Activity Coefficients from Cell Measurements

It has been emphasized repeatedly that the individual activity coefficients cannot be measured experimentally. However, these values are required for a number of purposes, e.g. for calibration of ion-selective electrodes. Thus, a conventional scale of ionic activities must be defined on the basis of suitably selected standards. In addition, this definition must be consistent with the definition of the conventional activity scale for the oxonium ion, i.e. the definition of the practical pH scale. Similarly, the individual scales for the various ions must be mutually consistent, i.e. they must satisfy the relationship between the experimentally measurable mean activity of the electrolyte and the defined activities of the cation and anion in view of Eq. (1.1.11). Thus, by using galvanic cells without transport, e.g. a sodium-ion-selective glass electrode and a Cl -selective electrode in a NaCl solution, a series of (NaCl) is obtained from which the individual ion activity aNa+ is determined on the basis of the Bates-Guggenheim convention for acr (page 37). Table 6.1 lists three such standard solutions, where pNa = -logflNa+, etc. 

Given a measured cell voltage of +0.798 3 V and using activity coefficients from Table 8-1, find Ag+lAg- Be sure to express PUi in bar in the reaction quotient. 

The computation of activity coefficients from A log j values will be illustrated for hydrochloric acid since in that case direct comparison can be made with the results of measurements on concentration cells without transference of the type described in Chapter 6. The relevant data are given in Table II and are from the work of Shedlovsky and Maclnnes.18 The emf data in the second column were obtained from a cell of the type illustrated in Fig. 4. The transference numbers in the third column were interpolated from the measurements of Longs-worth given in Table IV of Chapter 4. The A log f values in the fourth column were computed as described in the last paragraph. 

In principle the activity coefficients yb of solute substances B in a solution can be directly determined from the results of measurements at ven temperature of the pressure and the compositions of the liquid (or solid) solution and of the coexisting gas phase. In practice, this method fails unless the solutes have volatilities comparable with that of the solvent. The method therefore usually fails for electrolyte solutions, for which measurements of ye in practice, much more important than for nonelectrolyte solutions. Three practical methods are available. If the osmotic coefficient of the solvent has been measured over a sufficient range of molalities, the activity coefficients /b can be calculated the method is outlined below under the sub-heading Solvent. The ratio yj/ys of the activity coefficients of a solute B in two solutions, each saturated with respect to solid B in the same solvent but with different molalities of other solutes, is equal to the ratio m lm of the molalities (solubilities expressed as molalities) of B in the saturated solutions. If a justifiable extrapolation to Ssms 0 can be made, then the separate ys s can be found. The method is especially useful when B is a sparingly soluble salt and the solubility is measured in the presence of varying molalities of other more soluble salts. Finally, the activity coefficient of an electrolyte can sometimes be obtained from e.m.f. measurements on galvanic cells. The measurement of activity coefficients and analysis of the results both for solutions of a single electrolyte and for solutions of two or more electrolytes will be dealt with in a subsequent volume. Unfortunately, few activity coefficients have been measured in the usually multi-solute solutions relevant to chemical reactions in solution. 

Determination of the Activity Coefficient from Measured Cell Potential... 

Accuracy and Interpretation of Measured pH Values. The acidity function which is the experimental basis for the assignment of pH, is reproducible within about 0.003 pH unit from 10 to 40°C. If the ionic strength is known, the assignment of numerical values to the activity coefficient of chloride ion does not add to the uncertainty. However, errors in the standard potential of the cell, in the composition of the buffer materials, and ia the preparatioa of the solutioas may raise the uacertaiaty to 0.005 pH unit. 

The activity coefficients of sulfuric acid have been deterrnined independentiy by measuring three types of physical phenomena cell potentials, vapor pressure, and freeting point. A consistent set of activity coefficients has been reported from 0.1 to 8 at 25°C (14), from 0.1 to 4 and 5 to 55°C (18), and from 0.001 to 0.02 m at 25°C (19). These values are all based on cell potential measurements. The activity coefficients based on vapor pressure measurements (20) agree with those from potential measurements when they are corrected to the same reference activity coefficient. 

