Phase equilibrium, field-dependent

Snider is best known for his paper reporting what is now referred to as the Waldmann-Snider equation.34 (L. Waldmann independently derived the same result via an alternative method.) The novelty of this equation is that it takes into account the consequences of the superposition of quantum wavefunctions. For example, while the usual Boltzmann equation describes the collisionally induced decay of the rotational state probability distribution of a spin system to equilibrium, the modifications allow the effects of magnetic field precession to be simultaneously taken into account. Snider has used this equation to explain a variety of effects including the Senftleben-Beenakker effect (i.e., is, the magnetic and electric field dependence of gas transport coefficients), gas phase NMR relaxation, and gas phase muon spin relaxation.35... 

Compared with Equation (5.1), Equation (5.9) has the same form, except that the thermodynamic equilibrium magnetization is considered to be space-dependent, and that a space-dependent phase modulation has been introduced as a result of applying time-dependent magnetic field gradients. If the FID f(t) were constant in Equation (5.9), the spin density could be retrieved by inverse Fourier transformation of the single-pulse response s t), provided that the gradient modulation had been chosen in such a way that s(t) covers all values of the wave vector k, i.e., that the entire k space had been sampled. 

From the system of Eqs. (3.3) (3.4), it follows that at a given feed composition Zf and at a fixed field of phase equilibrium coefficients, Ki = fi(T, P, xi,... x ) separation products compositions xd and xb depend on only two parameters -relative withdrawal of one of the products D/Fand amount of theoretical plates N. At infinite reflux, the location of feeding plate does not influence the compositions of distillation products nor profile of concentrations. This is quite understandable -the external flow coming to the feeding plate is infinitely small in comparison with internal flows in the column. 

Equations (5.9), (5.10), (5.15), and (5.16) are necessary and sufficient conditions of trajectory tear-off from the boundary element of concentration simplex. Equations (5.9) and (5.10) can be called operating ones because they depend on separation mode, and Eqs. (5.15) and (5.16) can be called structural ones because they depend only on the structure of the field of phase equilibrium coefficients. 

For zeotropic mixtures, the main difficulty of the solution of synthesis task consists of the large number of alternative sequences that have to be calculated and compared with each other in terms of expenditures. This number greatly increases when the number of the products into which the mixture should be separated increases. The best sequence (or several sequences with close values of expenditures) depends on the concentrations of the components in the mixture under separation and on the field of phase equilibrium coefficients of the components in the concentration simplex. To ensure the solubility of the task of synthesis for multicomponent zeotropic mixtures, it is necessary to create a program system that would include as main modules programs of automatic design... 

Magnetostriction experiments were performed in the paramagnetic phase of different RSb compounds (Liithi et al. 1977). Such measurement enables one to determine size and sign of some of the magneto-elastic coupling constants. The effects result from the magnetic field dependence of the lattice equilibrium positions (see section 5.2.1). 

Fig. 2.37. Phase diagram for Ca0-Na20 Si02-(Al203)-H20 system in equilibrium with quartz at 400°C and 400 bars. Plagioclase solid solution can be represented by the albite and anorthite fields, whereas epidote is represented by clinozoisite. Note that the clinozoisite field is adjacent to the anorthite field, suggesting that fluids with high Ca/(H+) might equilibrate with excess anorthite by replacing it with epidote. The location of the albite-anorthite-epidote equilibrium point is a function of epidote and plagioclase composition and depends on the model used for calculation of the thermodynamic properties of aqueous cations (Berndt et al., 1989).

When an aqueous phase (noted w) is brought in contact with a second immiscible phase (noted o), the different species dissolved in one or the two phases spontaneously distribute depending on their hydrophilic-lipophilic balance until the thermodynamic equilibrium is reached. The distribution of the charged species generates an interfacial region, in which the electrical field strength differs from zero, so that an electrical Galvani potential difference, is established across the interface ... 

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

One can apply the MC technique to the same molecular model, as explored in MD. One can use the same box and the same molecules that experience exactly the same potentials, and therefore the results are equally exact for equilibrium membranes. However, MC examples of this type are very rare. One of the reasons for this is that there is no commercial package available in which an MC strategy is combined with sufficient chemistry know-how and tuned force fields. Unlike the MD approach, where the phase-space trajectory is fixed by the equations of motion of the molecules, the optimal walkthrough phase space in an MC run may depend strongly on the system characteristics. In particular, for densely packed layers, it may be very inefficient to withdraw a molecule randomly and to let it reappear somewhere else in... 

---