9868
Langmuir 2008, 24, 9868-9875
The Ionic Work Function and its Role in Estimating Absolute Electrode Potentials W. Ronald Fawcett* Department of Chemistry UniVersity of California DaVis, California 95616 ReceiVed December 18, 2007. ReVised Manuscript ReceiVed May 20, 2008 The experimental determination of the ionic work function is briefly described. Data for the proton, alkali metal ions, and halide ions in water, originally published by Randles (Randles, J. E. B. Trans Faraday Soc. 1956, 52, 1573) are recalculated on the basis of up-to-date thermodynamic tables. These calculations are extended to data for the same ions in four nonaqueous solvents, namely, methanol, ethanol, acetonitrile, and dimethyl sulfoxide. The ionic work function data are compared with estimates of the absolute Gibbs energy of solvation obtained by an extrathermodynamic route for the same ions. The work function data for the proton are used to estimate the absolute potential of the standard hydrogen electrode in each solvent. The results obtained here are compared with those published earlier by Trasatti (Trasatti, S. Electrochim. Acta 1987, 32, 843) and more recently by Kelly et al. (Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. B 2006, 110, 16066. Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. B 2007, 111, 408). A comparison of the ionic work function with the absolute Gibbs solvation energy permits an estimation of the surface potential of the solvent. The results show that the surface potential of water is small and positive whereas the surface potential of the nonaqueous solvents considered is negative. The sign of the surface potential is consistent with the known structure of each solvent.
Introduction The concept of the electronic work function is well known to all chemists. It is a measure of the work done to remove an electron from a metal that has a carefully prepared and clean surface. The concept of the work function of an ion in an electrolyte solution is much less familiar. The first experiments to measure ionic work functions were carried out by Kenrick1 working in the Ostwald laboratory at the University of Leipzig. By streaming mercury down the middle of a vertical tube and an electrolyte solution down the inner walls of the tube, the Volta potential difference between these phases was eliminated. Under these circumstances, the potential drop across the Kenrick cell can be used to estimate the work function of one of the ions in the electrolyte solution. Klein and Lange2 used ionizing radiation to eliminate the Volta potential drop between a metal and electrolyte solution separated by an air gap and thereby were able to perform the same experiment working with solid electrodes and quiescent solutions. These experiments were confirmed much later by Randles,3 Gomer and Tyson,4 and Farrell and McTigue.5 They were also extended to nonaqueous solvents by Case and Parsons,6 Parsons and Rubin,7 and Coetzee et al.8 As shown by Trasatti,9 the work function for the proton obtained in such an experiment may be used to estimate the absolute value of the potential of the standard hydrogen electrode. The majority of the data for ionic work functions was obtained over 40 years ago and is based on the thermodynamic properties for monatomic ions as summarized by Rosseinsky.10 These properties were recalculated on the basis of updated thermo* E-mail:
[email protected]. (1) Kenrick, F. B. Z. Phys. Chem. 1896, 19, 625. (2) Klein, O.; Lange, E. Z. Elektrochem. 1937, 43, 570. (3) Randles, J. E. B. Trans. Faraday. Soc. 1956, 52, 1573. (4) Gomer, R.; Tyson, G. J. Chem. Phys. 1977, 66, 4413. (5) Farrell, J. B.; McTigue, P. J. Electroanal. Chem. 1982, 139, 37. (6) Case, B.; Parsons, R. Trans. Faraday Soc. 1967, 63, 1224. (7) Parsons, R.; Rubin, B. T. J. Chem. Soc., Faraday Trans. I 1974, 70, 1636. (8) Coetzee, J. F.; Dollard, W. J.; Istone, W. K. J. Solution Chem. 1991, 20, 957. (9) Trasatti, S. Electrochim. Acta 1987, 32, 843. (10) Rosseinsky, D. R. Chem. ReV. 1965, 65, 467.
dynamic tables in 1999.11 The purpose of this article is to reexamine the ionic work function data using up-to-date estimates of ionic properties and to compare them with estimates of the Gibbs solvation energies of the same ions obtained by the mass spectrometric12 and other extra-thermodynamic methods. In addition, these data are used to reestimate the absolute potential of the SHE in water and four other solvents and to compare the results with those given earlier by Trasatti9 and more recently by Kelly et al.13,14
Background A condensed phase a is characterized by an inner potential φa. The inner potential15,16 arises for two reasons: phase a may have free charge on its surface that gives rise to an outer potential ψa; in addition, because of the special arrangement of the components of phase a at its surface, this phase has a surface potential χa. The relationship between these quantities is
φa ) ψa + χa
(1)
Although the outer potential of a phase can be measured, the inner and surface potentials cannot be determined experimentally. This terminology was originally introduced by Lange17,18 and is commonly used in modern electrochemistry.19,20 Unfortunately, these quantities are sometimes given different names by physicists and quantum chemists.21 (11) Fawcett, W. R. J. Phys. Chem. B 1999, 103, 11181. (12) Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. H.; Earhart, A. D.; Coe, J. V.; Tuttle, T. R. J. Phys. Chem. A 1998, 102, 7787. (13) Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. B 2006, 110, 16066. (14) Kelly, C. P.; Cramer, C. J.; Truhlar, D. G. J. Phys. Chem. B 2007, 111, 408. (15) Parsons, R. In Modern Aspects of Electrochemistry; Bockris, J. O’M., Conway, B. E., Eds.; Butterworths Scientific: London, 1954; Chapter 3. (16) Fawcett, W. R. Liquids, Solutions and Interfaces; Oxford University Press: New York, 2004; Chapters 8-10. (17) Lange, E.; Mischenko, K. Z. Phys. Chem. 1930, 149, 1. (18) Lange, E.; Go¨hr, H. Thermodynamische Elektrochemie; A. Hu¨thig: Heidelberg, 1962.
