar volume above 300 K. It is known that, at pressures up to several hundreds of atmospheres, the temperature dependence of the molar volume of pure water is not strongly affected by the change of pressure. Therefore, it was interesting to compare temperature dependencies of the molar volume of water in hydration shells with the molar volume of pure water at different pressures. The comparison showed that for the molar volume of water in hydration shells the temperature dependence is more pronounced than that for pure water (Figure 7). Such a result indicates that “electrostriction compression of water” affects the water structure stronger that the mere pressure increase. It can be assumed that the reason for this phenomenon lies in the decrease of the dielectric permeability of water in direct proximity to ions. As is seen from Figure 8, the change of average pressure in hydration shells of sodium chloride ions at 298.15 K is about 360 atm as compared with pure water. To be completely sure of the validity of the approach in question, above all at concentrations below the CSL, an additional verification was undertaken using an independent method that defines a solution as a totality of the solute and the solvent hydration complexes. Equation 1 for the molar volume includes only the formal molar fractions of the solute and solvent (x1 and x2). Their true values (Z1 and Z2) can be derived only through the formal ones taking into account the solvation numbers (h) that are concentration dependent:
Z1 )
n1 - hn2 x1 - hx2 ) , n1 - hn2 + n2 1 - hx2 x2 n2 Z2 ) ) (23) n1 - hn2 + n2 1 - hx2
Then, the molar volume (V true m ) and the molar compressibility of solution (βSV mtrue) will take on the following values:
Figure 12. Apparent molar volume as a function of hydration number at concentrations beyond the CSL (1, 273.15 K; 2, 278.15 K; 3, 283.15 K; 4, 288.15 K; 5, 293.15 K; 6, 298.15 K; 7, 303.15 K; 8, 313.15 K; 9, 323.15 K).
V mtrue )
Z1 · M1 + Z2(M2 + hM1) F
βSV mtrue )
Z1 · β1V /1 + Z2 βhVh F
(24)
(25)
true will be expressed by Consequently, Y K,S
true φK,S )
(Z1β1V /1 + Z2 βhVh - Z1β/1V /1) Z2
true true Y K,S ) φK,S - (Z1 /Z2)R/1V /1T ·
(
R/1 σ/1
-
)
(26)
Raq ) βhVh σaq
(27)
The use of hydration numbers calculated in this fashion makes the corrected apparent molar compressibility of the solute (Y true K,S ) equal to the molar adiabatic compressibility of the hydration
220 J. Phys. Chem. B, Vol. 113, No. 1, 2009
Afanasiev et al.
Figure 13. Dependence of the pressure derivative of apparent molar volume at constant solution entropy YK,S upon free solvent compressibility β1V /1 in a wide temperature interval (273.15-373.15 K).
complex. This makes Ytrue K,S independent of temperature (expressed through β1V 1/) at all studied conditions below the CSL. Figure 9 shows the dependence in question at different concentrations in the investigated range of temperatures up to 323.15 K. The rational activity coefficient of water in sodium chloride solution according to Lewis is equal to γ ) aW/x1. However, not all of the water in solution must be considered when calculating the activity coefficient because part of the solvent is involved in solvation shells of ions of the stoichiometric mixture. The concentration of water decreases with the increase of solute concentration, and only the free water (not included in hydration shells) can be regarded as a medium and be a true solvent as the definition implies: γR ) (aW/Z1) ) [aW(1 - hx2))/ (x1 - hx2)]. Figure 10 gives the dependence of the verified solvent activity coefficient γR upon hydration number h. Water activities are taken from ref 24. The dependencies γR ) f(h) are a typical example of a discontinuous function describing a spasmodic change of state of the solvent in solution. In the region of discontinuity, an abrupt change in γR is observed that is due to the solvent transferring to another state in solution. In this concentration region, the values of γR coincide with the region of the temperature inversion of the molar adiabatic compressibility of solution that characterizes the CSL. At small x2 (h being large), the verified activity coefficient of water grows from unity to its maximum value that corresponds to the transfer of the free solvent to coordination shells of ions. With further increase of salt concentration, starting from the point of minimum, γR begins to rise due to the removal of the solvent from hydrated ion pairs. At concentrations beyond the CSL,
Figure 14. Influence of NaCl concentration upon the temperature dependences of the molar adiabatic compressibility of free solvent at constant solution entropy.
