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The activity of individual ions. A conceptual discussion of the relation between the theory and the experimentally measured values
Fluid Phase Equilibria 312 (2011) 79–84
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The activity of individual ions. A conceptual discussion of the relation between the theory and the experimentally measured values Grazyna Wilczek-Vera a , Juan H. Vera b,∗ a b
Retired academic, Department of Chemistry, McGill University, Montreal, Canada Retired academic, Department of Chemical Engineering, McGill University, Montreal, H3A 2B2, Canada
a r t i c l e
i n f o
Article history: Received 19 July 2011 Received in revised form 6 September 2011 Accepted 8 September 2011 Available online 17 September 2011 Keywords: Ionic activities Electrolyte solutions Ion-selective-electrodes Electrochemical cells with liquid junction Junction potential
a b s t r a c t The difference between the definition of the chemical potential in mathematical terms and its evaluation from experimental data is emphasized. A detailed analysis of the treatments used for mixtures of non electrolytes is used as a guide for the understanding of the treatment required for electrolyte systems. In both cases, the activity coefficients are calculated from data obtained in isothermal closed systems. The perfect analogy between these two cases is demonstrated. Also in both cases, the conditions required for the interpretation of the chemical potential as partial molar property cannot be satisfied experimentally in the measurements. The solution to this apparent impasse is based on a clear distinction between the mathematical and the physical worlds of thermodynamics. Answers are given to the main arguments against the measurement of individual ionic activities. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Recent publications [1–3] have enriched the debate concerning the measurability or non-measurability of individual ionic activities. This discussion is not new. In a previous publication we have given details of its historical background [4]. In the present stage of this constructive debate, two main arguments are still standing in support of the paradigm of the impossibility of measuring these important properties of ions. Malatesta [5–7] has strongly argued that a publication by Taylor [8] demonstrated that the voltage response of ion-selective-electrodes is not a function of the individual activity of the ion but it is a function of the mean ionic activity coefficient of the electrolyte and the transference number of the particular ion. More recently, Zarubin [1] wrote that this paradigm is a “natural sequel to the definition of partial molar quantities which falls into contradiction with the condition of electrical neutrality when applied to ions in electrolyte solutions.” Thus, the main obstacle in reaching an understanding seems to be that the discussion is centered in the end result before clarifying the governing concepts. We realize now that the wording of our previous note [2] needs some further elaboration. In order to gain perspective, it is necessary to take a step back and make clear the connection between different concepts used for the treatment of electrolyte solutions. In doing so, thanks to the work of Zarubin [1], we also provide an additional answer to Malatesta’s queries.
∗ Corresponding author. Tel.: +1 514 398 4274; fax: +1 514 398 6678. E-mail address: [email protected] (J.H. Vera). 0378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2011.09.009
As discussed by Prausnitz et al. [9], one interesting and perhaps at times confusing aspect of the thermodynamics of mixtures is that it solves a physical problem using a mathematical formalism. The problem is to determine the equilibrium conditions of reacting or non-reacting phases. As the experimental determination of the variables for each possible case is impractical, the physical problem is translated into a mathematical problem. A mathematical solution is found and then this solution is expressed in physically meaningful terms. The key step is to differentiate the concepts used in the mathematical world from the elements operating in the physical world. Lewis and Randall [10] compared thermodynamics with a cathedral build with the efforts of a few architects and many workers. The architects traced the lines of the conceptual design. Then, following the indications of a few constructors, the many workers put in place the physical elements visible in the final product. While in the field of non electrolyte systems the distinction between the architects’ design, the builders’ craft and the workers’ toil is clear, in the treatment of electrolyte solutions there seem to be a confusion that has stopped the development of the field for more than eighty years. In this work, we use what we have learned from the treatment of non electrolyte systems to better understand the relation between theory and practice in the field of electrolyte solutions. 2. The mathematical abstraction In classical thermodynamics of phase equilibrium, the architect was Gibbs [11], the constructors indicating the steps for building the final structure were Lewis and Randall [10], and we, the
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Nomenclature Notation A a E F G H i, j m n P R S S T t V xi yi zi
Helmholtz function (J mol−1 ) activity potential (V) Faraday constant (C mol−1 ) Gibbs function (J mol−1 ) enthalpy (J mol−1 ) ionic or non ionic species molality (mol kg−1 ) number of moles pressure (Pa) gas constant (J K−1 mol−1 ) slope of the ISE potential reading versus the logarithm of the ion activity (V/decade) entropy (J K−1 mol−1 ) temperature (K) transference number total volume of the system (m3 ) liquid phase mole fraction of species i vapour phase mole fraction of species i charge of ion i
Greek letters activity coefficient ϕ fugacity coefficient refers to the standard state
constant value. The particular case in which the variables kept constant are the pressure, the temperature and the number of moles of all other compounds, while changing the number of moles of species i, is recognized as the partial molar contribution of species i to the total extensive property in question. Thus, in the ‘mathematical world’ the chemical potential is perfectly well defined for a particular geometric surface and it is the partial molar contribution of species i to the total Gibbs energy of the phase. In this mathematical world, Gibbs found the general solution to equilibrium problems. For phases in equilibrium, Gibbs stated [11] that the chemical potential for a species able to transfer from one phase to another has the same value in all phases in which the species is present. For chemical equilibrium in a single closed phase, De Donder [12] obtained that the Gibbs energy of the stoichiometric amounts of reactants is equal to the Gibbs energy of the stoichiometric amounts of products. At this point, we make a short digression. As discussed in the text of Prausnitz et al. [9], the chemical potential of species i can also be expressed as a partial derivative of other extensive thermodynamic properties with respect to the change of the number of moles of the compound, keeping fixed at a constant value the number of moles of all other compounds and two additional variables. From the possible choices of the internal energy, U, the enthalpy, H, or the Helmholtz function, A, the most interesting is the latter, as it opens the field to modeling with equations of state and also to computer simulations. In this case we write, A = U − TS
(3)
and the chemical potential is given by: Subscripts 0 refers to a constant potential + cation anion − ± mean ionic ionic or non ionic species i, j J liquid junction refers to sample solution at dimensionless molality k mk r refers to the filling solution in the reference electrode Superscripts E excess property L liquid phase property s property at saturation
workers, are still discussing their instructions even today. Gibbs did the two first steps necessary for the thermodynamic treatment of phase and chemical reaction equilibria. He made the abstraction of the physical problem of equilibrium into the world of mathematics and he also found the mathematical solution for the case of phase equilibrium. The mathematical solution was later extended to chemical equilibrium by De Donder [12]. Briefly stated, the Gibbs function can be written as: G = H − TS =
nj j
(1)
where i is the chemical potential that, for this purpose, is defined as
i =
∂G ∂ni
(2) T,P,nj = / i
In the mathematical framework, once a surface is defined it is possible to take partial derivatives with respect to any of its defining independent variables, keeping all the rest of the variables at a
i =
∂A ∂ni
(4) T,V,nj = / i
