Comment on The Role of Concentration Dependent Static Permittivity of Electrolyte Solutions in the Debye Hückel Theory

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1 Comment on The Role of Concentraton Dependent Statc Permttvty of Electrolyte Solutons n the Debye Hückel Theory Mónka Valskó and Dezső Boda Department of Physcal Chemstry, Unversty of Pannona, P. O. Box 158, H-81 Veszprém, Hungary E-mal: boda@almos.ven.hu Phone: /64/641. Fax: /64548 In ther recent paper, 1 Shlov and Lyashchenko (SL extended the Debye Hückel (DH theory (denoted as EDH n ths work for the case when the delectrc constant of the electrolyte can vary wth salt concentraton, ε(c, whch s a strong expermental evdence 5 (an dea advocated by Hückel 6 as early as 195. They dvde the thermodynamc functons nto two parts representng nteronc and on water (solvaton nteractons denoted by numbers 1 and, respectvely. These terms clearly correspond to the (on on and (on water terms n the excess chemcal potental ntroduced n our + model: 7 1 µ = µ + µ. (1 The term was computed from Grand Canoncal Monte Carlo (GCMC smulatons on the bass of the Prmtve Model (PM of electrolytes usng hard sphere ons wth the Paulng rad (Table 1, whle the term was To whom correspondence should be addressed computed from the Born equaton. 11 The excess chemcal potental s related to the actvty coeffcent through µ = kt lnγ, where k s Boltzmann s constant and T s temperature. The purpose of ths comment s fourfold. (1 We show that our term s equvalent wth the solvaton term of SL. ( We show that the Born rad can and should be used n t nstead of the Paulng rad. (3 We show that usng the ndvdual onc rad n the equatons (nstead of the mean, R ± = (R + + R / the theory of SL gves good results for the ndvdual onc actvtes too. (4 We also propose an alternatve equaton for the ndvdual actvty coeffcent elaborated n the Supplementary Informaton (SI. The term that we proposed n our 1 publcaton 7 s µ (c = z e 8πε R ( 1 ε(c 1, ( ε w where µ s the part of the excess chemcal potental of speces that s assocated wth the 1

2 nteractons, e s the unt charge, z s the valence of the onc charge, ε s the permttvty of vacuum, ε w s the delectrc constant of water (of the soluton at nfnte dluton, ε(c s the delectrc constant of the soluton at salt concentraton c, and R s a yet unspecfed onc radus. We brefly present SL s results keepng ther notaton (except that we drop the superscrpt E to denote excess. They started wth the Gbbs free-energy G = N z e 8πε ε w R ± τ (κ, (3 where N s the number of ons of speces and 1 τ (κ = λ dλ. (4 f (κ λ Functon f (κ s defned through the equaton ε(c = ε(κ = ε w f (κ (5 and expresses the c-dependence of the delectrc constant. In these equatons, κ s the nverse Debye screenng length defned through κ = j N j z j e ε ε w ktv (6 expressng the concentraton dependence (the relaton of densty and concentraton s N /V = 1N A c wth N A beng the Avogadro number and V the volume but stll contanng the water delectrc constant (ε w. The solvaton excess chemcal potental s obtaned from the dfferentaton µ, = G / N and s expressed as µ, = z e 8πε R ± ε w [χ (κ 1] (7 usng the nfntely dlute electrolyte as reference. Ths equaton corresponds to Eq. 9 of SL 1 n SI unt except that the second term of that equaton s omtted here (t s small and can be neglected. The functon χ (κ s gven by where χ (κ = τ (κ + σ (κ, (8 1 σ (κ = f (κ λλ dλ (9 f (κ λ wth f (κ λ = d f (κ λ/dλ (note the defnton of the dervatve slghtly dfferent from that used by SL. We can obtan an expresson for the ndvdual chemcal potental f we use an ondependent radus, R, nstead of the mean, R ±. We have two choces where to ntroduce the on-specfc radus. (1 We can ntroduce t n Eq. 7 after the dfferentaton µ, = G / N (route 1, or ( we can ntroduce t n the free energy: G = N z e 8πε ε w R τ (κ, (1 and perform the dfferentaton µ, = G / N afterwards (route. The second route results n a slghtly dfferent formula for the chemcal potental where µ, = z e 8πε ε w R χ (κ = τ (κ + 1 ( κ κ [ ] χ (κ 1, (11 σ (κ (1

