The size and structure of selected hydrated ions and implications for ion channel selectivity

Abstract

The part of the coordinated water around an ion in aqueous solution that is so strongly bound to the ion that it cannot be dissociated, known as hydration water, and the combination of the ion and hydration water, known as the hydrated ion, have been examined. An approach to determine the average volume and hydration number of hydrated ions has been developed and applied. Selected alkali metal and halogen hydrated ions were selected for analysis, according to their densities and the salt volume concentrations of aqueous solutions reported in the literature. The results of the new approach show that the effective radii and structures of hydrated Na⁺, K⁺, Rb⁺, and Cs⁺ play a key role in ion transport properties, such as ion channel selectivity.

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1. Introduction

Aqueous solutions in living systems contain appreciable quantities of ions, such as sodium and potassium ions. In studies of ion selectivity in membrane transport, the prevailing view¹ has been that, inside the selectivity filter, K⁺ ions are coordinated by oxygen atoms from proteins, which replace the water molecules that normally surround the ion. However, the field generated by ions, such as sodium ions, is 138.4 GV m⁻¹ near its surface, at 0.102 nm from its center.² As a result the coordination waters of an ion can be dissociated only partly, due to the high electric field and the strong polarity of water molecules. In fact, the unable-to-dissociate waters become part of the ion. Therefore, the total dissociation equilibrium of a coordinated ion, I(H₂O)_m, is I(H₂O)_m = I(H₂O)_n + (m − n)H₂O, where I(H₂O)_n is called a hydrated ion, and m and n (an integer from 0 to m) are the coordination number and the hydration number, respectively. The experimental observations that potassium ions and water in narrow ion channels are together transported^3,4 and the partial molar volumes of small and/or multivalent metal ions in aqueous solutions are negative values,⁵ support the existence of hydration water. Physically, the transport properties of ions in aqueous solution must be closely related to the average volume and hydration number of its hydrated ions. However, the scattering method^6,7 and molecular dynamics simulations^8–10 were only capable of telling us how many are in the first shell coordination number for water around an ion and the volume size of the coordinated ion. Thus far, absolute volumes of hydrated ions have not been reported in the literature. There are very important differences among hydration numbers of ions reported in the literature.^11–13 The hydration number reported by Stokes and Robinson¹¹ is 3.5, 4.2, 5.5, 1.9, 2.1 and 2.5 for NaCl, NaBr, NaI, KCl, KBr, and KI, respectively. Dielectric relaxation measurements¹² show that the hydration number for Na⁺ is 2.6, and for K⁺, Cs⁺, Cl⁻, Br⁻, and I⁻ is 0. Zavitsas¹³ indicated that the hydration number is 3.4, 3.5, 3.9, 1.3, 1.1, 1.4, and 0 for NaCl, NaBr, NaI, KCl, KBr, KI, and CsCl, respectively (the assigned hydration number for Cl⁻, Br⁻, and I⁻ was 0). However, experiments¹⁴ show that the B coefficients of these ions in the Jones–Dole expression for the viscosity of aqueous ionic solutions (in parentheses) are Cs⁺ (−0.047) < K⁺ (−0.009) < Na⁺ (0.085) and Cl⁻ (−0.005) > Br⁻ (−0.033) > I⁻ (−0.073) with dm³ mol⁻¹ units. Obviously the orders for the hydration numbers of these ions are not consistent with the ones for their B coefficients.

In this work, the partial molar volume of a salt in aqueous solution refers to the sum of the partial molar volumes of hydrated ions from the salt. First, we demonstrate that the collective formula of partial molar volume can be changed into a linear equation due to the fact that partial molar volumes of a salt and the solvent water in aqueous solution are independent of salt concentrations over a very large range. Second, an equation that correlates the activities of water in aqueous solution of a salt with the molarities of the salt is established, and the partial molar volume of the salt is a parameter of the equation. Third, applying the equations, the average volume and hydration number of some alkali metal- and halogen-hydrated ions are determined by the water activities. Densities and salt volume concentrations of aqueous solutions are reported in the literature. Finally, the relation between volumes and structures of hydrated ions and ion transport properties are investigated.