We observe that the chemical potential is not the partial molar contribution of species i to the value of the total Helmholtz function of the system as the total volume, instead of the pressure, is kept constant in the differentiation in the mathematical world. We reiterate that the definition of partial molar property requires that the pressure, the temperature and the number of moles of all other compounds are kept constant while changing the number of moles of species i. This is an important observation as we will see below. The fact is that Eq. (4) does not obey the conditions of partial molar property but it is potentially useful for testing the experimental results for individual ionic activities by means of computer simulations. 3. The step back to the physical world The mathematical solutions obtained by Gibbs to the problems of phase equilibrium are general and independent of the way we choose to evaluate the chemical potentials in the physical world. The confusion comes when one considers Eqs. (2) or (4) as part of the implementation of the mathematical solutions in the real world. For electrolyte systems, the condition of electroneutrality does not permit to change the number of moles of a single ion while keeping the number of moles of all other ions constant. Thus, considering equations of the type of (2) or (4) at this stage may wrongly suggest that the mathematical definition of chemical potential imposes a theoretically based impediment for the evaluation of the single ion chemical potentials. This is a self imposed limitation that has no theoretical support. On the other hand, for non electrolyte systems the number of moles of each species can be changed independently and this particular point does not create a problem of interpretation. However, as we discuss below, even for systems of
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non electrolytes one can find a situation in which the theory seems to conflict with reality. For clarity, we repeat that in the mathematical space all derivatives are permissible and the mathematical solution to the mathematical problem is general. This being the case, the next step is then to pass from the mathematical solution, obtained by Gibbs, to the physical solution of the equilibrium problem, i.e., to find the values of the dependent variables of the system in equilibrium. For this crucial step Lewis and Randall [10,13] introduced the concept of activity of species i, ai . Their idea was that the chemical potential of species i in a real mixture would have the same functional form as the chemical potential of the same species in an ideal gas mixture. Thus, they defined the activity of species i by the relation: i = i + RT ln(a1 (i, ))
(5)
For the case of a system formed by non electrolytes, it is usual to express the compositions in mole fractions. As this type of systems does not to have problems of interpretation, we concentrate here in the case of systems formed by electrolytes and a solvent. In this case, the composition is usually expressed in terms of molality of the ions present, i.e., moles of i per kg of solvent, and according to the definition introduced by Lewis and Randall [10], the activity of species i is related to its molality by: ai, ≡ i
mi m
(6)
In this identity, which in fact defines the activity coefficient of species i, i , the term m is the molality of the standard state of species i, equal to 1 mol kg−1 . Thus, the term i in Eq. (5) corresponds to the chemical potential of species i in ideal solution, i = 1, at its standard state composition of one mol per kg of solvent. This is clearly a hypothetical state of species i which has no physical counterpart in the physical world. For ionic species, the activity coefficient is normalized to unity at the physically meaningful reference state of infinite dilution of species i in the solvent. A step forward for the discussion of the measurability of ionic activities is based on a clear distinction between the nature of Eqs. (2) and (5), which belong to the two different worlds of thermodynamics. Erwin Schrödinger, Nobel Prize in Physics, presented the general case of the potential conflict between theory and practice in “The Tarner Lectures on Mind and Matter”, delivered at Trinity College, Cambridge, in 1956. We quote [14]: “Scientific theories serve to facilitate the survey of our observations and experimental findings. Every scientist knows how difficult it is to remember a moderately extended group of facts, before at least some primitive theoretical picture about them has shaped. It is therefore no wonder, and by no means to be blamed on the authors of original papers or of textbooks, that after a reasonable coherent theory has been formed, they do not describe the bare facts they have found or wish to convey to the reader, but clothe them in the terminology of that theory or theories. This procedure, while useful for our remembering the facts in a wellordered pattern tends to obliterate the distinction between the actual observation and the theory arisen from them.” The real problem of thermodynamics is that the theory formulated by Gibbs, the architect, is not just a primitive picture but it is a complete and forceful framework that sometimes tends to replace the facts. 4. Measurement of the activity of individual ions As indicated by Eq. (6), the determination of the activity of individual ions is reduced to the experimental measurement of the activity coefficient i . In a recent publication [15], we have shown that there is a perfect analogy in the way of experimentally ‘measuring’ activity coefficients for systems of non electrolytes and for the activity coefficients of individual ions in systems of electrolytes. Actually, in both cases these activity coefficients are not measured
81
directly but they are back-calculated from measurements of related data in a closed isothermal system. Nor the temperature, nor the number of moles of species i, or of any of the species, change during the measuring of a particular experimental point. For systems of non electrolytes, the activity coefficients are usually obtained from measurement of the total pressure P and the liquid and vapor phase mole fractions of species i, xi and yi , respectively, in equilibrium at a fixed temperature T. From the equality of the chemical potential for species i in a mixture in liquid–vapor equilibrium, we have [9]:
⎡ ⎤ P L v dP ⎢ ⎥ i xi i Pis ϕis exp ⎣ ⎦ = yi Pϕi RT
(7)
Ps i
In this equation, Pis , ϕis and vLi are the vapor pressure, the fugacity coefficient and liquid volume of species i at saturation at the temperature T of the system, while ϕi is the fugacity coefficient of species i in the gas mixture. Rearranging Eq. (7) we write
ln(xi i ) = ln
yi P Pis
⎡
⎢
− ⎣ln
ϕis ϕi
⎤
P vLi
+
dP ⎥ ⎦ RT
(7a)
Ps i
For the determination of the activity of ions in aqueous single electrolyte solutions, the voltage response Ei of an ion-selectiveelectrode (ISE), sensitive to ion i, is measured against a single junction reference electrode, both electrodes being immersed in a sample solution containing a single electrolyte at a dimensionless molal concentration (mi /m ). This voltage response is related to the activity ai of the ionic species i, given by Eq. (6), by:
m i
Ei = Ei,0 + Si ln i
m
+ EJ
or, rearranging ln
m
i m
i
1 = (Ei − Ei,0 ) − Si
(8)
EJ Si
(8a)
In these equations, Ei,0 is a constant value for limited time spans and Si is the slope of the voltage response of the ISE, which for a Nernstian electrode is equal to RT/zi F, where F is the Faraday constant and zi is the charge of the ion. The term EJ is the junction potential, i.e., the value of a difference in potential between the sample solution and the standard solution contained in the reference electrode. In both Eqs. (7a) and (8a), the first term of the right hand side gives the main contribution to the value of the activity of species i. The terms enclosed in square brackets in the right hand side of these equations are corrections introduced to improve the accuracy of the activity back-calculated from the experimental data. The evaluation of these correction terms may require making some reasonable assumptions. In the case of mixtures of non electrolytes under isothermal conditions, the correction term is normally estimated using empirical equations of state with empirical mixing rules for the mixture. For the liquid molar volume of pure compound i, iL , the usual assumption is to consider the liquid to be incompressible and to evaluate the liquid molar volume from the liquid density at the temperature T of the system. For the determination of the activity of individual ions, the situation is still more favorable as it has been shown that the method used to reduce the data largely cancels the effect of the correction term [4,15–19]. In addition, for the case of the activity of individual ions it is possible to implement an iteration procedure between independent equations and evaluate the correction term [4,15]. Again, the change in the values of the ionic activities obtained after an iteration procedure is not significant.
