Langmuir 2004, 20, 1871-1876
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Dehydration Energies of Alkali Metal Halides. A New Simulation Methodology Involving Mean Nearest Neighbor Distances and Thermodynamic Forces S. Harinipriya, V. Sudha, and M. V. Sangaranarayanan* Department of Chemistry, Indian Institute of Technology, Madras 600 036, India Received September 5, 2003. In Final Form: November 19, 2003 Dehydration energies of various alkali metal halides are estimated by employing a novel simulation methodology with the help of hydrated molecular radii. The hydration numbers at (i) infinite dilution and (ii) each movement of the molecules are calculated by employing random distribution of species in conjunction with ionic and molecular sizes. A satisfactory agreement with the reported estimates is noticed.
1. Introduction The estimation of the solvation free energies of ions and molecules plays a crucial role in diverse physicochemical processes such as heterogeneous charge transfer,1 bioavailability in drug design,2 analysis of phase equilibria in chemical engineering,3 etc. However, in view of the long-range nature of the electrostatic interactions along with the hydrogen bonding effects, the prediction of solvation energies even for simple inorganic ions and small organic compounds in a polar solvent (such as water) has remained formidable despite the present-day availability of intensive computational tools. Consequently, the evaluation of thermodynamic properties pertaining to hydration phenomena has been restricted to univalent4 and divalent5 cations or organic compounds with low molecular weights.6 While the computational strategies with the help of either Monte Carlo7 or Molecular Dynamics8 simulations have been extensively employed in this context, a phenomenological method by partitioning various components of solvation phenomena has served as a touchstone for the validity of the derived estimates from more involved (and expensive) computations. Among several sophisticated computational investigations hitherto investigated, mention may be made of the Widom particle insertion method,9 umbrella sampling,10 biased simulation procedure,11 extended ensemble method,12 etc. These methodologies provide new insights into the hydration phenomena per se; however, their extension to concentrated electrolytes remains formidable. It follows from the above that to evaluate the hydration energies of diverse systems, an entirely novel approach * To whom correspondence may be addressed. E-mail: mvs@ chem.iitm.ac.in. (1) See for example: Weaver, M. J. Int. J. Mass Spectrom. 1999, 182/183, 403. (2) Kellogg, G. E.; Abraham, D. J. Eur. J. Med. Chem. 2000, 35, 651. (3) Lin, C.-l.; Tseng, H.-c.; Lee, L.-s. Fluid Phase Equilib. 1998, 152, 169. (4) Lee, S. H.; Rasaiah, C. J. J. Phys. Chem. 1996, 100, 1420. (5) Obst, S.; Bradaczek, H. J. Phys. Chem. 1996, 100, 15677. (6) Sandberg, L.; Casemyr, R.; Edholm, O. J. Phys. Chem. B 2002, 106, 7889. (7) Jorgensen, W. L.; Ravimohan, C. J. Chem. Phys. 1985, 83, 3050. (8) Smith, D. E.; Dang, X. L. J. Chem. Phys. 1994, 100, 3757. (9) Widom, B. J. Chem. Phys. 1963, 39, 2804. (10) Ding, K.; Valleau, J. P. J. Chem. Phys. 1993, 98, 3306. (11) Frenkel, D.; Smit, B. Mol. Phys. 1992, 75, 983. (12) (a) Lyubartsev, A. P.; Laaksonen, A.; Vorontsov-Velyaminov, P. N. Mol. Phys. 1994, 28, 455. (b) Lyubartsev, A. P.; Martsinoskii, A. A.; Shevkunov, S. V.; Vorontsov-Velyaminov, P. N. J. Chem. Phys. 1992, 96, 1776.