ion-ion interactions result in a gradual displacement of the solvent from the inner coordination shells of ions, with this solvent being used for hydration of subsequent electrolyte portions. The solution exists both below and beyond the CSL. This is why, below the CSL, Z1 ) (x1 - hx2)/(1 - hx2) is the molar fraction of the uncoordinated water, and, beyond the CSL, it is the molar fraction of water displaced from hydration shells due to ion-ion interactions Z1′ ) (x1 - h′x2)/(1 - h′x2). Index (/) is used for parameters determined beyond the CSL. As the concentration is increased and the CSL is approached, the molar fraction of the free solvent approaches zero while the water activity in this composition range is still different from zero. Thus, the activity coefficient at CSL turns to infinity. It is this region that was used by us for determining the concentration dependence of hydration numbers. At CSL, Z1 ) 0 and therefore it becomes possible to determine it accurately from experiments. 3.4. Research at Concentrations beyond the Complete Solvation Limit. When the solute concentration approaches CSL, all solvent molecules penetrate hydration shells of ions,
TABLE 3: Volumetric Compression (V 1/ - V1h), Volume V1h, and Compressibility β1h of Water in Hydration Shells in Aqueous Solutions of NaCl, Volume V2h and Compressibility β2h of a Stoichiometric Mixture of Ions at Concentrations beyond the CSL T, K
(V /1 - V1h′) · 107, m3 · mol-1
V1h′ · 105, m3 · mol-1
β1h′ · 1010, Pa-1
V2h′ · 105, m3 · mol-1
β2h′ · 1010, Pa-1
273.15 278.15 283.15 288.15 293.15 298.15 303.15 313.15 323.15
3.051 2.763 2.516 2.307 2.132 1.990 1.876 1.727 1.665
1.771 1.774 1.777 1.780 1.783 1.787 1.791 1.798 1.806
4.095 4.020 3.957 3.903 3.857 3.820 3.790 3.749 3.730
2.262 2.277 2.290 2.300 2.309 2.318 2.325 2.350 2.354
7.985 7.442 6.983 6.594 6.263 5.981 5.741 5.360 5.083
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J. Phys. Chem. B, Vol. 113, No. 1, 2009 221
βhV h ) β2hV 2h + h · β1hV 1h
(29)
Since the solvation number beyond the CSL is described by expression 28, h′ ) (1/x2) - 1 ) (1 - x2)/x2 ) x1/x2, eq 29 can be brought to the form
βhV h ) β2hV 2h + (x1 /x2) · β1hV 1h
(29a)
Substituting h′ into eq 8, we get the equation
βSV m ) (x1 - hx2) · β/1V
/ 1
+
(x1 - hx2) · R/1V /1T ·
[
R/1 σ/1
-
]
Rm + x2 · βhV h (30) σm
in which expressions in parentheses will equal zero. Therefore, eq 30 can be written as
βSVm ) x2 · βhV h
(30a)
It follows from eq 30a that βh′Vh′ will be equal to the molar adiabatic compressibility of the solution divided by the concentration of the solute βh′Vh′ ) βSVm/x2; hence, eq 29a can be rewritten in the form
βSVm /x2 ) β2hV 2h + (x1 /x2) · β1hV 1h
(31)
Figure 15. Temperature dependence of YK,S at given concentrations.