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It is important to note that for systems of non electrolytes, the activity of species i depends on the nature of any other species present in the system. Similarly, for systems of electrolytes, the activity of species i depends on the nature of the solvent and on the nature of any other ion or other non ionic species present in the system. For a single electrolyte in a particular solvent, the individual activity of an ion i depends on the nature of the counter ion present in the system. 5. Methods used to treat the data For the treatment of the raw data collected for systems of non electrolytes, the individual activities, measured independently for the different species, are normally correlated with a parametric expression for the excess Gibbs energy of the mixture [9]. The values of the individual activity coefficients of the components of the mixture are then obtained from the analytical excess Gibbs energy form by partial differentiation 1 lni = RT
∂GE ∂ni
(9) T,P,nj = / i
This method ‘smoothes’ the data and it guarantees that the values of the individual activities satisfy the Gibbs–Duhem equation for the isothermal incompressible liquid phase. The method of partial differentiation of an excess Gibbs energy function was used by Pitzer [20] to obtain expressions for ionic activities from mean ionic activity data, without any consideration of the transport numbers of the ions. In a previous publication [21], we have argued that this way of generating individual ionic activities from mean ionic activities, seems inappropriate. In addition to ignoring the transference numbers of the ions, the algebra generated by this treatment is far from simple [22]. For the thermodynamic treatment of the raw data collected for systems of electrolytes, we have suggested [20] correlating the individual activities, measured independently for the different ionic species, and using the equation of Lin and Lee [23] for the calculation of the activity of water. Again, this method ‘smoothes’ the data and it guarantees that the values of the individual activities satisfy the Gibbs–Duhem equation for isothermal incompressible liquid phase. In addition, considering the state of affairs in the discussion of the measurability of individual ionic activities, for the reduction of data we prefer to avoid any step that could appear as violating the electroneutrality condition. At this stage, the use of expressions of the type of Eq. (9) could contribute to the misunderstanding regarding the need to ‘move in or out of the solution’ a single ionic species i in order to determine its individual activity. 6. Discussion and conclusions Notably, as discussed in detail elsewhere [15] and reiterated here, there is a perfect parallel between the experimental methods used for the determination of activities of ionic and non ionic species. In both cases the isothermal system is closed, i.e., there is no change in the temperature or in the number of moles of any of the species involved. In addition, in both cases once the data are collected one may or may not introduce the minor corrections shown in Eqs. (7a) and (8a) with the purpose of improving the accuracy of the results. Again, in both cases, experience has shown that the correction terms do not affect the final values significantly [15]. One additional point of similarity is that the activity coefficient of an individual ion depends on the nature of the counter-ion present in the system, exactly as the activity coefficient of ethanol depends if this compound is in a mixture with water or with methanol. Thus, one can have the legitimate question of why there is such an opposition to the possibility of measuring the individual ionic activities.
It seems to us that there are two potential reasons. One originates from a consideration of the independent equation for the junction potential EJ [16] RT EJ = − F j
k
tj zj
dln(mj j )
(10)
r
In this equation, the summation runs over all ions present in the solution in the reference electrode r and in the sample solution k, and tj is the transference number of ionic species j with charge zj , for which there is independent information in the literature. Taylor [8] demonstrated that by combining Eq. (8) with Eq. (10), the voltage response of a cell with liquid junction, Ei could be expressed as a function of the mean ionic activity of the electrolyte and the transference number of ion i. In a more recent publication [17], we have presented a simpler derivation of Taylor’s result. Combining the two independent Eqs. (8) and (10) one eliminates the variable EJ and obtains an expression for Ei in terms of the mean ionic activity of the electrolyte and the transference number of ion i. In our example [4,18], we discussed the case of the homoionic system of a sample solution of KCl in which the voltage response of a chloride ion-selective-electrode (i = −) was measured against a reference electrode containing a concentrated solution of KCl. In this case, the right hand side of Eq. (10) has only two terms and we obtained:
E
,k
=
E0,− −
RT ln a F
,r
+ S +
RT F
ln˛
,k
−
2RT F
(1 − t )d lna±
(11)
r
In this equation, a± is the mean ionic activity of the electrolyte and for a 1:1 electrolyte, (m+ = m− = m). For a 1:1 electrolyte, it takes the form: a2± = a+ a = (m+ + )(m ) = m2 ±2
(12)
where ± is the mean ionic activity coefficient of the electrolyte and all other symbols have their usual meanings [4,18]. Mutatis mutandi, a similar expression can be obtained for the voltage response of the potassium ion-selective-electrode. If the electrolyte in the sample is different from the electrolyte in the reference electrode solution, the expression for the voltage response of each ISE is also more complex [16], but it is formally similar to Eq. (11). The first term in square brackets of the right hand side of Eq. (11) is a constant for an experimental run. The second term cancels out for a perfectly Nernstian ISE as the slope for the response of the anion is negative. For common commercial chloride ISE’s, which are very closely Nernstian, this term is negligible. The third term of the right hand side shows that the voltage response of the ISE is in fact a function of the mean ionic activity coefficient of the electrolyte and the transference number of the ion. The explicit form of this dependence is not simple as the transference numbers are function of composition. Even for a homoionic system, there is a composition change through the liquid junction. The main limitation for the calculation of this term is that information of transference numbers is, in general, scarce and almost nonexistent for concentrated solutions [18]. However, even when all that it has been attained with the combination of Eqs. (8) and (10) is to hide the activity of the individual ion in the constant term of Eq. (11), there is no question about the fact that the voltage response of an ISE can be expressed as a function of the mean ionic activity coefficient of the electrolyte and the transference number of the ion. This conclusion can be attained without the unnecessary assumptions made by Zarubin [1]. Once the temperature and composition of the system are fixed, the values of all dependent variables such as mean ionic activity coefficient, activities of individual ions, density, refractive index, viscosity, etc., are also fixed and one can always express one of the variables in terms of others. Hence, in our view, the recent
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publication by Zarubin [1] provides a missing link and serves the purpose of answering also Malatesta’s arguments [24] in support of Taylor’s results. Zarubin reasoned that if the voltage response of an ion-selective-electrode can be shown to be a function of the activity of the individual ion and also it can be shown that it is a function of the mean ionic activity coefficient of the electrolyte and the transference number of ion i [1], then there must exist a relation of the form i = i (± , ti )
(13)
