Mean activity coefficient determination

In the above two equations, the former value is valid for basic SI units and the latter value for / in moles per cubic decimetre and a in nanometres. The parameter a represents one of the difficulties connected with the Debye-Hiickel approach as its direct determination is not possible and is, in most cases, found as an adjustable parameter for the best fit of experimental data in the Eq. (1.3.29). For common ions the values of effective ion radii vary from 0.3 to 0.5. Analogous to the limiting law, the mean activity coefficient can be expressed by the equation... 

It would appear from Eq. (3.2.8) that the pH, i.e. the activity of a single type of ion, can be measured exactly. This is not, in reality, true even if the liquid junction potential is eliminated the value of Eref must be known. This value is always determined by assuming that the activity coefficients depend only on the overall ionic strength and not on the ionic species. Thus the mean activities and mean activity coefficients of the electrolyte must be employed. The use of this assumption in the determination of the value of Eref will, of course, also affect the pH value found from Eq. (3.2.8). Thus, the potentiometric determination of the pH is more difficult than would appear at first glance and will be considered in the special Section 3.3.2. 

For a pure salt dissolved in water it is not feasible to determine the activity coefficient, V , for the cation and anion separately so that the mean activity coefficient concept has been defined for a salt s cation-anion pair as... 

These individual-ion activity coefficients have the desired property of approaching 1 at infinite dilution, because each ratio a,/(m,/m°) approaches 1. However, individual-ion activity coefficients, like individual-ion activities, cannot be determined experimentally. Therefore, it is customary to deal with the mean activity coefficient 7+ and the mean activity a which for a uni-univalent electrolyte can be related to measurable quantities as follows ... 

However, only the mean activity (a ) or mean activity coefficient (y ) of an electrolyte can be determined by measurements, since in all processes, the electroneutral condition prevails. Note that... 

Over the last 20-30 years not too much effort has been made concerning the determination of standard potentials. It is mostly due to the funding policy all over the world, which directs the sources to new and fashionable research and practically neglects support for the quest for accurate fundamental data. A notable recent exception is the work described in Ref. 1, in which the standard potential of the cell Zn(Hg)jc (two phase) I ZnS O4 (aq) PbS O4 (s) Pb(Hg)jc (two phase) has been determined. Besides the measurements of electromotive force, determinations of the solubility, solubiKty products, osmotic coefficients, water activities, and mean activity coefficients have been carried out and compared with the previous data. The detailed analysis reveals that the uncertainties in some fundamental data such as the mean activity coefficient of ZnS04, the solubility product of Hg2S04, or even the dissociation constant of HS04 can cause uncertainties in the f " " values as high as 3-4 mV. The author recommends this comprehensive treatise to anybody who wants to go deeply into the correct determination of f " " values. 

We cannot determine values of the activity coefficients of the individual ions, but by definition of the mean activity coefficients (Eq. (11.182)), we have... 

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... 

Also, we denote that the mean activity coefficients are determined at the ionic strengths of the solutions. 

The standard potential difference of the Ag/AgCl reference electrode E° is determined in cell (I) filled with HC1 at a fixed molality. For the molality of 0.01 mol kg-1, the values for the mean activity coefficient of the HC1 are given in [7] at various temperatures. 

Activity coefficients of anions and cations in an electrolyte solution cannot be determined independently. Thus, the mean activity coefficient, y is defined by ... 

The potential of such an electrode is, therefore, determined by the activity of its own ions in the electrolyte such an electrode is said to be reversible with respect to its own cations. It is well known that the activity + is expressed by the product of the molality (or molarity) and the activity coefficient of the respective ion. Since, however, the activity coefficients of individual ions are not known, they are being replaced by the mean activity coefficients y In sufficiently diluted solutions it is possible to substitute in the formula (VI-11) directly the concentrations for activities, since the activity coefficients in this instance would very nearly equal unity. 

Only mean activity coefficients can be experimentally determined for salts, not activity coefficients for single ions. The Maclnnes Convention is one method for obtaining single ion activity coefficients and states that because of the similar size and mobility of the potassium and chloride ions ... 

Determine the mean activity coefficient with respect to... 

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... 

V when AgBr is the electrolyte, and to be 0.8085 V at a mole fraction x2 - 0.5937. Determine the mean activity coefficient for AgBr. 

