Activity coefficients Debye-Htickel equation

For solutions dilute enough for the Debye-Htickel equation to be applicable, the plot of log (t /7r) + A c against Qog 7r + log (t /7r)]Vc should be a straight line, the intercept for c equal zero giving the required value of -- log 7r, by equation (39.68). The values of log (y /yB,) are obtained from equation (39.67), and log yr, which is required for the purpose of the plot, is obtained by a short series of approximations. Once log yn has been determined, it is possible to derive log 7db for oy solution from the known value of log (y /yn)> The mean ionic activity coefficient of the given electrolyte can thus be evaluated from the e.m.f. s of concentration cells with transference, provided the required transference number information is available. ... 

For higher concentrations, Aiassoc will have to be corrected for non-ideality by including activity coefficients calculated from the Debye-Htickel equation. 

For ionic strengths up to about 0.01, activity coefficients from the Debye-Htickel equation lead to results from equilibrium calculations that agree closely with experiment even at ionic strengths of 0.1, major discrepancies are generally not encountered. At higher... 

The situation for electrolyte solutions is more complex theory confimis the limiting expressions (originally from Debye-Htickel theory), but, because of the long-range interactions, the resulting equations are non-analytic rather than simple power series.) It is evident that electrolyte solutions are ideally dilute only at extremely low concentrations. Further details about these activity coefficients will be found in other articles. 

For gases, pure solids, pure liquids, and nonionic solutes, activity coefficients are approximately unity under most reasonable experimental conditions. For reactions involving only these species, differences between activity and concentration are negligible. Activity coefficients for ionic solutes, however, depend on the ionic composition of the solution. It is possible, using the extended Debye-Htickel theory, to calculate activity coefficients using equation 6.50... 

Equation (7.45) is a limiting law expression for 7 , the activity coefficient of the solute. Debye-Htickel theory can also be used to obtain limiting-law expressions for the activity a of the solvent. This is usually done by expressing a in terms of the practical osmotic coefficient

electrolyte solute, it is defined in a general way as... 

According to the Debye-Htickel theory, in the limit of the infinitely dilute solution, individual-ion activity coefficients are given by the equation... 

Activity coefficients of ions are determined using electromotive force, freezing point, and solubility measurements or are calculated using the theoretical equation of Debye and Htickel. 

The Debye-Htickel limiting law equation gives an expression for the activity coefficient of an ion as... 

Likewise, yAB also cannot be measured experimentally, although, like Qa and Qb, 7a and yB can be measured, and at first sight the conversion given above may seem to give little improvement. However, for ion-ion and ion-molecule reactions, the Debye-Htickel theory, see Equation (7.8), can calculate the activity coefficient for any charged species and convert Equation (7.17) into a useful form. For other reactions the approach is only qualitative, but for them the effects of non-ideality are much smaller. 

In the primitive Debye-Htickel theory—one that did not allow for the size of ions—the value for the activity coefficient is given by Eq. (3.60). The corresponding equation in the MSA is... 

Plot the values for the activity coefficient of the electrolyte as calculated in Problem 3 against the ionic strength. Then see what degree of match you can obtain from the Debye-Htickel law (one-parameter equation). Do a similar calculation with the equation in the text which brings in the distance of closest approach (a) and allows for the removal of water from the solution (two-parameter equation). Describe which values ofa,andn (the hydration number) fit best. Discuss the degree to which the values you had to use were physically sensible. 

The experimentally determined activity coefficients, based on vapor pressure, freezing-point and electromotive force measurements, for a number of typical electrolytes of different valence types in aqueous solution at 25 , are represented in Fig. 49, in which the values of log / are plotted against the square-root of the ionic strength in these cases the solutions contained no other electrolyte than the one under consideration. Since the Debye-Htickel constant A for water at 25 is seen from Table XXXV to be 0.509, the limiting slopes of the plots in Fig. 49 should be equal to —0.509 the results to be expected theoretically, calculated in this manner, are shown by the dotted lines. It is evident that the experimental results approach the values required by the Debye-Hiickel limiting law as infinite dilution is attained. The influence of valence on the dependence of the activity coefficient on concentration is evidently in agreement with theoretical expectation. Another verification of the valence factor in the Debye-Hiickel equation will be given later (p. 177). 

Utilize the results obtained from the data of Saxton and Waters, given in Problem 7 of Chap. Ill, together with the activity coefficients derived from the Debye-Htickel limiting equation, to evaluate the dissociation constant of a-crotonic acid. 

Equation (29.29) was derived by Langelier [3] on the assumption that K and K2 are based on concentrations (moles/liter) rather than on activities. For example, referring to (29.24), if K, is the true activity product, then Ks = Ksjl, where y refers to the mean ion activity coefficient for CaCOs. The activity coefficient was approximated by Langelier using the Debye-Htickel theory, -logy = where p is the ionic strength and z is the valence. Hence,... 

The Debye-Htickel theory is a cornerstone of electrolyte theory. It is always used in extrapolating data to infinite dilution, and must be embedded in any generalized treatment of activity coefficients as a function of concentration, as it is in the Pitzer equations. However, at concentrations beyond the validity of the limiting law (Equation 15.26), all attempts at predicting electrolyte behavior at higher concentrations are more or less empirical. 

Lastly, the uniformity of ionic strength provided to a solution by the presence of ample supporting electrolyte limits effects due to the non-ideality of the solution. According to the Debye-Htickel theory, the presence of electrostatic interactions between ions causes solution non-ideality because these forces are on average stabihsing. Therefore, activity, the quantity appearing in the Nernst equation, differs from concentration by a factor known as the activity coefficient, y, in a maimer which for a dilute (<0.01 M) solution is given by a simplified formula ... 

The question of the accuracy of Debye-HuckeTs theory has arisen since an ion s activity cannot be measured. Hence, calculations cannot be compared with experimental values. Fortunately, the Debye-Huckel equations can be indirectly checked. This is achieved by comparing the calculated values of mean activity coefficients with those determined experimentally. Mean activity coefficients can indeed be determined experimentally, unlike the activity coefficients of the sole ions. Moreover, they can be calculated from the values found for the individual ions of the electrolyte by using the Debye-Htickel laws. As a result, we can see that it is possible to compare the experimental and calculated values of some activity coefficients. 

Strong and long-range Coulombic forces acting between ions are primarily responsible for the departures from ideality (the activity coefficients are lowered) and dominate all other contributions. The effect has been evaluated in the Debye-Htickel theory and there exist several equations, which are useful in estimating the mean activity coefficient [68, 69]. The latter is related to the ionic strength of the solution ... 

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