is required that is simple and robust and has the ability to handle high concentrations without a corresponding increase in the computational resources. In this paper, we propose a methodology for evaluating the dehydration energies of different alkali metal halides by calculating the extent of dehydration of an entirely hydrated ionic pair at the initial stage. Apart from the computational simplicity, the procedure advocated here offers physical insights and quantitative reliability. One may wonder whether the postulate of a hydrated M+X- ionic pair is valid for strongly dissociated electrolytes of the type considered here. However, this visualization is consistent with the fact that the molecular radii (Rdesol) is written as the sum of the cationic and anionic radii based on the validity of the hard sphere model, and consequently, the procedure becomes realistic. 2. Methodology The essential input parameters of the present analysis are (i) the molecular radii of the alkali metal halides, considered as a sum of cationic and anionic radii, and (ii) the radius of the solvent (water) molecules. By use of these two parameters, an explicit expression for the hydration number of the molecules at infinite dilution is formulated. Subsequently, the hydration number at a chosen concentration is written, incorporating the random distribution of species. The specificity associated with the solvent arises via the dielectric constant, while that due to the ionic species arises from the charges as well as the diameters. In our simulation, the ionic pairs at the center of the cubic box shed their hydration sheaths progressively during the arrival at the surface of the cube and therefrom escapes to vacuum as dehydrated entities. Thus the simulation is employed for obtaining the number of molecules (ionic pairs) that (i) arrive at the surface of the cube and (ii) subsequently reach vacuum via random number generation employing volume ratio as the criterion. These values are then employed in the definition of the thermodynamic force so as to deduce the real and apparent dehydration energies, as shown below. 2.1. Simulation Details. The simulation was carried out in the NTP ensemble, and the system (consisting of hydrated alkali metal halides assumed as hard spheres) was placed in a cubic box of length l Å (cf. Figure 1). Boundary conditions were employed, and the cube was confined (rigidly fixed) along the z axis. The ionic pairs
10.1021/la035654a CCC: $27.50 © 2004 American Chemical Society Published on Web 01/23/2004
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and that of water molecules. In the present case, since we have considered the ionic pair (M+X-) as an entity, its hydration number at infinite dilution may be represented as
Nh∞ )
Figure 1. Actual displacement of hydrated M+X-. Red circle indicates the hydrated M+X- at the central region of the cube, blue protrusions are water molecules surrounding the hydrated M+X- (secondary and tertiary hydration sheath), yellow protrusions depict the partial formation of the hydration sheath when the hydrated M+X- moves to the next step, dotted orange circles are the hydrated M+X- after the first movement of the central molecule, and brown arrows denote the direction of movement of the hydrated molecule.
(M+X- molecules where M ) Li, Na, K, Rb, and Cs and X ) F, Cl, Br, and I) were initially placed at the center of the cube and allowed to move in any direction depending on the initial displacement of the ionic pair. The distance (dtotal) traveled by the molecules will be half the length of the cube (dtotal ) l/2). The simulation is performed at T ) 298 K for appropriate densities (g/cm3) of the halides. 2.2. Choice of Input Parameters. (a) Hydration Number at a Chosen Concentration. The hydration number at a particular concentration is denoted as Nhc. + For any ionic pair Mνz+Xνz- in water, Nhc < Nh∞, Nh∞ being the hydration number at infinite dilution. Hence
Nhc ) Nh∞ - ∆Nhtotal
∆Nhtotal )
Nh∞ds
(2)
〈r〉
Hence
(
Nhc ) Nh∞ 1 -
ds
)
〈r〉
(3)
In eq 3, as 〈r〉 f ∞ (limit of infinite dilution) the ratio {ds}/{〈r〉} becomes negligible and Nhc ≈ Nh∞. To obtain Nhc, the estimates of Nh∞ and 〈r〉 need to be evaluated. (ii) Hydration Number at Infinite Dilution (Nh∞). In general, the hydration number of an ion at infinite dilution is given by13
Nh∞ ) Vion/Vwater
(4)
where Vion and Vwater denote the volume of the bare ion (13) Bockris, J. O’M.; Conway, B. E. Modern Aspects of Electrochemistry; Academic Press: New York, 1954; Vol.1, Chapter 1.