10.1021/la7038976 CCC: $40.75 2008 American Chemical Society Published on Web 08/09/2008
Ionic Work Function
Langmuir, Vol. 24, No. 17, 2008 9869
When two condensed phases come into contact, there is usually a flow of charge between these phases so that the electrical properties are different before and after contact. The difference in inner potentials for phases a and b is called the Galvani potential difference, whereas the difference in outer potentials is called the Volta potential difference. The Galvani potential difference may be measured only when phases a and b have the same chemical composition. The most important example of such a measurement is the determination of the emf (electromotive force) of an electrochemical cell that is constructed from at least four phases in contact so that it terminates with the same metal at each end. When the thermodynamic properties of single ions are discussed, one makes use of the electrochemical potential. The electrochemical potential of ion i in phase a is given by
µ ˜ia ) µia,0 + RT ln ai + ziFφa
(2)
µa,0 i
where is the standard chemical potential for ion i, ai is its activity, and zi is its charge, and the other symbols have their usual meanings. This quantity was first introduced in electrochemical thermodynamics by Butler.23,24 It was discussed more fully by Guggenheim,25 who emphasized that the chemical and electrical contributions to the electrochemical potential cannot be determined separately by experiment. However, an experimental determination of the electrochemical potential is possible. In the case of an electrolyte solution, when one adds the electrochemical potentials for the component ions, the electrostatic terms in the inner potential cancel out because of electroneutrality. Thus, the distinction between the electrochemical potential and the chemical potential disappears when a system with no net charge is considered. The final quantity to be defined here is the real potential of species i. It is equal to the electrochemical potential under circumstances that phase a bears no net charge (ψa ) 0). The real potential is given by
Ria ) µia,0 + RT ln ai + ziFχa
(3)
This quantity is the work done to take species i from chargefree infinity into the bulk of condensed phase a. Obviously, it is the negative value of the work function that has been measured for electrons in many different metals. The experimental determination of the ionic work function involves measuring the potential difference between the terminals of an electrochemical cell that contains an air gap.1–8 A cell that can be used to determine the work function of the proton is
Cu | Hg | air | x m HCl | Pt, H2 | Cu′
(4)
The potential difference across this high-impedance system is called the compensation potential, ∆cφ. The Kenrick cell shown in Figure 1 is designed so that the Volta potential drop across the air gap is zero. This is achieved by streaming the mercury (19) Bockris, J. O’M.; Reddy, A. K. N.; Gamboa-Aldeco, M. Modern Electrochemistry, Fundamentals of Electrodics, 2nd ed.;Plenum Publishers: New York, 2000; Vol. 2A, Chapter 6. (20) Girault, H. H. Analytical and Physical Electrochemistry; EPFL Press: Lausanne, 2004. (21) Lamoureux and Roux22 refer to both the inner potential and the surface potential as the phase potential. The same problem is present in the discussion of Kelly et al.,14 who state that “the absolute solvation free energy differs from the real one by the charge on the ion times the potential of the phase”. This statement is correct only when the charge on the phase, that is, its outer potential, is zero. (22) Lamoureux, G.; Roux, B. J. Phys. Chem. B 2006, 110, 3308. (23) Butler, J. A. V. Proc. R. Soc. London, Ser. A 1926, 112, 129. (24) Butler, J. A. V. Electrical Phenomena at Interfaces; Methuen: London, 1951. (25) Guggenheim, E. A. J. Phys. Chem. 1929, 33, 129.
Figure 1. Diagram of the Kenrick cell used to determine the real potential of the proton in a 10-3 mol kg-1 HCl solution. The Hg flows in small droplets from the central reservoir down the center of the vertical tube. The dilute HCl solution flows from the surrounding reservoir down the inner walls of the same vertical tube. In this way, the Volta potential difference between the Hg and the HCl solution is maintained at zero.