3 and κ = j N j z j e ε ε w ktv R R j. (13 The dervaton of ths equaton s found n the SI wth some results that show that ths equaton gves slghtly larger values for the bgger on and slghtly smaller for the smaller on compared to route 1. In the man text of ths comment, therefore, we focus on the frst route and show that Eqs. and 7 (wth replacng R for R ± are equvalent. We can show ths equvalence f we prove that χ (κ = 1 f (κ. (14 The functon χ (κ can be wrtten n the form 1 λ f (κ λ λ f (κ λ χ (κ = f (κ λ dλ. (15 If we realze that ( λ = λ f λ f f f, (16 the ntegral can be wrtten smply as [ λ χ (κ = f ] 1 = 1 f (17 whch concludes the proof (the functon f s postve n the nterval λ 1. SL state that If f (κ λ = 1, then the functon τ (κ = 1 and we arrve at the expresson smlar to that obtaned n the Born theory of solvaton. If f = 1, the term s zero, because the on does not change ts delectrc envronment. They also state that Our solvaton contrbuton to the onc actvty coeffcent s dfferent from a plan Born-lke expresson used by Vncze, Valskó, and Boda 46,47. We have proven above that ths statement s wrong and that the solvaton contrbuton of SL s equvalent to the plan Born-lke expresson proposed by us. The dervaton of SL, therefore, justfes our ntutve consderatons that were based on dentfyng the solvaton part of the excess chemcal potental wth the electrostatc nteracton of a sngle on wth the surroundng delectrc medum n whch t has been nserted. Ths electrostatc potental s the same that SL use n ther charge-up process. Entropc terms are gnored n both approaches. To our best knowledge, other authors 1 14 also used the Born equaton wthout strct justfcaton. SL use the mean of the Paulng rad, R ± = (R + + R /, to compute the mean excess chemcal potental: µ ±, = z +z e 8πε ε w R ± [χ (κ 1]. (18 We use the ndvdual onc rad and compute the excess chemcal potentals of the ndvdual onc speces from whch the mean s obtaned as µ ± = ν +µ + + ν µ, (19 ν + + ν where ν s the stochometrc coeffcent (the same weghted average apples for the and components too. For pure electrolytes, ν + = z and ν = z +. In the SI, we show that the mean excess chemcal potentals computed from the ndvdual ones obtaned from the two routes are dentcal: µ ±, = µ ±,. In ths work, we show detaled results for NaCl and CaCl. For the delectrc constant, the followng equatons 4,5 were used: and ε NaCl (c = ε w 15.45c c 3/ ( ε CaCl (c = ε w 34c + 1c 3/ (1 3

4 ln(γ ± 1.5 (Paulng (Born -EDH (Paulng -GCMC (Born -EDH (Born Experment -GCMC -EDH ln(γ ± NaCl -4 CaCl c 1/ / M 1/ c 1/ / M 1/ Fgure 1: Mean actvty coeffcents of NaCl and ts and components (n molar scale. Sold and dot-dashed blue lnes show the term as computed from Eq. usng ether the Born (R B or the Paulng (R rad for R. The rad can be found n Table 1. Sold and dashed red lnes show the terms as computed from ether GCMC smulatons 15 or the EDH theory (Eq. 9 of SL 1. The black curves show the total actvty coeffcents as computed from dfferent combnatons sold: GCMC for and Born radus for ; dashed EDH for and Born radus for ; dot-dashed: EDH for and Paulng radus for. The expermental data are taken from Wlczek-Vera et al. 16 These notatons wll be used n other fgures, therefore, the legend wll not be repeated n those fgures for clarty. wth ε w = at T = K. The term was computed ether from GCMC smulatons or the EDH theory (Eq. 18 of SL usng the Paulng rad (Table 1. The mean term as calculated from Eq. 18 s a lttle bt smaller than the mean term calculated from Eq. 19. Therefore, we wll show only the latter data. It s a more mportant queston whch onc radus should be used Fgure : Mean actvty coeffcents of CaCl and ts and components. The notatons are the same as n Fg. 1. n Eq.. SL used the Paulng rad. Ths consderably overestmates the term. In Fgs. 1 and we show the mean actvty coeffcents of NaCl and CaCl, respectvely. The term as computed wth the Paulng rad (blue dotdashed lnes s consderably larger than that computed wth the Born rad (blue sold lnes. The total actvty coeffcents computed from the (Born curve (black sold and dashed lnes are n much better agreement wth experment than the one computed from the (Paulng curve (black dot-dashed. We have proposed n our papers 7 1 that the calculaton of the and terms can be decoupled. The only quantty that couples them s the concentraton dependent delectrc constant, ε(c. To compute a decent term, we must be as close to experments as possble. The Born radus can be consdered as an expermental parameter, because t s related to the expermental hydraton free energy through ( G s = z e 1 8πε R B 1. ( ε w 4

