Determination Ionic Activities of Metal Salts in Water

A thorough knowledge of thermodynamic properties of strong electrolytes is useful in leaching processes, purifying dissolved salts and precipitating out selected constituents in aqueous solutions. The data on ionic activities are essential in understanding thermodynamics and kinetics of hydrometallurgical processes. In particular, large changes in the activity coefficients, affected by addition of selected electrolytes, may cause precipitation of a pure salt ideally suited for the extraction of its metal component. There are very few sets of data available for metallurgically important salts in sufficiently large concentration ranges. It is therefore frequently necessary to estimate the ionic activities, particularly in multicomponent solutions, by using various types of empirical equations. A number of proposed equations for this purpose lead to the estimation of the activity coefficients with only moderate degrees of success. For multicomponent electrolytes, the methods of estimation require new and accurate data. The objectives of this report are therefore (1) to correlate the vapor pressure of water over solutions of electrolytes with the ionic activities , (2) to outline a series of experiments to obtain new and extensive data on the activities of metal salts in binary and multicomponent solutions, and (3) to examine the available methods of estimation of activities in concentrated strong electrolytic solutions, and to test their validity by using published data.

The methods and the results in this report are a part of the Federal Bureau of Mines program to provide basic thermodynamic data for extraction of strategic metals, pollution control, environmental preservation, and energy conservation.

In this Bureau of Mines report, appropriate equations correlating the activity of water with the activities of electrolytes in binary and multi-component solutions have been derived for computer calculation. It was found that the half-integer power series are far superior to the integer power series, for both types of series having the same number of terms. The proposed method of calculation is capable of accurately yielding the activities of all components from the measurements of the activity of a single component at various concentrations. This method replaces the cumbersome and objectionable method of graphical integration of the Gibbs-Duhem equation.

Two novel methods are proposed for determination of activities of electrolyte solutions important in hydrometallurgy: (1) Water vapor pressure measurement by differential pressure transducers having an accuracy of 0.001 pct, and (2) transpiration method with an inert gas bubbled through concentrated solutions followed by the analysis of the evaporated and redissolved electrolytes with specific ion electrodes having a sensitivity of ±10 -10 molal. The first method is suitable for nonvolatile electrolytes, and the second, for volatile electrolytes such as hydrogen halides, HNO3, H2SO3 , etc.

The existing methods of estimation of ionic activities have been surveyed, and the best method has been selected, discussed, and summarized.

Thermodynamic Background

A brief outline of the relationships for the activities in strong ionic solutions is useful in the discussions presented in this report. Consider a compound Av1, Bv2, designated as electrolyte 12, which consists of elements or radicals A and B with stoichiometric coefficients v1 and v2. Assume that this electrolyte dissociates into its ions Az1 and Bz2 with the charge numbers z1 and z2, as follows:

where z1 and z2 are taken to be positive for convenience. The principle of electroneutrality requires that v1z1 be equal to v2z2. The odd subscripts are used for cations, even subscripts are used for anions, and the appropriate combinations of the subscripts are used for electrolytes. By tradition, concentrations and activities are expressed in molalities m1, that is m1 moles of ion i in 1,000 grams of water (or 55.508 moles of water). The molality of Av1Bv2 is designated by m12, and it is evident that

where m12 refers to the number of moles of pure 12 used in preparing the solution. All the symbols used in this report are summarized in “Appendix— Nomenclature.” The activity of water, aw, is based on pure water as the standard state and on the mole fraction xw, expressed by

Substitution of the activities of ions, a1 and a2, in the Gibbs-Duhem relation for one electrolyte in solution gives

55.508 dln aw + m1 dln a1 + m2 dln a2 = 0,………………………………………..(4)

where ln is the natural logarithm. Substitution of equation 2 in equation 4 yields

Since a1v1 and a2v2 are not independently measurable, but their product is, the following definitions are introduced for brevity:

Equation 5 for c electrolytes in solution, obtained by the same procedure, is

where the summation gives c terms, one for every electrolyte ij used for preparing the solution. In this report, i will always denote a cation, and j, an anion. It can be shown by the detailed derivation outlined in equations 2-7 that equation 8 remains unchanged in its mathematical form whether some of the ions of component electrolytes are common or not.