from the left-hand side of the air gap down the center of a tube with a diameter of ∼1 cm. At the same time, the dilute HCl solution flows down the walls of the same tube. As a result of the movement of these liquids, no charge is built up on their surfaces so that
ψHg - ψs ) 0
(5)
where ψHg is the outer potential of the streaming mercury and ψs is that of the solution. A detailed thermodynamic analysis of cell 4 leads to the expression16
F∆cφ ) RHs + + RHg e -
µHg 2 2
(6)
Thus, the compensation potential for cell 4 gives the Gibbs energy change for the process
1 H (gas) f H+(soln) + e(Hg) 2 2
(7)
It follows that ∆cφ may be used to determine the real potential of the proton in a solution of finite concentration given the standard Gibbs energy of H2 as a gas and the real potential (work function) of an electron in mercury. However, the ultimate goal is to obtain the real potential of the proton in a solution of infinite dilution. The analysis of the experimental data to obtain this quantity is described below. The following cell can be used to define the standard hydrogen electrode (SHE):
9870 Langmuir, Vol. 24, No. 17, 2008
Fawcett
Cu | Pt, H2(1 bar) | x M HCl, H2O | AgCl | Ag | Cu′ (8) When the pressure of H2 is 1 bar and the activity of HCl (aHCl) is unity, the emf of this cell is 0.222 V. Under these conditions, the hydrogen electrode is the standard hydrogen electrode with a defined single electrode potential of 0.000 V. Therefore, the other electrode, the silver/silver chloride system, has a standard potential of 0.222 V. It is important to note that the activity of the H+ ion is not equal to that of the Cl- ion and that the concentration corresponding to unit activity must be determined experimentally.16 It is helpful to express the emf of cell 8 in terms of the Galvani potential drops at each interface. Starting with the spontaneous cathode
E ) (φ
- φ ) ) (φ
Cu′
Cu
Cu′
- φ ) + (φ - φ ) + (φs - φPt) + (φPt - φCu) (9) Ag
Ag
s
This expression may be considerably simplified on the basis of the electronic equilibria at each metal/metal junction. At the Pt/Cu junction Pt µCu,0 - FφCu ) µPt,0 e e - Fφ
(10)
and at the Cu′/Ag junction
µCu′,0 - FφCu′ ) µAg,0 - FφAg e e
(11)
Upon substituting eqs 10 and 11 into eq 9 and recognizing that the standard potential of Cu is independent of its position in the cell, one obtains the following expression for the emf of cell 8:
[
E ) φAg - φs -
][
µAg,0 µPt,0 e e - φPt - φs F F
]
(12)
The first term in square brackets on the right-hand side of eq 12 contains quantities connected only to the potential of the silver/silver chloride electrode, whereas the second contains quantities related to the potential of the SHE. It is reasonable to speculate that the absolute values of these potentials are obtained by adding the same constant to each of these terms. Now consider cell 8 with the central solution divided into two compartments separated by an air gap
Cu | Pt, H2 | HCl,H2O | air | HCl,H2O | AgCl | Ag | Cu′ (13) As pointed out by Trasatti,26 this configuration allows one to separate the thermodynamic terms associated with each electrode when the cell is operated so that there is no net charge on the solutions on either side of the air gap. The point of reference for each half cell is located in the air gap where a test electron has a potential of zero in the absence of an electrical field. The compensation potential for this cell may be written as
∆cφ ) (φ
Cu′
- φ ) + (φ - φ ) + (φ - φ ) + (φ - φ ) + Ag
Ag
s
s
a
a
s
(φs - φPt) + (φPt - φCu) (14) where φa is the inner potential of the air phase. Because this cell is operated so that the Volta potential drop across the air gap is zero,
ψ -ψ )0
(15)
φs - φa ) χs
(16)
s
a
and
The expression for the compensation potential now may be written as (26) Trasatti, S. Pure Appl. Chem. 1986, 58, 955.
[
][
]
µAg,0 µPt,0 e e s Pt s ∆cφ ) φ - φ +χ - φ -φ + χs F F (17) Ag
s
The first term in square brackets on the right-hand side of eq 17 gives the absolute potential of the silver/silver chloride electrode, and the second term gives the absolute potential of the hydrogen electrode. This route to defining the absolute electrode potential is particularly important because it shows the importance of the surface potential of the solvent when one wishes to compare these quantities in different solvents.27 On the basis of the equilibrium
H+(soln) + e-(Pt) H H2(gas) ⁄ 2 the potential drop
φPt
-
φs
(18)
is given by
F(φPt - φs) ) µHs + + µPt,0 e -
µH2
(19)
2
At unit activity of the proton in solution and 1 bar pressure, the expression for the absolute standard potential of the SHE becomes
µH0 2 o ESHE(abs) ) -
0
µH2 RHs,0+ µHs,0+ + + χs ) + F F F F
(20)
where RHs,0+ is the standard real potential of the H+ ion in solution. 0 The estimation of ESHE (abs) in water and several nonaqueous solvents is illustrated in the following section.
