Equilibrium flow

The scheme of the interaction mechanism (Equation 88) testifies to an electro-affinity of MeFe" ions. In addition, MeFe" ions have a lower negative charge, smaller size and higher mobility compared to MeF6X(n+1> ions. The above arguments lead to the assumption that the reduction to metal form of niobium or tantalum from melts, both by electrolysis [368] and by alkali metals, most probably occurs due to interaction with MeF6 ions. The kinetics of the reduction processes are defined by flowing equilibriums between hexa-and heptacoordinated complexes. 

The extraction is represented by a single perfectly mixed constant volume, continuous flow equilibrium stage. This may actually consist of separate mixer-settler units. 

PHASE VOLUMES INLET CONC. AND FLOWS EQUILIBRIUM CONSTANT... 

Osmosis is the passage of a pure solvent into a solution separated from it by a semipermeable membrane, which is permeable to the solvent but not to the polymeric solute. The osmotic pressure n is the pressure that must be applied to the solution in order to stop the flow. Equilibrium is reached when the chemical potential of the solvent is identical on either side of the membrane. The principle of a membrane osmometer is sketched in Figure 2. 

Fig. 9.2 Simple model for a flow equilibrium, also known as a steady state these states are an important property of living systems... 

The higher air partial pressure Pp2 at the condenser exit is produced by an accumulation of air, vi/hich, as long as it is present at the exit, results in a stationary flow equilibrium. From this accumulation of air, the (eventually throttled) gas ballast pump in equilibrium removes just so much as streams from the entrance (1) through the condenser. 

The charge is simultaneously in perfect energy flow equilibrium in 4-flow. It continuously receives EM energy from the time dimension (imaginary plane), transduces the energy into real 3-space, and radiates it radially outward in... 

In the syringe-type pump the liquid is enclosed in a cylinder. A piston moves at a constant speed to push the liquid. Eluent compressibility induces time-consuming flow equilibrium. Nevertheless, the flow from a syringe pump is pulse free. For micro LC, flow rates of 50 yuL/min are utilized in spite of the drawback of column pressurization. With very low flow rates (in the nanoliter range) the use of pumps is tedious, and split-flow techniques are required. 

The pronounced discrepancy between the measured dynamic 15 °C-elution curve and its extrapolated reversible-thermodynamic part, shown in Fig. 7, represents a direct proof of the inadequacy of the reversible Eq. (3) in the dynamic region of the column (PDC-effect). Moreover, the experiment shows immediately that the polymer of the mobile phase has to dissolve in the gel layer within the transport zone to a considerably higher extent than is allowed by the partition function (4) in a reversible-thermodynamic equilibrium between the gel and the sol at the same column temperature. As a consequence, a steady state, i.e. a flow-equilibrium, must be assumed in the system sol/gel within the considered transport zone, governing the polymer trans-... 

It is well known that a flow-equilibrium must be treated by the methods of irreversible thermodynamics. In the case of the PDC-column, principally three flows have to be considered within the transport zone (1) the mass flow of the transported P-mer from the sol into the gel (2) the mass flow of this P-mer from the gel into the sol and (3) the flow of free energy from the column liquid into the gel layer required for the maintenance of the flow-equilibrium. If these flows and the corresponding potentials could be expressed analytically by means of molecular parameters, the flow-equilibrium 18) could be calculated by the usual methods 19). However, such a direct way would doubtless be very cumbersome because the system is very complicated (cf. above). These difficulties can be avoided in a purely phenomenological theory, based on perturbation calculus applied to the integrated transport Eq. (3 b) of the PDC-column in a reversible-thermodynamic equilibrium. 

Fig. 13. Flow-equilibrium proposed for PDC to explain the measured PDC-efTect shown in Fig. 7...

It is well-known that the entropy production (150 is minimal in a flow-equilibrium where all mass-flow vanish, whereas the third flow of free energy remains 18,19). In the vicinity of the reversible-equilibrium, the relations... 

To find the time required until the flow-equilibrium in the transport zone is achieved, the eigenvalue problem of the system of Eqs. (14) with v = 1 must be solved. This was done in Ref. 4) the results show that this time is essentially longer than the time needed for conformational changes in the macromolecules transferred between sol and gel (e.g. = 10 ps for the P = 1082-mer at 15 °C). 