Equations 4-7 also justify the following definitions:

Note that m12+ is not equal to m12 when v1 ≠ v2. It should be emphasized that m1 and m2 in these equations refer to the total molalities of these ions when there are common ions in solution. The reference state for Y12 is defined by Y12 → 1 when m12 → 0. Substitution of equations 9 and 10 in 6 yields

a12 = (m12±) Y12………………………………………..(11)

Correlation of Activities

The mean ionic activity of 12 can be obtained either directly by measurements on solutions of 12, or indirectly from measurements of aw and integration of equations 5 or 8. The methods of graphical integration are tedious and objectionable for multicomponent electrolytes; therefore, a convenient method based on an appropriate power series is derived in this report. It may be emphasized at this point that a Margules type of simple power series, suitable for nonelectrolytes, was used for this purpose, but the results were found to be unsatisfactory.

The analytical forms satisfying equation 8 are obtained by the requirements that (1) the Debye-Huckel limiting law-that is, lnyi = bmij, where b is a constant for a given temperature-be obeyed as mij approaches zero for the binary system ij and water, and (2) ln aw, be a power series in (mij) to minimize the number of terms required to express precise data with a satisfactory degree of accuracy as compared with power series in (mij). The selection of power series, beyond the requirement that the Debye-Huekel limiting law be obeyed, is empirical, provided that the series is continuous and differentiable. It is appropriate to consider first a solution of one electrolyte in water, and then two or more electrolytes as necessary. The equations selected for this purpose are as follows:

where F(m12) is an appropriate function of the molality m12 of electrolyte 12.

In both equations, the terms after the equal signs and preceding the logarithmic terms are for the activity coefficients, and in equation 13 the last term is ln m12±v12 from equations 2 and 10. In equation 12, the functional form of F(m12) can be determined when equations 12 and 13 satisfy the Gibbs-Duhem
relation given by equation 5; thus, when equations 12 and 13 are substituted in equation 5 and dF(m12) is integrated from m12 = 0 to m12 to obtain F(m12) and then the result is inserted in equation 12, the last two terms in equation 12 cancel out, and A becomes -v12. Substitution of equations 12 and 13 in equation 5 also gives the relationships among the coefficients in equations 12 and 13; the results are as follows:

It is evident that the exponents of m12 in equations 14 and 15, corresponding to the same coefficients B, C, and D, differ by one and that the numerical coefficients before B, C, and D are given by minus the ratio of exponents of m12 in equation 14 and that in equation 15. For example, the next term in equation 14 is Em³12, and in equation 15, -(3/2)Em²12. The coefficient B is proportional to α based on the Debye-Huckel theory. The values of a at various temperatures are as follows:

The values of α can be substituted in the following Debye-Huckel limiting equation for any type of electrolyte to calculate B:

where the first equality is the Debye-Huckel limiting equation, and I, the ionic strength, is given by

Thus, for CaCl2 at 25° C, v1 = 1, v2 = 2, z1 = 2, z2 = 1, I = 3m12, v12 = 3, and from equation 16, -3(2) α3 m12 = -3Bm12, and B = 4.056. It is also possible to take B as an adjustable parameter in data correlation, particularly for multivalent electrolytes for which the limiting law is approached at molalities less than 10-6. Equations 14 and 15 are very important in determining the activities of electrolytes from the activity of water, which can easily be obtained from

where f is the fugacity of H2O(g); Z, the compressibility factor; and P, the pressure over the solution; f°, Z° , and P° are the corresponding quantities over pure water. For dilute solutions, aw is very close to P/P°, but for concentrated solutions at high values of P°, the correction required by Z/Z° must be made. A sufficient number of values of aw are then used to determine as many coefficients in equation 14 as desirable by the least squares method on a computer. The coefficients determined from equation 14 are then substituted in equation 15 to calculate the values of a12. It is evident that this method eliminates the old procedure for graphical integration of the Gibbs-Duhem equation, a procedure particularly objectionable for multicomponent solutions as discussed elsewhere in detail.