The left hand side of this equation is the value of the experimentally measured activity coefficient of the individual ion i, as it appears in Eq. (8), and the right hand side represents a function for which Zarubin with some simplifying assumptions obtains a value from a theoretical treatment of each type of electrochemical cell. There is an impressive agreement between the measured and the calculated values. One should note that the calculated values from Zarubin’s functions use experimental data for transport numbers and mean ionic activity coefficients that include their own experimental errors. Uncertainties in the calculated results are probably due more to uncertainty in the former than in the latter of these two pieces of experimental information. The agreement between the experimental values of i evaluated with Eq. (8) and the values obtained with the function of the right hand side of Eq. (13) should be treated as an example of the coherence of the approach used instead of interpreting it as a proof of nonexistence of individual activities of ions. The individual activity coefficients of ions can be expressed as functions of different properties of the system. If two parameters are used and one of them is representative of the system, as it is the case of the mean ionic activity coefficient, the other should necessary be a property characteristic of the individual ion, such as its transference number. This functionality does not make the activity of individual ions “conventional”, “non-existent” or “so called”, as they were denoted by Zarubin [1]. At this point the only standing objection to the possibility of measuring activity coefficients for the individual ions is that, as clearly stated by Zarubin [1], in the physical world their mathematical interpretation as partial molar quantities “falls into contradiction with the condition of electrical neutrality when applied to ions in electrolyte solutions”. For systems formed by non electrolytes, it would appear that one could picture the chemical potential or the activity coefficient of species i in the real world because one can change the number of moles of species i, while keeping the pressure, the temperature and the number of moles of all other species constant. This is just a mirage. It assumes that one can picture in the physical world the Gibbs energy, which is just an abstract entity in the mathematical world. An evidence of this mirage can be seen by considering a binary two-phase system of non electrolytes at equilibrium. For these systems, according to the Phase Rule, there are two degrees of freedom. Thus, if the temperature is fixed there is only one degree of freedom left and it is not possible to change the moles of one compound and keep the pressure constant. Similarly for isobaric measurements, if the pressure is fixed the temperature cannot be kept constant. This fact certainly does not affect the mathematical interpretation of the chemical potential as the change in Gibbs energy of a phase with the change of the number of moles of one single compound at constant pressure and temperature. Notably, these binary systems in which a vapor and a liquid phase coexist at equilibrium, either at a fixed temperature or at a fixed pressure, are exactly the type of systems used to evaluate the activity coefficients in systems of non electrolytes. Although Eq. (7) originates from the equality of the chemical potentials of species i in a mixture in vapor–liquid equilibrium, nobody ever objected its use arguing that one cannot change the number of moles of one compound at a time while
83
keeping the pressure, the temperature and the number of moles of all the other compounds constant. The fact is that, as we discussed previously [15], for the experimental determination of the activity coefficients of both non ionic and ionic species, it is not necessary to change the number of moles of the compound in question. In both cases, the activity coefficients are evaluated from measurements in closed systems in thermal equilibrium. Moreover, the similitude between the case of measurement of activities in systems of electrolytes and in systems of non electrolytes is complete if one considers the negative argument advanced in this study for the latter case. In both cases, it is not possible to keep constant the pressure, the temperature and the number of moles of all other compounds while changing the number of moles of species i. In other words, in both cases it is impossible to satisfy experimentally the conditions required by the right hand side of Eq. (2). Again here, this was never a cause for confusion in the case of systems of non electrolytes and it is about time to understand that this is not an impediment for measuring the activity of individual ions. It is important to realize that the misleading step is to put the wrong question or even to put the right question in the wrong world. This brings to mind the old trick for an undergraduate teasetest: “(A) Name three organic and three inorganic acids. (B) Name three organic and three inorganic alcohols.” In this context, the objections to the measurability of ionic activities are the ones that should be questioned. One must assume that well known scientists understood properly the concept of partial molar properties and the conditions of equilibrium obtained by Gibbs. Lewis and Randall [10], Pitzer and Brewer [20]; Harned [25] among others, all clearly stated that the activity of individual ions was experimentally obtainable if we could, in any way, calculate or estimate the effect of the junction potential in a cell with junction. Even Guggenheim [26] suggested the possibility of an iteration procedure between Eqs. (8) and (10) but without the availability of computer he declared this iteration to be a “vicious circle”. Few people would dare saying that these scientists did not understand the concepts involved. They obviously knew thermodynamics well and had a clear picture of the difference between the mathematical and the physical worlds. There is of course the question of how well the results from the theory agree with experimental phase equilibrium data. The fact is that the equality of the chemical potentials of a species present in phases in direct contact is only the starting point to represent reality. The result obtained from the mathematical framework needs the help of additional definitions of ideal system behaviour. Moreover, as the algebra for ideal systems gives only the functional dependence of the chemical potential on measurable variables, the representation of real data needs the introduction of fugacity and activity coefficients. These activity coefficients back-calculated from the experimental data are precisely the subject discussed in this manuscript. In special cases it is necessary to go still one step further and pay special attention to the selection of the proper standard state for the definition of the activity coefficients. Although this is not a problem for systems of electrolytes in a single solvent, it is important for the case of electrolytes in mixed solvents. For systems of non electrolytes, under isobaric conditions if the standard state has been chosen as the pure component liquid at the pressure and temperature of the system, the lighter compound does not physically exist under these conditions. This problem has been clearly discussed in the literature and a solution has been recommended [27]. To conclude, we observe that Zarubin [1] recently noted that experimental measurement of the voltage response of ISEs were not available for certain aqueous electrolyte systems of interest for testing his new treatment of the data. The fact is that any work reporting new ISE voltage response data, or for this purpose any
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attempt to advance our knowledge in the field of electrolytes, is not welcome. This prejudiced attitude discourages researchers to contribute to this field. It is time to consider with special attention the positive views of Lewis, Randall, Pitzer, Brewer, Harned and even Guggenheim, among others, supporting the measurability of individual ionic activities. Otherwise, this field will be stagnant for another hundred years while, in the mean time, the development of biotechnology will require the use of individual ionic activities for the treatment of Donnan equilibrium. The participation of young researchers gathering new experimental data and contributing fresh ideas should be encouraged not discouraged. Errors have been made and will be made. We should all participate in an open discussion of ideas keeping in mind the unsolicited advice that Frank Wilczek, Nobel Prize in Physics, gives to his graduate students [28]: “If you do not make mistakes, you’re not working on hard enough problems. And that’s a big mistake”. Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC) and to McGill University for financial support. References [1] D.P. Zarubin, J. Chem. Thermodyn. 43 (8) (2011) 1135–1152. [2] J.H. Vera, G. Wilczek-Vera, J. Chem. Thermodyn., to be published.
