Infinite dilution equation

As used here [rj] c denotes the increment in relative viscosity which results when a resin molecule is added to the solution at concentration c. Consequently, its value for a given resin can strongly depend on the concentration, and on the nature of the diluent. At infinite dilution Equation 1 gives Meff = M, and the intrinsic viscosity has its conventional meaning, i.e., the limiting value of (rj8 — r)o)/voc at infinite dilution. Here rj8 is the viscosity of a solution of concentration c (grams/dl) and 770 is the solvent viscosity. 

As discussed in Chapter 11, as the molality of a species i approaches zero, the activity coefficient approaches unity so that, at infinite dilutions, equation (2.22) becomes ... 

X+ and X ° being the ionic conductivities at infinite dilution. Equation (4.9) has been written for infinite dilution since it is only under such conditions, when ion-ion interactions are at a minimum, that the law strictly holds. It is then applicable to both strong and weak electrolytes. Its validity is demonstrated in the data of Table 4.2. 

Over a range of concentrations with [rj c < 1, but not at infinite dilution, equation (176) for the viscosity is sometimes expressed in the form ... 

VSTR is useful for estimating partial molar volumes at infinite dilution but is not used here because of Equation (4-17)... 

Activity coefficients for condensable components are calculated with the UNIQUAC Equation (4-15)/ and infinite-dilution activity coefficients for noncondensable components are calculated with Equation (4-22). ... 

The solute-solvent interaction in equation A2.4.19 is a measure of the solvation energy of the solute species at infinite dilution. The basic model for ionic hydration is shown in figure A2.4.3 [5] there is an iimer hydration sheath of water molecules whose orientation is essentially detemiined entirely by the field due to the central ion. The number of water molecules in this iimer sheath depends on the size and chemistry of the central ion ... 

With the knowledge now of the magnitude of the mobility, we can use equation A2.4.38 to calculate the radii of the ions thus for lithium, using the value of 0.000 89 kg s for the viscosity of pure water (since we are using the conductivity at infinite dilution), the radius is calculated to be 2.38 x 10 m (=2.38 A). This can be contrasted with the crystalline ionic radius of Li, which has the value 0.78 A. The difference between these values reflects the presence of the hydration sheath of water molecules as we showed above, the... 

From equation A2.4.38 we can, finally, deduce Walden s rule, which states that the product of the ionic mobility at infinite dilution and the viscosity of the pure solvent is a constant. In fact... 

As written, equation 6.5 is a limiting law that applies only to infinitely dilute solutions, in which the chemical behavior of any species in the system is unaffected by all other species. Corrections to equation 6.5 are possible and are discussed in more detail at the end of the chapter. 

If the mutual solubilities of the solvents A and B are small, and the systems are dilute in C, the ratio ni can be estimated from the activity coefficients at infinite dilution. The infinite dilution activity coefficients of many organic systems have been correlated in terms of stmctural contributions (24), a method recommended by others (5). In the more general case of nondilute systems where there is significant mutual solubiUty between the two solvents, regular solution theory must be appHed. Several methods of correlation and prediction have been reviewed (23). The universal quasichemical (UNIQUAC) equation has been recommended (25), which uses binary parameters to predict multicomponent equihbria (see Eengineering, chemical DATA correlation). 

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) ... 

For a binaiy system comprised of species p and q, Eqs. (4-232), (4-312), and (4-315) may be written for species p at infinite dilution. The three resulting equations are then combined to yield... 

Outlined below are the steps required for of a X T.E calciilation of vapor-phase composition and pressure, given the liquid-phase composition and temperature. A choice must be made of an equation of state. Only the Soave/Redlich/Kwong and Peng/Robinson equations, as represented by Eqs. (4-230) and (4-231), are considered here. These two equations usually give comparable results. A choice must also be made of a two-parameter correlating expression to represent the liquid-phase composition dependence of for each pq binaiy. The Wilson, NRTL (with a fixed), and UNIQUAC equations are of general applicabihty for binary systems, the Margules and van Laar equations may also be used. The equation selected depends on evidence of its suitability to the particular system treated. Reasonable estimates of the parameters in the equation must also be known at the temperature of interest. These parameters are directly related to infinite-dilution values of the activity coefficients for each pq binaiy. 

Hayduk-Laudie They presented a simple correlation for the infinite dilution diffusion coefficients of nonelectrolytes in water. It has about the same accuracy as the Wilke-Chang equation (about 5.9 percent). There is no explicit temperature dependence, but the 1.14 exponent on I compensates for the absence of T in the numerator. That exponent was misprinted (as 1.4) in the original article and has been reproduced elsewhere erroneously. 

Concentrated, Binary Mixtures of Nonelectrolytes Several correlations that predict the composition dependence of Dab. re summarized in Table 5-19. Most are based on known values of D°g and Dba- In fact, a rule of thumb states that, for many binary systems, D°g and Dba bound the Dab vs. Xa cuiwe. CuUinan s equation predicts dif-fusivities even in hen of values at infinite dilution, but requires accurate density, viscosity, and activity coefficient data. 

Gordon Typically, as the concentration of a salt increases from infinite dilution, the diffusion coefficient decreases rapidly from D°g. As concentration is increased further, however, D g rises steadily, often becoming greater than D°g. Gordon proposed the following empirical equation, which is apphcable up to concentrations of 2N ... 

The general equation can be further reduced to the case of infinite dilution limit, a binary mixmre, ionic solutions, and so on. These equations are supplemented by closure relations such as the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. 

Here Q is the solute concentration and R the gas constant. This is in fact obeyed over a rather wide range of concentrations, almost up to solute mole fractions of 0.61, with an error of only 25 percent. This is remarkable, since the van t Hoff equation is rigorous only in the infinitely dilute limit. Even in the case of highly nonideal solutions, for example a solution with a ratios of 1.5 and e ratios of 4, the van t Hoff equation is still obeyed quite well for concentrations up to about 6 mole percent. It appears from these results that the van t Hoff approximation is much more sensitive to the nonideality of the solutions, and not that sensitive... 

If a single particle is falling freely under gravity in an infinitely dilute suspension, it will accelerate until it reaches a steady-state velocity. This final velocity is known as the terminal settling velocity (t/t) and represents the maximum useful superficial velocity achievable in a fluidised bed. Thus, the contained particles will be elutriated from the column if the superficial velocity is above Ut, the value of which can be predicted using the Stokes equation... 

For gas-liquid solutions which are only moderately dilute, the equation of Krichevsky and Ilinskaya provides a significant improvement over the equation of Krichevsky and Kasarnovsky. It has been used for the reduction of high-pressure equilibrium data by various investigators, notably by Orentlicher (03), and in slightly modified form by Conolly (C6). For any binary system, its three parameters depend only on temperature. The parameter H (Henry s constant) is by far the most important, and in data reduction, care must be taken to obtain H as accurately as possible, even at the expense of lower accuracy for the remaining parameters. While H must be positive, A and vf may be positive or negative A is called the self-interaction parameter because it takes into account the deviations from infinite-dilution behavior that are caused by the interaction between solute molecules in the solvent matrix. 

In the limit of infinitely dilute solutions, where equation (6.112) holds, m2 —f2/k2. If we maintain this ratio as our definition of activity, a2. then a2 = m2 in these solutions. For solutions which are not in the limiting region, we writecc... 

As indicated by the final equation, the dilution steps are continued until the infinitely dilute solution is approached. The sum of all of the steps represents the change to infinite dilution from the given starting solution. Thus, the sum of all... 

---