Determination of Activities

Several methods are available for measurement of ionic activities as discussed in a number of publications. The advent of electronic devices for pressure measurement appears to be promising in developing a new method of investigation. As a highly accurate novel method, it is proposed here that P and P° in equation 18 be determined by using very precise pressure transducers capable of attaining an accuracy of 0.0001 to 0.001 pct. The precision in this case is necessary because small errors in aw give relatively large errors in a12 because of the large factor, 55.508, in equation 14 relative to equation 15. Very small and corrosion-resistant transducers with convenient electronic indicating and recording devices are available for this purpose. In addition, if the electrolyte is sufficiently volatile, such as HF, HCl, HBr, HI, and HNO3, it is possible to use a transpiration technique for measuring the equilibrium partial pressure of the electrolyte over a given solution. For this purpose an inert gas of known volume is bubbled first through the solution, and the saturated gas is then bubbled through slightly alkaline water to recover the electrolyte vapor in the gas. The analysis and the weight of water yield the partial pressure of electrolyte in the inert gas.

With the advent of specific ion electrodes, it is now possible to determine accurately the concentrations of electrolytes in water in the range of 10 -10m (m = molal after a number in a sentence) without difficulty. This method supplements, or extends, the range of measurements as will be seen later.

It may be noted that if the activity of electrolyte a12 is known, the constants in equation 15 can be determined first, and then substituted in equation 14 to obtain the activity of water, aw, and the vapor pressure of water, P, over solutions of a given electrolyte.

Equations 14 and 15 can now be tested by using the data for NaCl, for which the values of aw have been listed in the range of 0.1 to 6.0 molal concentration. The calculations show that, with four parameters determined from aw at 0.2, 1, 2.6, and 6m, the resulting equation for Y12 at 25° C is

The values of a12 calculated from this equation agree within approximately 1 pct with the tabulated values in the range of 0.1 to 6m. With six parameters , however, the calculated values are as precise as the most precise values obtainable by experiment. In fact, the limitation on the number of parameters depends on the number of data points and on the precision of measurements. The least squares method of determining these parameters is recommended whenever possible.

Multicomponent Solutions

The equations corresponding to two electrolytes dissolved in water, that is, a ternary solution, can be obtained by the same procedure. For the activity of water, the following series is adopted:

In this equation, m12 = m and m34 = n are used to avoid writing the subscripts. Note that the first set of four terms corresponds to the contribution by (water plus 12) and the second set, by (water plus 34). The cross terms (mh/2) (nk/2) correspond to the mutual interaction of the electrolytes. Substitution of this equation in the Gibbs-Duhem equation with the corresponding series for the second and third components yields

It should be observed that the terms containing V34, B’, J for electrolyte 34 are not present in this equation, and that the cross terms are obtained from the following generating function:

where i = h/2 and j = k/2 are in the exponents of m and n in equation 20, and h and k are chosen as integers so that they could be used as subscripts, as will be seen later in conjunction with equation 23. Thus, for the term containing V in equation 20, i = 1.5, j = 1, and the corresponding term in equation 21 is -(1.5/1.5)m0.5n. Actually, equation 22 is also applicable for deriving equation 15 where k is zero; therefore equation 22 can be used to express equations 20 and 21 in compact forms as follows with h and k as positive integers:

The additional restrictions are required to make the ratios preceding Bhk finite. These equations are not recommended for use in computer programing because multisubscript coefficients are inconvenient as compared with plain coefficients B, C, D, in equations 20 and 21.

Equation 21 corresponding to the second electrolyte, 34, can be obtained by a similar procedure; the result is

The foregoing equations establish the basis for obtaining the activities of electrolytes from the activity of water and lead to the description of the proposed investigations in the next section.

The activities of electrolytes can also be expressed in terms of mole fractions. The results require that equations 2-11 retain their forms with mi and mij replaced by mole fractions xi and xij. Equation 8, however, becomes simpler because xw dlnxw + ∑xidlnxi for all values of i becomes zero because the sum of all mole fractions is equal to one; therefore,

where Yi j is based on mole fraction. The equations corresponding to equations 23 to 24, with x12 = x, x34 = y, and xw = 1 – x – y, are then given by

Coefficients b3o and bo3 are related to B and B’, respectively, as required by the Debye-Huckel theory. For very large concentrations of electrolytes these equations have advantages over the previous equations, particularly because x and y remain in the range of 0 to 1, whereas m or n in equations 21 and 25 become infinity when x or y become 1, respectively. The relationships for converting Yi j and mi j± into Yi j and x i j± is

where xi j± is obtained from an equation similar to equation 10; that is, xt j ±Vi j = xivi · xjvj, and xi and xj are obtained by converting mi and mj into mole fractions. No attempt will be made in this report to utilize equations 28-30 since very high concentrations of electrolytes, that is, in excess of x = 0.50, are seldom encountered in hydrometallurgy. The foregoing equations can readily be extended to quaternary and multicomponent solutions.