rs3
(5)
where r+ and r- represent the bare crystallographic radii of the ionic species M+ and X-, while rs is the radius of water molecules. Since extensive simulation studies have been performed for NaCl solutions, it is of interest to verify the prediction of eq 5 as regards the estimation of hydration numbers at infinite dilution (Nh∞). For NaCl, the values of r+, r-, and rs are 1.39,14 1.81,14 and 0.85 Å15, respectively, and hence Nh∞ is evaluated as 14.03 from eq 5, which is in satisfactory agreement with the value of 13.7 by Marti et al. using the stochastic transition path sampling method.16 2.3. Mean Nearest Neighbor Distance 〈r〉. In several physicochemical processes such as hopping conduction,17 electron-transfer reactions,18 random walk phenomena,19 etc., the distribution of species in the dimensionality of interest plays a crucial role and the evaluation of the mean nearest neighbor distance becomes essential. Among several representations pertaining to 〈r〉 for size-dependent species, the explicit equation reported by Faulkner et al. is especially convenient, according to which 〈r〉 for hard spheres is given by18
〈r〉 )
(1)
where ∆Nhtotal indicates the total change in hydration number of the ionic pair between infinite dilution and a chosen concentration. (i) Total Change in the Hydration Number. In general, ∆Nhtotal depends on (i) the diameter of the solvent molecules, ds, (ii) the mean nearest neighbor distance between two hydrated ionic pairs, 〈r〉, and (iii) the magnitude of the hydration number at infinite dilution. ∆Nhtotal is directly proportional to ds (since the extent of desolvation increases with the size of the solvent molecules) and inversely proportional to 〈r〉 (when 〈r〉 f ∞, more solvent molecules will surround an ionic pair and hence the number of solvent molecules shed by an ionic pair decreases). Consequently, we may write
(r+3 + r-3)
3 (4πc )
1/3
exp(η)[Γ(4/3) - b]
(6)
where ∞
b)
(-1)nη(m+4/3)
∑ m!(m + 4/3)
(7)
m)0
and the dimensionless density η is defined as 3 /3 η ) 4πcddesol
(8)
c is taken as the concentration of the ionic pair (M+X-) in molecules/Å3 and Γ(4/3) denotes the Gamma function, ddesol being the bare crystallographic diameter of the species M+X-. Employing eqs 2-8 in (1), the hydration number of the species M+X- at a chosen concentration can be easily evaluated. For example, in the case of 5 M NaCl solutions, using ddesol as 6.4 Å20 and rs as 0.85 Å15 as mentioned earlier, Nhc is obtained as 10.6 in good agreement with the value of 10.4 deduced from the mean potential force method.16 2.4. Hydrated Radii of M+X-. The hydrated radius hyd R is always greater than the bare radius (Rdesol) of the ionic pair M+X-. Hence
Rhyd ) Rdesol + Rexcess
(9)
where Rexcess denotes the excess radius gained by the (14) Lide, D. R. CRC Handbook of Chemistry and Physics, 68th ed.; CRC Press, Inc., Boca Raton, FL, 1987. (15) Bockris, J. O’M.; Conway, B. E. Modern Aspects of Electrochemistry; Academic Press, Inc.: New York, 1954; Vol. 1, Chapter 2. (16) Marti, J.; Csajka, F. S. J. Chem. Phys. 2000, 113, 1154. (17) See for example: Shklovskii, B. I.; Eforos, A. L. Electronic properties of doped semiconductors; Springer-Verlag: Berlin, 1984. (18) Faules Fritsch, I.; Faulkner, R. L. A. J. Electroanal. Chem. 1989, 263, 237. (19) Mclendon, G.; Miller, J. R. J. Am. Chem. Soc. 1985, 107, 7811.