5 1 Na + 1 Ca + ln(γ.5 ln(γ ln(γ.5 Cl ln(γ 1 Cl c 1/ / M 1/ c 1/ / M 1/ Fgure 3: Indvdual actvty coeffcents of Na + (top panel and Cl (bottom panel and ther and components n NaCl solutons of varyng concentratons. The notatons are the same as n Fg. 1. Fgure 4: Indvdual actvty coeffcents of Ca + (top panel and Cl (bottom panel and ther and components n CaCl solutons of varyng concentratons. The notatons are the same as n Fg. 1. It s mportant to note that the Born radus s not a physcal radus. It s an effectve parameter to make the results of a smple theory match the expermental data of a complex system. As a matter of fact, we can elmnate R B from Eqs. and to obtan µ (c = G s ε(c ε w ε(c (ε w 1, (3 whch equaton contans only expermental parameters (see Table 1. It mght be surprsng that ths smple electrostatc equaton descrbes the term so well. The explanaton, n our opnon, s that the man contrbuton to the change n solvaton free energy whle the on s gettng from the nfntely dlute solu- ton (ε w to a concentrated soluton (ε(c s purely electrostatc. Both solutons are hghdensty and crowded, so entropc terms may cancel. The change n the delectrc envronment s descrbed by Eq. 3 seemngly well. The solvaton free energy, G s, establshes the ampltude of the change, whle ε(c establshes the c-dependence. The problem wth the procedure of SL s that ths decouplng s not made n ther work so they used the Paulng rad n the term too. The same thng was done by Abbas et al. 1 SL cte the dscouragng opnon of Fawcett and Tkanen. 17 The falure of the works of Fawcett and Tkanen 4,17,18 s due to the fact that they, smlarly to earler 5

6 Table 1: Expermental parameters of ons studed n ths work: the valence, z, the Paulng radus, R, the hydraton Gbbs free energy, 3 G s, and the Born radus, RB. Ion z R /Å G s /kjmol 1 R B /Å Na Ca Cl works, 19 1 gnored the term. Inchekel et al., 13 on the other hand, used the Born rad (they used a complex equaton of state to estmate the term. Lu and Esenberg 14 used a concentraton-dependent Born radus n ther Posson-Ferm theory. Usng the ndvdual onc rad n Eq. (or, equvalently, n Eq. 7 makes t possble to compute the ndvdual onc actvty coeffcents from the EDH theory. Fgures 3 and 4 show the results for NaCl and CaCl, respectvely. As far as the expermental data 16 are relable (ths queston s strongly debated n the lterature; see the dscusson n our earler paper 1 the agreement wth them s qute good. Results for µ, obtaned from Eq. 11 are found n the SI. Two man conclusons can be drawn from Fgs (1 The curves computed wth the Paulng rad are off, whch justfes the proposal of computng the term usng purely expermental data (Eq. 3. ( The estmate of the EDH theory for the term (agreement of red sold and dashed curves s surprsngly good. Ths s especally stunnng for the :1 system. The EDH theory s a mean feld theory for pont ons, namely, t gnores electrostatcs correlatons between ons and, partly, volume excluson effects between ons n the onc cloud, whle onc sze can be bult nto the theory wth dstance parameters between the central on and the ons around t. The explanaton of the relatvely good agreement s probably cancellaton of errors from gnorng these two effects. 4 The EDH results are smaller than the GCMC results for NaCl, whle the opposte trend s observed for CaCl. We hypothesze that n the case of NaCl the mssng hard sphere correlatons (that are postve domnate, whle n the case of CaCl the mssng onc correlatons n Ca + Cl Cl trplets (that are negatve domnate. In any case, the EDH theory used together wth Eq. 3 to compute the term seems lke an attractve tool to estmate actvty coeffcents of electrolytes. We obtaned results for other electrolytes (data are avalable from the authors smlar to NaCl and CaCl. All these data have been obtaned wthout usng any adjustable parameter (a fact that we cannot emphasze enough. SL proposed a modest parameter adjustment by keepng parameter a = R + + R (that appears n the term and adjusted the mean radus by R ± = ηa, where η s a fttng parameter. They obtaned that η =.7 and.8 gve good results for NaCl and KCl, respectvely. Such a fttng procedure was never our ntenton, 8 but f someone wants to ft, we would rather suggest the followng procedure: use the Born rad n Eq. 7 (or, equvalently, use Eq. 3 and rather ft n the term playng wth the value of a that can be dfferent for the two ons. Adjustable dstance parameters can be bult nto the DH theory wth whch vrtually perfect agreement wth experments can be acheved (see the crtcal assessment of varous modfcatons of the DH theory n the work of Fraenkel 5. Supportng Informaton The Supportng Informaton s avalable free on the ACS Publcatons webste at It contans the dervaton and dscusson of route 6