2. Approach

Since the hydration waters of an ion become effectively part of the ion without increasing the number of ions, the number of free water molecules that are called the solvent water should be decreased. According to the thermodynamic principle,where v is the volume of aqueous solution of salts, n_w, n_i, v_w, and v_i are the mole numbers and the partial molar volumes of the solvent water and the hydrated ion i, respectively. For a single salt, eqn (1) is replaced by

v = n_wv_w + n_sv_s (2)

where n_s and v_s are the mole number and the partial molar volume of the salt respectively. If H is the average hydration number of a salt, eqn (2) is changed towhere m_w and m_s comprise the total mass of the solvent water and the hydration water with the molar mass M_w and the mass of salt with the molar mass M_s, respectively, and d is the density of the solution. A large number of diffraction experiments^15–24 have shown that the length for ion-O (from water) distance and the average coordination number of ions are almost independent of the H₂O–salt molar ratio. This fact implies that v_s and H in eqn (3) are independent of salt concentration over a very large range. According to the Gibbs–Duhem equation, x_wdv_w + x_sdv_s = 0, where x_w and x_s are the mole fractions of solvent water and the salt, and v_w is also independent of the salt concentration. Interestingly, recent measurements^25,26 have shown that ions have no effect on water structure beyond the first sphere of surrounding water molecules. Similarly, this fact implies that v_w is nearly independent of salt concentration, and supports the notion that v_s and H are independent of salt concentration. Since v_s, H, and v_w are constant in the aqueous solution of a salt over a very large range of concentration, eqn (3) can be changed into linear equations incorporating the terms (the volume of solution per mole salt, VMS) and (the number of water per mole salt, NWMS) or (the volume of solution per mole water, VMW) and (the number of salt per mole water, NSMW),

VMS = v_w × NWMS + (v_s − H × v_w) (4)

or

VMW = (v_s − H × v_w) × NSMW + v_w (5)

Once v_s is determined, the H will be obtained by the intercept (v_s − H × v_w) of eqn (4) or the slope (v_s − H × v_w) of eqn (5), and the molar volume of pure water v^*_w.

Statistically, at the liquid–gas equilibrium there must be a few water molecules on the hydrated ions at the surface of the aqueous solution of a salt. Therefore, vapor pressure results from evaporation of surface water and water on the surface-hydrated ions. The former is nearly in direct proportion to the surface quantities , where c_w is the volume concentrations of the solvent water and the latter are in direct proportion to the surface quantities , where c is the volume concentration of the salt. Thus

where p_w, p^*_w, and c^*_w are the vapor pressure over the solution, the saturated vapor pressure and the volume concentration of pure water, respectively, and b is the parameter that measures the volatility of water on the surface of hydrated ions. Generally the relative vapor pressure is the activity a of the solvent water in the aqueous solution of a salt. Due to the field generated by hydrated ions, the volatility of the water at the side of the surface hydrated ions is somewhat smaller than that of the surface water away from the hydrated ions. Thus the first term on the right side in eqn (6) is slightly larger than the real contribution of the surface water to pressure, but the deviation is reduced when the salt concentration increases. The parameter b also is dependent on the salt concentration and increases when the salt concentration increases. In theory, if there is no interaction between hydrated ions and solvent water molecules, b should vanish to zero.

Based on the previous analysis, eqn (6) is empirically changed into

where b₁ and b₂ are salt-dependent parameters. Substituting from eqn (2) into eqn (7), and combining v_w = v^*_w and c^*_wv^*_w = 1, eqn (7) is finally changed into

When the coordination shells of cation and anion interpenetrate, eqn (8) is probably no longer valid, due to water molecules pushing each other on the surface-hydrated ions. Therefore, to determine the volume v_s of a salt in aqueous solution by fitting experimental data of and c with eqn (8), the should be the ones below a critical concentration c_cr (see the ESI†).

3. Results and discussion

Using the densities^27–33 for aqueous solutions of NaCl, KCl, NaBr, KBr, RbCl, CsCl, RbBr, CsBr, RbI, and CsI at 25 °C, we determined their NWMS, VMS, NSMW and VMW at molalities (m) in the very large ranges, successively from 0.01 to 5.95, 0.1 to 4.4, 0.03 to 5.2, 0.06 to 4.4, 0.02 to 6.8, 0.01 to 5.9, 0.04 to 6.0, 0.04 to 4.7, 0.01 to 4.7, and 0.02 to 3.1 (see the ESI†). The VMS, NWMS, VMW, and NSMW data for each salt solution were fitted with eqn (4) and (5), respectively; the multiple fitted v_w and (v_s − Hv_w) are shown in Table 1. The R-squares of the resulting linear regression with eqn (4) were generally unity with a confidence level of 99% and the fitted slopes v_w were generally 18.07 cm³ mol⁻¹; and the v^*_w from the density³⁴ 997.047 kg m⁻³ of pure water at 25 °C is 18.0687 cm³ mol⁻¹. Fig. 1 shows plots of VMS versus NWMS for NaCl and KCl solutions, respectively.

