Homogeneous-field approximation

In the homogeneous field approximation, the field interacts with the dipole moment of the molecule (cf. Chapter 12)... 

Chemical thermodynamics and kinetics provide the formalism to describe the observed dependencies of chemical-conformational reactions on the external physical state variables temperature, pressure, electric and magnetic fields. In the present account the theoretical foundations for the analysis of electrical-chemical processes are developed on an elementary level. It should be remarked that in most treatments of electric field effects on chemical processes the theoretical expressions are based on the homogeneous-field approximation of the continuum relationship between the total polarization and the electric field strength (Maxwell field). When, however, conversion factors that account for the molecular (inhomogeneous) nature of real systems are given, they are usually only applicable for nonpolar solvents and thus exclude aqueous solutions. Therefore, in the present study, particular emphasis is placed on expressions which relate experimentally observable system properties (such as optical or electrical quantities) with the applied (measured) electric field, and which include applications to aqueous solutions. 

Fermi, Nudear Physics, University of Chicago Press, Chicago, 1950, p. 142. the homogeneous field approximation, the field interacts with the dipole moment of the molecule (cf. Chapter 12) V x,t) = = —fi where S denotes the electric field intensity of the... 

Chapter 15 - It was shown, that the reesterification reaction without catalyst can be described by mean-field approximation, whereas introduction of catalyst (tetrabutoxytitanium) is defined by the appearance of its local fluctuations. This effect results to fractal-like kinetics of reesterification reaction. In this case reesterification reaction is considered as recombination reaction and treated within the framework of scaling approaches. Practical aspect of this study is obvious-homogeneous distribution of catalyst in reactive medium or its biased diffusion allows to decrease reaction duration approximately twofold. 

To parameterize the polymersome model, the identification of the virial coefficients, vaij and wapv is driven by the requirement that the amphiphiles described by (10) and (12) should create a stable bilayer with given material properties. Assuming that the hydrophobic interior should be in a melt state, the coefficients vaa and Waaa are determined such that (12) enforces the A-blocks to create a melt in equilibrium with its vapor which, in a solvent free model, represents the surrounding water. It can be shown, from (12), that the equation of state of such a homogeneous melt, within mean-field approximation, is [138]... 

We consider further the case in which the system is subject to the action of an external homogeneous field and spatial quantisation exists (which is approximately true for weak fields). The alterations of m and the polarisation of the light are then subject to the rules derived above. It is easy to see that the transition possibilities Aj = —1, 0, +1, which are valid for a free system, remain true for j. 

If we go further and assume an extreme form of the mean-field approximation, namely that the liquid is homogeneous even on the scale of length of u(r), then we can put g(r) = 1 in (4.112). An integration by parts then returns us to Laplace s equation for [Pg.92]

With this identity, either of (5.9) or (5.10) could have been inferred from the other. Indeed, with given by (4.69) and UIV by (4.73), it follows from (5.11) that the only thermodynamically consistent that is uniform and temperature-independent is one which is proportional to the density , as in (5.7). This shows that the two aspects of the mean-field approximation in a homogeneous fluid, (5.3) and (5.7), are not independent. It B important always, in any application of the ideas of the mean-field theory, to verify its thermodynamic consistency in this way. ... 

We continue here to treat the model of attracting hard spheres in mean-field approximation, but we no longer assume the fli to be homogeneous. Taking the more general (5.6) instead of (5.7), and using the inhomogpneous form of (5.1), we shall find a functional equation for the density profile of the liquid-gas interface. 

To this point we have outlined the properties of the homogeneous phases in the lattice-gas model and in (5.25) or (5.27), and (5.31) have applied the mean-field approximation. We turn now to the inhomogeneous lattice gas, and treat the liquid-vapour interface in the same approximation. 

The probabilities q. and must still be determined as functions of p, and Pb- We cannot simply identify q. with p. and qt with Pb, even though the analogous identification was made successfully in the mean-field approximation to the one-component lattice gas in 5.3, where it led fo u ,t = -cep in the homogeneous fluid ((5.7) and (5.24), or (5.33) taken in the limit in which the fluid is homogeneous). We verified the thermodynamic consistency of the earlier mean-field theory in the discussion following (5.11). In the present model, had we identified q. with p. and qb with Pb, the thermodynamic identity (Appendix 1)... 

The free-energy density of the homogeneous fluid can be found in the mean-field approximation by integrating the Gibbs-Helmholtz equation, since we know the energy density < (p), (5.82). 

In 5.6 we studied the interface in a symmetric mixture model—the two-component, primitive version of the penetrable-sphere model— which we treated in a mean-field approximation. Thm is a three-component version of that model due to Guerrero e( of., with which we may illustrate many of the ideas of this section. Again, like molecules do not interact, while unlike molecules repel each other as hard spheres. If the components are a, 6, c, then in any homogeneous phase, in mean-field approximation, the densities p., etc., and activities etc., all in units of tlte volume Vo the exdusion sphere, are related by ... 

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