Optically active molecules show circular dichroism. Their extinction coefficients f l and are different and change as a function of wavelength. Using a suitable spectroelectrochemical cell, Af = fl -which is usually small compared to conventional extinction coefficients, can be measured. Combined with the special properties of a thin layer cell kinetic data can be extracted from CD-data [01 Liu]. (Data obtained with this method are labelled CD.)... 

The review of Martynova (18) covers solubilities of a variety of salts and oxides up to 10 kbar and 700 C and also available steam-water distribution coefficients. That of Lietzke (19) reviews measurements of standard electrode potentials and ionic activity coefficients using Harned cells up to 175-200 C. The review of Mesmer, Sweeton, Hitch and Baes (20) covers a range of protolytic dissociation reactions up to 300°C at SVP. Apart from the work on Fe304 solubility by Sweeton and Baes (23), the only references to hydrolysis and complexing reactions by transition metals above 100 C were to aluminium hydrolysis (20) and nickel hydrolysis (24) both to 150 C. Nikolaeva (24) was one of several at the conference who discussed the problems arising when hydrolysis and complexing occur simultaneously. There appear to be no experimental studies of solution phase redox equilibria above 100°C. 

Delay et al (Ref 12) detd IR absorption spectra in the range 3 to l9u and from the intensities of the bands concluded that the sym form was more abundant in the azides of Ag, Cu, Hg Na but the reverse was true for the azides of Pb Tl. Gray Wad ding ton (Ref 18) stated th at TlNj crysts are isomor-phous with those of Na Rb azides. The elec conductivity of TIN, is 5.9 x 10 s mho at 275° (Ref 18). Brouty (Ref 10) detd the mean activity coefficient of TIN, by EMF+ measurements and calcd ionic radii of Ti Nj. Conductivity measurements by Brouty (Ref 11) did not agree with Onsager s theory deviations were found at very high dilutions. An electro-chem cell used by Suzuki (Ref 16) gave a Ap1 29S° value of 59.17 kcal/mol for... 

This equation provides a means of determining the transference number of the negative ions from measurement of the emf of the cell with the conditions that all of the assumptions made in obtaining the equation are valid and that the values of the mean activity coefficients in the solutions are known. An equation can be derived by use of the same methods for the case in which the solutions contain several solutes. When the electrodes are reversible with respect to the M+ ion, the equation is... 

The left-hand side of this equation can be calculated from measurements of cell voltage as a function of concentration. The second term on the right-hand side becomes zero at infinite dilution. However, because no meaningful measurements can be made at zero concentration of reactants, we must extrapolate the equation to infinite dilution using the known concentration behavior of activity coefficients. In approaching infinite dilution, it is sufficient to use the Debye-Huckel... 

The activity a2 of an electrolyte can be derived from the difference in behavior of real solutions and ideal solutions. For this purpose measurements are made of electromotive forces of cells, depression of freezing points, elevation of boiling points, solubility of electrolytes in mixed solutions and other characteristic properties of solutions. From the value of a2 thus determined the mean activity a+ is calculated using the equation (V-38) whereupon by application of the analytical concentration the activity coefficient is finally determined. The activity coefficients for sufficiently diluted solutions can also be calculated directly on the basis of the Debye-Hiickel theory, which will bo explained later on. 

In this table, P represents anions of protein and organic phosphate. The membrane is permeable to the group represented by P. The mean values of the charge on P are -6.7 and -1.08 for the interior and the exterior of the cell, respectively. An electrical potential difference of At// = i/t, t// = 90 mV is measured, i and o denote the intracellular and extracellular, respectively. The activity coefficients of components inside and outside the cell are assumed to be the same, and pressure and temperature are 1 atm and 310 K. Assume that the diffusion flows in from the surroundings are positive and the diffusion flows out are negative. Using tracers, the unidirectional flows are determined as follows ... 

