Limits dilute solution

Define fC = Xm/XaXb and = c /caCb, convert molar concentrations to mole fractions using the dilute solution limit, Eq. (6-25), thus obtaining Using Eqs. (6-21) and (6-22) yields... 

Sidebar 7.7 describes how the independent solute particle assumption (a) leads to expected Henry s law behavior in the dilute-solution limit. The saturation assumption (b) leads to the expectation (in agreement with observations) that... 

In the assumed dilute-solution limit, we can also assume that VA V, the total volume of solution. Under these conditions, (7.72) reduces to the Van t Hoff osmotic equation... 

Xv denotes that x is also defined on a volume fraction basis, the same as gv. The equation (1.6) would be not strictly applicable to the dilute solution limit, but it can be interpreted as the definition of x for the whole range of concentrations [6],... 

The case of infinite dilution of the polymer is the one for which total and preferential sorption have been most extensively studied. In this dilute solution limit the interpretation of the preferential adsorption coefficient,, requires knowledge of the g interpretation parameters at infinite dilution, g°, for the polymer in each one of the pure liquids [9],... 

The ionization of electrolytes is clearly manifest in the thermodynamic properties of their solutions. For example, in the ideally dilute solution limit, a solution of a strong electrolyte behaves as ions, rather than molecules, interacting with solvent molecules. A NaCl solution of molality m behaves, in the limit of infinite dilution, as an ideally dilute solution of concentration 2m, as 2 mol of ions are produced from each mole of NaCl dissolved in solution. A general strong electrolyte, dissociating by the equation... 

It is of practical as well as ideological importance that the modern theory of van der Waals forces reduces to the older forms derived for the interaction of individual small molecules in dilute gases. The modern approach can in fact be used to derive new expressions for the interaction between pairs of solutes in dilute solutions. The essential property of e in the dilute-gas or dilute-solution limit is that the dielectric response is strictly proportional to the number density of gas or solute molecules. That is, an electric field applied to a dilute gas or solution acts on each dilute species without distortion of the field by other gas or solute molecules. 

Other investigators (1 ) have obtained results similar to those of Kerner. The results of Dewey (ll) which are valid for a dilute solution agree with the Kerner equation in the dilute solution limit. Christensen ( 12) reviews and rederives the effective modulus calculations for spherical inclusions. The three models which are... 

FIGURE 11.15 Vapor pressures above a mixture of two volatile liquids. Both ideai (biue lines) and non-ideai behaviors (red curves) are shown. Positive deviations from ideal solution behavior are illustrated, although negative deviations are observed for other nonideal solutions. Raoult s and Henry s laws are shown as dilute solution limits for the nonideal mixture the markers explicitly identify regions where Raoult s law and Henry s law represent actual behavior. 

The results of all three methods are summarized below along with the results of the dilute solution limiting case, which is clearly inapplicable here. 

SOLUTION The composition profiles in the Stefan tube are given by Eq. 8.6.2. Before we can compute the profiles we must determine the rates of evaporation of acetone and methanol. Since the evaporating species are present in low concentrations at the top of the tube (although not at the bottom) we shall use the dilute solution limit for the effective diffusivities... 

Note that each second-law property diverges in the dilute-solution limit (x,- -> 0). Note also that each expression in (5.1.8)-(5.1.13) has the same functional form as the corresponding expression for an ideal-gas mixture (cf. 4.1.3). ... 

The Margules expressions for activity coefficients are based on the Lewis-Randall standard state (5.1.5), and therefore they must obey the pure-component limit (5.4.12). In addition, as with Porter s equations, the parameters A and A2 are simply related to the activity coefficients at infinite dilution. In particular, when we apply the dilute-solution limit (5.4.13) to (5.6.12) and (5.6.13), we obtain... 

To get a value for the slope of the ideal-solution straight line, we use the slope of the real fugacity curve in the dilute-solution limit. 

The normalization occurs in the dilute-solution limit, taken with T, P, and solute-free mole fractions held fixed. 

The intrinsic viscosity ( /] is obtained by extrapolation of reduced viscosity of the dilute polymer solution (q-qs)lcqs> to zero polymer concentration, c—>0 (here rj is the viscosity of the polymer solution and the viscosity of pure solvent). In the nondraining limit of large N, the coils behave in a shear flow as impermeable for the solvent particles of effertive radius J ,. In dilute-solution limit, the Einstein equation i/ = i/s[l+ (5/2) ] applies, where is the volume fraction of particles in the solution. Hence, the intrinsic viscosity [i/] measures the (inverse) average intramolecular concenttation of the monomer units assuming that they are confined within a sphere of radius J ,. 

Suppose a polymer chain consisting of N spheres of diameter b (pearl-necklace model see Fig. 1.34). We consider the dilute solution limit in which each chain is isolated from the other chains in the solution. When the chain dimension is R, these N spheres are contained in a cube of volume close to R, but no other spheres... 

Thus we find that (N) is the number average of Ni. The molecular weight estimated from the measurement of the osmotic pressure and Eq. 2.20 in the dilute solution limit is therefore the number-average molecular weight. 

In Chapter 3, we will learn about the dynamics of an isolated polymer chain in the dilute solution limit and the first-order change in the dynamics with polymer concentration. We will also learn typical experimental methods to investigate the dynamics—dynamic light scattering and viscosity. The dynamics of polymer solutions above the overlap concentration will be discussed in Chapter 4, along with their thermodynamics. 

Self-Diffusion and Mutual Diffusion When each suspension or solute molecule is moving independently, the diffusion is a single-particle phenomenon. The latter is observed in the dilute solution limit where there are no other solute molecules in the neighborhood. When other solute molecules are nearby, the diffusion is strongly affected by the other solute molecules. The second terms in Eqs. 3.17 and 3.45 are not negligible any more. What DLS measures is 5(k, t), not 5i(k, r). Only when c c, 5(k, t) is equal to 5i(k, t). Otherwise, the apparent diffusion coefficient D estimated from the slope of gi(r) depends on c. We will learn how the apparent D depends on c. 

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