Table 1 Fitted v_w and (v_s − H × v_w) for aqueous solutions of selected alkali halides with eqn (4) and (5), respectively

Salt	Equation	vw/cm^3 mol^−1	(vs − H × vw)/cm^3 mol^−1	R-square
NaCl	(4)	18.07 (18.07, 18.07)	18.97 (18.42, 19.53)	1
	(5)	18.04 (18.03, 18.06)	20.8 (20.43, 21.17)	0.9985
KCl	(4)	18.07 (18.06, 18.07)	29.66 (29.08, 30.24)	1
	(5)	18.04 (18.03, 18.05)	31.23 (30.92, 31.54)	0.9996
NaBr	(4)	18.06 (18.07, 18.07)	25.68 (24.97, 26.39)	1
	(5)	18.05 (18.04, 18.06)	27.48 (27.2, 27.75)	0.9998
KBr	(4)	18.07 (18.07, 18.07)	35.37 (34.43, 36.32)	1
	(5)	18.06 (18.04, 18.07)	37.14 (36.75, 37.53)	0.9997
RbCl	(4)	18.07 (18.07, 18.07)	34.41 (33.74, 35.08)	1
	(5)	18.04 (18.02, 18.06)	36.68 (36.19, 37.17)	0.9993
CsCl	(4)	18.07 (18.07, 18.07)	41.45 (40.77, 42.13)	1
	(5)	18.04 (18.02, 18.05)	43.72 (43.26, 44.18)	0.9997
RbBr	(4)	18.07 (18.07, 18.07)	41.07 (40.52, 41.63)	1
	(5)	18.04 (18.03, 18.06)	42.93 (42.54, 43.32)	0.9997
CsBr	(4)	18.07 (18.07, 18.07)	48.21 (47.74, 48.68)	1
	(5)	18.05 (18.04, 18.06)	49.73 (49.38, 50.08)	0.9998
RbI	(4)	18.07 (18.07, 18.07)	51.92 (51.45, 52.38)	1
	(5)	18.05 (18.04, 18.06)	53.46 (53.13, 53.78)	0.9999
CsI	(4)	18.07 (18.07, 18.07)	59.12 (58.77, 59.46)	1
	(5)	18.06 (18.06, 18.06)	60.12 (59.93, 60.32)	1

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Fig. 1 Plots of VMS versus NWMS at 25 °C. (a) Open circles: the experimental data from aqueous solutions^27,28 of NaCl at molalities (m) from 0.01 to 5.95 m. Solid line: plot of eqn (4), in which the slope v_w and the intercept (v_s − H × v_w) are taken as 18.07 and 19.53 (mL mole⁻¹), respectively. (b) Open circles: experimental data from aqueous solutions^29,30 of KCl at molalities from 0.1 to 4.4 m. Solid line: plot of eqn (4), in which the slopes v_w and the intercepts (v_s − H × v_w) are taken as 18.07 and 29.66 (mL mol⁻¹), respectively.

Obviously, the experimental data lay almost within the fit lines for eqn (4). The results strongly indicate that the partial molar volumes of a salt and the solvent water are independent of salt concentration over a very large range. The poorest and best R-squares of the resulting linear regression with eqn (5) are 0.9985 for NaCl and 1 for CsI, respectively, and the fitted intercepts v_w closest to 18.07 cm³ mol⁻¹ are in the range from 18.05 to 18.07 cm³ mol⁻¹. Fig. 2 shows plots of VMW versus NSMW for NaCl and CsI solutions, respectively.

Fig. 2 Plot of VMW versus NSMW at 25 °C. (a) Open circles: the experimental data from aqueous solutions^27,28 of NaCl at molalities from 0.01 to 5.95 m. Solid line: plot of eqn (5), in which the slopes (v_s − H × v_w) and intercepts v_w are taken as 20.8 and 18.06 (mL mol⁻¹), respectively. (b) Open circles: experimental data from aqueous solutions^27,33 of CsI at molalities from 0.02 to 3.15 m. Solid line: plot of eqn (5), in which the slope (v_s − H × v_w) and the intercept v_w are taken as 59.93 and 18.06 (mL mol⁻¹), respectively.