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species on account of hydration which may be written as
Rexcess ) γrs
(10)
Scheme 1. Evaluation of the Number of M+XMolecules that Reach Vacuum from the Center of Simulation Box
and γ is the dimensionless factor that depends on the hydration number of M+X- at infinite dilution (Nh∞) as well as that at a chosen concentration (Nhc). It can be represented as the ratio between the hydration number at a chosen concentration (Nhc) and the total change in hydration number (∆Nhtotal). Thus
γ ) Nhc/∆Nhtotal
(11)
Hence
Rexcess ) rs
Nhc ∆Nhtotal
(12)
and the hydrated radius of the ionic pair M+X- is written as
Rhyd ) Rdesol + rs
Nhc ∆Nhtotal
(13)
Equation 13 can now be employed to obtain the hydrated radius of M+X- at a chosen concentration. For 5 M NaCl solutions, using the earlier mentioned values of Rdesol, Nh∞, Nhc, and rs, Rhyd is deduced as 5.83 Å in contrast to the value of 4.95 Å reported earlier.16 Although the difference between the two estimates is 0.88 Å, it may be noted that our value is higher since it includes not only the distance between the centers of Na+ and Cl- ions but also the associated ion-pairing effects. 2.5. Justification of the Postulated Expressions for Nhc and Rhyd. a. Hydration Number at a Chosen Concentration. In general, any electrolyte solution of desired concentrations can be prepared by adding it to the solvent (cf. water in the present analysis). The initial addition of the electrolyte to water leads to infinite dilution of the species (corresponding to hydration number at infinite dilution, Nh∞), further addition leads to concentrated solutions (corresponding to hydration numbers at different concentrations, Nhc). Thus, with the help of the variation of hydration numbers while passing from infinite dilution to each concentration (∆Nhtotal), Nhc can be obtained. Moreover, Nh∞ is always higher thanNhc, thus leading to eq 1. The rationale behind the expression for ∆Nhtotal is as follows: When the concentration of the ionic pair increases, the number of nearest neighbors also increases, thus altering the number of water molecules surrounding it. The extent to which a water molecule can be shared by the neighboring electrolyte species depends on the corresponding dimensions. Further, ds/〈r〉 reflects the extent of sharing of water molecules by neighboring electrolytes. Since the variation in hydration number is referred with respect to the infinite dilution, the term Nh∞(ds/〈r〉) corresponds to the total variation in the hydration number from infinite dilution to a chosen concentration. b. Hydrated Radii of the Ionic Pair. Since the hydrated radius is always greater than the bare radius of a species, the expression for the excess radius gained due to hydration (Rexcess) can be justified as follows: The only parameter that influences Rexcess is the size of the water molecules, thus Rexcess ∝ rs. The proportionality (20) ddesol is calculated as 2(r+ + r-), where r+ and r- represent the bare radii of cation and anion deduced from the tabular compilation of ref 14.
constant γ, can now be written as
γ)
〈r〉 - ds ds
(14)
where the ratio (〈r〉 - ds)/ds denotes the variation in the number of nearest neighbors caused due to hydration of the bare species. Upon rearrangement of the above equation in conjunction with eq 2, γ can be expressed as in eq 11. 3. Methodology To obtain the number of M+X- molecules that reach the surface of the cube and ultimately become dehydrated by reaching vacuum, the central molecule is allowed to move in steps. Random numbers are generated for estimating the number of molecules that have reached the subsequent steps. To generate random numbers pertaining to each step, the parameters such as hydration number of M+X(Nhcn), the hydrated radius (Rhyd n ), and the corresponding decrease in the hydration number due to movement of the molecule (∆Nhn) need to be evaluated. Since the surface of the cube is confined, we may write
dtotal(n) ) dtotal(n-1) - xn
(15)
) Rhyd Rhyd n sur ≈ Rdesol + 1
(16)
where dtotal(n-1) and dtotal(n) denote the distance covered by M+X- at (n - 1)th and nth steps, respectively (at n ) 1, dtotal(n-1) ) dtotal), while xn represents the actual displacement of the molecule in the nth step and Rhyd sur is the hydrated radius of those molecules that reach the surface of the cube (cf. Scheme 1). 3.1. The Hydration Number and Hydrated Radius at Each Movement of the Molecule. The variation in the hydration number of the ionic pair M+X- during each movement (∆Nhn), the hydration number, and the hydrated radius in each step (Nhcn and Rhyd n ) are required