Results Water. Farrell and McTigue5 determined the compensation potential for cell 4 for 19 molalities of HCl in the range from 0.001 to 0.506 mol kg-1. The corresponding change in ∆cφ is from -0.2359 to -0.0746 V. On the basis of eq 6, the variation in ∆cφ is due to the change in RT ln aH+. However, the single ion activity coefficient needed to estimate the hydrogen ion activity from its molality is not known except at very low ionic strengths where Debye-Huckel theory in its limiting form is valid. The strategy used to obtain ∆cφ in the limit of zero ionic strength is to plot ∆cφ - RT/F ln a( against the square root of the ionic strength, where a ( is the mean ionic activity of HCl calculated using the experimentally measured activity coefficient. The results obtained by these authors for molalities up to 0.018 mol kg-1 are shown in Figure 2. These data are easily extrapolated to zero ionic strength where the ordinate is equal to -55.9 ( 0.2 mV. On the basis of this analysis, from eq 6 in the limit of infinite dilution
∆cφ - RT ln 0
Hg a( ) RHW,0 + + Re -
µHg,02
(21)
2
Using data from a recent compilation,16 the work function of -1 mercury, -RHg e , is 4.50 eV or 434.2 kJ mol . The Gibbs energy change associated with the reaction
1 H f H· 2 2
(22)
is 203.3 kJ mol-1, and that associated with the reaction
H · f H++eis 1313.45 kJ
mol-1.28,29
Thus, the value of
(23) 1/ µ g,0 2 H2
is -1517
(27) Trasatti, S. Electrochim. Acta 1990, 35, 269. (28) NBS Tables of Chemical Thermodynamic Properties J. Phys. Chem. Ref. Data 1982, 11, Supplement 2.
Ionic Work Function
Langmuir, Vol. 24, No. 17, 2008 9871 W,0 RHW,0 - µiW,0(con.) + ) Ri
(25)
µW,0 i (con.)
Figure 2. Plot of the compensation potential corrected for the activity of HCl, ∆cφ - (RT/F) ln a (, against the square root of the ionic strength, I1/2, using the results of Farrell and McTigue.5
where is the Gibbs energy of solvation of ion i on the conventional scale. The average value of RH+ s,0 is -1091 ( 3 kJ mol-1. Using mass spectrometric data, Tissandier et al.12 estimated W,0 that µH+ , the absolute value of the standard Gibbs energy of solvation of the proton in water at 25 °C and 1 bar pressure, is -1104.5 kJ mol-1. They argued that small ion-water clusters do not have a surface structure. Thus, the extrapolation procedure to obtain the thermodynamic properties of a large cluster yielded the properties of a system with no surface potential. This is clearly an extrathermodynamic assumption. W,0 It follows that µH+ together with the standard real potential of the proton may be used to estimate the surface potential of water:
χW ) (RHs,0+ - µHs,0+) ⁄ F ) (-1091 + 1104.5) ⁄ 96.485 ) 0.14 V (26)
Figure 3. Plots of RiS,0 for the alkali metal cations (∆) and the halide ions (3) against the reciprocal of the Shannon and Prewitt radius, ri-1, for data obtained in water. The data were estiinated from the results of Randles3 but were corrected using up-to-date thermodynamic data. For the sake of clarity, the halide ion results have been shifted vertically by 100 kJ mol-1.
kJ mol-1. From equation 21
RHW,0 + ) -0.0559 × 96.485 + 434.2 - 1517 ) -1088 kJ mol-1 (24) where 96.485 is the Faraday constant expressed in kC mol-1, W,0 RH+ is is the standard real potential of the proton in water at 25 °C and 1 bar pressure. Randles3 measured the real potentials of the alkali metal and halide ions in water at 25 °C. Fortunately, his results are presented in such a way that the values of RW,0 may be recalculated using i up-to-date thermodynamic data.28,29 Plots of RW,0 against the i reciprocal of the Shannon and Prewitt radius of the ion are shown in Figure 3 for the alkali metal and halide ions. These plots are good straight lines and provide a reasonable way of assesing the results. By comparing the value of RW,0 with the Gibbs energy i of ion solvation on the conventional scale, one obtains additional W,0 estimates of RH+ . Because the Gibbs energy of proton solvation is zero on the conventional scale, (29) Bard, A. J., Parsons, R. Jordan, J., eds. Standard Potentials in Aqueous Solutions; Marcel Dekker: New York, 1985.
The positive value is interpreted as resulting from the strong hydrogen bonding in water that tends to leave excess water dipoles with their oxygen atoms pointing toward the gas phase. As a result, a test charge moving from charge-free infinity into the bulk of water experiences a positive dipolar layer at the water surface. This result is confirmed by the observation30 that the temperature coefficient of χW is negative, indicating that χW becomes smaller with increasing temperature. This is due to a decrease in the extent of dipolar alignment with increasing temperature. The present result agrees well with the estimate made by Trasatti9,26 (+0.13 V). It also agrees well with a recent estimate by Krishtalik31 (+0.14 V). It is interesting that quantum chemical estimates of χW are all large in magnitude and negative.22 It is well known that estimating bulk dielectric properties from the dipolar properties of individual water molecules is an extremely difficult task. This may also be W,0 true for the surface potential. However, the estimate of µH+ by Tissandier et al.12 may also be incorrect. For example, if the true value of χW is -0.54 V as assumed by Lamoureux and Roux,22 W,0 then the value of µH+ would be -1036 kJ mol-1. This is well outside of the range of error ((3 kJ mol-1) estimated by Tissandier W,0 et al.12 and Kelly et al.13 for µH+ . Moreover, the magnitude of the quantum chemical estimate is so large that it is hard to accept as physically realistic. W,0 The value of µH+ obtained by Tissandier et al.12 may now be used to estimate the absolute Gibbs energy of solvation for any ion on the absolute scale for which a value exists on the conventional scale.16 Values of µW,0 for the alkali metal and i halide ions in water at 25 °C and 1 bar pressure are given in Table 1. The corresponding values of the standard real potentials are found by adding ziFχW to the data in Table 1. Real potential data for the same ions are recorded in Table 2, assuming that ziFχW is 13.5 kJ mol-1. The magnitude of the real potential is smaller than the Gibbs energy of solvation for cations; of course, the opposite is true for anions. The equation describing the real potential data for the alkali metal cations is W,0 -1 -RM + ) 91.2 + 37.13rM+
(27)
and that describing the data for the halide ions is (30) Randles, J. E. B.; Schiffrin, D. J. J. Electroanal. Chem. 1965, 10, 480. (31) Krishtalik, L. I. Russ. J. Electrochem. 2008, 44, 43.