[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]
D.P. Zarubin, J. Chem. Thermodyn., to be published. G. Wilczek-Vera, J.H. Vera, Fluid Phase Equilib. 236 (2005) 96–110. F. Malatesta, Fluid Phase Equilib. 295 (2010) 244–248. F. Malatesta, Chem. Eng. Sci. 65 (2010) 675–679. F. Malatesta, J. Chem. Eng. Data 5 (2009) 2978. P.B. Taylor, J. Phys. Chem. 31 (1927) 1478–1500. J.M. Prausnitz, R.M. Lichtenthaler, A.E. de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria, third ed., Prentice Hall, 1999. G.N. Lewis, M. Randall, Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill, N.Y., 1923. J.W. Gibbs, in: J.W Gibbs, W.R. Longley, R.G. Van Name (Eds.), The Collected Works, Yale University Press, New Haven, reprinted 1957. T. De Donder, Thermodynamic Theory of Affinity: A Book of Principles, Oxford University Press, Oxford (England), 1936. G.N. Lewis, M. Randall, J. Am. Chem. Soc. 43 (1921) 1112–1154. E. Schrödinger, What is Life? With Mind and Matter and Autobiographical Sketches, Canto, Cambridge University Press, N.Y., 2010. G. Wilczek-Vera, J.H. Vera, Chem. Eng. Sci. 66 (2011) 3782–3791. G. Wilczek-Vera, E. Rodil, J.H. Vera, AIChE J. 50 (2004) 445–462. E. Rodil, J.H. Vera, Fluid Phase Equilib. 205 (2003) 115–132 (211 (2003) 289). G. Wilczek-Vera, E. Rodil, J.H. Vera, Fluid Phase Equilib. 244 (2006) 33–45. G. Wilczek-Vera, E. Rodil, J.H. Vera, Fluid Phase Equilib. 241 (2006) 59–69. K.S. Pitzer, L. Brewer, in: M. Randall, G.N. Lewis (Eds.), Thermodynamics, 2nd ed., McGraw-Hill, NY, 1961, revised. G. Wilczek-Vera, J.H. Vera, Ind. Eng. Chem. Res. 48 (2009) 6436–6440. E. Cheluget, G. Wilczek-Vera, J.H. Vera, Can. J. Chem. Eng. 70 (1992) 313–319. C.-l. Lin, L.-s. Lee, Fluid Phase Equilib. 205 (2003) 69–88. F. Malatesta, J. Chem. Eng. Data 54 (2009) 2978. H.S. Harned, J. Phys. Chem. 30 (1926) 433–456. E.A. Guggenheim, J. Phys. Chem. 33 (1929) 842–849. J. Wisniak, H. Segura, Ind. Eng. Chem. Res. 47 (2008) 5758–5760, Erratum: Ind. Eng. Chem. Res., 48 (2008) 2738. F. Wilczek, B. Devine, Longing for the Harmonies. Themes and variations from Modern Physics, W.W. Norton & Company, N.Y. and London, 1987.
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The activity of individual ions. A conceptual discussion of the relation between the theory and the experimentally measured values Grazyna Wilczek-Vera a , Juan H. Vera b,∗ a b
Retired academic, Department of Chemistry, McGill University, Montreal, Canada Retired academic, Department of Chemical Engineering, McGill University, Montreal, H3A 2B2, Canada
a r t i c l e
i n f o
Article history: Received 19 July 2011 Received in revised form 6 September 2011 Accepted 8 September 2011 Available online 17 September 2011 Keywords: Ionic activities Electrolyte solutions Ion-selective-electrodes Electrochemical cells with liquid junction Junction potential
a b s t r a c t The difference between the definition of the chemical potential in mathematical terms and its evaluation from experimental data is emphasized. A detailed analysis of the treatments used for mixtures of non electrolytes is used as a guide for the understanding of the treatment required for electrolyte systems. In both cases, the activity coefficients are calculated from data obtained in isothermal closed systems. The perfect analogy between these two cases is demonstrated. Also in both cases, the conditions required for the interpretation of the chemical potential as partial molar property cannot be satisfied experimentally in the measurements. The solution to this apparent impasse is based on a clear distinction between the mathematical and the physical worlds of thermodynamics. Answers are given to the main arguments against the measurement of individual ionic activities. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Recent publications [1–3] have enriched the debate concerning the measurability or non-measurability of individual ionic activities. This discussion is not new. In a previous publication we have given details of its historical background [4]. In the present stage of this constructive debate, two main arguments are still standing in support of the paradigm of the impossibility of measuring these important properties of ions. Malatesta [5–7] has strongly argued that a publication by Taylor [8] demonstrated that the voltage response of ion-selective-electrodes is not a function of the individual activity of the ion but it is a function of the mean ionic activity coefficient of the electrolyte and the transference number of the particular ion. More recently, Zarubin [1] wrote that this paradigm is a “natural sequel to the definition of partial molar quantities which falls into contradiction with the condition of electrical neutrality when applied to ions in electrolyte solutions.” Thus, the main obstacle in reaching an understanding seems to be that the discussion is centered in the end result before clarifying the governing concepts. We realize now that the wording of our previous note [2] needs some further elaboration. In order to gain perspective, it is necessary to take a step back and make clear the connection between different concepts used for the treatment of electrolyte solutions. In doing so, thanks to the work of Zarubin [1], we also provide an additional answer to Malatesta’s queries.
∗ Corresponding author. Tel.: +1 514 398 4274; fax: +1 514 398 6678. E-mail address: [email protected] (J.H. Vera). 0378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2011.09.009
As discussed by Prausnitz et al. [9], one interesting and perhaps at times confusing aspect of the thermodynamics of mixtures is that it solves a physical problem using a mathematical formalism. The problem is to determine the equilibrium conditions of reacting or non-reacting phases. As the experimental determination of the variables for each possible case is impractical, the physical problem is translated into a mathematical problem. A mathematical solution is found and then this solution is expressed in physically meaningful terms. The key step is to differentiate the concepts used in the mathematical world from the elements operating in the physical world. Lewis and Randall [10] compared thermodynamics with a cathedral build with the efforts of a few architects and many workers. The architects traced the lines of the conceptual design. Then, following the indications of a few constructors, the many workers put in place the physical elements visible in the final product. While in the field of non electrolyte systems the distinction between the architects’ design, the builders’ craft and the workers’ toil is clear, in the treatment of electrolyte solutions there seem to be a confusion that has stopped the development of the field for more than eighty years. In this work, we use what we have learned from the treatment of non electrolyte systems to better understand the relation between theory and practice in the field of electrolyte solutions. 2. The mathematical abstraction In classical thermodynamics of phase equilibrium, the architect was Gibbs [11], the constructors indicating the steps for building the final structure were Lewis and Randall [10], and we, the
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Nomenclature Notation A a E F G H i, j m n P R S S T t V xi yi zi
Helmholtz function (J mol−1 ) activity potential (V) Faraday constant (C mol−1 ) Gibbs function (J mol−1 ) enthalpy (J mol−1 ) ionic or non ionic species molality (mol kg−1 ) number of moles pressure (Pa) gas constant (J K−1 mol−1 ) slope of the ISE potential reading versus the logarithm of the ion activity (V/decade) entropy (J K−1 mol−1 ) temperature (K) transference number total volume of the system (m3 ) liquid phase mole fraction of species i vapour phase mole fraction of species i charge of ion i
Greek letters activity coefficient ϕ fugacity coefficient refers to the standard state
constant value. The particular case in which the variables kept constant are the pressure, the temperature and the number of moles of all other compounds, while changing the number of moles of species i, is recognized as the partial molar contribution of species i to the total extensive property in question. Thus, in the ‘mathematical world’ the chemical potential is perfectly well defined for a particular geometric surface and it is the partial molar contribution of species i to the total Gibbs energy of the phase. In this mathematical world, Gibbs found the general solution to equilibrium problems. For phases in equilibrium, Gibbs stated [11] that the chemical potential for a species able to transfer from one phase to another has the same value in all phases in which the species is present. For chemical equilibrium in a single closed phase, De Donder [12] obtained that the Gibbs energy of the stoichiometric amounts of reactants is equal to the Gibbs energy of the stoichiometric amounts of products. At this point, we make a short digression. As discussed in the text of Prausnitz et al. [9], the chemical potential of species i can also be expressed as a partial derivative of other extensive thermodynamic properties with respect to the change of the number of moles of the compound, keeping fixed at a constant value the number of moles of all other compounds and two additional variables. From the possible choices of the internal energy, U, the enthalpy, H, or the Helmholtz function, A, the most interesting is the latter, as it opens the field to modeling with equations of state and also to computer simulations. In this case we write, A = U − TS
(3)
and the chemical potential is given by: Subscripts 0 refers to a constant potential + cation anion − ± mean ionic ionic or non ionic species i, j J liquid junction refers to sample solution at dimensionless molality k mk r refers to the filling solution in the reference electrode Superscripts E excess property L liquid phase property s property at saturation