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for obtaining the real and apparent dehydration energies. ∆Nhn depends on (i) the total change in the hydration number of M+X- between infinite dilution and chosen concentration at each step (∆Nhtotal(n-1) ) Nh∞ - Nhc(n-1) and at n ) 1, Nhc(n-1) ) Nhc); (ii) the distance traveled by the hydrated molecule from the center of the cubic box toward the surface (dtotal(n-1)) in each step, and (iii) charges of the ions constituting the ionic pair, concentration of the species, and dielectric constant of water. An exact incorporation of all the above factors is a tedious endeavor; however, a heuristic method of analysis is rendered possible by employing the traditional DebyeHuckel theory21 (TDH) as applicable to the present simulation procedure. It may be emphasized that improvements to the TDH have been sought in several recent studies22 so as to describe the electrostatic hole correction terms in a more satisfactory manner. The expression for the classical Debye length is given by
LD )
(
(z+2
)
0kBT +
1/2
(17)
z-2)NBe2
where the dielectric constant of water and permittivity of vacuum are represented by and0, respectively. NB denotes the number of molecules in the solution, and other symbols have their usual significance. In the methodology here, NB varies at each step on account of the shedding of the hydration sheath progressively and the expression for the Debye Length should reflect this feature. Hence we surmise that a discretized Debye length of the form
LD(n-1) )
(
0kBT
)
(
)
LD(n-1) + rs dtotal(n-1)
(19)
At n ) 1, ∆Nhtotal(n-1) ) ∆Nhtotal. Analogously, Nhcn and Rhyd can be written as n
Nhcn ) Nhc(n-1) - ∆Nhn
(20)
and
) Rdesol + Rhyd n
rsNhcn ∆Nhn
RT(N(n-1) - Nn) Nn(Rn-1hyd - Rnhyd)
(21)
(22)
where Nn and N(n-1) denote the number of M+X- molecules present at the nth and (n - 1)th steps, respectively, Rhyd n hyd and Rn-1 are the corresponding hydrated radius, and hyd Rn-1 equals Rhyd when n ) 1. R denotes the universal gas constant and T refers to the absolute temperature (298 K here). Further, on the basis of considerations emerging from Figure 1, the parametric relation for computing xn is given by
xn ) 2〈r〉 + Nhcnrs
(23)
The apparent dehydration energy of M+X- follows as nfinal
∑
nfinal
wn ) -
n)1
(18)
is appropriate, where n varies from 1 to nfinal. In eq 18, LD(n-1) denotes the Debye length in the (n - 1)th step and at n ) 1, LD(n-1) ) LD. N(n-1) denotes the number of electrolyte molecules in the (n - 1)th step and at n ) 1, N(n-1) equals the initial electrolyte molecules (Ninitial), Nwater being the number of water molecules. Further, the variation in the hydration number of the ionic pair is given as
∆Nhn ) ∆Nhtotal(n-1)
Fn )
Wtotal ) ∆Gapp dehyd )
1/2
(z+2 + z-2)(N(n-1) + Nwater)e2
21, the random numbers can be generated for each step. Thus the number of molecules arriving at the surface of the cube and those subsequently reaching vacuum are obtained. 3.2. Thermodynamic Force and Associated Energies. If Fn denotes the thermodynamic force and xn represents the actual displacement of M+X- at the end of the nth step, then Wtotal denotes the total work involved in the displacement of M+X- from the center of the cube to the surface, en route to dehydration. In analogy with the definition of the thermodynamic force for continuum models,23 it is appropriate to write Fn as
∑ Fnxn
(24)
n)1
However, the real dehydration energy of a species consists of the apparent dehydration energy as well as the excess energy required to cross the barrier at the solution/vacuum interface.15 This excess energy is customarily denoted as the surface potential of the species (χ). To evaluate the same, it becomes essential to compute the number of molecules that possess the required energy to cross the barrier in a single step and escape to vacuum among the molecules arriving at the surface of the cube (Nsur) by employing the random number criterion as depicted in Scheme 1. In analogy with eq 22, the thermodynamic force for this process (Fsur) is given by
Fsur )
RT(Nsur - Ndesol) Ndesol(R′ - Rdesol)
(25)
where Ndesol denotes the number of molecules reaching vacuum, R′ being the remaining distance the molecules have to travel for reaching vacuum. From geometrical considerations, R′ is given by nfinal
R′ )
2Rhyd sur
+ (dtotal -
∑ dtotal(n))
(26)
n)1
Although eqns 13 and 21 involve the estimation of hydrated radii for the species M+X- at (i) a chosen concentration and (ii) each movement, respectively, they are not identical, since eq 13 deals with the total change in the hydration number (∆Nhtotal) whereas eq 21 is derived for the evaluation of discretized parameter (∆Nhn). Using from eqns 20 and the above estimates of Nhcn and Rhyd n (21) (a) Debye, P.; Huckel, E. Phys. Z. 1923, 24, 185. (b) Debye, P.; Huckel, E. Phys. Z. 1924, 25, 97. (22) Abbas, Z.; Gunnarsson, M.; Ahlberg, E.; Nordholm, S. J. Phys. Chem B 2002, 106, 1403 and references therein.
