Infinite dilution, extrapolation

Infinite dilution (extrapolation to zero concentration) means so small a concentration that the possibility of two ions meeting, and thence associating, is tiny to non-existent. 

Thus, if the activities of the various species can be detennined or if one can extrapolate to infinite dilution, the measurement of the emf yields the standard free energy of the reaction. 

Extrapolation to infinite dilution requites viscosity measurements at usually four or five concentrations. Eor relative (rel) measurements of rapid determination, a single-point equation may often be used. A useful expression is the following (eq. 9) (27) ... 

Experimentally deterrnined equiUbrium constants are usually calculated from concentrations rather than from the activities of the species involved. Thermodynamic constants, based on ion activities, require activity coefficients. Because of the inadequacy of present theory for either calculating or determining activity coefficients for the compHcated ionic stmctures involved, the relatively few known thermodynamic constants have usually been obtained by extrapolation of results to infinite dilution. The constants based on concentration have usually been deterrnined in dilute solution in the presence of excess inert ions to maintain constant ionic strength. Thus concentration constants are accurate only under conditions reasonably close to those used for their deterrnination. Beyond these conditions, concentration constants may be useful in estimating probable effects and relative behaviors, and chelation process designers need to make allowances for these differences in conditions. 

It has already been mentioned that in an aqueous solution of KC1 at a concentration of 3.20 X 10-6 mole per liter, the equivalent conductivity was found to have a value, 149.37, that differed appreciably from the value obtained by the extrapolation of a series of measurements to infinite dilution. We may say that, even in this very dilute solution, each ion, in the absence of an electric field, does not execute a random motion that is independent of the presence of other ions the random motion of any ion is somewhat influenced by the forces of attraction and repulsion of other ions that happen to be in its vicinity. At the same time, this distortion of the random motion affects not only the electrical conductivity but also the rate of diffusion of the solute, if this were measured in a solution of this concentration. 

In the remainder of this book, in discussing the mobility of any species of ion, we shall be concerned only with the characteristic value obtained by extrapolation to infinite dilution. In the following chapters we shall accordingly use the letter w to denote the mobility at infinite dilution. 

Shortly after the formulation of the Debye-Huckel theory, a survey of the data on ionic mobilities from this point of view was made, extrapolating the values to infinite dilution.1 Table 4 gives values of Cl for atomic and molecular ions for 7 = 0°C and T2 = 18°C. 

In Figs. 31 and 32 the ordinates give the equivalent conductivity of HC1, each point being the result of a series of measurements extrapolated to infinite dilution.1 For comparison with similar diagrams given in a later... 

The quantity U is the only part of dF/dn that is characteristic of the species that are taking part in the reaction and if the concentrations of all the species can be measured in sufficiently dilute solution, an accurate value of U can be obtained by inserting the values of xlt zi. . . in (72). In this way, a value of U correct to less than one part in 1000 can often be obtained directly. If higher accuracy is desired, or if for some reason measurements cannot be made in sufficiently dilute solution, measurements are made in a series of dilute solutions, and the value of 2 In Xi at infinite dilution is obtained by extrapolation (see the example of an extrapolation in Fig. 33, in Sec. 63). 

The e.m.f. would thus be 0.048 volt, and its sign would be such that the spontaneous current transfers HC1 from ethanol to water. This is the sign that the resultant e.m.f. of equimolal cells, placed back to back, is found to have the value, extrapolated to infinite dilution, is either 0.296... 

In one of the two cells placed back to back, the solvent, as mentioned above, was pure water in each case. When the mixed solvent in the other cell contains only a small percentage of methanol, the resultant e.m.f. will obviously be small, and it should progressively increase with increasing difference between the solvents. In Fig. 61 abscissas are values of 1/e for the mixed solvent, running from 0.0126 for pure water to 0.0301 for pure methanol. Ordinates give the unitary part of the e.m.f. extrapolated to infinite dilution. It will be seen that for KC1, NaCl, and LiCl the curves differ only slightly from straight lines, but the curve for HC1 has quite a different shape. From the experimental results on the electrical conductivity depicted in Fig. 31 we expect the curve for HC1 to take this form. In Sec. 115 we shall discuss this result for HC1, and in Sec. 116 we shall return to the interpretation of the results obtained with the alkali chlorides. 

When the added water has a molarity n, let a fraction g of positive ions be alcoholic ions, while the fraction (1 — g) is in the form of (HjO)+ ions, On extrapolating to infinite dilution, the equilibrium constant of the reaction (43) may be written... 

