Indian Journal of Chemistry Vol. 39A, June 2000, pp. 584- 588

A simple method to predict the solvation free energy and enthalpy of electrolytes in aqueous and.non-aqueous solutions

Rohini Badarayani & Ani[ Kumar • Physical Chemistry Division, National Chemical Laboratory, Pune 4 11 008, India

Received 28 July 1999; revised 6 October 1999

A simple method is proposed to predict the solvation free energy of electrolytes in the aqueous and non-aqueous solutions upto 623 K. The solvation free energies are calculated by a suitable combination of scaled particle theory and modified Born equation. The method is applied to a variety of solutions and at various temperatures. The temperature derivative of solvation free energy with respect to temperature offers reasonably accurate estimations of entropy and enthalpy of solvation.

Precise knowledge of solvation free energy, 11Gs and enthalpy, M-fs of solutions involving ions is a prerequisite for developing solvation models to estimate thermodynamic properties of these systems from ambient conditions of temperature and pressure to the elevated ones.I-4 In addition, such theoretical estimates are important for understanding chemical reactions, spectroscopically determinable parameters ' biochemical and biomimetic reactions 5

• In the past , several attempts 6

-8 have been made to calculate /1G5

and M-15 •

In this work, our sole objective is to present a simple method to estimate /1G5 and M-!5 from the knowledge of the properties of constituent components.

Methodology The development of our approach begins with the

process of introducing the solute species with a sphere of macroscopic size with diameter, cr2 into the solvent. In doing so, all the interactions between the sphere and the solvent are "switched off'; the sphere becoming a hard sphere . Thus , this step involves the creation of a cavity in the solvent of suitable size to accommodate the solute molecules. The reversible work of non-electrical nature or the partial molar free energy , t'1Gnei required to do this is equivalent to one required to introduce 1 mol of hard spheres or cavities of appropriate radius such as to produce a mole of cavities in the solution . Now this is followed by the introduction into the cavity of a solute molecule interacting with the solvent or the molar

reversible work , 11Ge1 is identical with that of charging the hard spheres or cavities introduced in the earlier i>tep to the required potential. This work is associated with giving each cavity or hard sphere the proper charge distribution and polarisibi lity to stimulate a reat solute molecule. Now since the process of dissolution of gaseous ions in solvents is called ion solvation , one can then write an expression for solvation free energy as :

11Gs = t'1Gnei + 11Gei + R TIn (R T I VI ) + RT In (MI/1000) . . . (1)

In Eq. (I), the last term on the right hand side accounts for the change in the standard states -hypothetical 1 atm gas to hypothetical I molal solution. The quantities, VI and MI ind icate the molal volume and molar mass, respectively of the solvent. For calculating the term, t'1Gnei we propose to use the scaled particle theory (SPT) originally developed by Reiss et. a/.9

· I0 for dense hard sphere fluids . This

statistical mechanical theory based on properties of exact radial distribution functions has been applied to several sttuations pertaining to the mixtures of hard spheres , disks and rods, real fluids , determination of surface tension, sol ubili t ies, etc., to mention a few among its many applications. The SPT and its significance have been highlighted by Reiss.I I Justification to use the SPT in the present context comes from the consideration that the solute particle irrespective of being charged or neutral would always need a cavity to fit in a solvent. It may be

BADARA Y ANI eta/.: A SIMPLE METHOD FOR PREDICTING SOLVATION FREE ENERGY 585

Table 1-!1 G s rei (!1 G s T- !1 G s 298) at different temperatures as predicted by the present method for some representative aqueous ' · systems ; values given in parentheses are the experimental values23

!1 G,, rei I kJ mol - I

i 0 E -, .:.c. ......

T/K 298 323

373

423

473

523

573

623

NaCI 0

-2.67 (-2.82) -8.25

(-8.08) -12.19

(-12.79) -16.05

(-16.90) -20.04 (-19.86) -21.0 I

(-20.33) -16.99 (-16.80)

T/K

ll--b.-..a.. ___ _

CaCI2

0 -1.49 ( -1.42) -3.42

(-3 .58) -4.43

(-4.66) -4.29

(-4.10) -0.38

(-0.32) 10.57

(11.34) 38.1

(39.3)

3~

.. (/) <l

Fig. !-Experimental !lG, ,rei and !15, values as a function of T. For flG, ,rei : NaCI (o), CaCI2 (!1); AIC13 (•); for !1Ss : NaCI (o), CaCI2 (!1); GdCI3 (0 .) Points Experimental ; lines predicted , all aqueous systems.

assumed that the neutral part of ion solvation ts related to the process of cavity formation of appropriate size.