The Nemst equation applies (if we neglect the activity coefficients of the ions, in keeping with PB theory) to the emf (electromotive force) of an electrochemical cell. The emf of such a cell and the surface potential of a colloidal particle are quantities of quite different kinds. It is not possible to measure colloidal particle with a potentiometer (where would we place the electrodes ), and even if we could, we have no reason to expect that it would obey the Nemst equation. We have been at pains to point out that all the experimental evidence on the n-butylam-monium vermiculite system is consistent with the surface potential being roughly constant over two decades of salt concentration. This is clearly incompatible with the Nemst equation, and so are results on the smectite clays [28], Furthermore, if the zeta potential can be related to the electrical potential difference deviations from Nemst behavior, as discussed by Hunter... 

One notes that a knowledge of the mean molal activity of HCi in a solution of molality m and the tabulation of standard emfs enables one to calculate the value for the schematized cell. Normally, however, the procedure is used in reverse i.e., from a measurement of emfs the mean activity coefficients for ions in solution may then be determined. The procedure is now... 

Measurements of emf (electromotive force) are to be made with this cell under reversible conditions at a number of concentrations c of HCl. From these measurements relative values of activity coefficients at different concentrations can be derived. To obtain the activity coefficients on such a scale that the activity coefficient is unity for the reference state of zero concentration, an extrapolation procedure based on the Debye-Huckel limiting law is used. By this means, the standard electrode emf of the silver-silver chloride electrode is determined, and activity coefficients are determined for all concentrations studied. 

The measured electrical potential of the central vacuole relative to the external solution, EM> is -155 mV, which is very close to the calculated Nernst potential for K+. Thus K+ in this case may be in equilibrium between the external solution and the central vacuole. The K+ concentration in the central vacuole of these Chara cells is 60 mol m-3. The activity coefficient for K+ in the central vacuole (yK) equals (cij = yff Eq. 2.5), so yKis (48 mol m 3)/(60 mol m-3), or 0.80, and the K+ activity coefficient in the bathing solution (y ) is about 0.96. For this example, in which there are large differences in the internal and the external concentrations, the ratio y /yK is (0.96)/(0.80), or 1.20, which differs appreciably from 1.00. If concentrations instead of activities are used in Equation 3.6, then Enk is —162 mV, which is somewhat lower than the measured potential of —155 mV. Calculating from the concentration ratio, the suggestion that K+ is in equilibrium from the bathing solution to the vacuole could not be made with much confidence, if at all. 

The standard potential of the silver-silver bromide electrode has been determined from emf measurements of cells with hydrogen electrodes and silver-silver bromide electrodes in solutions of hydrogen bromide in mixtures of water and N-methylacetamide (NMA). The mole fractions of NMA in the mixed solvents were 0.06, 0.15, 0.25, and 0.50, and the dielectric constants varied from 87 to 110 at 25°C. The molality of HBr covered the range 0.01-0.1 mol kg 1. Data for the mixed solvents were obtained at nine temperatures from 5° to 45°C. The results were used to derive the standard emf of the cell as well as the mean ionic activity coefficients and standard thermodynamic constants for HBr. The information obtained sheds some light on the nature of ion-ion and ion-solvent interactions in this system of high dielectric constant. 

The activity coefficients can thus be determined directly from this equation, using the measured values of the e.m.f. of the cell depicted above. 

Determination of Transference Numbers.—Since activity coefficients can be derived from e.m.f. measurements if transference numbers are known, it is apparent that the procedure could be reversed so as to make it possible to calculate transference numbers from e.m.f. data. The method employed is based on measurements of cells containing the same electrolyte, with and without transference. The e.m.f. of a concentration cell without transference E) is given by equation (11), and if the intermediate electrodes are removed so as to form a concentration cell with transference, the e.m.f., represented by Et, is now determined by equation (25), provided the transference numbers may be taken as constant within the range of concentrations in the cells. It follows, therefore, on dividing equation (25) by (11), that... 

The product of the activity coefficients can be estimated from the Debye-Hiickel equations, and mcr and E are known hence nin in the given solution can be derived from the measured e.m.f. of the cell. 

Thus, from a tabulation of standard emfs, from the specification of the Us for pure phases as in Section 3.7, and from use of the Debye-Hiickel limiting law one can calculate the emf of the particular cell of interest. Usually, the procedure is used in reverse, whereby a measurement of yields the activity or activity coefficient for the solutes of interest. 

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