The experimental data of CsI lay almost within the fit line, but some of the experimental data for NaCl lay outside the fit line. The numerical value comparison of NWMS with NSMW and VMS with VMW in Fig. 1a and 2a indicates that the magnitudes of NSMW and VMW were much smaller than NWMS and VMS, respectively; relatively speaking then, the larger experimental error results led to a somewhat poor fit with eqn (5).

Using the densities²⁸ of aqueous CaCl₂ solutions from 0.05 to 6.46 m at 25 °C, the NWMS and VMS are determined, and fitting the data of VMS and NWMS with eqn (4) with confidence level 99%, the multiple fitted v_w and the R-square are 18.06 (18.05, 18.07) cm³ mol⁻¹ and 1, respectively (see the ESI†). Eqn (4) also fits for a multivalent salt over a very wide range of concentration.

Correlating the experimental densities^27–33 with molalities, we determined that the molarities c corresponding to over the aqueous solutions^35–38 at molalities from 0.2 to 5.2 for NaCl, from 0.2 to 4.4 molality for KCl, from 0.1 to 4.5 for NaBr, from 0.1 to 4.0 for KBr, RbCl, CsCl, RbBr, CsBr, from 0.1 to 3.5 for RbI and from 0.1 to 3.0 for CsI at 25 °C (see the ESI†) and fitting the data of and the molarities c for each salt solution with eqn (8), the multiple fitted v_s are shown in Table 2. The R-squares of the resulting multiple nonlinear regressions with confidence level 99% are from 0.9996 to 1. Fig. 3 shows plots of versus c for aqueous solutions of NaCl and KCl, respectively.

Table 2 Fitted v_s for aqueous solutions of selected alkali halides with eqn (8)

Salt	vs/dm^3 mol^−1	R-square
NaCl	0.08327 (0.07994, 0.08661)	0.9998
KCl	0.06534 (0.06256, 0.06811)	0.9999
NaBr	0.08892 (0.08422, 0.09361)	0.9996
KBr	0.06627 (0.06380, 0.06875)	0.9999
RbCl	0.06168 (0.05964, 0.06372)	0.9999
CsCl	0.06395 (0.06111, 0.06679)	0.9999
RbBr	0.06224 (0.05981, 0.06467)	0.9999
CsBr	0.06617 (0.06209, 0.07025)	0.9998
RbI	0.06484 (0.06206, 0.06761)	0.9999
CsI	0.05677 (0.05409, 0.05944)	1

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Fig. 3 Plot of a versus c at 25 °C. (a) Open circles: a from aqueous solution³⁶ of NaCl at molalities from 0.2 to 5.2 m and corresponding c obtained by correlating the experimental densities^27,28 with the molalities. Solid line: plot of eqn (8), in which the v_s, b₁, and b₂ are taken as v_s = 0.08327 dm³ mol⁻¹, b₁ = 0.02852 molarity^−2/3, and b₂ = 1.532 molarity⁻¹, respectively. (b) Open circles: a from aqueous solution³⁶ of KCl at molalities from 0.2 to 4.4 m and corresponding c obtained by correlating the experimental densities^29,30 with the molalities. Solid line: plot of eqn (8), in which the v_s, b₁, and b₂ are taken as v_s = 0.06534 dm³ mol⁻¹, b₁ = 0.01368 molarity^−2/3, and b₂ = 1.713 molarity⁻¹, respectively.

The experimental data lay almost within the fit lines. The results strongly indicated that eqn (8) can be used to obtain the volume of a salt in aqueous solution over a large range of concentrations.

Correlating the experimental densities²⁸ with the molalities, we determined the molarities c corresponding to a over CaCl₂ aqueous solution³⁹ at molalities from 0.1 to 4.5 at 25 °C, and fitting the data of and c with eqn (8), the R-square of the resulting multiple nonlinear regression with confidence level 99% is 0.9996 (see the ESI†). Results show that eqn (8) also is fitted to a multivalent salt over a wide range of concentrations.