while the corresponding work term becomes (cf. eq 25 above)
Wsur ) χ ) -Fsur(R′ - Rdesol)
(27)
The above equation enables the evaluation of the surface potential, whose maximum value in general is ca. 10% of the apparent dehydration energy. Further, the real (23) Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: London, 1994.
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Table 1. The Number of Electrolyte Molecules, Length of the Simulation Box, and Concentration and Densities of Alkali Metal Halides Employed in the Analysisa no. of box bare radius of electrolyte14 electrolyte length molecules (Å) electrolyte Rdesol in Å LiF LiCl LiBr LiI NaF NaCl NaBr NaI KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI
2.25 2.73 2.88 3.12 2.72 3.20 3.35 3.59 3.45 3.60 3.84 3.05 3.53 3.68 3.92 3.21 3.69 3.84 4.08
3098 2987 2147 1712 3001 1852 1953 2065 1977 1913 2677 1932 1878 1994 2774 1556 2271 2123 2896
480 486 720 634 620 640 640 600 620 600 560 702 574 560 550 660 590 520 474
Table 2. Dehydration Energies for Different Electrolytes at the Concentrations Indicated in Table 1a
concn (M)
density (g/cm3)
electrolyte
0.0580 0.0380 0.0150 0.0190 0.0290 0.0156 0.0200 0.0300 0.0200 0.0250 0.0600 0.0150 0.0280 0.0380 0.0780 0.0155 0.0400 0.0600 0.1500
0.0061 0.0065 0.0023 0.0036 0.0031 0.0027 0.0033 0.0049 0.0033 0.0042 0.0073 0.0025 0.0048 0.0062 0.0091 0.0032 0.0057 0.0092 0.0168
LiF LiCl LiBr LiI NaF NaCl NaBr NaI KCl KBr KI RbF RbCl RbBr RbI CsF CsCl CsBr CsI
a It is essential to display four decimals for the concentration, since the number of molecules, box size, and hydrated radii are interdependent.
dehydration energy becomes15,24 app ∆Greal dehyd ) ∆Gdehyd - χ
(28)
3.3. Validity of the Postulated Expressions for ∆Nhn and xn. The validity of the postulated expressions for ∆Nhn and xn may be demonstrated as follows: (a) The Variation in the Hydration Number in Each Step, ∆Nhn. We note that the initial step in the shedding of water molecules surrounded by an ionic pair while moving to the next step constitutes its total variation in hydration number from infinite dilution, since the number of electrolyte molecules satisfying the volume criteria and reaching the next step decreases as the number of steps increases. Thus, ∆Nhn is directly proportional to ∆Nhtotal(n-1); (ii) as the distance traveled by the hydrated molecule increases, the extent of shedding of water molecules decreases, therefore ∆Nhn is inversely proportional to dtotal(n-1); (iii) with the movement of the electrolyte molecules, the Debye length decreases and as a consequence, the shedding of water molecules also decreases and hence ∆Nhn is directly proportional to (LD(n-1) + rs), the addition of rs takes into account, the variation in LD(n-1) with the nature and size of the solvent under consideration, thus leading to eq 19 of the text. (b) Actual Displacement of the Ionic Pair in Each Step, xn. In each step, a bare ionic pair can move a distance of twice its nearest neighbor distance, whereas hydrated molecules are displaced a length of Nhcnrs, in addition to 2〈r〉. The term Nhcnrs accounts for the displacement of the hydration sheath in each step. Thus, the expression xn ) 2〈r〉 + Nhcnrs takes into account the displacement of the ionic pair along with its hydration sheath. 4. Results and Discussion The number of water molecules (Nwater) is fixed as 18000. On the other hand, the chosen concentration in conjunction with the density of the system dictates the number of molecules of metal halides (Ninitial). As shown in Table 1, (24) See for example: Schmickler, W. Interfacial Electrochemistry; Oxford University Press: London, 1996.