9872 Langmuir, Vol. 24, No. 17, 2008
Fawcett
-1 -RXW,0 - ) 4.5 + 52.23rX-
(28)
where ri is the Shannon and Prewitt radius of ion i. Finally, the absolute potential of the SHE may now be estimated using eq 20: o ESHE (abs) ) (-1091 + 1517) ⁄ 96.485 ) 4.42 V (29)
This result agrees well with the value obtained earlier by Trasatti9,26 (4.44 V). It differs significantly from the value obtained by Kelly et al.,13 namely, 4.24 V. This difference is discussed in detail below. Methanol. The Gibbs solvation energies for simple ions in nonaqueous solvents are easily estimated from the value in water using the Gibbs energy of transfer defined as
∆trµiS,0 ) µiS,0 - µiW,0
(30)
Here, µS,0 i is the S, and µW,0 is the i
standard Gibbs energy of solvation in solvent corresponding value in water. Values of the Gibbs energy of transfer have been estimated in a large number of solvents for the simple ions considered here and have been tabulated by Marcus.32–34 Single ion values are usually estimated using the so-called TATB assumption discussed in an earlier paper.35 The TATB assumption can be criticized on several grounds.14,35
Figure 4. Same as in Figure 3 but for methanol. For the sake of clarity, the results for the halide ions have been shifted vertically by 100 kJ mol-1.
The real potential for an ion in a nonaqueous solvent can, in principle, be obtained by experiment. Case and Parsons6 used the cell
Ag | AgCl | NaCl, MeOH | air | NaCl, H2O | AgCl | Ag′ (31) Table 1. Standard Gibbs Energy of Solvation for Simple Monoatomic Ions -µiS,0/
-1
kJ mol
solvent
watera
MeOHb
EtOHb
AcNb
DMSOb
ion H+ Li+ Na+ K+ Rb+ Cs+
1104 529 424 352 329 306
1096 523 418 340 322 297
1093 517 411 333 315 288
1064 501 407 337 321 297
1124 545 439 361 343 316
FClBrI-
429 304 278 243
408 291 266 235
402 284 260 229
265 250 230
269 250 230
a Estimated from the conventional Gibbs energy of solvation16 assuming that the absolute value of the standard Gibbs energy of solvation of the proton12 is -11.04.5 kJ mol-1. b Initially estimated using the Gibbs energy of transfer from water to the given solvent as reported by Marcus32–34 and then smoothed as described in the text.
in the Kenrick configuration to determine the standard real potential of the Cl- ion in methanol. The resulting value for MeOH,0 RCl is -273 kJ mol. They then estimated RMeOH,0 for a series i of monovalent and divalent ions using available thermodynamic data. Using their results for H+, the alkali metal cations, and the halide ions and comparing these with the corresponding value of µMeOHs,0 , one may estimate the surface potential of methanol. i The result is χMeOH ) -0.186 V. The negative value of this parameter indicates that the net orientation of the solvent dipoles involves the methyl group pointing to the gas phase and the polar -OH group remaining in the liquid. The quantity ziFχMeOH was then added to the estimates of µMeOHs,0 to obtain a second estimate i of RMeOH,0 . The value of RMeOH,0 for the proton is -1113 kJ i i mol-1, and that for µMeOHs,0 is -1096 kJ mol-1. i Plots of RMeOH,0 against the reciprocal of the ionic radius for i the alkali metal and halide ions are shown in Figure 4 for the present data. Excellent linear plots are obtained for both cations and anions. In the case of the cations, the equation of the best straight line is
Table 2. Standard Real Potentials for Simple Monoatomic Ions
MeOH,0 -1 -RM ) 105.0 + 38.39rM + +
-RiS,0/ kJ mol-1 solvent
water a
MeOHb
EtOHb
AcNb
DMSOc
ion H+ Li+ Na+ K+ Rb+ Cs+
1091 516 411 339 316 293
1114 541 436 358 340 313
1105 531 424 345 327 300
1074 510 416 346 331 307
1147 569 463 384 367 340
FClBrI-
443 318 292 257
390 273 248 217
390 271 246 215
256 240 220
243 227 206
a Initially estimated from the data reported by Randles3 and recalculated on the basis of up-to-date thermodynamic data28,29 but smoothed on the basis of eqs 27 and 28. b Initially estimated from the data of Case and Parsons6 and recalculated on the basis of up-to-date thermodynamic data but smoothed as described in the text. c Estimated from the data of Parsons and Rubin7 and Coetzee et al.8 as described in the text.
