Chemical potential increment

Anywhere a chemical potential increment or gradient exists, an elementary separation step can occur. Anywhere random flow currents exist, separation is dissipated. Thus random flow currents are parasitic in regions where incremental chemical potential is used for separation. These currents should thus be eliminated, insofar as possible, in regions where electrical, sedimentation, and other continuous (c) fields are generating separations. Likewise, they should not be allowed to transport matter over discontinuous (d) separative interfaces such as phase boundaries or membrane surfaces. However, they are nonparasitic in bulk phases (removed from the separative interface) where only diffusion occurs. Here, in fact, they aid diffusion and speed the approach to equilibrium. This positive role is recognized in the following category of flow. 

The chemical potential increments of fields and interfaces provide the selective influence generally needed for separation. The selectivity exhibits itself in the form of unequal concentration distributions along the axis of the field or across the interface. However, separation along this coordinate is often subdued by the intrinsic limitations of two-phase systems or by the inherent weakness of certain fields applied to particular classes of molecules (see Chapter 8). 

The negative chemical potential increment associated with migration from radius rx to r2 is the force integrated over the rx to r2 interval. This becomes... 

For the liquid to remain in equilibrium with its vapor, the chemical potential p" of the latter should experience an increase by exactly the same increment, i.e. Ap"=Ap, which means that the equilibrium vapor pressurep(r) over the curved interface should be higher than that over the flat interface p0. If the vapor follows the ideal gas law, its chemical potential increment can be written as... 

The external (induced) chemical potential increment is not only inversely proportional to the cluster radius, but can also depend on the nature of adsorbed species of type i, in particular the dimension of reactive molecules and the adsorption strength. These are included in the following way as a first approximation... 

Equation (A2.1.23) can be mtegrated by the following trick One keeps T, p, and all the chemical potentials p. constant and increases the number of moles n. of each species by an amount n. d where d is the same fractional increment for each. Obviously one is increasing the size of the system by a factor (1 + dQ, increasing all the extensive properties U, S, V, nl) by this factor and leaving the relative compositions (as measured by the mole fractions) and all other intensive properties unchanged. Therefore, d.S =. S d, V=V d, dn. = n. d, etc, and... 

Before pursuing the diffusion process any further, let us examine the diffusion coefficient itself in greater detail. Specifically, we seek a relationship between D and the friction factor of the solute. In general, an increment of energy is associated with a force and an increment of distance. In the present context the driving force behind diffusion (subscript diff) is associated with an increment in the chemical potential of the solute and an increment in distance dx ... 

As will be seen later (Section V.l), meaningful molecular weights in multicomponent systems can be determined, if the specific refractive index increment appertains to conditions of constant chemical potential of low molecular weight solvents (instead of at constant composition). Practically, this can be realised by dialysing the solution against the mixed solvent and then measuring the specific refractive index increment of the dialysed solution. The theory and practice have been reviewed4-14-1S> 72>. 

It is fortunate that theory has been extended to take into account selective interactions in multicomponent systems, and it is seen from Eq. (91) (which is the expression used for the plots in Fig. 42 b) that the intercept at infinite dilution of protein or other solute does give the reciprocal of its correct molecular weight M2. This procedure is a straightforward one whereby one specifies within the constant K [Eq. (24)] a specific refractive index increment (9n7dc2)TiM. The subscript (i (a shorter way of writing subscripts jUj and ju3) signifies that the increments are to be taken at constant chemical potential of all diffusible solutes, that is, the components other than the polymer. This constitutes the osmotic pressure condition whereby only the macromolecule (component-2) is non-diffusible through a semi-permeable membrane. The quantity... 

Fig. 54. Specific refractive index increments at constant composition (o) and constant chemical potential ( ) for solutions of nylon-6 in 2,2,3,3-tetrafluoropropanol/l-chlorophenol binary mixtures, is the volume fraction of l-chlorophenol and filled circles refer to the two pure single solvents161)...

Fig. 24. Contour plot of the structure factor (the kinematic LEED intensity) of a x y/i monolayer in a triangular lattice gas with nearest-neighbor repulsion, at a temperature k TIi = 0.355 (about 5% above T ) and a chemical potential // = 1.5 (0c = 0.336 at the transition temperature.) Contour increments are in a (common) logarithmic scale separated by 0.1, starting with 3.2 at the outermost contour. Center of the surface Brillouin zon is to the left k, and k the radial and azimuthal components of kH, are in units of nlXla, a being the lattice spacing. Data are based on averages over 2x10 Monte Carlo steps per site. (From Bartelt et...