7 wth results for NaCl and CaCl. Acknowledgement The authors acknowledge the support of the Hungaran Natonal Research Fund (OTKA NN11357 n the framework of ERA Chemstry. The fnancal and nfrastructural support of the State of Hungary and the European Unon n the frame of the TÁMOP-4...B- 15/1/KONV-15-4 project s gratefully acknowledged. We are grateful for the nsprng dscussons wth Doug Henderson, Drk Gllespe, and Chrs Outhwate. References (1 Shlov, I. Y.; Lyashchenko, A. K. The Role of Concentraton Dependent Statc Permttvty of Electrolyte Solutons n the Debye Hückel Theory. J. Phys. Chem. B 15, n press. ( Hasted, J. B.; Roderck, G. W. Delectrc propertes of aqueous and alcoholc electrolytc solutons. J. Chem. Phys. 1958, 9, (3 Barthel, J.; Schmthals, F.; Behret, H. Untersuchungen zur Dsperson der Komplexen Delektrztätskonstante Wäßrger und Nchtwäßrger Elektrolytlösungen. Z. Phys. Chem. 197, 71, (4 Tkanen, A. C.; Fawcett, W. R. Role of solvent permttvty n estmaton of electrolyte actvty coeffcents for systems wth on parng on the bass of the Mean Sphercal Approxmaton. Ber. Bunsenges. Phys. Chem. 1996, 1, (5 Buchner, R.; Hefter, G. T.; May, P. M. Delectrc relaxaton of aqueous NaCl solutons. J. Phys. Chem. A 1999, 13, 1 9. (6 Hückel, E. Zur Theore konzentrerterer wässerger Lösungen starker Elektrolyte. Phys. Z. 195, 6, (7 Vncze, J.; Valskó, M.; Boda, D. The nonmonotonc concentraton dependence of the mean actvty coeffcent of electrolytes s a result of a balance between solvaton and on-on correlatons. J. Chem. Phys. 1, 133, (8 Vncze, J.; Valskó, M.; Boda, D. Response to Comment on The nonmonotonc concentraton dependence of the mean actvty coeffcent of electrolytes s a result of a balance between solvaton and on-on correlatons [J. Chem. Phys. 134, (11]. J. Chem. Phys. 11, 134, (9 Valskó, M.; Boda, D. The effect of concentraton- and temperaturedependent delectrc constant on the actvty coeffcent of NaCl electrolyte solutons. J. Chem. Phys. 14, 14, (1 Valskó, M.; Boda, D. Unravelng the behavor of the ndvdual onc actvty coeffcents on the bass of the balance of on-on and on-water nteractons. J. Phys. Chem. B 15, 119, (11 Born, M. Volumen und Hydratatonswarme der Ionen. Z. Phys. 19, 1, (1 Abbas, Z.; Ahlberg, E.; Nordholm, S. From restrcted towards realstc models of salt solutons: Corrected Debye- Hückel theory and Monte Carlo smulatons. Flud Phase Equlb. 7, 6, (13 Inchekel, R.; de Hemptnne, J.-C.; Fürst, W. The smultaneous representaton of delectrc constant, volume and actvty coeffcents usng an electrolyte equaton of state. Flud Phase Equlb. 8, 71, (14 Lu, J.-L.; Esenberg, B. Posson Ferm model of sngle on actvtes n aqueous 7

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