(32)
The corresponding equation for the anions is
-RXMeOH,0 ) -19.43 + 48.96rX-1-
(33)
Equations 32 and 33 were used to obtain the results for the standard real potentials recorded in Table 2. The standard Gibbs solvation energies reported in Table 1, were found by subtracting ziFχMeOH from the results for RMeOH,0 in Table 2. i Kelly et al.14 used values of the Gibbs energy associated with ion-methanol cluster formation in the gas phase and estimated that µH+MeOH,0 is -1095 kJ mol-1 with a precision of ( 8 kJ mol-1. Their estimate36 is in excellent agreement with that obtained from the Gibbs energy of transfer.33 The value of 6.
(32) MarcusY. Ion SolVation; Wiley-Interscience: New York, 1985; Chapter (33) Kalidas, C.; Hefter, G.; Marcus, Y. Chem. ReV. 2000, 100, 819. (34) Marcus, Y. Chem. ReV. 2007, 107, 3880. (35) Fawcett, W. R. J. Phys. Chem. 1993, 97, 9540.
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Langmuir, Vol. 24, No. 17, 2008 9873
Figure 5. Same as in Figure 3 but for ethanol. For the sake of clarity, the results for the halide ions have been shifted vertically by 100 kJ mol-1.
RH+MeOH,0 in Table 2 may be used to estimate the absolute potential of the SHE in methanol:
o ESHE (abs) ) (-1114 + 1517) ⁄ 96.485 ) 4.17 V (34)
This agrees quite well with the estimate made by Trasatti9 (4.19 V). Ethanol. Data are also shown in Tables 1 and 2 for simple monatomic monovalent ions in ethanol (EtOH). Estimates of µEtOH,0 were obtained from the corresponding values in water i using the Gibbs energy of transfer cited by Marcus.32–34 Case and Parsons6 also reported real potential data in ethanol, but for fewer ions than in methanol. Comparing values of µEtOH,0 obtained i from the Marcus data with the available values of REtOH,0 , the i best estimate of ziFχEtOH is -15 kJ mol-1. The corresponding surface potential of ethanol is -0.155 V. The real potentials for the ions considered here were then estimated by adding ziFχEtOH to µEtOH,0 . Plots of the real potential data obtained using the i Gibbs energy of transfer are shown in Figure 5. Excellent linear plots were obtained with the following equations: EtOH,0 -1 -RM ) 93.80 + 38.59rM + +
(35)
Figure 6. Plots of RiS,0 for the alkali metal cations (∆) and halide anions (3) in acetonitrile using the corrected data of Case and Parsons6 against the reciprocal of the Shannon and Prewitt radius, r-1 i . The data designated (O, )) were calculated using the Gibbs energy of transfer32–34 and by assuming ziFχS ) -1.0 kJ mol-1.
Acetonitrile. Case et al.37,38 used the following cell in the Kenrick configuration to determine the standard real potential of Ag+ in acetonitrile:
Ag | AgNO3, H2O | air | AgNO3, AcN | Ag′
They also measured the standard potentials for a series of aromatic hydrocarbon molecules that can be both oxidized and reduced in acetonitrile. Using data for the ionization energies and electron affinities of these molecules, they then estimated the real potentials for the corresponding cation and anion radicals that are reasonably stable in acetonitrile. Finally, they assumed that the Gibbs solvation energy of the cation radical is equal to that of the corresponding radical anion because of their large sizes. Thus, by subtracting the real potential of the cation from that of the anion they obtained an estimate of the surface potential χAcN. The average value based on data for six large organic molecules is -0.10 ( 0.06 V. Now, using the values of ∆trµAcN,0 given by Marcus,32–34 the absolute Gibbs i energies of solvation for the ions considered here were estimated. These data were converted to standard real energies of solvation by adding ziFχAcN to each value of µAcN,0 . Plots of these data and i those reported by Case and Parsons6 are shown in Figure 6. The acetonitrile data show more scatter than those for MeOH and EtOH. The equations of the best straight lines estimated using all data are
and
-RXEtOH,0 ) -23.94 + 48.96rX-1-
(36)
REtOH,0 i
Values of estimated according to eqs 35 and 36 are reported in Table 2. The values of µEtOH,0 in Table 1 were obtained i by subtracting ziFχEtOH from the corresponding value of REtOH,0 . i The absolute potential of the SHE in ethanol is
o ESHE (abs) ) (-1108 + 1517) ⁄ 96.485 ) 4.24 V (37)
The estimate by Trasatti9 is 4.21 V. (36) Kelly et al.13,14 report values of µiS,0 using a standard state of 1 M for the gas phase. Their data are easily converted to the IUPAC standard state of 1 bar by adding 7.95 kJ mol-1 to their results.