The chemical potential of species i, is expressed in terms of the Gibbs free energy added to a system at constant T and P, as well as relative to the mole fraction of each added increment of i. When adding an incremental number of molecules of i, free energy is introduced in the form of internal energies of i as well as by the... 

In this formula, v and v are the increments of the refractive index for the polymer to the increments measured at constant molarity and at constant chemical potential, respectively. (dn/dOs ) is the refractive index increment of the monomer in pure solvent... 

In this paper we have used the quantity (1 — vp0) in writing equations for sedimentation equilibrium experiments. Some workers prefer to use the density increment, 1000(dp/dc)Tfn, instead when dealing with solutions containing ionizing macromolecules. This procedure was first advocated by Vrij (44), and its advantages are discussed by Casassa and Eisenberg (39). Nichol and Ogston (13) have used the density increment in their analysis of mixed associations. The subscript p. means that all of the diffusible solutes are at constant chemical potential in the buffer... 

A thermodynamic approach was put forward by one of us (10), based on the Flory-Huggins lattice theory of a polymer solution the chemical potentials of each monomer must be equal in each phase copolymerization increment causes a little change in the chemical potential in the particles diffusion of monomers from the water phase will reequilibrate the system and in turn diffusion from droplets to water phase takes place. For instance, expression from monomer 1 in the particles is ... 

The thermodynamics approach to macromolecules in solution and their interactions with solvent components have been reviewed by Eisenberg (1990). Briefly, the density increment of a solution containing three components—(1) water, (2) macromolecules, and (3) salt—due to an increase in the concentration of component 2 at constant chemical potential of the other components, is given by... 

VFT behavior is obtained by equating z = T/(T — Tv ft) and noting that increment of chemical potential Ap — Vft [65], Fitting the VFT model to the experimental results of iH in the paraelectric phase gives Tv ft = 228 and Ap 0.02 eV. Identifying the temperature at which xB deviates from the Arrhenius model with the onset of cooperativity yields a minimum cluster size given by... 

Table 1. Relationship between X and the physical solute properties using different FFF techniques [27,109] with R=gas constant, p=solvent density, ps=solute density, co2r=centrifugal acceleration, V0=volume of the fractionation channel, Vc=cross-flow rate, E=electrical field strength, dT/dx=temperature gradient, M=molecular mass, dH=hydrodynamic diameter, DT=thermal diffusion coefficient, pe=electrophoretic mobility, %M=molar magnetic susceptibility, Hm=intensity of magnetic field, AHm=gradient of the intensity of the magnetic field, Ap = total increment of the chemical potential across the channel...

The routine approach to studying the effect of the thermodynamics of intermediate state Kj (reaction complex reactant—active center ) on the overall reaction rate is to find the functional dependence of the reaction rate on standard chemical potential of this intermediate component (or increment Ak of this standard value with respect of the reference state = Pg° + Akj ) and to determine the maximum of this functional dependence. The influence of the intermediate state on the characteristics of transition complexes of both steps is expressed, in accordance with (4.76), as follows ... 

A decrease in size of a condensed phase particle results in an increase in the chemical potential of the substance due to excess surface energy and the elevation of the Laplace pressure inside the particle (see Section 1.1). With a spherical particles of radius r, one can estimate the increment Aj of the chemical potential of component i of the continuous phase by the expression... 

The increment relates predominantly to the enthalpy contribution to the Gibbs potential and, therefore, contributes to the standard chemical potential of the condensed phase components ... 

Figure 8.1 The process of computing the incremental chemical potential involves adding one extra segment to an M - 1 segment chain moving in the solvent. The tangent hard sphere model of a (M — l)-mer (M = 5) is shown here. The dashed circles enclose the volume excluded to the centers of the solvent spheres.

Consider computing the excess chemical potential for a chain molecule composed of M monomers, denoting such a molecule type by a - The longer the chain, the more difficult the computation. It is easier to compute the incremental chemical potential for addition of a single segment. Following the notation discussed in Section 1.3, p. 16,... 

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