Since the partial molar volumes of a salt are independent of the salt concentration over very wide ranges, the additivity is also applicable at high salt concentrations. On application of the additivity, the v_s of NaCl minus the v_s of KCl should equal the v_s of NaBr minus the v_s of KBr and the v_s of NaBr minus the v_s of NaCl should equal the v_s of KBr minus the v_s of KCl or the v_s of CsBr minus the v_s of CsCl. Therefore, 0.08327, 0.06534, 0.08422, 0.06627, 0.06372, 0.06111, 0.06467, 0.06209, 0.06206 and 0.05944 dm³ mol⁻¹ in Table 2 are the closest to the absolute molar volume of NaCl, KCl, NaBr, KBr, RbCl, CsCl, RbBr, CsBr, RbI and CsI in aqueous solution, respectively.

The multiple calculated hydration number by the v_s for each salt and the multiple intercepts (v_s − Hv_w) of eqn (4) in Table 1 (v_w is all taken as v^*_w, 18.07 cm³ mol⁻¹) are presented in Table 3. Again, upon applying the additivity, it is known that the hydration number H of NaCl, KCl, NaBr, KBr, RbCl, CsCl, RbBr, CsBr, RbI and CsI in aqueous solution is 3.5, 2.0, 3.2, 1.7, 1.6, 1.1, 1.3, 0.8, 0.5 and 0, respectively. Consequently, the average hydration numbers H_i for each hydrated ion is (in parentheses) Na⁺ (2.4) > K⁺ (0.9) > Rb⁺ (0.5) > Cs⁺ (0) and Cl⁻ (1.1) > Br⁻ (0.8) > I⁻ (0), respectively. The hydration number sequences of Na⁺ > K⁺> Rb⁺ > Cs⁺ and Cl⁻ > Br⁻ > I⁻ are consistent with sequences for absolute hydration enthalpies,⁴⁰ and the B-coefficient, the B_NMR CaCl₂ and the negative water structural entropy −ΔS_struct (ref. 14) of Na⁺ > K⁺ > Rb⁺ > Cs⁺ and Cl⁻ > Br⁻ > I⁻. At 298.15 K, the absolute hydration enthalpies⁴⁰ of Na⁺, K⁺, Rb⁺, Cs⁺, Cl⁻, Br⁻, and I⁻ are −391, −308, −283, −258, −392, −361, and −321 (kJ mol⁻¹), and the B-coefficients¹⁴ are 0.085, −0.009, −0.033, −0.047, −0.005, −0.033, and −0.073 with dm³ mol⁻¹ units, respectively. The crystal radii⁴¹ of the above ions are Na⁺ (0.102) > K⁺ (0.138) > Rb⁺ (0.152) > Cs⁺ (0.167) and Cl⁻ (0.181) > Br⁻ (0.196) > I⁻ (0.220) with nanometer units, respectively. Comparison between the orders of the B-coefficients and crystal radii indicated that ion transport properties depend strongly on the ion hydration number.

Table 3 Calculated H of alkali halides by (v_s − H × v_w) and v_s

Salt	H
NaCl	3.5584 (3.5888, 3.5274)
KCl	1.9745 (2.0066, 1.9424)
NaBr	3.2396 (3.2789, 3.2003)
KBr	1.7100 (1.7620, 1.6574)
RbCl	1.6220 (1.6591, 1.5849)
CsCl	1.0880 (1.1256, 1.0504)
RbBr	1.3060 (1.3362, 1.2750)
CsBr	0.7681 (0.7941, 0.7421)
RbI	0.5612 (0.5872, 0.5357)
CsI	0.0177 (0.0371, −0.0011)

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Further analysis of H_i is summarized in Table 4. According to Table 4, it is known that Cs⁺ and I⁻ ions are bare and have a one-fold spherical structure, 60% and 40% of Na⁺ ions are hydrated with two and three water molecules, 90% and 10% of K⁺ ions are hydrated with one and zero water molecules, 50% of Rb⁺ ions are hydrated with one and zero water molecules, 90% and 10% of Cl⁻ ions are hydrated with one and two water molecules, and 80% and 20% of Br⁻ ions are hydrated with one and zero water molecules, respectively. As compared with the structures of hydrated potassium ions, the complex structures of hydrated sodium ions lead to Na⁺ limiting molar conductivity.⁴²

Table 4 Hydration number (H_i) and percentage (P) of hydrated ions

Aqueous ions	H1 and P1	H2 and P2
Na^+	2 and 60	3 and 40
K^+	1 and 90	0 and 10
Rb^+	1 and 50	0 and 50
Cs^+	0 and 100
Cl^−	1 and 90	2 and 10
Br^−	1 and 80	0 and 20
I^−	0 and 100

