7468
J. Phys. Chem. B 2000, 104, 7468-7470
The Conventional Method of the Determination of Enthalpy Change at Ion Solvation in Solutions: Is It Correct? N. B. Zolotoy* and G. V. Karpov SemenoV Institute of Chemical Physics of RAS, 117977, Kosygin str., 4, Moscow, Russia ReceiVed: January 4, 2000; In Final Form: May 4, 2000
It is shown that the conventional (using the Born-Haber cycle) method of the determination of enthalpy change at ion solvation in solutions gives incorrect values for this fundamental parameter. A new experimental method free from the defects of the conventional one is presented.
1. Introduction Solvation of ions in solutions, that is, formation of solvated ions (ion clusters), is the fundamental process controlling all phenomena in electrolyte solutions. The enthalpy change at ion solvation, ∆H, is a fundamental characteristic because ∆H is a measure of the interaction between an ion and solvent molecules. Therefore, it is important to understand the correctness of the conventional tabulated values of ∆H. In this paper, by analysis of the theoretical base of the conventional method, it is shown that this method gives incorrect ∆H values. More correct results can be obtained by the new experimental mass spectrographic method using field evaporation of ions from solutions (FEIS). The FEIS method is free from the defects of the conventional one. 2. Analysis of the Conventional Method of Determination of Enthalpy Change at Ion Solvation in Solutions The conventional method of determination of enthalpy change at solvation of cations, ∆Hc, and anions, ∆Ha, was developed in ref 1. The method uses the Born-Haber cycle for a (C+A-)cr compound
1. ((C+A-)cr)g f C+g + A-g 2. C+g f C+solv 3. A-g f A-solv 4. C+solv + A-solv f ((C+A-)cr)s
(∆Hcr) (∆Hc) (∆Ha) (-∆Hd ) Ld)
}
(1)
and Hess’s law for this cycle
∆Hcr + ∆Hc + ∆Ha + Ld ) 0.
(2)
The subscripts denote cr, “crystal”; g, “gas”; solv, “solvated”; and s, “solution”. The enthalpy changes corresponding to the cycle steps are shown in parenthesis: ∆Hcr corresponds to the dissociation of a (C+A-)cr compound into the cations C+ and the anions A- in gas; ∆Hc and ∆Ha correspond to the solvation of the cations and the anions; Ld is the heat of dissolution of the (C+A-)cr compound in the solvent. Expression 2 may be rewritten as
∆Hc + ∆Ha ) -∆Hcr - Ld.
(3)
Expression 3 is used for calculation of ∆Hc + ∆Ha through
the values of ∆Hcr and Ld, which are determined experimentally. The method has the following disadvantages: 1. The well-known disadvantage consists of the possibility of determination of a total value, ∆Hc + ∆Ha, only. To separate this sum into cation and anion terms, additional assumptions should be made. These assumptions can be incorrect, and therefore, the resulting ∆Hc and ∆Ha can be also incorrect. As example, ∆Hc and ∆Ha values are given in Table 1.2 The manners of separating ∆Hc + ∆Ha are decribed in refs 2 and 3. 2. So far, the enthalpy change at the transfer of unsolvated ions from the gas phase into solution was always ascribed to only the ion solvation in solution. In reality, this transfer consists of two steps: the transition of the unsolvated ion from the gas phase into solution across the phase boundary and the solvation of this ion in the solution; each step must be accompanied by the corresponding enthalpy change. 3. Generally speaking, cycle 1 is unclosed and, therefore, the left part of expression 2 must not be equal to zero. To close the cycle, a step of the (C+A-)cr transfer from the solution into the gas phase with a corresponding enthalpy change ∆Hsg must be included in cycle 1. Obviously, in ref 1, it was silently assumed that the ∆Hsg value is negligible against the other ∆H values of cycle 1 and, therefore, this step was also neglected. If, in accordance with the second remark, each of the steps 2 and 3 of cycle 1 is divided into two stages and the third remark is neglected, cycle 1 may be rewritten as
1. ((C+A-)cr)g f C+g + A-g 2a. C+g f C+s 2b. C+s f C+solv 3a. A-g f A-s 3b. A-s f A-solv 4. C+solv + A-solv f ((C+A-)cr)s
(∆Hcr) (-Qc) (∆Hc)r (-Qa) (∆Ha)r (-∆Hd ) Ld)
}
(1′)
where Qc and (∆Hc)r are, respectively, the heat of evaporation and the real enthalpy change at solvation of the cations in solution, and Qa and (∆Ha)r are the same for the anions. Hess’s law for cycle 1′ is
(∆Hc)r + (∆Ha)r ) -Ld - ∆Hcr + Qc + Qa ) ∆Hc + ∆Ha + Qc + Qa. (3′)
10.1021/jp000024s CCC: $19.00 © 2000 American Chemical Society Published on Web 07/18/2000
Determination of ∆H in Solutions
J. Phys. Chem. B, Vol. 104, No. 31, 2000 7469
TABLE 1: The Values of Enthalpy Change, ∆Hc and ∆Ha, at Hydration of some Cations and Anions in Infinitely Diluted Aqueous Solutions under 298 K cation
Li+
Na+
K+
Cs+
Mg2+
Ca2+
Al3+
-∆Hc, kcal mol
132
106
86
72
472
393
1141
anion
F-
Cl-
Br-
I-
(SO4)2-
(CO3)2-
-∆Ha, kcal mol
104
81
78
64
265
332
Expression 3′ can be naturally divided into cation and anion components:
}
(∆Hc)r ) ∆Hc + Qc (∆Ha)r ) ∆Ha + Qa .
