Osmotic coefficients

The osmotic pressure (20.86), the lowering of freezing point (22.9), and the elevation of boiling point (21.59) are all proportional to the logarithm of the activity of the solvent (c/. 20.9), given by 

Suppose we have a solution obtained by mixing moles of solvent and 7I2 moles of an electrolyte. Then if is the number of moles of 2 undissociated we have 

To evaluate from freezing point measurements it is thus necessary to have a knowledge of e. If this is not known then we can do no more than calculate an apparent osmotic coefficient (j a, which is calculated as though the substance 2 were not dissociated (e = 0) or a coefficient cf)a by assuming complete dissociation (e = 1). These are related by the equations 

In very dilute solutions, the lowering of freezing point 6 is given by cf. 22.10-22.12), 

For example at 0 C, in an aqueous solution of acetic acid of molality 0-002138, the three osmotic coefficients have the values  

When a solution only contains one solute, Eq. (2.49) becomes 

By inclusion of the expression for the activity coefficient (log y) given in Eq. (2.43) and then by making the substitution u = (m l l+Bajm ), it can be shown that m = u l(l-Baju) from which, in turn, leads to the following expression from which the osmotic coefficient can be calculated 

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As pointed out earlier, the contributions of the hard cores to the thennodynamic properties of the solution at high concentrations are not negligible. Using the CS equation of state, the osmotic coefficient of an uncharged hard sphere solute (in a continuum solvent) is given by... 

The themiodynamic properties calculated by different routes are different, since the MS solution is an approximation. The osmotic coefficient from the virial pressure, compressibility and energy equations are not the same. Of these, the energy equation is the most accurate by comparison with computer simulations of Card and Valleau [ ]. The osmotic coefficients from the virial and compressibility equations are... 

The solutions to this approximation are obtained numerically. Fast Fourier transfonn methods and a refomuilation of the FINC (and other integral equation approximations) in tenns of the screened Coulomb potential by Allnatt [M are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [ ]. 

Figure A2.3.12 The osmotic coefficient of a 1-1 RPM electrolyte compared with the Monte Carlo results of...

Figure A2.3.14 Osmotic coefficients for 1-1, 2-1 and 3-1 RPM electrolytes according to the MS and HNC approximations.

The osmotic coefficients from the HNC approximation were calculated from the virial and compressibility equations the discrepancy between ([ly and ((ij 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-Htickel limiting law for the energy and osmotic coefficient of a 2-2 RPM electrolyte according to several theories. 

Figure A2.3.16. Theoretical HNC osmotic coefTicients for a range of ion size parameters in the primitive model compared with experimental data for the osmotic coefficients of several 1-1 electrolytes at 25°C. The curves are labelled according to the assumed value of a+- = r+ + r-...

Figure A2.3.17 Theoretical (HNC) calculations of the osmotic coefficients for the square well model of an electrolyte compared with experimental data for aqueous solutions at 25°C. The parameters for this model are a = r (Pauling)+ r (Pauling), d = d = 0 and d as indicated in the figure.

In principle, simulation teclmiques can be used, and Monte Carlo simulations of the primitive model of electrolyte solutions have appeared since the 1960s. Results for the osmotic coefficients are given for comparison in table A2.4.4 together with results from the MSA, PY and HNC approaches. The primitive model is clearly deficient for values of r. close to the closest distance of approach of the ions. Many years ago, Gurney [H] noted that when two ions are close enough together for their solvation sheaths to overlap, some solvent molecules become freed from ionic attraction and are effectively returned to the bulk [12]. 

Table A2.4.4. Osmotic coefficients obtained by various methods.

Osmotic coefficient Molality basis K [Pg.94]

The activity coefficient of water is related to the osmotic coefficient by the formula ... 

This difficulty can be overcome by writing equivalent expressions whose variables do not go to infinity at the limit of. vi—>0. A function known as the practical osmotic coefficient is one that can be used in a graphical method to obtain In 7 4.2- The practical osmotic coefficient expressed in terms of mole fraction is defined as... 

A form of equation (6.181) using the molality m instead of vt is often used. The osmotic coefficient expressed in terms of molality is given by... 

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... 

The osmotic coefficient is often used as a measure of the activity of the solvent instead of a because a is nearly unity over the concentration range where 7 is changing, and many significant figures are required to show the effect of solute concentration on a. The osmotic coefficient also becomes one at infinite dilution, but deviates more rapidly with concentration of solute than does a. ... 

The osmotic coefficient 4> and activity coefficient are related in a simple manner through the Gibbs-Duhem equation. We can find the relationship by writing this equation in a form that relates a and 2-... 

P7.5 The osmotic coefficients of aqueous CaCf solutions at 298.15 K are as follows ... 

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

The changes in osmotic coefficients with temperature and concentration make it difficult to solve the above equations accurately, but accurate determinations of the composition and relative amounts of the concentrated liquid and ice can be made from phase diagrams which are plots of the freezing points of solutions versus their concentration. From these, it is possible to determine the exact NaCl concentration at any temperature. Examples are shown in Figure 9 for solutions of 0 to 2.0 M glycerol in 0.15 M NaCl. This figure nicely illustrates how the presence of glycerol reduces the concentration of NaCl in the residual unfrozen solution. 

Osmotic coefficient data measured by Park (Park and Englezos, 1998 Park, 1999) are used for the estimation of the model parameters. There are 16 osmotic coefficient data available for the Na2Si03 aqueous solution. The data are given in Table 15.1. Based on these measurements the following parameters in Pitzer s... 

Table 15.1 Osmotic Coefficient Data for the Aqueous Naf>iO Solution...

Molality Osmotic Coefficient ([Pg.269]

The calculated osmotic coefficient is obtained by the next equation... 

A, is the Debye-Hiickel osmotic coefficient parameter m is the molality of solute... 

There are 26 experimental osmotic coefficient data and they are given in Table 15.2 (Park and Englezos, 1999 Park, 1999). Two sets of the binary parameters for the NaOH and Na2Si03 systems and two mixing parameters. 

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