For our purpose, the relevant equation describing

~Gnelis :

AICI3 LaCI3 Na2S04 0 0 0

3.84 1.32 -4.40 (3 .77) (1.17) ( -4.32) 13.41 4.78 -9.00

(13 .76) (4.86) (-9. 18) 27.25 12.01 -14.04

(27.36) ( 11.06) (-13 .52) 46.21 20.54 -15.58

(45 .80) (20.61) (-14.63) 72.01 35.44 -11.52

(72.22) (36.62) (-10.51) 114.54 68.40 -0.34

(113.41) (68 .01) ( -0.30) 189.4 118.10 6.2

(191.2) (119.2) (6.5)

where, Ko = R T (-In x + 4.5 y I x 2

) - 1t P cr1 3 I 6

Kt =(-RTicrt) { 6ylx+ 18ylx2} +1tPcr1

2

K2 = (RT I cr1 2) { 12 y I x + 18 y I x2

} - 2 1t p <Jt K3 = 4 1t PI 3

cr12= (crt+ cr2) I 2 " . (3)

In the above, x = 1- y; y being the reduced number density, popularly called as packing fraction of solvent. The packing fraction, y is calculated from 1t N cr 1

3 I 6 V 1 , where N is Avagadro's number. The diameter cr1 of solvent can be calculated from isothermal compressibility 12 or from molar volume data. 13

For ~Get we propose to employ a simple pheno­menological model, like classical Born equation 14 as modified by us although detailed atomic models are useful for full understanding of solvation. In the classic Born model, ~Ge1 is evaluated using classical electrostatics and treating solvent as a structureless dielectric continuum. Since the original Born equation has been found to yield unreasonably high magnitude of ~Ge1 , several modifications incorporating the use of adjustable parameters have been published and reviewed elsewhere.15

-17 In order to arrive at a more

reasonable value of ~Get, we modified the Born equation in such a way that fundamental constants and properties of solute and solvent only are included in the revised model.

In the modified Born equation, we adopted a different approach following Conway17

, where it is reasonable to divide the polarisibility of the solvent

586 INDIAN J CHEM, SEC. A, JUNE 2000

Table 2-The neutral (non-electrical= ~Gs,nel or ~Gs,cav) and electrical (~Gs,el) contributions to ~G, in the case of aqueous

NaCI at different temperatures

TIK ~G s.nel or ~G '· caj -~Gsell k J mor 1 k J mo.r 1

298 61.93 788.23 323 60.24 789.23 373 56.11 790.00 423 52.03 791.44 473 47.68 791.45 523 42.73 789.57 573 37.33 784.27 623 34.65 777.50

medium into two components: (a) purely electronic part associated with the deformation of the electron cloud and (b) total polarisation, which includes the inertia polarisation associated with the displacement of nuclei. The screening effect of the surroundings of a solvated ion is defined by appearance of the corresponding polarisation charges. It has been described earlier18 that the density of polarisation charge due to deformation of clouds is expressed as given in Eq. (4)

P pol ( r) = q L ( I \IIi 12

- I \IIi 0 12

)

i ~ . .

~ q L. ( \j/j 0 \j/j) + \jlj 0 \j/j) ) • . • ( 4) where, the one-electron wave functions of ith electron of ligand in electric field of the central ion and without the field are denoted by \jl i c r l and \jl i o,

respectively. \jl i ( r ) is the change in \jl i 0 due to electric field of central ion; it is, therefore reasonable to suggest that polarisation change is localised mainly on the boundary of the electron cloud of ligand.

In the case of water the dipole moment appears due to partial electron transfer from hydrogen to the free orbitals of ground states of the oxygen atom. Thus, the polarisation charges can be considered as directly attached to the centres of oxygen and hydrogen atoms. It may be noted that the dipole moment on solvent molecules is directly associated with the polarisation charge of inertia component of the polarisation. The dielectric properties of a solvent medium at short distances from central ion are determined by the electronic component of the polarisability of ligands. It is clear that the screening of field of solvated ion by total polarisation of the solvent begins at the sphere passing through nuclei of nearest atoms carrying excess charge. The theoretical implications of this model have been discussed earlier18

The expression to calculate the LlGet is given by Eq. (5):

I

0 E ..,

.lC

' ... i

-~ Ss exp I kJ mol -1 K-1

6·r-~OT·4~--------~0;·2~--------~

17

0 ·2

0 ·4

4 6 -~ Hs up I k J mol-1

....... ~ i :.:::

-; 0 E ..., .X

' ... • .. ... (/)

(/)

<l I

Fig 2- Experimental versus predicted t:J/s for aqueous systems; (hollow symbols) : I LiCl, 2 NaCI, 3 KCl , 4 KI , 5 CsBr, 6 CaClz, 7 MgClz, 8 AICI3, 9 NaOH, I 0; and experimental versus predicted t.Ss; (solid symbols): 10 LiCl-methanol; II LiCl­ethanol; 12 LiCl-dimethylformamide; 13 dimethyl sulphoxide; 14 NaCl -: methanol ; 15 NaCI- ethanol; 16 NaCl- dimethyl­sulphoxtde; 17 Csl- methanol; 18 Rbl-propylene carbonate; 19 Csl- formamide; 20 KI-dimethyl sulphoxide.