(4)
Since Qc and Qa > 0, then |(∆Hc)r| < |∆Hc| and |(∆Ha)r| < |∆Ha|. Let us, as an example, evaluate the value of (∆Hc)r. As we know, experimental data concerning the values of the heats of the evaporation of ions from solutions are absent at present. However, the estimation of the lower limit value of the heat of evaporation of Na+ ions from aqueous solutions, 58 kcal/mol, was made in ref 4 on the basis of the experimental data. Using the tabular value of ∆HNa+ ) -106 kcal/mol, we obtain from (4) the upper limit of the enthalpy change at hydration of Na+ ions in aqueous solutions: (∆HNa+)r ) -106 + 58 ) -48 kcal/ mol. Thus, the absolute value of the enthalpy change at hydration of sodium ions is about half the tabulated value. Such a situation may be also expected for other ions. 3. Determination of Enthalpy Changes by the Gibbs Function The ∆H calculations were also made with the help of the Gibbs function, G. At the transfer of 1 mol of ions from vacuum into solvent, the change of Gibbs function, ∆G, is5
∆G ) -NAq2(1 - 1/)/2R
(5a)
and the corresponding change of enthalpy is6
∆H ) -NAq2(1 - 1/ - (T/2)(∂/∂T)p)/2R
(5b)
here R and q are the radius and the charge of the ion, NA is the Avogadro number, and T and are the solvent temperature and dielectric permeability. For example, this method gives for Na+ ions ∆H ) -176 kcal/mol and for F- ions ∆H ) -123 kcal/ mol.7 This method, unlike the conventional one, allows the separate calculation of ∆H for cations and anions, but it is also incorrect because the heat of evaporation of ions was not taken into account. 4. The Field Evaporation of Ions from Solutions (FEIS): Separate Determinations of Enthalpy Changes for Cations and Anions It was shown above that methods used for determination of the enthalpy change give incorrect ∆H values. In this part the new experimental method, giving correct ∆H values, will be described. This method is the version of the mass spectrometric method of electrohydrodynamic ionization developed in 1974.8 We used our method version (FEIS method) for study of ion solvation in water and volatile organic solvents.4,9-15 It consists of introducing an electrolyte solution directly into the highvacuum camera of a mass spectrograph, evaporating ions and
TABLE 2: The Enthalpy Changes (-∆Hn-k,n) (kcal/mol) for Solvation Reactions Na+Etn-k(H2O)m + kEt T Na+Etn(H2O)ma n-k m
k
0
1
2
3
4
0 0 0 0 0 1 1 1 1 2 2 2 2 3 3 3 4 4 4 5 5 5
1 2 3 4 5 1 2 3 4 1 2 3 4 1 2 3 1 2 3 1 2 3
7.4 10.3 13.5 16.9 18.9 4.9 8.9 13.8 16.5 8.6 13.4 16.7 17.9 9.7 18.4 23.3 12.6 19.9 23.7 3.4 5.7 8.1
2.9 6.1 9.5 11.5
3.2 6.6 8.6
3.4 5.4
2.0
4.0 8.9 11.6
4.9 7.6
2.7
4.8 8.1 9.3
3.3 4.5
1.2
8.7 13.6
4.9
7.3 11.1
3.8
2.3 4.7
2.4
a
Et is an ethanol molecule.
ion clusters into vacuum from the solution with the help of a strong electric field and following mass analysis of the evaporated ions and ion clusters. It was shown that ions and ion clusters are just evaporating from the solution but not breaking away with an accidental number of solvent molecules.4,15-18 It was also shown that the contribution of ionmolecule reactions in the gaseous phase to FEIS mass spectra was practically not observed. In addition, in the mass spectra, there were the ion clusters containing the ion pairs C+A- (C+ and A- are the cation and the anion of the electrolyte, respectively) and other nonvolatile components of the solution. This fact can be considered as one more piece of evidence of evaporation of these ion clusters just from the solution. The sign of the charge of the evaporated ions and ion clusters is the same as the sign of the potential creating the strong electric field. It gives a possibility to study ion solvation in solutions separately for cations and anions in contrast to the conventional method. Using the FEIS method, it was found that sets of equilibrium concentrations of ions Fz and ion clusters Fz(S)n (n ) 0, 1, 2, ..., S is a solvent molecule) exist in electrolyte solutions. The equilibrium is sustained by the equilibrium processes of solvation-desolvation
}
Fz + S T Fz(S) Fz(S) + S T Fz(S)2 Fz + 2S T Fz(S)2 l Fz(S)n1 + S T Fz(S)n l Fz + nS T Fz(S)n .