Once the standard potential of Cell I has been determined precisely, calculations of the mean activity coefficient, y , of HC1 and the primary and secondary medium effects using well-known relations are relatively simple tasks. Using empirical equations of the type E = a - - bT + cT2 and E° = a0 + b0T + c0T2, it is possible to calculate the molal enthalpies and heat capacities. These types of calculations are demonstrated in many... 

Unfortunately, the calculated values of yt cannot be confirmed by direct experiment, because in principle all experimental methods yield the mean activity coefficient y rather than the individual ionic values. By use of the definition given in (2-16), the experimentally determined value can be apportioned to give nd y. This procedure is theoretically justified only at high dilution, where the DHLL is valid because the limiting slope of log y plotted against /n is found experimentally to be O.SZ Zb, as required by (2-17). At higher values of n the ion-size parameter a must be introduced. 

It is thus possible in principle to determine the mean activity coefficient of the electrolyte by dividing the actual vapour pressure of the electrolyte over the solution by that over an ideal solution which is obtained by finding by extrapolation to very dilute solutions (c/. 21.27). In fact the methods outhned in the two following paragraphs are more precise and more easily applied. 

The mean activity coefficient of a sparingly soluble salt in any solution, containing other electrolytes, can thus be evaluated provided the solubility product and the mean molality of the ions of the salt in the given solution are known. In order to obtain X, the values of are determined from the experimentally observed solubilities of the sparingly soluble salt in the presence of various amounts of other electrolytes, and the results are extrapolated to infinite dilution (Fig. 27). In the latter case the activity coefficient is unity, in accordance with the chosen standard state, and hence, by equation (39.71), KV"" is equal to the extrapolated value of... 

Combine these data with the results obtained in Exercise 6 to determine the mean activity coefficient of the hydrochloric acid at molalities of 0.05, 0.1, 0.5 and 1.0 at 25 C. 

Experimental determination of single-ion activity coefficients such as those shown in Table 10-2 is unfortunately impossible because all experimental methods give only a mean activity coefficient for the positively and negatively charged ions... 

Based on agreement between calculated and experimental values of mean ionic activity coefficients, we can infer that the Debye-Hiickel relationship and the data in Table 10-2 give satisfactory activity coefficients for ionic strengths up to about 0.1 M. Beyond this value, the equation fails, and we must determine mean activity coefficients experimentally. 

Values of/x = Ac/A may be calculated from Kohlrausch s measurements of electrical conductivity of hydrochloric acid solutions. /h and fci can be evaluated from the potentiometric measurements on hydrochloric acid solutions performed by Scatchaed. These data are very reliable since the concentration chain was so arranged as to eliminate diffusion potentials. In this way, ScATCHARD determined the mean activity coefficient V/h/ci instead of the individual ion activities. If we assume that in a potassium chloride solution/ = /ci— which is plausible when we recall that both ions have the same structure—and that fci is the same in hydrochloric acid solutions and potassium chloride solutions of the same concentration, then we can calculate/h and fci in hydrochloric acid solutions. Naturally these values are not strictly correct since the effect of the potassium ions on the activity of the chloride ions probably is different from that of the hydrogen ions at the same ionic strength. In the succeeding table are given values of /x, /h, and fci calculated by the above method. 

The experimental results for cell (9.5.1) can be used to determine the mean activity coefficient of the electrolyte HCl. Expressing the individual ionic activities in terms of the mean activity coefficient and the electrolyte molality, equation (9.5.3) can be rewritten as... 

Fig. 1 0.29 Plots of the change in ionic activity coefficients determined by specific ion electrodes against acetamide concentration cam for 0.25 M NaF Na ion activity coefficient ( ) ion activity coefficient (A) mean activity coefficient for NaF( ) [53],...

The treatment of the solubility data in the paper for the determination of the standard enthalpy of dissolution could not be fully understood and the following treatment was resorted to by the evaluator. The data in the temperature interval 283 to 313 K were selected. Approximate activity coefficients were taken from the data for MgS04 in [50HAR/OWE]. A second order polynomial was fitted to these data and mean activity coefficients for CaSe04 in the saturated solutions obtained by interpolation. No attempt was made to correct for the temperature variation of the activity coefficient. 

Eq. (B.l) will allow fairly accurate estimates of the aetivity coefficients in mixtures of electrolytes if the ion interaction coefficients are known. Ion interaction coefficients for simple ions can be obtained from tabulated data of mean activity coefficients of strong electrolytes or from the corresponding osmotic coefficients. Ion interaction coefficients for complexes can either be estimated from the charge and size of the ion or determined experimentally from the variation of the equilibrium constant with the ionic strength. 

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