so there is no free solvent left in solution. Therefore, in this case, the solvation number of the stoichiometric mixture of ions at concentrations beyond the CSL means the ratio of the number of solvent molecules to the number of solute molecules or the relationship between their molar fractions h′ ) N1/N2′ ) x1/x2. Considering that x1 ) 1 - x2, the last expression can be written as
h )
1 -1 x2
(28)
This corresponds to the balance between the number of solvent and solute particles, and coincides with the indicated relationship.25 Beyond the CSL, due to interactions of the solute ions with solvent dipoles, solvent molecules are gradually forced out of the inner coordination shells of ions and are involved in solvation of additional portions of the solute. In this connection, the question arises of how to determine such solvation parameters as the molar adiabatic compressibility without hydration ′ V2h ′ and the volume V2h ′ of the stoichiometrical mixture shell β2h ′ V1h ′ , and the of ions, the molar adiabatic compressibility β1h ′ of solvation shells of ions beyond the CSL. Since volume V1h below the CSL the values β2hV2h and β1hV1h are determined from eq 15, where h is calculated by formula 21, it would be logical to try applying an expression similar to eq 15 to determine the ′ V2h ′ and β1h ′ V1h′ beyond the CSL. If we substitute values β2h indexed values for all the values in eq 15 (they correspond to the values determined beyond the CSL), we obtain
It should be noted that expression 31 is just another form of writing expression 29. By analogy with the calculations for concentrations below the CSL, we construct a relationship between the function βSVm/x2 and variable h′ ) x1/x2. From Figure 11, it is seen that beyond the CSL the molar adiabatic compressibility is still linearly dependent on the hydration number but, at the same time, becomes temperature dependent. This fact in no way disagrees with the results obtained for concentrations below the CSL. The described behavior of β′hV ′h may be explained by further overlap of inner hydration shells in inner coordination spheres beyond the CSL which results in hydrated water acquiring the properties of a free solvent which, in their turn, depend on temperature. This explains the fact that the molar adiabatic compressibility of hydration complexes becomes temperature dependent beyond the CSL. The linearity of the function βSVm/x2 ) βh′Vh′ ) f (h′) makes it possible to ′ V2h ′ and β1h ′ V1h ′ are independent of conclude that the values β2h the solute concentration and we can assume them to depend on temperature alone. By analogy with the results obtained at concentrations below ′ and V1h ′ , we are going to the CSL, to determine the values V2h use eq 19. Substituting in it indexed values for all values depending on concentration, we obtain the relationship φV ) ′ - h′(V 1/ - V1h ′ ) that, in view of eq 28, will have the form of V2h
φV ) V2h - (x1 /x2) · (V /1 - V1h)
(32)
Figure 12 demonstrates linear dependence of the apparent molar volume of solute on hydration numbers at concentrations beyond the CSL, with correlation coefficient Rcorr > 0.9999. Thus, φV/ is a linear function of hydration number h′ ) (1/x2) ′ and V1h ′. - 1, and from eq 32, one can determine the values V2h
222 J. Phys. Chem. B, Vol. 113, No. 1, 2009 It is necessary to note that the linearity of function φV/ ) f(h′) ′ and V1h ′ are dependent on temperature means that the values V2h but not on concentration. As eq 31 makes it possible to define the molar adiabatic compressibility of a stoichiometric mixture ′ V2h ′ as well as the molar of ions without hydration shells β2h ′ V1h ′ at adiabatic compressibility of water in hydration shell β1h concentrations beyond the CSL (when using both eqs 31 and ′ ) β2h ′ V2h ′ / 32), it becomes possible to calculate the values β2h ′ , β1h ′ ) β1h ′ V1h ′ /V1h ′ . The results of the calculations are listed V2h in Table 3. 3.5. Results Obtained at High Temperatures. When the temperature rose above 323.15 K, dependence YK,S ) f(β1V 1/) (used to calculate solvation numbers and the molar compressibility of hydrated complexes in the temperature interval 273.15-323.15 K) changes its direction: at first, it increases and then decreases nonlinearly at higher temperatures (Figure 13). This makes it impossible to calculate solvation numbers using the correlations given in the paper. The dependence shown in Figure 13 of the derivative of apparent molar volume with respect to pressure at constant solution entropy YK,S ) -(∂φV/ ∂p)Sm on the compressibility of free solvent β1V /1 in a wide range of temperatures (273.15-373.15 K) indicates