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The hydration numbers of Cl⁻ (0), Br⁻ (0), K⁺ (0) and Na⁺ (2.6) from dielectric relaxation measurements¹² do not contradict the structures of hydrated ions in Table 4. There can still be two movements for hydration water: circular motion on the surface of the ion and rotation around the axis of the bond between the ion and hydration water. According to Table 4, Na⁺(H₂O)₃ is trigonal planar molecular geometry and Na⁺(H₂O)₂ is linear molecular geometry. The external field will make dipoles (hydration water) on the surface of Na⁺ round in opposite direction, while the solvent water molecules between the hydration waters must stop the circular motion, so that the two or three dipoles on the surface of Na⁺ cannot freely move around. In addition, owing to symmetry, the rotation of hydration waters around the Na⁺–O axis cannot generate the change in dipole orientation. The results imply that the Na⁺ hydration number equals the number of irrotationally bound solvent molecules.²⁶ In contrast to Na⁺, the sole hydration water on the surface of K⁺ can freely move around and the movement can lead to changes in dipole orientation, so the number of irrotationally bound solvent molecules is zero. The sole hydration water on the surface of Br⁻ or 90% Cl⁻ can freely rotate around the Br⁻–H axis and the movement must lead to changing dipole orientation, so the number of irrotationally bound solvent molecules is zero.

Eqn (8) can represent the vapor pressures over aqueous solutions of alkali halides up to several molarities with a confidence level of 99% as shown in Fig. 3. In contrast, Raoult's law, , can only represent vapor pressure over very dilute aqueous salt solutions. In the ideal condition, physically the vapor pressure over the solution should be proportional to the surface quantity , not the bulk quantity x_w. However, the correctness of eqn (8) does not exclude the colligative properties of dilute solutions derived from Raoult's law, because at the low concentration limit, the values of calculated with eqn (8) were practically equal to the values of x_w. When c is 0.1 M, is 0.9964, and the calculated with eqn (8) is 0.9953 and 0.9961 for NaCl and KCl solutions, respectively (where v_s, b₁, and b₂ of NaCl and KCl are given, respectively, in Fig. 3 caption). When c are 0.01 M, x_w are 0.9996, and calculated is 0.9995 and 0.9996 for NaCl and KCl solutions, respectively. According to Zavitsas,¹³ when the activity a of solvent water in aqueous solution of a salt a = (55.509 − mH_T)/(55.509 − mH_T + mi_e) over wide concentration ranges (where m, H_T, and i are the molality, the thermodynamic hydration number, and the stoichiometric number of particles produced per mole of the salt, respectively), the hydration number from the freezing-point depressions of aqueous solutions of salt were NaI (3.9) > NaBr (3.5) > NaCl (3.4). The order numbers are not in agreement with the order B-coefficients for these anions. There are several major defects in Zavitsas's work. First, the strategy of using freezing-point depressions of the “ideal” aqueous solution of a salt to determine the thermodynamic hydration number in reality implies that the hydration number of an ion is independent of temperature! It is hard to believe the hydration number at 25 °C is same with as at −25 °C. Secondly, the equilibrium constant K_e for the ion pair was miscalculated as K_e = α²/(1 − α)² to support the notion that degree α of association to ion pairs is independent of salt concentration over wide concentration ranges. However, according to the definition of K_e, K_e = (1 + c_w/αc)α²/(1 − α)². In addition, the existence of the univalent ion pair require the relative permittivity of solvent be below 30,⁴³ so it is questioned that 31% NaCl is in the form of ion pairs in water. Thirdly, the H_T = 0 of CsCl is incompatible with the hypothesis: ion pair formation would replace at least one of the water molecules strongly bound to the cation by the anion to form an ion pair.

According to thermodynamic principles, the hydration numbers for NaCl (3.5), NaBr (4.2), NaI (5.5), KCl (1.9), KBr (2.1), and KI (2.5) reported by Stokes and Robinson¹¹ should be consistent with the ones found in the present approach. However, this is not the case. Although Stokes and Robinson also assumed that some molecules of solvent are strongly bound to the ions and then the remaining “free” water molecules are considered to be the solvent, they disregarded hydration in the process of deriving the Debye–Hückel equation with two parameters.