(6)
A dependence of FEIS mass spectra of the evaporated ions and ion clusters on the solution temperature, T, gives a possibility of a direct determination of the value of the enthalpy change, ∆Hn-k,n, at solvation of an ion cluster containing n k solvent molecules by k solvent molecules.4,13-15 To do this,
7470 J. Phys. Chem. B, Vol. 104, No. 31, 2000 it is necessary to measure the slope of the dependence of ln(In/In-k) on T-1, where In and In-k are the intensities of the mass spectral lines corresponding to ion clusters with n and n - k solvent molecules. In this manner, the values of ∆Hn-k,n at the solvation of ion clusters of Na+Etn(H2O)m by ethanol molecules in water-ethanol solution of NaI salt were determined in the 219-248 K temperature range (Na+ is a sodium ion, Et and H2O are an ethanol and a water molecule, n, m ) 0, 1, 2, ...).4,13-15 The values of ∆H for the various reactions (6) are presented in Table 2. The expression Na+Etn-k(H2O)m + kEt T Na+Etn(H2O)m is the generalized form of (6). Some data of Table 2 were published in refs 4 and 13-15. These data are revised. The mean estimated spread for data of Table 2 is about 30%. Acknowledgment. The authors are very grateful to Prof. Dr. H. Heydtmann (J. W. Goethe-Universita¨t, Frankfurt-amMain, Germany) for photoplates Ilford Q2 and Dr. N. I. Butkovskaya for help in preparing the manuscript for publication. The authors also thank Prof. Yu. I. Petrov for helpful discussions. References and Notes (1) Bernall, J.; Fowler, R. J. Chem. Phys. 1933, 1, 515.
Zolotoy and Karpov (2) Gordon, J. E. The Organic Chemistry of Electrolyte Solutions; A Wiley Interscience Publication, John Wiley & Sons: New York, London, Sidney, Toronto, 1975. (3) Desnoyers, J. E.; Jolicoeur, C. In Modern Aspects of Electrochemistry; Bockris, J. O’M., Convey, B. E., Eds.; New York: Plenum: 1969; No. 5, Chapter 1. (4) Zolotoy, N. B.; Karpov, G. V. J. Chem. Phys. 1998, 109, 4938. (5) Born, M. Z. Phys. 1920, 1, 45. (6) Bjerrum, N.; Larsson, E. Z. Phys. Chem. 1927, 127, 358. (7) Damaskin, B. B.; Petrii, O. A. OsnoVy teoreticheskoi elektrokhimii (Bases of Theoretical Electrochemistry); Vysshaya Shkola: Moskva, 1978 (in Russian). (8) Simons, D. S.; Colby, B. N.; Evans, C. A., Jr. Int. J. Mass Spectrom. Ion Phys. 1974, 15, 291. (9) Zolotoi, N. B.; Karpov, G. V. Dokl. Chem. 1988, 303, 329. (10) Zolotoy, N. B.; Karpov, G. V. Chem. Phys. Lett. 1995, 232, 43. (11) Zolotoy, N. B.; Karpov, G. V. Chem. Phys. Lett. 1995, 239, 158. (12) Zolotoi, N. B.; Karpov, G. V. Dokl. Phys. Chem. 1995, 344 (13), 215. (13) Zolotoi, N. B.; Karpov, G. V. IzV. Akad. Nauk. Ser. Fiz. (Bull. Russ. Acad. Sci.: Phys.) 1998, 62, 1808 (in Russian). (14) Zolotoi, N. B.; Karpov, G. V. Dokl. Phys. Chem. 1996, 348 (46), 137. (15) Zolotoy, N. B.; Karpov, G. V. In Physics of Clusters; Lakhno, V. D., Chuev, G. N., Eds.; World Scientific: Singapore, New Jersey, London, Hong Kong, 1998. (16) Zolotoy, N. B.; Karpov, G. V. Abstracts of Papers of 11th International Mass Spectrometry Conference; Bordeaux, 1988, TUM-42. (17) Zolotoi, N. B. Tech. Phys. 1995, 40, 1175. (18) Zolotoi, N. B.; Karpov, G. V. Dokl. Chem. 1996, 348 (1-3), 129.