that this function deviates from linearity at all concentration values, from 0.05 to 6.0 m. This correlation was derived from temperature dependencies of β1V /1 and YK,S of aqueous NaCl solutions which showed that the function β1V /1 crosses the concentration isotherm twice, at 278.15 and 343.15 K (Figure 14), whereas the function YK,S changes monotonously (Figure 15). It follows from the above that the method described in the present paper is not suitable for determining solvation numbers at high temperatures only due to peculiarities of water as a solvent and depends neither on solute concentration nor on its nature. We believe that, for nonaqueous solvents, whose temperature dependencies of compressibility do not pass through turning points, the method will be appropriate for high temperatures as well. 4. Conclusions Thus, the undertaken research makes it possible to conclude that reliable data on density, heat capacity, and the speed of ultrasound in solutions, as well as the use of correct thermodynamic correlations discussed in the present paper, enable solvation parameters in aqueous electrolyte solutions to be calculated over quite a wide concentration interval. Acknowledgment. The authors express gratitude to the Russian Fund of Basic Researches for supporting the present study (Grant No. 05-03-32251). List of Symbols Vm x1 x2 h h′ V 1/ Vh p Sm S 1/ V1h V2h T h0 k
molar volume of the solution mole fraction of the solvent mole fraction of the solute hydration number hydration number beyond the CSL molar volume of the solvent molar volume of hydration complexes pressure solution entropy entropy of the pure solvent molar volume of water in hydration shells molar volume of the stoichiometric mixture of ions temperature hydration number at infinite dilution constant characteristic of the given electrolyte
Afanasiev et al. Rcorr Z1 Z2 true Y K,S aW ′ V2h ′ V1h
coefficient of linear correlation true molar fraction of solvent true molar fraction of hydration complexes true value of the derivative of the apparent molar solute volume with respect to pressure at constant solution entropy water activity molar volume of the stoichiometric mixture of ions beyond the CSL molar volume of water in hydration shells beyond the CSL
Greek Symbols F βS β1 βh β1/ φK,S β1h β2h φV R1/ Rm σ 1/ σm F1/ γR ′ β2h ′ β1h
solution density coefficient of adiabatic compressibility of solution coefficient of adiabatic compressibility of free solvent coefficient of adiabatic compressibility of hydration complexes coefficient of adiabatic compressibility of pure solvent apparent adiabatic compressibility of solute coefficient of adiabatic compressibility of water in hydrated spheres coefficient of adiabatic compressibility of the stoichiometric mixture of ions without hydrated environment apparent volume of the solute coefficient of thermal expansion of pure solvent coefficient of thermal expansion of solution isobaric thermal capacity of solvent per its unit volume isobaric thermal capacity of solution per its unit volume density of pure solvent rational activity coefficient coefficient of adiabatic compressibility of the stoichiometric mixture of ions without hydration environment beyond the CSL coefficient of adiabatic compressibility of water in hydration shells beyond the CSL
References and Notes (1) Water Structure, State, SolVation; Kutepov, A. M., Ed.; Nauka: Moscow, 2003; p 404. (2) Ohtaki, H. Monatsh. Chem. 2001, 132, 1237–1253. (3) Koneshan, S.; Rasaiah, Jayendran, C.; Lynden-Bell, R. M.; Lee, S. H. J. Phys. Chem. B 1998, 102, 4193–4204. (4) Max, J.-J. J. Chem. Phys. 2001, 115, 2664–2675. (5) Max, J.-J. J. Chem. Phys. 2007, 126, 184507. (6) Chizhik, V. I. NMR-relaxation; Pub. Of St.-Pb. State Univ.: St.Petersburg, 2004; p 388. (7) Afanasiev, V. N.; Tunina, E. Y. Pap. Acad. Sci. RF, Ser. Chem. 2003, 2, 322–328. (8) Afanasiev, V. N.; Zaitsev, A. A. J. Struct. Chem. 2006, 47, 94– 101. (9) Onori, G. J. Chem. Phys. 1988, 89, 510–516. (10) Afanasiev, V. N.; Ustinov, A. N.; Vashurina, I. Y. J. Solution Chem. 2006, 35, 1477–1491. (11) Afanasiev, V. N.; Ustinov, A. N. J. Solution Chem. 2007, 36, 853– 868. (12) Blandamer, M. J. J. Chem. Soc., Faraday Trans. 1998, 94, 1057– 1062. (13) Millero, F. J.; Vinokurova, F.; Fernandez, M.; Hershey, J. P. J. Solution Chem. 1987, 16, 269–283. (14) Connaughton, L. M.; Hershey, J. P.; Millero, F. J. J. Solution Chem. 1986, 15, 989–1002. (15) Colin, E.; Clarke, W.; Glew, D. N. J. Phys. Chem. Ref. Data 1985, 89, 489–610. (16) Afanasiev, V. N.; Panenko, E. S.; Kutepov, A. M. Pap. Acad. Sci. RF 2001, 377, 212–215.
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