Proposed Investigations

Three types of investigations are proposed in this report: (1) Measurement of the vapor pressure of water over a given solution, (2) measurement of partial pressure of the volatile electrolyte over a given solution by transpiration method, and (3) determination of solubility of a given salt in water at various concentrations of other electrolytes. Each method has its own advantages and disadvantages, depending on the selected electrolytes. Brief discussions of these investigations will now be presented.

Measurement of Vapor Pressure of Water

The measured property in this case is the pressure P in equation 18. For this purpose, a simple apparatus
shown in figure 1 is proposed. A Pyrex bulb, A, contains a known solution of nonvolatile electrolytes in water, stirred with a magnet bar, B, sealed in Pyrex. Bulb A is attached to Pyrex arm, C, by a ball-and-socket joint lubricated with a suitable grease. Arm C and the vacuum connection tubes are all Pyrex construction, joined to a differential transducer, Tr, by Kovar glass joints. Bulb D is identical with bulb A, but it contains deionized double distilled water. Various stopcocks and connections permit evacuation of each bulb, as indicated. The entire assembly is immersed in a thermostat controlled to ±0.001° C.

A novel feature of this apparatus is the differential transducer, Tr, gold-plated for protection against electrolytes. The transducer can measure a small pressure difference, P, with an accuracy of better than 0.001 pct. The vapor pressure of pure water, P° in D, is obtained from appropriate tables by using the thermostatic temperature. The activity of water in A is therefore

If, for example, P° = 25 torr (exact), and P = 2 torr is known with an accuracy of 0.01 pct, then aw is known with an accuracy of 0.0009 pct. Therefore, even a transducer of 0.1 pct accuracy is capable of yielding results within 0.009 pct, which is adequately precise for most purposes. The transducer and a quartz-type high precision thermometer readings can both be separately and continuously recorded for convenience.

Additional bulbs, such as E, can be connected to C, as shown on the right side of A, to take data for solutions of different molalities. It is possible to have three or more additional bulbs to obtain several sets of data at various temperatures during one experimental run. Determination of the activities at two or three temperatures permits calculation of heat of vaporization, and construction of vapor-liquid phase diagram for each system.

Transpiration Method

The transpiration method is capable of measuring the activities of volatile electrolytes in sufficiently high ranges of concentration. For this purpose, a known amount of an inert gas, such as helium, is bubbled through a solution of electrolytes. The gas emerging from the solution is saturated with the volatile electrolyte if the bubbling is sufficiently slow and the gas is passed through an immersed fritted glass to generate small bubbles. The emerging gas can be passed through an absorbing solution to redissolve the volatilized electrolyte. The absorbing solution can then be analyzed by using specific ion electrodes capable of measuring 10 -10 molality with a high degree of accuracy. The result of analysis is then used to compute the partial pressure, Pij , of volatile electrolyte in the inert gas. The activity, aij, is then computed from

where Kp is the equilibrium constant for the volatilization of electrolyte from a solution of known concentration. Since Kp is independent of concentration, it is sufficient to determine Kp from a value of Pij and the corresponding value of aij, as measured by the other methods in the low range of Pij .

Thus, for HCl, the vaporization reaction is H+ + Cl- = HCl(g). At 25° C and mi j = 5, Pi j = 7.00 x 10 -5 and aij = 11.9 from other types of measurements, such as the depression of the freezing point measurement, from which KP = 4.94 x 10 -7. This value of Kp may now be used to obtain the activity of HCl at higher concentrations. This method must replace the method presented in the preceding section for the vapor pressure of water because the vapor phase contains water as well as the volatile electrolyte.

