Homogeneous systems, defined

For homogeneous systems, the average number density is n = (N) / V=v Let us define a local number... 

A very simple version of this approach was used in early applications. An alchemical charging calculation was done using a distance-based cutoff for electrostatic interactions, either with a finite or a periodic model. Then a cut-off correction equal to the Born free energy, Eq. (38), was added, with the spherical radius taken to be = R. This is a convenient but ill-defined approximation, because the system with a cutoff is not equivalent to a spherical charge of radius R. A more rigorous cutoff correction was derived recently that is applicable to sufficiently homogeneous systems [54] but appears to be impractical for macromolecules in solution. 

It may be recalled that in homogeneous reactions all reacting materials are found within a single phase, be it gas, liquid or solid if the reaction is catalytic, then the catalyst must also be present within the phase. Thus, there are a number of means of defining the rate of a reaction the intensive measure based on unit volume of the reacting volume (V) is used practically exclusively for homogeneous systems. The rate of reaction of any component i is defined as... 

The work of Crank [38] provides a review of the mathematical analysis of well defined component transport in homogeneous systems. These mathematical models and measured concentration profile data may be used to estimate diffu-sivities in homogenized samples. The use of MRI measurements in this way will generate diffusivities applicable to models of large-scale transport processes and will thereby be of value in engineering analysis of these processes and equipment. 

It is convenient to approach the concept of reaction rate by considering a closed, isothermal, constant pressure homogeneous system of uniform composition in which a single chemical reaction is taking place. In such a system the rate of the chemical reaction (r) is defined as ... 

The above sections have focused upon homogeneous systems with a constant composition in which tracer diffusion coefficients give a close approximation to selfdiffusion coefficients. However, a diffusion coefficient can be defined for any transport of material across a solid, whether or not such limitations hold. For example, the diffusion processes taking place when a metal A is in contact with a metal B is usually characterized by the interdiffusion coefficient, which provides a measure of the total mixing that has taken place. The mixing that occurs when two chemical compounds, say oxide AO is in contact with oxide BO, is characterized by the chemical diffusion coefficient (see the Further Reading section for more information). 

It is thus entirely expressed in terms of the zero wave number Fourier coefficient pQ(p t). Similarly, the pair correlation function in a spatially homogeneous system is defined by17... 

For the IEM model, it is well known that for a homogeneous system (i.e., when pn and .) remain constant and the locations ((). ) move according to the rates rn. Using the matrices defined above, we can rewrite the linear system as... 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

We have defined the chemical potential of a component as the partial derivative of the Gibbs free energy of the system (or, for a homogeneous system, of the phase) with respect to the number of moles of the component at constant P and T—i.e.,... 

In our assumptions, system (27) has the finite number of roots (by Lemma 14.2 in Bykov et al., 1998), so that the product in Equation (26) is well defined. We can interpret formula (26) as a corollary of Poisson formula for the classic resultant of homogeneous system of forms (i.e. the Macaulay (or Classic) resultant, see Gel fand et al., 1994). Moreover, the product Res(R) in Equation (26) is a polynomial of R-variable and it is a rational function of kinetic parameters fg and Tg (see a book by Bykov et al., 1998, Chapter 14). It is the same as the classic resultant (which is an irreducible polynomial (Macaulay, 1916 van der Waerden, 1971) up to constant in R multiplier. In many cases, finding resultant allows to solve the system (21) for all variables. ... 

With increasing flow rate, the orientational state in the nematic solution should change. Larson [154] solved numerically Eqs. (39) and (40b) with Vscf(a) given by Eq. (41) for a homogeneous system (T[f ] = 0) in the simple shear flow to obtain the time-dependent orientational distribution function f(a t) as a function of k. The non-steady orientational state in the nematic solution can be described in terms of the time-dependent (dynamic) scalar order parameter S[Pg.149]

Ion-exchange equilibrium can be considered to be analogous to chemical equilibrium. From that point of view, the mass-action law can be used to express the state of equilibrium despite the fact that this law is defined exclusively for homogeneous systems. Derived this way, the so-called pseudo-equilibrium constant Ke is not really a constant, since it depends on the total concentration ... 

For the production of chemicals, the rate of the reaction is a key parameter for the productivity defined in Equation (5) as the number of molecules produced per time. In homogeneous systems, the reaction rate depends on temperature, pressure, and composition [1]. In the case of solarthermal cycles, a metal oxide is used for the C02-splitting reaction rendering the reaction medium a heterogeneous two-phase system consisting of a solid (metal, metal oxide) and a fluid (CO2, CO, or carrier gas with O2). Therefore, the reaction kinetics becomes much more complex. Whereas microscopic kinetics only deals with time-dependent progress of the reaction, macroscopic kinetics additionally takes the heat- and mass-transport phenomena in heterogeneous systems into account. The transfer of species from one phase to the other must be considered in the overall mass balance [1]. The reaction of a gas with a porous solid consists of seven steps ... 

The traditional cocatalyst, diethylaluminumchloride or triethylaluminum, shows only a pure polymerization activity and was used as a homogeneous system to understand the polymerization, which is simpler in soluble than in heterogeneous systems. Kinetic studies and applications of various methods have helped to define the nature of the active centers, to explain aging effects, to establish the mechanism of the interaction with olefins, and to obtain quantitative evidence of some elementary steps [12,13]. 

Some of the earliest catalysts for this transformation were heterogeneous metal oxides (typically of tungsten, molybdenum or rhenium) on a support such as silica or alumina. These are still the catalysts employed for all current large scale industrial processes. Ill-defined homogeneous systems comprising alcohol solutions of a metal halide in conjunction with a promoter were also employed extensively in the early years, and are currently still used for the... 

Method of Undetermined Coefficients If Q x) is a product or linear combination of products of the functions a , x p a positive integer or zero) cos cx and sin cx, this method may be used. The "families [a ], [e " ], [sin cx, cos cx] and [x, xf, . .., x, 1] are defined for each of the above functions in the following way The family of a term f is the set of all functions of whichand all operations of the form cos c x + y), sin c x + y), (x + yf onf and their linear combinations result in. The technique involves the following steps (1) Solve the homogeneous system. (2) Construct the family of each term. (3) If the family has no representative in the homogeneous solution, assume i/J is a linear combination of the families of each term and determine the constants so that the equation is satisfied. (4) If a family has a representative in the homogeneous solution, multiply each member of the family by the smallest integral power of x for which all such representatives are removed and revert to step 3. 

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