The phenomenological concept described above allows to find the partition function Q(P) = (cg/cs)flow of the flow-equilibrium by means of a perturbation calculus applied to Eq. (3 b) the reversible partition function K(P) = cjcs in Eq. (3 b) is replaced by Q(P) Q(P) is set equal to K(P) multiplied by an exponential factor containing the free enthalpy of deformation of the coils transported from the sol into the gel through the gel front, where a strong and steep velocity gradient of the column liquid deforms the coil chain with this a new non-linear integrated transport equation... 

This was the idea behind concept (14b) in Ref.4). The corresponding formulation of Eq. (15i) of Ref.4), however, was unhappily chosen Eq. (15f)ofthis paper should have been used. If the kinetics of separation were explicitly introduced into the transport Equation of PDC (instead of the implicit concept (I7a-c) of the flow-equilibrium), an integrodifferential equation more complicated than (41 a-b) would be obtained, which could hardly be solved analytically... 

The calculation of the phenomenological function a(P T) of the flow-equilibrium from the measured calibration curves, shown in Figs. 7 and 8, is based on a nonlinear fit of Eq. (19) to these curves. It proceeds by the same method 4) as applied to the calculation of the reversible-thermodynamic data from Table 2 in Section 3.1 the phenomenological function a(P T), obtained in this way, is shown in Table 3 b. With this, the relative perturbation, 8Q/K, of the thermodynamic equilibrium by the transport can be calculated according to Eq. (20). 

The temperature dependence of the function 5Q/K is shown in Fig. 14 for four typical weight average degrees of polymerization, as indicated. The shape of these curves resembles the shape of the standard deviations cr of the corresponding elution curves, as shown in Fig. 153. This agreement represents a direct proof of the correctness of the flow-equilibrium concept proposed, because the function ct — crP = 0 represents an independent measure for the column resolution not based on any concept for the resolution mechanism. 

T o calculate all kinetic constants of the flow-equilibrium, only the (P, T)-dependence of the rate constant ks of a spontaneous polymer diffusion from the sol into the gel must be investigated independently. The rate constant kg of the reversible rediffusion of this P-mer from the gel into the sol follows from Eqs. (16b) and (4) and from Table 2,... 

The rate constant k of the corresponding retarded rediffusion in the flow-equilibrium is obtained from Eqs. (21) and (20) and Table 3. The reduced axial rate x of the axial... 

Fig. 14. Temperature dependence of the perturbation function 8Q(P)/K(P) of the flow-equilibrium calculated from PDC-measurements for four typical weight average degrees of polymerization Pw of the injected polystyrene sample 3), as indicated...

According to this picture, some distribution of diffusion distances must exist in the gel, from which the above mentioned rediffusion into the sol is started. After forming a mean value, a mean depth of penetration, A, due to the coll diffusion, can be defined in the gel, representing some part of the gel thickness, 1, and depending both on the partition function K(P) and on its relative perturbation, 5Q(P)/K(P), by the polymer transport in the flow-equilibrium. This mean depth of penetration A and the diffusion coefficient Dg of the P-mer in the gel evidently represent the main factors contributing to the expression for the rate constant ks to be found. 

Figure 17 and Table 4 show the dependence of the rate constant k on the weight average of the degree of polymerization as indicated. This dependence was calculated from Eq. (28c) and assumed to be independent of the temperature in the range 10 to 30 °C of the measurements with the system PS/CHX (cf. Ref.5), p. 2853). The P-dependence of the axial transport rate x, calculated from Eq. (25) for the mean overall volume rate w = 15 cm3/h of CHX at three typical column temperatures is also shown in Fig. 17 and Table 4. Fig. 18 and Table 5 show the dependence of the corresponding rate constants kg for the reversible rediffusion, and kg for the retarded rediffusion of the polymer in the flow-equilibrium, at the column temperature for four typical average degrees of polymerization. These values were calculated from Eqs. (24) and (21), respectively, using ks from Fig. 17. It can be seen that the functions kg(T P) (dashed in Fig. 18) represent asymptotes to the functions k T P) (full lines), as expected.

Table 6. Example for the efficiency of the flow-equilibrium calculated from the measurements3> according to the theory of PDC, cf. also Fig. 7. (Zero column volume = 160.54 cm3 volume rate...

The high resolution of the PDC-column, as demonstrated in Table 6, can well be explained by the assumption that the flow-equilibrium is fully active for the 1082-mer at 15 °C in the system PS/CHX, whereas it is only weakly active for the 353-mer at this temperature. This assumption can additionally be supported by an investigation of the deformability of these macromolecules by the velocity gradient near the gel front (see Sect. 3.4). But first the energetics of the flow-equilibrium need to be investigated. 

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