(38)
AcN,0 -1 -RM ) 120.46 + 34.30rM + +
(39)
-RXAcN,0 ) 69.85 + 31.04rX-1-
(40)
and
The data reported in Tables 1 and 2 were obtained using these equations and by assuming that the surface potential χAcN is -0.1 V. Finally, the standard real potential of the proton reported here is an average of the value obtained using the Gibbs energy of transfer (-1069 kJ mol-1) and that reported by Case and Parsons6 (-1079 kJ mol-1). The resulting estimate of the standard potential of the SHE in acetonitrile is o ESHE (AcN) ) (-1074 + 1517) ⁄ 96.485 ) 4.59 V (41)
The estimate given by Trasatti9 is 4.60 V.
9874 Langmuir, Vol. 24, No. 17, 2008
Fawcett Table 3. Absolute Potential of the Standard Hydrogen Electrode o ESHE (abs)/V
solvent
present results
Kelly et al.13,14
Trasatti9,26
water methanol ethanol acetonitrile dimethyl sulfoxide
4.42 4.17 4.24 4.59 3.83
4.24 4.33
4.44 4.19 4.21 4.60
4.66 4.04
calculation of E0SHE(abs), not the real potential, RH+S,0. The second reason36 is that the standard state for the quantities involving gases is chosen to be 25 °C and 1 M rather than the IUPAC standard state of 25 °C and 1 bar pressure. In the comparisons made here, the results of Kelly et al.13,14 are presented using the usual standard state of 25 °C and 1 bar pressure. Thus, the equation 0 used by these authors to estimate ESHE (abs) is Figure 7. Same as in Figure 3 but for dimethyl sulfoxide.
µAcN,0 i
14
The values of estimated by Kelly et al. using ioncluster formation data are higher than those given in Table 1. For example, their estimate of µAcN,0 for the proton is -1081 kJ i mol-1 compared to the present estimate of -1064 kJ mol-1. Dimethyl Sulfoxide. The final solvent considered here is dimeyhyl sulfoxide (DMSO) which is a strong Lewis base. As a result, the standard Gibbs energies of solvation of cations in this solvent are larger in magnitide than in water (Table 1). Very few experimental data exist for the real potentials of ions in DMSO. Parsons and Rubin7 found that the real potential change for the transfer of Cl- from water to DMSO is 77.0 kJ mol-1 so that RDMSO,0 is equal to -241 kJ mol-1. Coetzee et al.8 measured Clthe corresponding quantity for Ag+; the resulting value of RDMSO,0 Ag+ is -547 kJ mol-1. By comparing the real potentials with the corresponding absolute Gibbs energies of solvation, µDMSO,0 , i one can estimate the value of FχDMSO. The result for FχDMSO is -23.5 ( 2 kJ mol-1 so that the surface potential of DMSO is -0.24 V. The quantity ziFχDMSO was added to the estimates of µDMSO,0 to obtain the values of RDMSO,0 plotted in Figure 7. i i The equations describing the dependence of RDMSO,0 on the i reciprocal of the ionic radius are DMSO,0 -1 -RM ) 130.62 + 38.55rM + +
(42)
for the alkali metal cations and
-RXDMSO,0 ) 47.20 + 32.75rX-1-
(43)
for the halide ions. Excellent linear plots are obtained in both cases. Finally, the estimate of the real potential of the proton is -1147 kJ mol-1. The corresponding value of the absolute potential of the SHE is o ESHE (DMSO) ) (-1147 + 1517) ⁄ 96.485 ) 3.83 V (44)
Kelly et al.14 estimated that the absolute value of the Gibbs solvation energy of the proton is -1136 kJ mol-1, which is somewhat higher than the value obtained here (-1124 kJ mol-1). F. Absolute Potential of the SHE. The values of the absolute potential of the SHE reported recently by Kelly et al.13,14 are quite different from those reported here and those estimated by Trasatti.9,26 There are two reasons for the differences. The first is that these authors assume that the Gibbs energy of solvation of the proton, µH+S,0, is the appropriate quantity to be used in the (37) Case, B.; Hush, N. S.; Parsons, R.; Peover, M. E. J. Electroanal. Chem. 1965, 10, 360. (38) Case et al.37 refer to the real potential as the real solvation energy.
µH0 2 0 ESHE(abs) ) -
µHS,0+ + F F
(45)
They estimate that µH0 2 is -1513.4 kJ mol-1 on the basis of integrated heat capacity data, a result that agrees fairly well with the value used here (-1517 kJ mol-1) that was obtained from thermodynamic tables.28,29 On the basis of the analysis of cell o 13, the estimates of ESHE (abs) by Kelly et al.13,14 are different from those presented here because they do not consider the surface potential of the solvent χS. In their discussion,14 these authors point out that the quantity estimated by eq 45 is of theoretical interest. It should then be given a different name, for example, 0 the hypothetical value of ESHE . Very recently, Donald et al.39 estimated the absolute potential of the SHE using thermodynamic data obtained from mass spectrometric results for nanodrops of water containing [Ru(NH3)6]3+ and other transition metal ions. By studying the effects of reducing these ions with thermally generated electrons, they were able to estimate the standard potential for each redox couple on the absolute scale. Using data for six redox couples, 0 they estimated that ESHE (abs) is equal to 4.2 ( 0.4 V. Their result agrees well with that of Kelly et al.13 but not with that obtained here. This is probably due to the fact that they were working with nanodrops without a developed surface structure so that their 0 estimate of ESHE (abs) corresponds to that obtained by eq 45. o The present results for ESHE (abs) together with those reported 9,26 earlier by Trasatti and Kelly et al.13,14 are summarized in Table 3. For all of the solvents considered, the present estimates agree with the earlier work of Trasatti within the estimated o experimental error ((0.5 V). The estimate of ESHE (abs) by Kelly 13 et al. for water is considerably lower than the previous estimate. However, for all other solvents, their estimate14 is higher. As a result, the correct ordering of EoSHE(abs) with respect to the nature of the solvent is lost.
