Equilibria concentration profiles

The balancing of these opposing transport processes yields the equilibrium concentration profile of the analyte given by the well-known exponential relationship [4]... 

In a sedimentation equilibrium experiment the cell is rotated at a relatively low speed (typically 5000-10000 rpm) until an equilibrium is attained whereby the centrifugal force just balances the tendency of the molecules to diffuse back against the concentration gradient developed. Measurements are made of the equilibrium concentration profiles for a series of solutions with different initial polymer concentrations so that the results can be extrapolated to c = 0. A rigorous thermodynamic treatment is possible and enables absolute values ot Mwand Mz, to be determined. The principal restriction to the use of sedimentation equilibrium measurements is the long time required to reach equilibrium, since this is at least a few hours and more usually is a few days. 

Field-flow fractionation experiments are mainly performed in a thin ribbonlike channel with tapered inlet and outlet ends (see Fig. 1). This simple geometry is advantageous for the exact and simple calculation of separation characteristics in FFF Theories of infinite parallel plates are often used to describe the behavior of analytes because the cross-sectional aspect ratio of the channel is usually large and, thus, the end effects can be neglected. This means that the flow velocity and concentration profiles are not dependent on the coordinate y. It has been shown that, under suitable conditions, the analytes move along the channel as steady-state zones. Then, equilibrium concentration profiles of analytes can be easily calculated. 

An ultracentrifuge is found to have a lysozyme concentration of 1 mg/g of water 2 cm from the axis of rotation. Determine the equilibrium concentration profile over the range from 2 cm to 4 cm from the rotation axis if the temperature is 25° C and /... 

Figure 21.5. Illustration of the influence of local nonequilibrium on band dispersion. Dashed lines equilibrium concentration profile solid lines actual concentration profile.

After some time equilibrium will appear with a time constant Tp = Z /D. The equilibrium concentration profile is given by the first term of Eq. 2 ... 

Any modification in the double layer potential affects the concentration profile of ions that reside within the double layer structure. In the case of an enzyme-substrate reaction, in which a nonequilibrium contribution to the potential arises due to differences in substrate and product diffusion coefficient, as seen from Eq. (16), variation in the equilibrium concentration profiles of not only charged reactant and charged product but also variations in the concentration profiles of nonreacting ionic species present is possible.The concentration profiles of all ionic species adjust according to changes in the reaction rate, In order to... 

One particular component may be a flotsam with respect to some components in the bed, while simultaneously it is also a jetsam relative to other components. Thus the distinction between a jetsam and a flotsam is less important in a multicomponent system with density differences. The component distribution at equilibrium is of primary concern. There are mathematical models available in the literature which enable the calculation of this equilibrium concentration profile in a gas fluidized bed, to be discussed later. 

Mathematical Models for Prediction of Equilibrium Concentration Profiles... 

An initially non-linear pressure increase in the transient state is followed by a linear increase in the steady state when an equilibrium concentration profile in the film is reached. From the slope of the steady state pressure increase it is possible to calculate the permeability of a penetrant through a polymeric sample.The time to reach the stationary flux conditions is called lag-time and it allows us to determine the diffusivity of the system. Alternatively, in the pressure decay methods, a polymer sample and a gas are closed in a eonstant volume. The pressure decreases in time due to the sorption of the gas into the polymer are monitored. 

The scan rate, u = EIAt, plays a very important role in sweep voltannnetry as it defines the time scale of the experiment and is typically in the range 5 mV s to 100 V s for nonnal macroelectrodes, although sweep rates of 10 V s are possible with microelectrodes (see later). The short time scales in which the experiments are carried out are the cause for the prevalence of non-steady-state diflfiision and the peak-shaped response. Wlien the scan rate is slow enough to maintain steady-state diflfiision, the concentration profiles with time are linear within the Nemst diflfiision layer which is fixed by natural convection, and the current-potential response reaches a plateau steady-state current. On reducing the time scale, the diflfiision layer caimot relax to its equilibrium state, the diffusion layer is thiimer and hence the currents in the non-steady-state will be higher. 

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

Fig. 7. Constitutional supercooling, (a) impurity concentration profile during solidification (b) actual temperature T and equilibrium freezing temperature T...

The concentration profiles of the solute in both the mobile and stationary phases are depicted as Gaussian in form. In due course, this assumption will be shown to be the ideal elution curve as predicted by the Plate Theory. Equilibrium occurs between the mobile phase and the stationary phase, when the probability of a solute molecule striking the boundary and entering the stationary phase is the same as the probability of a solute molecule randomly acquiring sufficient kinetic energy to leave the stationary phase and enter the mobile phase. The distribution system is continuously thermodynamically driven toward equilibrium. However, the moving phase will continuously displace the concentration profile of the solute in the mobile phase forward, relative to that in the stationary phase. This displacement, in a grossly... 

The profile of the concentration of a solute in both the mobile and stationary phases is Gaussian in form and this will be shown to be true when dealing later with basic chromatography column theory. Thus, the flow of mobile phase will slightly displace the concentration profile of the solute in the mobile phase relative to that in the stationary phase the displacement depicted in figure 1 is grossly exaggerated to demonstrate this effect. It is seen that, as a result of this displacement, the concentration of solute in the mobile phase at the front of the peak exceeds the equilibrium concentration with respect to that in the stationary phase. It follows that there is a net transfer of solute from the mobile phase in the front part of the peak to the... 

It should be stressed that in the case of linear isotherm, the peak broadening effect results from eddy diffusion and from resistance of the mass transfer only, and it does not depend on Henry s constant. In practice, such concentration profiles are observed for these analyte concentrations, which are low enough for the equilibrium isotherm to be regarded as linear. 

To simulate the empirical concentration profiles, an appropriate mass-transfer model has to be used. One of the simplest models is the model based on the equilibrium-dispersive model, frequently used in column chromatography [1]. It can be given by the following equation ... 

As indicated before, the columns and the rows of a bilinear or trilinear dataset have a particular meaning, e.g., a spectrum and a chromatogram or the concentration profiles of reactants and the reaction products in an equilibrium or kinetic study. The resulting data table is made up by the product of the tables of these pure factors, e.g., the table of the elution profiles of the pure compounds and the table of the spectra of these compounds. One of the aims of a study of such a table is the decomposition of the table into its pure spectra and pure elution profiles. This is done by factor analysis (Chapter 34). 

The D-A model predicts the distribution of uranium and U-series isotopes across a bone section (Figs. 3 and 4). Under constant conditions Uranium is diffusing from the inner and outer surfaces of the bone, giving a u-shaped Uranium concentration profile that gradually flattens with time to a uniform uranium distribution when the bone reaches equilibrium with the uranium in the groundwater. Because the uranium is equilibrating with the outer portions of the bone section first, closed system U-series dates approach the true age of the bone towards the surfaces, but are underestimated towards the centre. Further details of the D-A model are given in the Appendix. 

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