Infinite-dilution limit

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... 

Further simphfication of the SPM and RPM is to assume the ions are point charges with no hard-core correlations, i.e., du = 0. This is called the Debye-Huckel (DH) level of treatment, and an early Nobel prize was awarded to the theory of electrolytes in the infinite-dilution limit [31]. This model can capture the long-range electrostatic interactions and is expected to be valid only for dilute solutions. An analytical solution is available by solving the Pois-son-Boltzmann (PB) equation for the distribution of ions (charges). The PB equation is... 

This equation has the expected behavior that AG< becomes more positive with decreasing solubility of the solute. However, free energies of solvation for different solutes cannot be related to their relative solubilities unless the vapor pressures of the different solutes are similar or one takes account of this via Equation 76. Furthermore, if the solubility is high enough that Henry s law does not hold, then one must consider finite-concentration activity coefficients, not just the infinite-dilution limit. 

To obtain K, inspect Figure 1 which plots the flowing phase concentration against the stationary phase concentration of component 1. In the infinite dilution limit, we have... 

This reduces to Eq. (156) in the infinite dilution limit go). In the Rouse... 

Figure 3.15 Plot of integral heat of solution Aifsoln(n) versus n (= moles H20/moles acid), showing the infinite-dilution limit A/fsoln(oo), the heat of dilution AHdn(ti, n2) from nx to n2, and the differential heat of solution (slope of tangent line) 8H(n ), 8H(n2) for representative concentrations...

The vertical bar and subscript (L) denote that the partial derivatives are to be evaluated in the infinite-dilution limit of the left-hand chamber.] In order that (7.68b) be satisfied for all sufficiently small AP and Axa, the two first-order correction terms in (7.69) must cancel, i.e. (omitting the evaluation limit for simplicity),... 

As in the analogous case of gases (Section 2.4), corrections for nonideality can be obtained by measurements of osmotic pressure at different solute concentrations, with extrapolation toward the infinite-dilution limit. For electrolytes, the correction for ionic dissociation is important. 

The group contribution method UNIFAC [18] is based on the UNIQUAC thermodynamic model as well. It thus suffers from the same thermodynamic approximations as UNIQUAC, especially for strong interactions in the infinite dilution limit. 

Here, the intrinsic viscosity [r ] is defined as the infinite dilution limit of the reduced viscosity and kH is Huggins coefficient. This result is valid up to the second order in concentration. The PFPE solvent interaction is related to kfj [r ]2. For flexible polymer chains in a theta solvent, kn has been found to vary from 0.4 to 1.0 [110]. From the intercept and slope, we obtain [r ] and kH. These values... 

The mean ionic and single-ion activity coefficients are conceptually different parameters, but both must conform to the Debye-Hiickel infinite-dilution limit. This theoretical constraint on activity coefficients takes on a particular mathematical form, depending upon the way in which an electrolyte solution is characterized. In a strictly thermodynamic picture of aqueous solutions, the Debye-Hiickel limit can be expressed as follows 9... 

The single-chain structure factors calculated in the previous sections correspond to the infinite dilution limit. This limit also corresponds to zero scattering intensity and is not useful so that concentration effects have to be included in the modeling of polymer solutions. First, Zimm s single-contact approximation [5] is reviewed for dilute polymer solutions then, a slight extension of that formula which applies to semidilute solutions, is discussed. 

Equation 7.19 shows that the electrochemical potential of the ion is constant and equal to its value at an infinite distance from the plates, as it must be at thermal equilibrium in the infinite dilution limit. 

Brownian motion of a single noninteracting particle can be described in terms of self-diffusion characterized by Do, the particle self-diffusion coefficient in the infinite dilution limit. The probability / (Ar. r) of a particle displacement Ar in time r satisfies the diffusion equation... 

Thus the limiting values of G /ji A 2i r are equal to tlie infinite-dilution limits of In yi and lny2. Tins result is illustrated in Fig. 2.5 b). 

The expressions of (9 In Ya/ x )p T,x and of their infinitely dilute limits are important for the thermodynamics of dilute ternary solutions [20,21], especially for dilute supercritical ternary solutions [9-12], The limiting expressions (17) and (20) were already derived in a different way by Jonah and Cochran [12] and Chailvo [11], In the next section of the paper, the above expressions will be applied to ternary supercritical solutions. 

The superscript ° signifies the infinite dilution limit. Equations 2.4.13 for the diffusion... 

There are four dimensionless parameters in eqns. (4.3.7 to 4.3.9) o, r,y, xji, and kjj. This version of the WS mixing rule can be used as a four-parameter model to correlate the behavior of complex mixtures or in several ways that have fewer adjustable parameters. For example, one can solve the two relations obtained from eqn. (4.3.8) in the infinite dilution limit bi... 

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