Equilibrium single solute

Figure 8.2 Schematic illustrations of AGm versus X2 showing how jUj -may be determined by the tangent drawn at any point, (a) The polymer-solvent system forms a single solution at all compositions, (b) Compositions between the two minima separate into equilibrium phases P and Q.

Adsorption and ion exchange share so many common features in regard to apphcation in batch and fixed-bed processes that they can be grouped together as sorption for a unified treatment. These processes involve the transfer and resulting equilibrium distribution of one or more solutes between a fluid phase and particles. The partitioning of a single solute between fluid and sorbed phases or the selectivity of a sorbent towards multiple solutes makes it possible to separate solutes from a bulk fluid phase or from one another. 

In Chapter 13 we dealt with the equilibrium established when a single solute, either a weak acid or a weak base, is added to water. This chapter focuses on the equilibrium established when two different solutes are mixed in water solution. These solutes may be—... 

As before ft s without superscript refer to the clathrate for convenience we neglect the vapor pressure of Q, but this is not essential. Equilibrium between clathrate and gas was implicitly assumed in our derivation it led to Eq. 12. Equilibrium between clathrate and Qa is governed by Eq. 21 with the = sign. Repeating these equations for a single solute A we have... 

The most important application to be considered under this heading is the calculation of intrinsic defect concentrations in dilute solid solutions. If the solution is so dilute that only the leading terms in the various cluster expansions need be retained then the results required are slight generalizations of those above and follow at once from the notation for the general results. For example, the equilibrium concentration of vacancies in a dilute solution of a single solute, s, is found from Eqs. (74a) and (75) to be... 

To calculate the multicomponent vapor-liquid equilibrium, equilibrium constants for chemical reactions 1-9 are taken from literature in comparison to the original publication, in the present work different numerical values for the second dissociations of hydrogen sulfide and sulfur dioxide were chosen (cf. Appendix III). Henry s constants are evaluated from single solute solubility data without neglecting Poynting corrections ... 

A number of attempts have been made to modify the IAS model (Eqs. 22-29) to improve its accuracy and reduce computational efforts. Using the IAS model, DiGiano et al. [80] derived a Simplified Competitive Equilibrium Adsorption Model (SCAM). This model, which is based on the Freundlich isotherm, assumes the single-solute isotherms of all the components are equal and it utilizes average isotherm constants when this assumption is not valid. The IAS model equations have been reduced to a single expression ... 

We consider a transfer at constant temperature of an infinitesimal amount of any single solute i from a position r in the cell in the centrifugal field to a second position r + dr. For this transfer at equilibrium, at constant T and external P, dG = 0, so we write in place of Equation (21.12),... 

A wide class of analytic second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter r], possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system s free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r e 0 (for T < TCI) bifurcating from it (Fig. 2.3.4a). In the (F, rf) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). 

The equilibrium concentrations in the solid and liquid phases at the surface of the adsorbent particle are qsfc end Cs and afc, bk and 8k are isotherm constants for each solute corresponding to the single solute isotherm equation (4). 

The system of equations (1) to (10) provide the basis for predicting multicomponent rate profiles. The input parameters required are the mass transfer and diffusion coefficients for each solute, the single solute isotherm constants, and the mixture equilibria correlation coefficients. Estimation of these equilibrium and rate parameters are discussed in the following sections. 

The equations (20) to (30) provide the basis for predicting the adsorption rate profiles for the binary system. The input parameters required for the model are the single-solute film transfer and surface diffusion coefficients, the single-solute isotherm constants and the mixture equilibria coefficients. The rate parameters were obtained from single solute rate data (20), and the equilibrium parameters were obtained from single and multi-solute equilibrium data. 

When S(t) is calculated from computer simulation on a system containing a single solute molecule, the overbar is interpreted as an average over statistically independent nonequilibrium trajectories. The total Stokes shift can also be obtained via equilibrium statistical mechanical theory or simulation,... 

The important question for the designer of a continuous countercurrent extraction process is how many equilibrium stages are needed, given the flow rates and inlet and outlet concentrations A simple graphical procedure is possible for a single-solute extraction when the two... 

D. If all reactants and products are initially 1 m, what is the concentration of each at equilibrium in a single solution ... 

Order-disorder, or rod-to-coil , transitions in dilute solution have been reported for polydiacetylenes (2, 5-11), polysilylenes (12-15), and alkyl-substituted polythiophenes (16). The interpretation of the experimental observations has been the subject of considerable controversy with respect to whether the observations represent a single-polymer-molecule phenomenon or a many-chain aggregation or precipitation process (3-16). Our own experimental evidence (12, 13) and that of others (5-8, 10, 16) weigh heavily in favor of the single-chain interpretation. In our theoretical interpretation, we will assume that the order-disorder transitions observed in dilute pol-ysilylene solutions represent equilibrium, single-chain phenomena. 

A single solution at chemical equilibrium containing the components of more than one half-reaction can have only one electrode potential hence the value of E in (12-43) must equal that in (12-42). Therefore... 

Distribution of a single solute Consider a series of tubes numbered 0,1,2,3,..., r, each containing (for the sake of simplicity) equal volumes of an aqueous phase and an immiscible organic liquid of lower density. A solute is placed in the aqueous phase of tube 0. After equilibrium is reached, the organic phase of tube 0 is transferred to tube 1 as that of tube 1 is transferred to tube 2, and so forth. The process of equilibration and transfer is repeated n times each time, all upper organic layers are transferred to the next higher tube in sequence. The mechanics of connecting the vessels and of the transfer process need not concern us here. ... 

Theory The Freundlich isotherm developed in 1924 by Freundich (16) is widely used to study oil processing adsorption. It is applied as an empirical expression to describe the reversible adsorption of a single solute from aqueous solution at equilibrium at a fixed temperature, and it is expressed as... 

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