Osmotic coefficient Debye-Hiickel theory

Debye-Hiickel theory 333-50 in electrochemical cells 481-2, 488 and osmotic coefficient 345-8 parameters 342... 

For an ideal solution, Jq = I and is unity. Then Eq. (9) is consistent with Eq. (10 11), since the total molality of all solute species is vm for a completely dissociated solute of molality m. For ionic solutions, the Debye-Hiickel theory predicts a value of yo different from unity and therefore a deviation of g from unity. A treatment of this aspect of the Debye-Hiickel theory is beyond the scope of this book, and we shall merely state the result. The osmotic coefficient g at 0°C for dilute solutions of a single strong electrolyte in water is given by... 

For hydrochloric acid, a strong electrolyte, calculate an experimental value of with Eq. (9) for each of the concentrations studied. In addition, use Eq. (13) to obtain a value of the osmotic coefficient based on the Debye-Hiickel theory for each concentration. Compare these experimental and theoretical values. 

They note that the Debye-Hiickel theory relates osmotic behavior to the effective ionic diameter. It would be expected then that species containing the same anions and having cations of similar diameter would have nearly equal osmotic coefficients unless there was a "chemical" difference between the two. This difference could be interpreted as incomplete dissociation or complex formation. 

The osmotic coefficient is again either set to unity (that is the common approach) or taken from the Debye—Hiickel theory for an aqueous solution containing ris moles of salt S and naoles of counterions dissociated from the... 

Electrostatic and statistical mechanics theories were used by Debye and Hiickel to deduce an expression for the mean ionic activity (and osmotic) coefficient of a dilute electrolyte solution. Empirical extensions have subsequently been applied to the Debye-Huckel approximation so that the expression remains approximately valid up to molal concentrations of 0.5 m (actually, to ionic strengths of about 0.5 mol L ). The expression that is often used for a solution of a single aqueous 1 1, 2 1, or 1 2 electrolyte is... 

Another attempt to go beyond the cell model proceeds with the Debye-Hiickel-Bjerrum theory [38]. The linearized PB equation is used as a starting point, however ion association is inserted by hand to correct for the non-linear couplings. This approach incorporates rod-rod interactions and should thus account for full solution properties. For the case of added salt the theory predicts an osmotic coefficient below the Manning limiting value, which is much too low. The same is true for a simplified version of the salt free case. 

Figure 3.48 shows two ways of expressing the results of Mayer s viriai coefficient approach using the osmotic pressure of an ionic solution as the test quantity. Two versions of the Mayer theory are indicated. In the one marked DHLL + B2, the authors have taken the Debye-Hiickel limiting-law theory, redone for osmotic pressure instead of activity coefficient, and then added to it the results of Mayer s calculation of the second viriai coefficient, B. In the upper curve of Fig. 3.48, the approximation within the Mayer theory used in summing integrals (the one called hypernetted chain or HNC) is indicated. The former replicates experiment better than the latter. The two approxi-... 

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between (tiy and ( )( is a measure of the accuracy of the approximation. The osmotic coefficients calculated via the energy equation in the MS approximation are comparable in accuracy to the HNC approximation for low valence electrolytes. Figure A2 3.15 shows deviations from the Debye-Hiickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

Recently, Manning [19] has developed a theory in which the real polyelectrolyte chain is replaced by an infinite line charge. Statistical mechanical considerations lead to the conclusion that sufficiently many counter-ions have to condense on the polyion to lower the charge-density parameter to unity. The non-condensed counterions may by treated by the Debye-Hiickel approximation. Thus, a limiting law has been developed which is supposed to be valid at high dilution. For the osmotic coefficient this limiting law is... 

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