Theoretical expression

In the last section we noted that Simha and others have derived theoretical expressions for q pl(p for rigid ellipsoids of revolution. Solving the equation of motion for this case is even more involved than for spherical particles, so we simply present the final result. Several comments are necessary to appreciate these results ... 

This simple model illustrates how the fraction K and, through it, Vj are influenced by the dimensions of both the solute molecules and the pores. For solute particles of other shapes in pores of different geometry, theoretical expressions for K are quantitatively different, but typically involve the ratio of solute to pore dimensions. 

Comparing this to the theoretical expression based on the steady state approximation suggests that the mechanism is not straightforward ... 

When a gas is blown steadily through an orifice into an essentially inviscid liquid, a regular stream of bubbles is formed. A theoretical expression that relates the bubble volume V to volumetric gas flow rate G and gravitational acceleration g is the following ... 

This equation of state applies to all substances under all conditions of p, and T. All of the virial coefficients B, C,. .. are zero for a perfect gas. For other materials, the virial coefficients are finite and they give information about molecular interactions. The virial coefficients are temperature-dependent. Theoretical expressions for the virial coefficients can be found from the methods of statistical thermodynamic s. 

There is a clear one-to-one correspondence between the theoretical expressions and the computational implementation in terms of one- and two-electron matrix elements. Implementations of the expressions are therefore facilitated. 

In contrast to points (l)-(3) of discussion, the effect of ion association on the conductivity of concentrated solutions is proven only with difficulty. Previously published reviews refer mainly to the permittivity of the solvent or quote some theoretical expressions for association constants which only take permittivity and distance parameters into account. Ue and Mori [212] in a recent publication tried a multiple linear regression based Eq. (62)... 

The theoretical expression for the relations at the maximum is obtained by putting (tPn/dP) = 0. Differentiating Eq. (8), substituting into the result for Nm the proper expression from Eq. (8), and equating the relationship obtained to zero, the maximum condition for order x results in the form... 

As we have previously seen the theoretical expression for the average waiting time is given by... 

The equations resulting from both these studies are extremely complex, and contain several reaction parameters not readily evaluated from separate experiments. Figure 15 shows a comparison between experimental data and the two theoretical presentations. The theoretical curve was obtained by curve-fitting through adjusting the unknown parameters. This comparison shows that the theoretical expressions have sufficient flexibility to adequately correlate experimental burning rates. However, the value of the activation... 

Solutions for diffusion with and without chemical reaction in continuous systems have been reported elsewhere (G2, G6). In general, all the parameters in this model can be determined or estimated, and the theoretical expressions may assist in the interpretation of mass-transfer data and the prediction of equipment performance. 

These parameters can be determined and predicted, and the theoretical expressions may thus assist in interpretation of mass-transfer data and in prediction of equipment performance. The case of mass transfer without chemical reaction is reported elsewhere (G5). 

No exact mathematical analysis of the conditions within a turbulent fluid has yet been developed, though a number of semi-theoretical expressions for the shear stress at the walls of a pipe of circular cross-section have been suggested, including that proposed by BLAS1US.(6)... 

By comparing impedance results for polypyrrole in electrolyte-polymer-electrolyte and electrode-polymer-electrolyte systems, Des-louis et alm have shown that the charge-transfer resistance in the latter case can contain contributions from both interfaces. Charge-transfer resistances at the polymer/electrode interface were about five times higher than those at the polymer/solution interface. Thus the assignments made by Albery and Mount,203 and by Ren and Pickup145 are supported, with the caveat that only the primary source of the high-frequency semicircle was identified. Contributions from the polymer/solution interface, and possibly from the bulk, are probably responsible for the deviations from the theoretical expressions/45... 

Some progress has been made in developing theoretical expressions for rj(6) for deactivation processes such as coking. Deactivation by loss of active sites can be modeled as a chemical reaction proceeding in parallel with the main reaction. It may be substantially independent of the main reaction. Site sintering, for example, will depend mainly on the reaction temperature. It is normally modeled as a second-order reaction ... 

The theoretical expressions for the three-state E/E/B model are shown in Table II. The two-site E/B model can be... 

It should be emphasized that Eq. (52) is empirical in origin. However, the more complicated theoretical expressions to be discussed in Chapter XIV can be approximated quite closely by this simple equation over ranges of as much as a hundredfold in M. The convenience of application of the empirical relationship assures its continued use for correlating intrinsic viscosities and molecular weights. 

Gee and Orr have pointed out that the deviations from theory of the heat of dilution and of the entropy of dilution are to some extent mutually compensating. Hence the theoretical expression for the free energy affords a considerably better working approximation than either Eq. (29) for the heat of dilution or Eq. (28) for the configurational entropy of dilution. One must not overlook the fact that, in spite of its shortcomings, the theory as given here is a vast improvement over classical ideal solution theory in applications to polymer solutions. 

Thermodynamics describes the behaviour of systems in terms of quantities and functions of state, but cannot express these quantities in terms of model concepts and assumptions on the structure of the system, inter-molecular forces, etc. This is also true of the activity coefficients thermodynamics defines these quantities and gives their dependence on the temperature, pressure and composition, but cannot interpret them from the point of view of intermolecular interactions. Every theoretical expression of the activity coefficients as a function of the composition of the solution is necessarily based on extrathermodynamic, mainly statistical concepts. This approach makes it possible to elaborate quantitatively the theory of individual activity coefficients. Their values are of paramount importance, for example, for operational definition of the pH and its potentiometric determination (Section 3.3.2), for potentiometric measurement with ion-selective electrodes (Section 6.3), in general for all the systems where liquid junctions appear (Section 2.5.3), etc. 

The lack of quantitative success of the theory presented in the previous section is primarily the result of inadequacies in the mixing and elasticity terms. Sophisticated theoretical expressions are available which afford much improved predictions of swelling behavior. However, even these theories are not particularly successful when applied to hydrogels. Here the limitations of the theory presented in Section III.A.l will be identified. 

Since the conductivity coefficient, Fc, is defined by J = FC(P - P"), the theoretical expression for Fc is obtained from this definition and Eq. (82) to yield... 

Ag+ + e Ag is the simplest reaction which can be studied easily in a potassium nitrate solution. In any metal ions metal reaction, Cr C0o, and the theoretical expression in Eq. (9) reduces to43... 

The method has been successfully applied to PPT samples having different orientation levels. It is noteworthy that the factor (3+6RiSO) is not a constant throughout the whole spectral range since Riso is a priori different for each band. This factor is then particularly important if one wants to compare the relative intensities of various vibrational modes. Sourisseau and Talaga [58] have expanded the theoretical expression of orientation-independent intensity sums for systems with biaxial symmetry by making use of the K2 Raman invariant. 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

A variety of theoretical expressions, as well as experimental values, for the correction factor A as a function of the power law flow index ( ) were summarized by Chhabra (1992). 

For example, if the entering solids fraction tp is 0.4, the corresponding values of the local solids fraction relative slip velocities (Vr) of 0.01, 0.1, and 0.5 are 0.403, 0.424, and 0.525, respectively. There are many theoretical expressions for slip, but practical applications depend on experimental observations and correlations (which will be presented later). In gas-liquid or gas-solid flows, (pm will vary along the pipe, because the gas expands as the pressure drops and speeds up as it expands, which tends to increase the slip, which in turn increases the holdup of the denser phase. 

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