Discussion It is apparent from the above results that the standard Gibbs energy of solvation and the real energy of solvation are significantly different numerically. They are also significantly different conceptually. The work function for an ion in an infinitely dilute solution, namely, -RS,0 i , is an experimentally measurable quantity as demonstrated above. However, the corresponding Gibbs energy of solvation, namely, µS,0 i , cannot be measured experimentally. Conceptually, -µS,0 corresponds to the work i function for an ion in an infinitely dilute solution whose surface region is structurally the same as the bulk. Such a solution (liquid) has not been found to date.
Ionic Work Function
The linear plots used to assess the real potential data in Figures 3-7 do not imply that that there is a Born contribution to the values of RS,0 i . In fact, it is known that the corresponding data for the standard Gibbs energy of solvation in water do not obey the Born equation.40 The fact that plots of RS,0 against the i reciprocal of the ionic radius r-1 i are linear is simply a useful observation that provides a very convenient way of comparing data from different sources. The magnitude and sign of the surface potential of water was the subject of considerable discussion 30 years ago.41,42 As discussed by Conway,43 there were several extrathermodynamic methods of varying reliability for estimating the standard Gibbs energy of solvation of the proton at that time. In this regard, the analysis of mass spectral data for ion-water clusters to obtain W,0 a more reliable estimate of µH+ by Tissandier et al.12 represented an important breakthrough in the thermodynamics of ion solvation. The estimate of χW obtained here, namely, 140 mV, is quite close to the earlier estimates of 130 mV.41,42 The sign of χW is consistent with the known structure of water from both neutron diffraction experiments44 and Raman spectroscopy.45 Thus, in the bulk each oxygen atom is surrounded on average by four hydrogen atoms as a result of strong hydrogen bonding. At the surface of water, this is no longer possible, and some oxygen atoms are oriented toward the gas phase with fewer hydrogen atom neighbors. The other interesting result of the present analysis is that the surface potential for all nonaqueous solvents is negative. Similar results were obtained by Krishtalik31 for six nonaqueous solvents. (39) Donald, W. A.; Leib, R. D.; O’Brien, J. T.; Bush, M. F.; Williams, E. R. J. Am. Chem. Soc. 2008, 130, 3371. (40) Blum, L.; Fawcett, W. R. J. Phys. Chem. 1992, 96, 408. (41) de Battisti, A.; Trasatti, S. Croat. Chem. Acta 1976, 48, 607. (42) Randles, J. E. B. Phys. Chem. Liq. 1977, 7, 107. (43) Conway, B. E. J. Solution Chem. 1978, 7, 721. (44) Soper, A. K. Chem. Phys. 2000, 258, 121.
Langmuir, Vol. 24, No. 17, 2008 9875
The present group includes two protic solvents (MeOH and EtOH) and two aprotic solvents (AcN and DMSO). It suggests that the polar portion of the solvent molecule remains associated with other molecules whereas the alkyl group prefers to orient toward the gas phase. This is consistent with the behavior of these solvents when they are present as dilute solutes in water.16 It is also consistent with the structure of the pure solvent as determined by X-ray and neutron diffraction.46–48 Recently, Pethica49 has discussed limitations on the assumed additivity of the chemical and electrostatic components of the electrochemical potential implicit in eq 2. However, he has also emphasized that the application of this equation to the measurement of electronic work functions and ionic real potentials is valid. As seen from the data presented in this article, the electrostatic contribution to the real potential is a small fraction of the total energy for the systems considered here so that the concerns raised by Pethica are not important. Finally, the data presented in Tables 1 and 2 for some solvents, especially AcN and DMSO, must be considered to be tentative. For these solvents, the estimates of µS,0 i obtained from gas-phase ion-solvent clusters differed significantly from estimates based on experimental values of RS,0 and estimates of µS,0 obtained i i using the TATB assumption. More experimental work to expand the existing mass spectrometric data would be welcome. Acknowledgment. I thank Grace J. Chavis for her help with the Figures. The financial support of the National Science FoundationthroughgrantCHE0451103isgratefullyacknowledged. LA7038976 (45) Walrafen, G. E. J. Chem. Phys. 1964, 3249. (46) Yamaguchi, T.; Hidaka, K.; Soper, A. K. Mol. Phys. 1999, 97, 603. (47) Radnai, T.; Itoh, S.; Ohtaki, H. Bull. Chem. Soc. Jpn. 1988, 61, 3845. (48) Luzar, A.; Soper, A. K.; Chandler, D. J. Chem. Phys. 1992, 96, 8460. (49) Pethica, B. A. Phys. Chem. Chem. Phys. 2007, 9, 6253.