In Chapter 7 we found it convenient to distinguish between proton transfers involving a solvent molecule and those involving only solute particles but this difference will lose its significance when the distinction between solvent and solute begins to break down. We recall that in Sec. 54 the mole fraction of the solvent did not differ appreciably from unity and could be omitted from (72). In investigating concentrated solutions, however, there is no question of extrapolating to infinite dilution the mole fraction of the solvent will differ from unity and will have to be retained in all formulas. At the same time each of the mole fractions will need to be multiplied by its activity coefficient. 

For strong electrolytes the molar conductivity increases as the dilution is increased, but it appears to approach a limiting value known as the molar conductivity at infinite dilution. The quantity A00 can be determined by graphical extrapolation for dilute solutions of strong electrolytes. For weak electrolytes the extrapolation method cannot be used for the determination of Ax but it may be calculated from the molar conductivities at infinite dilution of the respective ions, use being made of the Law of Independent Migration of Ions . At infinite dilution the ions are independent of each other, and each contributes its part of the total conductivity, thus ... 

It was found that the law of proportionality (12) holds good only if the solution is dilute, and in the determination of

different concentrations in dilute solutions, plotting these against the concentrations, and extrapolating to zero concentration, or infinite dilution (H.M., 67). 

We can generalize this result by stating that extrapolating enthalpy changes in solution to infinite dilution gives the enthalpy change AHa for the process5... 

Relative partial molar enthalpies can be used to calculate AH for various processes involving the mixing of solute, solvent, and solution. For example, Table 7.2 gives values for L and L2 for aqueous sulfuric acid solutions7 as a function of molality at 298.15 K. Also tabulated is A, the ratio of moles H2O to moles H2S(V We note from the table that L — L2 — 0 in the infinitely dilute solution. Thus, a Raoult s law standard state has been chosen for H20 and a Henry s law standard state is used for H2SO4. The value L2 = 95,281 Tmol-1 is the extrapolated relative partial molar enthalpy of pure H2SO4. It is the value for 77f- 77°. 

In Chapter 5 we saw how to determine Cp, 2 from heat capacity measurements. 07 is easily obtained from these same measurements using equation (7.106), after the value of C° 2 is obtained by extrapolation of Cp. 2 to infinite dilution. 

Extrapolate the data to infinite dilution to obtain a value for the molecular weight of the polymer. (Note that an average molecular weight is obtained since the polymer consists of a mixture of molecules of different chain lengths.)... 

Finally, they measured the enthalpy of solution of C HsO in water as a function of concentration and extrapolated to infinite dilution to get a value of -5.84 kJ-mol-1 for the reaction... 

Ais obtained from the enthalpies of solution of HCl(g) in water, extrapolated to infinite dilution. 

The hydrodynamic radius reflects the effect of coil size on polymer transport properties and can be determined from the sedimentation or diffusion coefficients at infinite dilution from the relation Rh = kBT/6itri5D (D = translational diffusion coefficient extrapolated to zero concentration, kB = Boltzmann constant, T = absolute temperature and r s = solvent viscosity). 

The value obtained by Hamm et alm directly by the immersion method is strikingly different and much more positive than others reported. It is in the right direction with respect to a polycrystalline surface, even though it is an extrapolated value that does not correspond to an existing surface state. In other words, the pzc corresponds to the state of a bare surface in the double-layer region, whereas in reality at that potential the actual surface is oxidized. Thus, such a pzc realizes to some extent the concept of ideal reference state, as in the case of ions in infinitely dilute solution. 

In physical chemistry, we apply the term colligative to those properties that depend upon number of molecules present. The principal colligative properties are boiling point elevation, freezing point depression, vapour pressure lowering, and osmotic pressure. All such methods require extrapolation of experimental data back to infinite dilution. This arises due to the fact that the physical properties of any solute at a reasonable concentration in a solvent are... 

We determine the value of [rj] experimentally by making measurements on a series of polymer solutions and plotting a graph of (p - iift/riQ c against c. This gives a straight line with intercept at [p]. The value of [p] is characteristic of the isolated polymer molecule in solution, because of the extrapolation to infinite dilution, and is a fimction of temperature, pressure, polymer type, solvent, and (most important of all in the present context) relative molar mass. 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

The change in the sedimentation constant with concentration enters solely from the change in 1//, and it is customary therefore to extrapolate a plot of 1/s against c to infinite dilution. The results of sedimentation studies by Newman and Eirich on several polystyrene... 

---