Coordination numbers for Na⁺, K⁺, and Cl⁻ are 4.5, 3.5 and 2.0 from standard molar ionic compression⁴⁴ and within the range of 5.0–5.8, 6.0–7.0, and 6.0–6.5 from molecular dynamics simulations,^8–10 respectively. According to the present theory, the coordination number of an ion must be larger than the hydration number of the ion. The smaller the radius of an ion and/or the higher the valence, the larger the hydration number; then the hydrated ions seem to be a cluster of water. Because hydration water can strongly shield the ion field, even the field of a multivalent hydrated ion is not too strong. In addition, hydration waters can block the concentric contraction of the coordination water; the coordination waters simultaneously also block the concentric contraction. These facts cause the volume of coordination water to continue nearly unchanged, as in the bulk, so that the molar volume of free water is the same as that of pure water until the hydrated ions interpenetrate.

The Cs⁺–H₂O bond length from neutron diffraction^6,7 is 0.295 nm, and the radius of a water molecule is generally taken as 0.138 nm,⁴⁵ so the radius of Cs⁺ in aqueous solution is taken as 0.157 nm. If the molar volume of a spherical-shaped ion is taken as the sum of the volume of closely packed spheres and the void space between closely packed spheres, according to cubic closet packing, the molar volume of Cs⁺ in aqueous solution should be 0.0132 dm³ mol⁻¹. Applying the additivity, the molar volumes of Na⁺, K⁺, Rb⁺, Cl⁻, Br⁻, and I⁻ in aqueous solution are about 0.0354, 0.0174, 0.0159, 0.0479, 0.0489, and 0.0462 dm³ mol⁻¹, and their effective radii are 0.218, 0.172, 0.167, 0.242, 0.243, and 0.239 nm.

Because the hydration numbers of Na⁺, K⁺, Rb⁺, and Cs⁺ in aqueous solution are 2.4, 0.9, 0.5, and 0, respectively, in contrast to the order of their crystal radii,⁴¹ the order of their effective radii in aqueous solution is Na⁺ > K⁺ > Rb⁺ > Cs⁺. In spite of this, according to Table 4, the transverse radii of hydrated K⁺, Rb⁺ and Cs⁺ are still their crystal radii. These facts are closely related to ion channel selectivity.^46,47 The structures of hydrated sodium ions with larger volume are more complicated, so these ions do not as easily pass into the narrow potassium channel.⁴⁸ In addition, the existence of hydrated ions does not exclude crown ethers of higher affinity than water; some cations (M) strongly bind, forming complexes, M(H₂O)_n + crown = M(H₂O)_ncrown = Mcrown + nH₂O. Similarly, shedding some or all hydration water, hydrated ions, such as Na⁺(H₂O)₃ or Na⁺(H₂O)₂, are coordinated by oxygen atoms from suitable protein due to the entropy gain arising through the release of water molecules from the hydrated ion, that is, Na⁺(H₂O)_2.4 + protein = Na⁺(H₂O)₁protein + 1.4H₂O and Na⁺(H₂O)_2.4 + protein = Na⁺protein + 2.4H₂O, where 2.4 is the average hydration number of hydrated sodium ions. According to these reactions, it is known that a significant effect of entropy gain also restricts movement of the sodium ion hydrated with one or no water molecule inside the selectivity filter. In contrast to hydrated sodium ions, the structures of hydrated potassium, rubidium and cesium ions with smaller volumes are simpler, so they can all enter into the K⁺ channels. However, to pass through the selectivity filter, an ion must shed a certain number water molecules from the first coordination shell of the ion. Because the greater the effective radius of an ion, the smaller the decoordination energy, all K⁺ channels show a selectivity sequence⁴⁶ of K⁺ > Rb⁺ > Cs⁺. In the same way, Na⁺ channels show a selectivity sequence⁴⁷ of Na⁺ > K⁺.

4. Conclusions

The measurement of physical quantities, such as water activities in aqueous solutions of a salt and densities of the solution, yield the average volume and hydration number of hydrated ions. Differences in volume and structure among hydrated ions result in different behaviors, such as ion transport properties. In addition, the existence of hydration water implies that the interactions between hydration water and solvent water are important component parts of the hydrated ion–solvent water interaction.

Acknowledgements

The work is supported by a grant from the National Natural Science Foundation of China (21173112).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c4ra10987b

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