For concentration dependence

As with the diffusion coefficient, sedimentation coefficients are frequently corrected for concentration dependence and reduced to standard conditions ... 

The signal tram gas chroaatographic detectors can be further characterized as nass or concentration dependent. For concentration-dependent detectors the aost notable feature is that the detector response is dependent upon the flow rate through the detector (carrier gas and makeup gas, if any) and, therefore, the sensitivity of the detector is usually defined as the product of the pe dc area and flow rate divided by the wei t of the sanple. For B ss-dependent detectors sensitivity is defined as the product of the peak area divided by the sanple wel t in grans or noles and is independent of flow rate through the detector. 

Fig. 2 Determination of Bt values (amount of functional immobilized ligand in the column) for the immobilized Erythrina cristagalli agglutinin. / -Nitrophenyl, (pNP)-lactose, diluted to various concentrations (8 to 50 pM), was used for concentration-dependence analysis. (A) The solid and dotted lines demonstrate elution profiles of pNP-lactose and control sugar (pNP-mannose), respectively. (B) Woolf-Hofstee-type plot was made by using V-V0 values. Adapted from 47 with permission.

Nernst Equation for Concentration Dependence of RedOx Potential. Equation (5.9) applied to the general RedOx electrode (5.16) yields... 

The higher exponent for the hydrogen bonded compound was consistent with the Cates model for concentration-dependent association (Cates 1987). 

The square root of time dependence also holds for concentration-dependent diffusivity and the D value in the above equation would be a kind of average D across the profile. 

Derive the explicit numerical algorithm for solving a diffusion equation for concentration-dependent D. 

Figure 18.12 Vacuum electrochemical cell and assorted glassware that allows for concentration-dependent electrochemical studies using a dropping mercury electrode. [From Ref. 26, with permission.]...

Various methods have been proposed for the evaluation of D as a function of penetrant concentration from sorption measurements. They all are applicable only for sorption data of the Fickian type, and may be classified into two groups. Methods belonging to one group utilize data for the initial slope of the reduced sorption curve, while the ones belonging to the other group resort to rates at which M (<) approaches the equilibrium value Since little work has yet been done to adapt the approach-to-equihbrium data for concentration-dependent D, the subsequent discussion will be confined to methods of the former group only. 

Hence, in this case, the measurement of either absorption or desorption allows straightforward calculation of D of the given system. For concentration-dependent D the following approximation was first suggested [Crank and Park (1949)] ... 

Analytic solutions to Eq. (1) subject to these auxiliary conditions can be obtained only for the special case in which D is independent of concentration. For concentration-dependent D even numerical work has not as yet been attempted. It is fortunate that without recourse to actual calculations we can deduce necessary relations and important features basic to permeation curves for systems in which D is a function of concentration alone. Those are as follows ... 

First Aris Number Anx for Concentration-dependent Effective Diffusion Coefficients and Simple Reactions... 

The difference in the apparent reaction orders using the two techniques is probably due to the fact that the concentration profiles of reactants and intermediates depend upon the time scale. Thus, differences can be expected for concentration dependent competing mechanisms. In general terms where A represents a diactivated olefin, the major mechanism of dimerization in solvents of low water content was proposed to be given by eqns (121)-(122)... 

Most laboratory analysis methods measure concentration. The choice of units for concentration depends in part on the medium and in part on the process that is being measured or described. In water, a common expression of concentration is mass of chemical per unit volume of water. Many naturally occurring chemicals in water are present at levels of a few milligrams per liter (mg/liter). The fundamental dimensions associated with such a measurement are [M/L3]. The letters M, L, and T in square brackets refer to the fundamental dimensions of mass, length, and time, which are discussed further in the Appendix. For clarity in this book, specific units, such as (cm/hr) or (g/m3), either are free-standing or are indicated in parentheses, not in square brackets. 

The final boiling point for concentration depends on the water content required in the finished product. 106-110°C is characteristic for jellies (solid content 76-78%). When gums are produced the basic solution is heated to 120-130°C. As it cools to below 100°C, the gelatine, previously dissolved at 60-65°C, will be added. 

A discussion of various models for concentration dependent diffusion is available (21). and compilations of solutions to Pick s laws for some of these cases exist (l3, 14). 

Fig. 1 Calculated and measured ( ) swelling curves for PNIPAAm of different molecular weight-of the net chains in g/mol. The Flory-Huggins interaction parameter % was calculated using (12). The dashed curve was calculated considering only a temperature-dependent x (X2 = 0)-The sharp decrease of the volume degree of swelling with temperature is only observed for concentration-dependent interactions. The sharpness of the curve is influenced by Me...

FIGURE 20.3-6 Schematic representation of typical forms for concentration-dependent diffusion coefficients. 

These, and all other equations for concentration-dependent diffusion, consist of an infinite dilution diffusivity and a thermodynamic correction term. The thermodynamie correction term in all cases is equivalent to the derivative dGildx. The definition of the thermodynamic metastable limit (the spinodal curve) is the locus of points where dG2ldx = 0. This means that concentration-dependent diffusion theory predicts a diffusivity of zero at the spinodal. Thermodynamics tells us that the diffusivity goes from some finite value at saturation to zero at the spinodal. Unfortunately, it does not tell us how the diffusion coefficient declines. In addition, lack of thermodynamic data makes prediction of the spinodal difficult. We are, therefore, left with only the fact that as the concentration is increased in the supersaturated region, the diffusivity should decline towards zero but we do not know at what concentration the diffusivity becomes zero. 

It is rare to find diffusivity data of most species at any concentration near saturation. It is, therefore, necessary to first estimate the diffusivity at saturation after which the diffusivity in the supersaturated solution can be estimated. To estimate the diffusivity at saturation from low concentration data requires the use of an equation for concentration-dependent diffusion coefficients that can be used with solid solutes dissolved in liquid solvents. One such equation that can be used for nonelectrolytes is the... 

Using the relationships (2.29)-(2.33) together with Eqs. (2.20) and (2.22) for concentration dependencies of the potential, one can obtain the equations representing the experimentally measured properties of an equilibrium many-electron system via the equilibrium constants for any kind of stationary methods. 

For concentration-dependent detectors the implications of this, therefore, are ... 

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