Approximations, equilibrium phenomena

The approximate solution of the evolution equation for P (x, y t) based on the hypothesis of low coupling between cells shows that the transition between two homogeneous stationary states occurs via an inhomogeneous transient state. Transient inhomogeneous structures may be also responsible for the stabilisation of an unstable state. Such kinds of phenomena are reminiscent of nucleation processes (Blanche, 1981 Frankowicz, 1984) and might be considered as a nonequilibrium analogue of an equilibrium phenomenon. 

This is the first example we have had of the phenomenon of resonance, which we shall discuss at some length in the next section. It should be pointed out now, however, that the hydrogen molecule has one structure which is described by one wave function, ip. However, it may be necessary because of our approximations, to write p as a combination of two or more wave functions, each of which only partially describes the hydrogen molecule. Table 5.1 lists values for the energy and equilibrium distance for the various stages of our approximation, together with the experimental values. 

The second type of resolution by direct crystallization is knovm as entrainment. Here, the differences in the rate of crystallization of the enantiomers in a supersaturated solution give rise to a separation. Strict control of the conditionsforthe crystallization are required, with the system of crystals and solution not being allowed to come to equilibrium and time playing an important role. The occurrence of conglomerates has been estimated to be approximately 10% of all racemic compounds. We will now illustrate this phenomenon with some pertinent examples. 

An approximate treatment of the phenomenon of the capillary rise can be easily made in terms of Laplace s equation. If the liquid wets the wall of the capillary, the liquid surface is forced to lie parallel to the wall, and the liquid surface has to be concave in shape. The pressure in the liquid below the surface is less than that in the gas phase above the liquid surface. If the capillary is circular in cross section and not too wide in radius, the meniscus will be approximately hemispherical, as is illustrated in Figure 6.5. Such a case is described well by Eq. (6.11). If h denotes the height of the meniscus above the flat liquid surface, then at equilibrium, AP must also be equal to the hydrostatic pressure of the liquid column inside the capillary. Thus AP = Apgh, where Ap is the difference in density between the liquid and gas phases and g is the acceleration due to gravity. Equation (6.11) then becomes... 

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