Solubility Method

When a solid electrolyte 12, having r moles of water of crystallization saturates a solution, the corresponding reaction and its equilibrium constant are

where a12 is the activity of solid. The solid phase may be pure (with r = 0), as for most of the 1:1 type electrolytes, or with a fixed but fairly large value of r as for most of the polyvalent electrolytes. Addition of other electrolytes changes the solubility of 12 and may cause very large variations in r. Some solid electrolytes form complex salts with electrolytes added in solution. If the solid has a fixed composition with changing composition of solution, then it is permissible to take a12 as unity; if not, then must be evaluated from the standard Gibbs energy change corresponding to changes in hydration, and changes by complex solid electrolyte formation. It is important that the composition of the solid, as well as that of the liquid, be determined from the solubility measurements in the absence of other electrolytes, if the values of aw and Y12 are known or can be obtained by extrapolation of data from lower concentrations of 12 in water. Since KP is independent of composition, addition of other electrolytes would only change the values of Y12 m1, m2, and, aw assuming that a12 remains constant. Thus, the effect of added electrolytes on Y12 can be determined from the solubility measurements. When the variation of Y12 is known from such measurements, the resulting data can be substituted in equations 20, 21, and 26 to obtain all the related thermodynamic properties of other components. An interesting application of this concept on the solubility of AlCl3·6H2O will be presented by Brown, Daut, Mrazek, and Gokcen in a forthcoming report.

It should be emphasized that the measurements on solubilities must be very accurate, particularly for polyvalent electrolytes, to obtain fair values of activities. The required accuracy becomes more stringent as the moles of hydration, r, increase from one salt to another. In general, very precise condensed phase boundaries (solubilities) are required to obtain acceptable activities, whereas mediocre values of activities are capable of yielding highly accurate phase boundaries.

Estimation of Activities

Accurate estimation of activity coefficients is a difficult task since both coulombic and noncoulombic forces play important roles in affecting the properties of ions in concentrated solutions. The best existing method of estimation of the activity coefficients has been developed by Meissner and his associates in a series of eight papers, although an earlier attempt by Bromley has its own merits. Since the degree of success attained by Meissner and his coworkers is significant, a summary of the work is given here in four sections with the present author’s discussions.

Activity Coefficients in Binary Solutions

The Meissner method is based on the observation that when log Y12 of only one electrolyte in each solution is plotted versus ionic strength I, the curves for various electrolytes cross one another, but when log P12 = (1/z1 z2)log Y12 is plotted versus I, very few curves cross one another (9).

In fact, P12 varies with I as shown in figure 2. The property P12 is called the reduced activity coefficient of one electrolyte in water and defined by

For more than one electrolyte in solution, the reduced activity coefficient is similarly defined, that is,

where Y12 is the activity coefficient in a ternary of multicomponent solution. In principle, a single value of P12 at a sufficiently high value of I determines the location of the entire curve for a given temperature. The value of the empirical parameter q in figure 2 depends on the temperature and on the electrolyte, and it is obtained from

The terms in equation 37 prior to F constitute empirical corrections to the Debye-Huckel equation given by equation 38. These equations have been obtained by trial and error on a computer to calculate the best coefficients capable of representing the activity coefficients of as many electrolytes as possible. Actually, each q value is an average of various values based on Y12 from experimental data points in the literature. If the value of q has been determined for one temperature, such as q25 for 25° C, the value at any other temperature, qt , can be calculated from

where t is the temperature in degrees Celsius, and a = -0.0079, b = 0.0029 for sulfates, and a = -0.005 and b = +0.0085 for all other electrolytes. Sulfuric acid, some sulfates of multivalent elements, thorium nitrate, and halides of zinc and cadmium do not obey the foregoing equations with sufficient degrees of accuracy. Table 1 shows the selected values of q from Kusik and Meissner. It is evident that except for high ionic concentrations of HClO4, LiCl, LiBr, LiNO3, and NaOH, the values of P12 for 1:1 electrolytes are represented with standard deviations less than 10 pct. This is also true for higher valence electrolytes, with the exception of CaCl2 , MgCl2, UO2SO4, and ZnSO4. However, for multivalent electrolytes a small error in T°s causes a larger error in P12, for example, an error of 5.2 pct in P12 for MnSO4 is equivalent to 22.5 pct error in y12 since z1 – z2 = 2 for this electrolyte. Therefore, the success of this method decreases with increasing values of the product z1 · z2 for the same error in P12.

Activity of Water in Binary Solutions

The activity of water for 1:1 electrolytes, designated as (aw), is represented in the Kusik-Meissner diagram by the dotted lines as shown in figure 3. The activity (aw) for electrolytes higher than 1:1 is computed from the following equation:

For example, at 25° C, P12 = 1.7 for NiCl2 when m12 = 4 and I = 12, and from P12 and I, figure 3 gives (aw)’ = 0.52. Substitution of I and (aw) in equation 40 gives (aw) = 0.645 in good agreement with the experimental value of 0.635.