workers, are still discussing their instructions even today. Gibbs did the two first steps necessary for the thermodynamic treatment of phase and chemical reaction equilibria. He made the abstraction of the physical problem of equilibrium into the world of mathematics and he also found the mathematical solution for the case of phase equilibrium. The mathematical solution was later extended to chemical equilibrium by De Donder [12]. Briefly stated, the Gibbs function can be written as: G = H − TS =
nj j
(1)
where i is the chemical potential that, for this purpose, is defined as
i =
∂G ∂ni
(2) T,P,nj = / i
In the mathematical framework, once a surface is defined it is possible to take partial derivatives with respect to any of its defining independent variables, keeping all the rest of the variables at a
i =
∂A ∂ni
(4) T,V,nj = / i
We observe that the chemical potential is not the partial molar contribution of species i to the value of the total Helmholtz function of the system as the total volume, instead of the pressure, is kept constant in the differentiation in the mathematical world. We reiterate that the definition of partial molar property requires that the pressure, the temperature and the number of moles of all other compounds are kept constant while changing the number of moles of species i. This is an important observation as we will see below. The fact is that Eq. (4) does not obey the conditions of partial molar property but it is potentially useful for testing the experimental results for individual ionic activities by means of computer simulations. 3. The step back to the physical world The mathematical solutions obtained by Gibbs to the problems of phase equilibrium are general and independent of the way we choose to evaluate the chemical potentials in the physical world. The confusion comes when one considers Eqs. (2) or (4) as part of the implementation of the mathematical solutions in the real world. For electrolyte systems, the condition of electroneutrality does not permit to change the number of moles of a single ion while keeping the number of moles of all other ions constant. Thus, considering equations of the type of (2) or (4) at this stage may wrongly suggest that the mathematical definition of chemical potential imposes a theoretically based impediment for the evaluation of the single ion chemical potentials. This is a self imposed limitation that has no theoretical support. On the other hand, for non electrolyte systems the number of moles of each species can be changed independently and this particular point does not create a problem of interpretation. However, as we discuss below, even for systems of
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non electrolytes one can find a situation in which the theory seems to conflict with reality. For clarity, we repeat that in the mathematical space all derivatives are permissible and the mathematical solution to the mathematical problem is general. This being the case, the next step is then to pass from the mathematical solution, obtained by Gibbs, to the physical solution of the equilibrium problem, i.e., to find the values of the dependent variables of the system in equilibrium. For this crucial step Lewis and Randall [10,13] introduced the concept of activity of species i, ai . Their idea was that the chemical potential of species i in a real mixture would have the same functional form as the chemical potential of the same species in an ideal gas mixture. Thus, they defined the activity of species i by the relation: i = i + RT ln(a1 (i, ))
(5)
For the case of a system formed by non electrolytes, it is usual to express the compositions in mole fractions. As this type of systems does not to have problems of interpretation, we concentrate here in the case of systems formed by electrolytes and a solvent. In this case, the composition is usually expressed in terms of molality of the ions present, i.e., moles of i per kg of solvent, and according to the definition introduced by Lewis and Randall [10], the activity of species i is related to its molality by: ai, ≡ i
mi m
(6)
In this identity, which in fact defines the activity coefficient of species i, i , the term m is the molality of the standard state of species i, equal to 1 mol kg−1 . Thus, the term i in Eq. (5) corresponds to the chemical potential of species i in ideal solution, i = 1, at its standard state composition of one mol per kg of solvent. This is clearly a hypothetical state of species i which has no physical counterpart in the physical world. For ionic species, the activity coefficient is normalized to unity at the physically meaningful reference state of infinite dilution of species i in the solvent. A step forward for the discussion of the measurability of ionic activities is based on a clear distinction between the nature of Eqs. (2) and (5), which belong to the two different worlds of thermodynamics. Erwin Schrödinger, Nobel Prize in Physics, presented the general case of the potential conflict between theory and practice in “The Tarner Lectures on Mind and Matter”, delivered at Trinity College, Cambridge, in 1956. We quote [14]: “Scientific theories serve to facilitate the survey of our observations and experimental findings. Every scientist knows how difficult it is to remember a moderately extended group of facts, before at least some primitive theoretical picture about them has shaped. It is therefore no wonder, and by no means to be blamed on the authors of original papers or of textbooks, that after a reasonable coherent theory has been formed, they do not describe the bare facts they have found or wish to convey to the reader, but clothe them in the terminology of that theory or theories. This procedure, while useful for our remembering the facts in a wellordered pattern tends to obliterate the distinction between the actual observation and the theory arisen from them.” The real problem of thermodynamics is that the theory formulated by Gibbs, the architect, is not just a primitive picture but it is a complete and forceful framework that sometimes tends to replace the facts. 4. Measurement of the activity of individual ions As indicated by Eq. (6), the determination of the activity of individual ions is reduced to the experimental measurement of the activity coefficient i . In a recent publication [15], we have shown that there is a perfect analogy in the way of experimentally ‘measuring’ activity coefficients for systems of non electrolytes and for the activity coefficients of individual ions in systems of electrolytes. Actually, in both cases these activity coefficients are not measured
81
directly but they are back-calculated from measurements of related data in a closed isothermal system. Nor the temperature, nor the number of moles of species i, or of any of the species, change during the measuring of a particular experimental point. For systems of non electrolytes, the activity coefficients are usually obtained from measurement of the total pressure P and the liquid and vapor phase mole fractions of species i, xi and yi , respectively, in equilibrium at a fixed temperature T. From the equality of the chemical potential for species i in a mixture in liquid–vapor equilibrium, we have [9]:
⎡ ⎤ P L v dP ⎢ ⎥ i xi i Pis ϕis exp ⎣ ⎦ = yi Pϕi RT
(7)
Ps i
In this equation, Pis , ϕis and vLi are the vapor pressure, the fugacity coefficient and liquid volume of species i at saturation at the temperature T of the system, while ϕi is the fugacity coefficient of species i in the gas mixture. Rearranging Eq. (7) we write
ln(xi i ) = ln
yi P Pis
⎡
⎢
− ⎣ln
ϕis ϕi
⎤
P vLi
+
dP ⎥ ⎦ RT
(7a)
Ps i
For the determination of the activity of ions in aqueous single electrolyte solutions, the voltage response Ei of an ion-selectiveelectrode (ISE), sensitive to ion i, is measured against a single junction reference electrode, both electrodes being immersed in a sample solution containing a single electrolyte at a dimensionless molal concentration (mi /m ). This voltage response is related to the activity ai of the ionic species i, given by Eq. (6), by:
m i
Ei = Ei,0 + Si ln i
m
+ EJ
or, rearranging ln
m
i m
i
1 = (Ei − Ei,0 ) − Si
(8)
EJ Si
(8a)
In these equations, Ei,0 is a constant value for limited time spans and Si is the slope of the voltage response of the ISE, which for a Nernstian electrode is equal to RT/zi F, where F is the Faraday constant and zi is the charge of the ion. The term EJ is the junction potential, i.e., the value of a difference in potential between the sample solution and the standard solution contained in the reference electrode. In both Eqs. (7a) and (8a), the first term of the right hand side gives the main contribution to the value of the activity of species i. The terms enclosed in square brackets in the right hand side of these equations are corrections introduced to improve the accuracy of the activity back-calculated from the experimental data. The evaluation of these correction terms may require making some reasonable assumptions. In the case of mixtures of non electrolytes under isothermal conditions, the correction term is normally estimated using empirical equations of state with empirical mixing rules for the mixture. For the liquid molar volume of pure compound i, iL , the usual assumption is to consider the liquid to be incompressible and to evaluate the liquid molar volume from the liquid density at the temperature T of the system. For the determination of the activity of individual ions, the situation is still more favorable as it has been shown that the method used to reduce the data largely cancels the effect of the correction term [4,15–19]. In addition, for the case of the activity of individual ions it is possible to implement an iteration procedure between independent equations and evaluate the correction term [4,15]. Again, the change in the values of the ionic activities obtained after an iteration procedure is not significant.