-LlGd = ( q 2 I 2 ) ( I - II De ) ( I I r2 - l I r3 ) + ( cf I 2 ) ( 1 - 1 I D)( II r 3 ) . . • (5)

In Eq. (5), De is the dielectric constant due to electron polarisation of the atoms nearest to the central ion. The radius of the central ion and the distance to the centre of the nearest atom of the solvent are denoted by r2 and r3 , respectively. D is dielectric constant of bulk solvent and q is the charge. Note that the electronic polarisation of water molecules is identified with the high frequency dielectric constant (De). Value of r3 is obtained as r2

+ 8 ; 8 being the interatomic distance listed for several solvents elsewhere. 19

Differentiation of Eq. (I) with respect to temperature at constant pressure gives solvation entropy , M s which then via LlGs yields the final expressions for Wnet and Wet as

Ws = Wnet +Wei + RT (a T- I ) ... ( 6) where a is thermal expansivity of solvent and

LV/rrl=O>+Q1 cr 12 +(hcr 1/+Q3cr 123

with

Qo = aR r y lx Ql = 3 a Rr y I cr I x2

... (7)

BADARA Y ANI et al.: A SIMPLE METHOD FOR PREDICTING SOLVATION FREE ENERGY 587

Q2 = 3 a. Rf y [ ( 1 + 2 y ) I x3 ] I cr1

2

Q3 = 1t PI 6 ... (8)

and for LllieJ. we obtain after assuming & D. I & T s::::sO,

- Lllle1 = (q 2 I 2) (1 I r2 - 1 I r3) [ 1 + 1 I D.] +(<[12)[1-11D-(TIIY)&DI &T] ... (9)

Results and Discussion Equations (1) and (6) can be used to predict the

ll.Gs and Ml. or Ms in any solvent and at any temperature of interest. We note that the terms involved in both the equations can be readily calculated from the fundamental properties of ions and solvent in question. We used the Pauling crystal radii for ions and calculated hard core diameters of solvents as a function of temperature from the methods described elsewhere 12'13 using the standard source of data on solvents?0

.21 Dielectric constants

and their temperature dependence were taken from Bottcher22

• Although there are several experimental data on ll.Gs in literature , we preferred to test the above approach with systematic and standard source of Tremine and Goldman.23 Our predicted values of ll.G5 • re1,i.e., the difference between solvation free energy with respect to that at 298.15 K (ll.Gs.rel =

ll.Gs,T - ll.Gs. 298) are in good agreement with those obtained from the experiments. A careful survey of Table I, where the calculated ll.Gs,rei values for aqut~ous NaCI, CaCh, AlC13, LaCb and Na2S04 systems are compared with those obtained from the experiments, confirms the application of the present method. For the purpose of illustration, in Fig. I are plotted the experimental ll.Gs,rel values versus temperature (K) for three different charge types of salts, i.e., NaCI, CaCh and AICb in water. The numerical contributions of the neutral and ionic components for aqueous NaCI from 298 to 623K are listed in Table 2. A sim ilar agreement between calculated and experimental6

•24 values is witnessed for

M •. The average root mean squares deviation (rmsd) value for 1 0 aqueous salt systems was noted to be 5 per cent, which is a fairly good achievement. In the case of M s and Mfs (Fig 2) , the average rmsd values of 7 and 9 per cent, respectively were obtained for 9 aqueous systems with a variety of charges . An application of the approach to the non-aqueous solvent systems25-27 yields good estimates of s. values ( rmsd = 9 %) for II systems in several solvents, like methanol, ethanol, n-propanol , dimethyl sulphoxide, formamide, etc. An exhaustive comparison of

predicted results with experimental data is not presented here in view of the scope of this article, but the method is very simple for generating the required properties. The details of sample systems studied are shown in the figures. It is interesting to note the range of dielectric constant of solvents (e.g. minimum D =

5.62 for chlorobenzene; maximum D = 182.4 for N­methylformamide ), to which the present approach is applicable. In view of the scarce experimental data on non-aqueous systems at high temperatures, the proposed formulation could not fully be tested for those data. The above analysis, however, confirms that the proposed combination of the SPT with the revised Born model offers an effective tool to estimate the solvation free energies , enthalpies and entropies both in aqueous and a variety of non­aqueous solvents with varying dielectric constants upto temperatures as high as 623K_ The estimations of solvation free energies can further be improved if the saturation of orientation polarisation in strong fields can be incorporated in the above equation. Further, any contribution due to overlap energy of the electron cloud of the central ion, of the ligands and the departure from the spherical symmetry of electric field of the central ion have also been ignored in current treatment. Our input information comprises size of components in the form of radius or diameter, density and dielectric constant of solvent.

Acknowledgement The authors thank the anonymous reviewer for

making constructive suggestions on the original draft of this paper.

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