calcd apparent dehydration energy (eV) 102 103 104 9.85 8.55 8.44 7.81 8.97 7.42 7.06 6.72 6.83 6.58 6.25 7.77 6.42 6.25 6.24 7.78 6.23 6.17 4.96
9.53 8.39 8.12 7.65 8.74 7.25 6.90 6.61 6.73 6.68 6.22 7.53 6.31 6.18 6.22 7.51 6.16 6.10 4.93
9.55 8.43 8.15 7.65 8.76 7.26 6.90 6.62 6.76 6.58 6.23 7.57 6.31 6.19 6.23 7.51 6.16 6.12 4.94
calcd mean apparent dehydration energy (eV)
reported apparent dehydration energy25 (eV)
9.64 8.46 8.24 7.70 8.82 7.31 6.98 6.65 6.77 6.63 6.23 7.62 6.37 6.21 6.23 7.65 6.20 6.13 4.94
9.74 8.45 8.19 7.77 8.60 7.31 7.05 6.63 6.58 6.32 5.91 7.67 6.37 6.11 5.70 7.41 6.11 5.86 5.44
a The mean surface potentials of KI, RbCl, RbBr, RbI, CsBr, and CsI are 0.07, 0.03, 0.04, 0.09, 0.07, and 0.06 eV, respectively. All the other halides do not have any surface potentials.
Nel varies from 1556 to 3098. The Monte Carlo seeds (different number of seeds) 102, 103, and 104 were employed. The calculations were carried out using MATLAB version 6.0 in a Pentium IV personal computer with the help of the simulation methodology shown in Scheme 1. The listing of a typical MATLAB program (∼60 lines) is provided in the Supporting Information. The estimation of real and apparent dehydration energies of each metal halide consumes about 4 h for 104 Monte Carlo seeds after the box sizes have been fixed. In several trial runs, the results obtained using 105 seeds were not significantly different from that for 104 seeds. Consequently, seeds higher than 104 were not employed. As can be seen from Table 2, the dehydration energies obtained for 103 and 104 seeds differ from each other only to a minor extent, in contrast to that obtained from 102 seeds. In Table 2, the estimated apparent dehydration energies of various alkali metal halides using the present method are compared with the reported data.25 A satisfactory agreement is noticed in all the cases except RbI and CsI, the origin of which is elusive, at present. The surface potential evaluated using eq 27 is zero for all halides except KI, RbCl, RbBr, RbI, CsBr, and CsI. This implies that other alkali metal halides inherently possess the excess energy required to cross the barrier of solution/vacuum interface leading to the vanishing of the surface potential. This nonzero value of the surface potential is consistent with the chemical principles whereby the larger the size of the alkali metal halide, the more energy is required to cross the solution/vacuum interface. The response of the proposed methodology to the variation in the system parameters (e.g., Rdesol) is remarkably high. For a difference in Rdesol of 0.08 Å, in the case of LiI and NaCl, the variation in the apparent dehydration energy is 0.39 eV (approximately five times the variation in Rdesol), thus indicating the sensitivity of the methodology toward the input parameters. rep (25) (a) Noyes, M. R. J. Am. Chem. Soc. 1962, 84, 513. ∆Gdehyd ) rep rep ∆Gdehyd,cation + ∆Gdehyd,anion . The solvation free energy of the electrolyte is evaluated as the sum of the contribution from the corresponding cations and anions. (b) Marcus, Y. J. Chem. Soc., Faraday Trans. 1991, 87, 2995.