Activity Coefficients in Multicomponent Solutions

The activity coefficients in mixtures of electrolytes can be estimated from the following equation proposed by Kusik and Meissner:

The values of Pij for binary solutions are determined at the temperature and the total ionic strength I of the multicomponent mixture (total ionic strength is always without subscript). The terms V and I with various subscripts are defined by

As an example, consider m12 = 5.30 for NaNO3, and m32 = 5.27 for Ca(NO3)2 so that I = 21.11; from equation 42, V12 = 2.0 and V32 = 2.25; from table 1 and figure 2, FP12 = 0.28 and P32 = 0.87; substitution in equation 41 gives P12 = 0.49 in fair agreement with 0.38, based on experiments.

Activity of Water in Multicomponent Solutions

The activity of water (aw)mix for the multicomponent mixtures can be calculated (9) from

The activity of water in the binary solution (aw)ij in this equation must be calculated at the total ionic strength of the multicomponent solution. It is important to remember that (aw)ij for multivalent electrolytes must be computed from (aw) for 1:1 electrolytes by equation 40. The terms Wij, Xi, and
Yj are defined by

For a solution in which all zi are equal, and all zj, are equal, but zi is not necessarily equal to zj, equation 43 simplifies into the following form:

Estimations of Pij and q for Binary Solutions

The foregoing methods of calculation are based on the existence of at least one experimental result for Pij of an electrolyte in a binary solution, so that q can be calculated. If such a result is not available for an electrolyte , it can be estimated from the following empirical equation:

where i is the common cation and j is the anion for which Pij is unknown, and Aj and Bj are constants for j, listed in table 2. An example would illustrate the method used to obtain Aj and Bj . Let Picj be for alkaline chlorides, and the values of Picl are known, and Pij be for alkaline bromides (j = Br) for which some but not all values of Pij are known. A linear plot of Pij versus Picl yields the values of Aj and Bj. Since the values of Picl for most chlorides are known, substitution of Picl in this equation yields the value of Pij. The values of P for both types of ions refer to the same value of.the ionic strength, I, and for convenience, I = 2 has been chosen by Meissner and Tester. The results, adapted from their work, are listed in table 2. The application of equation 46 is illustrated by the following example. Let i and j be Li + and Br , respectively, so that Pij be for LiBr whose reduced activity coefficient is to be estimated. Table 2 gives Aj 1.259, Bj = -0.121, and PLiCl = 0.921; therefore, for LiBr Pij = PLiBr is given by Pij = 1.259 x
0.921 – 0.121 = 1.039, whereas the experimental value is 1.015. After Pij is estimated, the unknown value of q can be computed from equation 37 with I = 2, by successive approximation.

Concluding Remarks

Methods of estimation and correlation of activities, particularly at low concentrations, are presented and discussed elsewhere. Another method of estimation for fairly concentrated solutions is due to Bromley. This method is useful for binary solutions at 25° C usually for I ≤ 6, (although in some cases I ≤ 3, 4, 10, or 15). Bromley’s equation is as follows:

The values of B for various electrolytes have been tabulated by Bromley. For example, B = 0.1433 for HCl, and at I =4, this equation yields P12 = 1.756, in very close agreement with the experimental value of 1.762 and with 1.751 from the Meissner equation. A value of B for HCl at I > 6 is not given by Bromley. Likewise for AlCl3 , at m = 1, or I = 6, the values of P12 are 0.798 and 0.832 by the Bromley method and the Meissner method, respectively, bracketing the experimental value of 0.814. The author recommends using the Bromley method in conjunction with the Meissner method to gain confidence in estimated values for binary solutions. Numerous other computations show that in general, both methods of estimation yield concordant values for binary solutions in the range of 0 < I > 6.

Excellent theoretical bases for equations having a form somewhat similar to equation 47, but quadratic in molality, have been proposed by Pitzer and coworkers. For concentrations in excess of 6m, however, all such equations require additional terms or readjustment of their parameters for realistic representations of available data. Some electrolytes fail to obey such equations at concentrations an order of magnitude lower than 6m, as expected. In general, all such equations, including equations 37 and 47, are very useful for interpolation or extrapolation for short ranges of concentrations for which experimental results are available. The adjustable parameters for these purposes must be obtained from the immediate vicinity of the data that form the basis of interpolation or extrapolation.