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It is important to note that for systems of non electrolytes, the activity of species i depends on the nature of any other species present in the system. Similarly, for systems of electrolytes, the activity of species i depends on the nature of the solvent and on the nature of any other ion or other non ionic species present in the system. For a single electrolyte in a particular solvent, the individual activity of an ion i depends on the nature of the counter ion present in the system. 5. Methods used to treat the data For the treatment of the raw data collected for systems of non electrolytes, the individual activities, measured independently for the different species, are normally correlated with a parametric expression for the excess Gibbs energy of the mixture [9]. The values of the individual activity coefficients of the components of the mixture are then obtained from the analytical excess Gibbs energy form by partial differentiation 1 lni = RT
∂GE ∂ni
(9) T,P,nj = / i
This method ‘smoothes’ the data and it guarantees that the values of the individual activities satisfy the Gibbs–Duhem equation for the isothermal incompressible liquid phase. The method of partial differentiation of an excess Gibbs energy function was used by Pitzer [20] to obtain expressions for ionic activities from mean ionic activity data, without any consideration of the transport numbers of the ions. In a previous publication [21], we have argued that this way of generating individual ionic activities from mean ionic activities, seems inappropriate. In addition to ignoring the transference numbers of the ions, the algebra generated by this treatment is far from simple [22]. For the thermodynamic treatment of the raw data collected for systems of electrolytes, we have suggested [20] correlating the individual activities, measured independently for the different ionic species, and using the equation of Lin and Lee [23] for the calculation of the activity of water. Again, this method ‘smoothes’ the data and it guarantees that the values of the individual activities satisfy the Gibbs–Duhem equation for isothermal incompressible liquid phase. In addition, considering the state of affairs in the discussion of the measurability of individual ionic activities, for the reduction of data we prefer to avoid any step that could appear as violating the electroneutrality condition. At this stage, the use of expressions of the type of Eq. (9) could contribute to the misunderstanding regarding the need to ‘move in or out of the solution’ a single ionic species i in order to determine its individual activity. 6. Discussion and conclusions Notably, as discussed in detail elsewhere [15] and reiterated here, there is a perfect parallel between the experimental methods used for the determination of activities of ionic and non ionic species. In both cases the isothermal system is closed, i.e., there is no change in the temperature or in the number of moles of any of the species involved. In addition, in both cases once the data are collected one may or may not introduce the minor corrections shown in Eqs. (7a) and (8a) with the purpose of improving the accuracy of the results. Again, in both cases, experience has shown that the correction terms do not affect the final values significantly [15]. One additional point of similarity is that the activity coefficient of an individual ion depends on the nature of the counter-ion present in the system, exactly as the activity coefficient of ethanol depends if this compound is in a mixture with water or with methanol. Thus, one can have the legitimate question of why there is such an opposition to the possibility of measuring the individual ionic activities.
It seems to us that there are two potential reasons. One originates from a consideration of the independent equation for the junction potential EJ [16] RT EJ = − F j
k
tj zj
dln(mj j )
(10)
r
In this equation, the summation runs over all ions present in the solution in the reference electrode r and in the sample solution k, and tj is the transference number of ionic species j with charge zj , for which there is independent information in the literature. Taylor [8] demonstrated that by combining Eq. (8) with Eq. (10), the voltage response of a cell with liquid junction, Ei could be expressed as a function of the mean ionic activity of the electrolyte and the transference number of ion i. In a more recent publication [17], we have presented a simpler derivation of Taylor’s result. Combining the two independent Eqs. (8) and (10) one eliminates the variable EJ and obtains an expression for Ei in terms of the mean ionic activity of the electrolyte and the transference number of ion i. In our example [4,18], we discussed the case of the homoionic system of a sample solution of KCl in which the voltage response of a chloride ion-selective-electrode (i = −) was measured against a reference electrode containing a concentrated solution of KCl. In this case, the right hand side of Eq. (10) has only two terms and we obtained:
E
,k
=
E0,− −
RT ln a F
,r
+ S +
RT F
ln˛
,k
−
2RT F
(1 − t )d lna±
(11)
r
In this equation, a± is the mean ionic activity of the electrolyte and for a 1:1 electrolyte, (m+ = m− = m). For a 1:1 electrolyte, it takes the form: a2± = a+ a = (m+ + )(m ) = m2 ±2
(12)
where ± is the mean ionic activity coefficient of the electrolyte and all other symbols have their usual meanings [4,18]. Mutatis mutandi, a similar expression can be obtained for the voltage response of the potassium ion-selective-electrode. If the electrolyte in the sample is different from the electrolyte in the reference electrode solution, the expression for the voltage response of each ISE is also more complex [16], but it is formally similar to Eq. (11). The first term in square brackets of the right hand side of Eq. (11) is a constant for an experimental run. The second term cancels out for a perfectly Nernstian ISE as the slope for the response of the anion is negative. For common commercial chloride ISE’s, which are very closely Nernstian, this term is negligible. The third term of the right hand side shows that the voltage response of the ISE is in fact a function of the mean ionic activity coefficient of the electrolyte and the transference number of the ion. The explicit form of this dependence is not simple as the transference numbers are function of composition. Even for a homoionic system, there is a composition change through the liquid junction. The main limitation for the calculation of this term is that information of transference numbers is, in general, scarce and almost nonexistent for concentrated solutions [18]. However, even when all that it has been attained with the combination of Eqs. (8) and (10) is to hide the activity of the individual ion in the constant term of Eq. (11), there is no question about the fact that the voltage response of an ISE can be expressed as a function of the mean ionic activity coefficient of the electrolyte and the transference number of the ion. This conclusion can be attained without the unnecessary assumptions made by Zarubin [1]. Once the temperature and composition of the system are fixed, the values of all dependent variables such as mean ionic activity coefficient, activities of individual ions, density, refractive index, viscosity, etc., are also fixed and one can always express one of the variables in terms of others. Hence, in our view, the recent
G. Wilczek-Vera, J.H. Vera / Fluid Phase Equilibria 312 (2011) 79–84
publication by Zarubin [1] provides a missing link and serves the purpose of answering also Malatesta’s arguments [24] in support of Taylor’s results. Zarubin reasoned that if the voltage response of an ion-selective-electrode can be shown to be a function of the activity of the individual ion and also it can be shown that it is a function of the mean ionic activity coefficient of the electrolyte and the transference number of ion i [1], then there must exist a relation of the form i = i (± , ti )
(13)