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Table 3. Significance of Various Parameters in Other Physicochemical Contexts parameters
physical significance
variation in the hydration number during each step (∆Nhn) variation in the Debye length in each step (LD(n-1)) variation in the hydrated radius in each step (Rhyd n )
The fraction of water molecules shed during each movement of the hydrated alkali metal halide has been calculated for the first time from the present methodology. ∆Nhn obtained here has profound significance in diverse aspects of solvation.27 The “instantaneous” change in the Debye length in each movement of the alkali metal halide arising for the first time from this approach will be of immense utility for the evaluation of double layer capacitance in metal/solution24 and liquid/liquid interfaces.28 The decrement in the hydrated radius of the alkali metal halides in each movement helps in the determination of the size of the molecules at every movement. This has extensive applications in the field of biological sciences: the quadruplex structures of the nucleobases29 present in the hairpin loops of DNA and RNA are stabilized by alkali metal halides of particular size that can get accommodated in the cage formed.30 In general, hydrated sodium and potassium chlorides perform the above phenomena. The hydrated metal halides of size similar to NaCl or KCl can also stabilize the quadruplex structures thereby possessing high potentiality in the sequencing of DNA or RNA.
Figure 2. Variation of the mean apparent dehydration energies of alkali metal halides with the corresponding molecular radius. The line is drawn as a guide to the eye.
It is well-known26 that the lattice energy of M+Xincreases as M varies periodically from Li to Cs. In contrast, the hydration energies show a reverse trend. Further, the extent of hydration decreases with increase in the size of the metal halide. Since hydration involves the penetration of water molecules into the lattice of the halide, the hydration energy should exceed the lattice energy.26 Consequently, LiF with the least molecular radius has the most negative hydration energy while CsI having the largest molecular radius among the metal halides possess the least negative hydration energy. Figure 2 depicts the correlation of dehydration energies with the corresponding molecular radii. Since accurate values of the cationic and anionic radii constituting the electrolytes are available14 in general, no error bars pertaining to the calculated apparent dehydration energies are provided in Figure 2. It may be emphasized here that the estimation of the dehydration energies for higher concentration essentially requires the proper choice of number of molecules of the electrolyte and solvent, along with the appropriate box (26) Lee, J. D. A New Concise Inorganic Chemistry, 3rd ed.; English Language Book Society and Van Nostrand Reinhold Company, Ltd., London, 1977.
size. Hence, the estimation to higher concentrations for any electrolyte is fairly straightforward. 4.1. Physical Significance of the Present Methodology. The physical significance of the parameters derived from present methodology and their implications in other fields of chemical and biological sciences are listed in Table 3.27-30 In summary, a simple computational strategy to estimate the dehydration energies of alkali metal halides is proposed here which does not require high-end computing resources. Further, the method has an built-in flexibility, being amenable for extension to diverse types of electrolytes and solvents. While a small subset of inorganic compounds have been considered here, there is no prima facie limitation in extending the approach to other systems too. The handling of solutions with higher densities poses no particular computational difficulty, in contrast to the hitherto-available methodologies. The main success of this methodology may be attributed to the precise incorporation of all system parameters such as ionic sizes, solvent dimensions, extent of dehydration, etc. from the perspective of the physical chemistry of ionic solutions. On the other hand, the hitherto-available prescriptions employ idealized potential functions. Nevertheless, some of the parametric relations for hydration numbers and their variations pertaining to each step require an extensive analysis. Acknowledgment. The helpful comments of the reviewers are gratefully acknowledged. This work was supported by the Council of Scientific and Industrial Research, Government of India. Supporting Information Available: The listing of the program for estimating the dehydration energy of 0.06 M CsBr as an illustrative example is provided for MATLAB version 6.0. This material is available free of charge via the Internet at http://pubs.acs.org. LA035654A (27) See for example: Rode, B. M.; Schwenk, C. F.; Tongraar, A. J. Mol. Liq., in press. (28) See for example: Pereira, C. M.; Schmickler, W.; Silva, A. F.; Sousa, M. J. Chem. Phys. Lett. 1997, 268, 13. (29) See for example: Lehninger, A. L.; Nelson, L. D.; Cox, M. M. Principles of Biochemistry, 2nd ed.; CBS Publishers: New Delhi, 1993; Chapter 4 and references therein. (30) de Levie, R. Chem. Rev. 1988, 88, 599 and references therein.