The left hand side of this equation is the value of the experimentally measured activity coefficient of the individual ion i, as it appears in Eq. (8), and the right hand side represents a function for which Zarubin with some simplifying assumptions obtains a value from a theoretical treatment of each type of electrochemical cell. There is an impressive agreement between the measured and the calculated values. One should note that the calculated values from Zarubin’s functions use experimental data for transport numbers and mean ionic activity coefficients that include their own experimental errors. Uncertainties in the calculated results are probably due more to uncertainty in the former than in the latter of these two pieces of experimental information. The agreement between the experimental values of i evaluated with Eq. (8) and the values obtained with the function of the right hand side of Eq. (13) should be treated as an example of the coherence of the approach used instead of interpreting it as a proof of nonexistence of individual activities of ions. The individual activity coefficients of ions can be expressed as functions of different properties of the system. If two parameters are used and one of them is representative of the system, as it is the case of the mean ionic activity coefficient, the other should necessary be a property characteristic of the individual ion, such as its transference number. This functionality does not make the activity of individual ions “conventional”, “non-existent” or “so called”, as they were denoted by Zarubin [1]. At this point the only standing objection to the possibility of measuring activity coefficients for the individual ions is that, as clearly stated by Zarubin [1], in the physical world their mathematical interpretation as partial molar quantities “falls into contradiction with the condition of electrical neutrality when applied to ions in electrolyte solutions”. For systems formed by non electrolytes, it would appear that one could picture the chemical potential or the activity coefficient of species i in the real world because one can change the number of moles of species i, while keeping the pressure, the temperature and the number of moles of all other species constant. This is just a mirage. It assumes that one can picture in the physical world the Gibbs energy, which is just an abstract entity in the mathematical world. An evidence of this mirage can be seen by considering a binary two-phase system of non electrolytes at equilibrium. For these systems, according to the Phase Rule, there are two degrees of freedom. Thus, if the temperature is fixed there is only one degree of freedom left and it is not possible to change the moles of one compound and keep the pressure constant. Similarly for isobaric measurements, if the pressure is fixed the temperature cannot be kept constant. This fact certainly does not affect the mathematical interpretation of the chemical potential as the change in Gibbs energy of a phase with the change of the number of moles of one single compound at constant pressure and temperature. Notably, these binary systems in which a vapor and a liquid phase coexist at equilibrium, either at a fixed temperature or at a fixed pressure, are exactly the type of systems used to evaluate the activity coefficients in systems of non electrolytes. Although Eq. (7) originates from the equality of the chemical potentials of species i in a mixture in vapor–liquid equilibrium, nobody ever objected its use arguing that one cannot change the number of moles of one compound at a time while
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keeping the pressure, the temperature and the number of moles of all the other compounds constant. The fact is that, as we discussed previously [15], for the experimental determination of the activity coefficients of both non ionic and ionic species, it is not necessary to change the number of moles of the compound in question. In both cases, the activity coefficients are evaluated from measurements in closed systems in thermal equilibrium. Moreover, the similitude between the case of measurement of activities in systems of electrolytes and in systems of non electrolytes is complete if one considers the negative argument advanced in this study for the latter case. In both cases, it is not possible to keep constant the pressure, the temperature and the number of moles of all other compounds while changing the number of moles of species i. In other words, in both cases it is impossible to satisfy experimentally the conditions required by the right hand side of Eq. (2). Again here, this was never a cause for confusion in the case of systems of non electrolytes and it is about time to understand that this is not an impediment for measuring the activity of individual ions. It is important to realize that the misleading step is to put the wrong question or even to put the right question in the wrong world. This brings to mind the old trick for an undergraduate teasetest: “(A) Name three organic and three inorganic acids. (B) Name three organic and three inorganic alcohols.” In this context, the objections to the measurability of ionic activities are the ones that should be questioned. One must assume that well known scientists understood properly the concept of partial molar properties and the conditions of equilibrium obtained by Gibbs. Lewis and Randall [10], Pitzer and Brewer [20]; Harned [25] among others, all clearly stated that the activity of individual ions was experimentally obtainable if we could, in any way, calculate or estimate the effect of the junction potential in a cell with junction. Even Guggenheim [26] suggested the possibility of an iteration procedure between Eqs. (8) and (10) but without the availability of computer he declared this iteration to be a “vicious circle”. Few people would dare saying that these scientists did not understand the concepts involved. They obviously knew thermodynamics well and had a clear picture of the difference between the mathematical and the physical worlds. There is of course the question of how well the results from the theory agree with experimental phase equilibrium data. The fact is that the equality of the chemical potentials of a species present in phases in direct contact is only the starting point to represent reality. The result obtained from the mathematical framework needs the help of additional definitions of ideal system behaviour. Moreover, as the algebra for ideal systems gives only the functional dependence of the chemical potential on measurable variables, the representation of real data needs the introduction of fugacity and activity coefficients. These activity coefficients back-calculated from the experimental data are precisely the subject discussed in this manuscript. In special cases it is necessary to go still one step further and pay special attention to the selection of the proper standard state for the definition of the activity coefficients. Although this is not a problem for systems of electrolytes in a single solvent, it is important for the case of electrolytes in mixed solvents. For systems of non electrolytes, under isobaric conditions if the standard state has been chosen as the pure component liquid at the pressure and temperature of the system, the lighter compound does not physically exist under these conditions. This problem has been clearly discussed in the literature and a solution has been recommended [27]. To conclude, we observe that Zarubin [1] recently noted that experimental measurement of the voltage response of ISEs were not available for certain aqueous electrolyte systems of interest for testing his new treatment of the data. The fact is that any work reporting new ISE voltage response data, or for this purpose any
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attempt to advance our knowledge in the field of electrolytes, is not welcome. This prejudiced attitude discourages researchers to contribute to this field. It is time to consider with special attention the positive views of Lewis, Randall, Pitzer, Brewer, Harned and even Guggenheim, among others, supporting the measurability of individual ionic activities. Otherwise, this field will be stagnant for another hundred years while, in the mean time, the development of biotechnology will require the use of individual ionic activities for the treatment of Donnan equilibrium. The participation of young researchers gathering new experimental data and contributing fresh ideas should be encouraged not discouraged. Errors have been made and will be made. We should all participate in an open discussion of ideas keeping in mind the unsolicited advice that Frank Wilczek, Nobel Prize in Physics, gives to his graduate students [28]: “If you do not make mistakes, you’re not working on hard enough problems. And that’s a big mistake”. Acknowledgements The authors are grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC) and to McGill University for financial support. References [1] D.P. Zarubin, J. Chem. Thermodyn. 43 (8) (2011) 1135–1152. [2] J.H. Vera, G. Wilczek-Vera, J. Chem. Thermodyn., to be published.
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