Volume fictitious

Raoult expressed his results in terms of the molecular depression , for a mol. of solute in 100 grams of solvent. The volume of the solvent is 100/p, and this may be taken as the volume of the dilute solution. The corresponding osmotic pressure P, on the assumption that the law of proportionality holds good at this concentration (which is only a fictitious extrapolation) is given by ... 

This field of the centrifugal force, unlike the attraction field, is fictitious, and correspondingly, we observe a volume distribution of fictitious sources with a density proportional to co. A summation of the first and second Equations (2.62 and 2.63) gives the system of equations of the gravitational field at regular points... 

Note that packing contributions are most obvious in the outer shell contribution to Eq. (33) because the indicator function 0 (1 — bj) constrains that term to the case where no occupancy of the defined inner shell is permitted. These excluded volume interactions are the essense of packing contributions. To study those contributions, we consider a fictitious solute that does not interact with the solvent at all e AU/kT = 1. In that case, of course, p7x is zero and we write... 

Here fi2 = 1/Lv(t + r), L is the mean free path of radicals at thermal velocity v, and the initial spur radius r0 and the fictitious time T are related by r2 = Lvr. On random scattering, the probability per unit time of any two radicals colliding in volume dv will be ov/dv, where <7 is the collision cross section. The probability of finding these radicals in dv at the same time t is N(N - 1 )p2 dv2, giving the rate of reaction in that volume as crvN(n - 1 )p2 dv. Thus,... 

The film thickness is a fictitious quantity and cannot be measured directly. Its magnitude is usually of the order 10 3-10 2 cm, depending on agitation. This criterion is valid only for counterions of equal mobility and infinite solution volume. However, it is a useful approximation despite its limited precision. 

The two principle modes of the magnetostriction (A -° and As or Xy-2) introduced above are illustrated in fig. 2. With respect to the non-magnetic fictitious state, a spherical, isotropic sample exhibits a relative volume change AV/V = A" 0, when it becomes magnetic. In addition, when one forces the moments to be directed along an applied magnetic field B, an anisotropic deformation is induced, which transfers the sphere into an ellipsoid with the same volume. 

The introduction of the superscript (n, V) implies the adoption of a Gibbs convention algebraically expressed by Equations 5, which state that we are comparing the real system with a fictitious one consisting of two bulk phases in contact in the absence of an interfacial region, and with the added specification that the numbers of moles n and the total volume V of the fictitious system shall be the same as in the real one. The quantity S(r 7) (which is called SA° by Guggenheim (3) and sf by Defay (1)) is the entropy change per unit area of interface created when... 

The Gibbs convention (Equation 17) states that we compare the real system with a fictitious one having the same total volume and total numbers of moles of all constituents as the real system. Under this convention the total moles of individual components 1 and 2 will differ between the real and the fictitious systems, but because the total of all moles of both components is the same, the surface excesses of each must sum to zero, and this is the meaning of Equation 20. 

Inserting the geometrical factor 4 r/3 for the volume of a sphere we find that s 1 corresponds to a fictitious system filled with excluded volume coils ... 

Marchi and co-workers [27,28] have applied Equation (1.79) in the context of classical MD by using a Fourier pseudo-spectral approximation of the polarization vector field. This approach provides a convenient way to evaluate the required integrals over all volume at the price of introducing in the extended Lagrangian a set of polarization field variables all with the same fictitious mass. They also recognized the cmcial requirement that both the atomic charge distribution and the position-dependent dielectric constant be continuous functions of the atomic positions and they devised suitable expressions for both. 

In the presence of reaction, the molar flux of A at a position y within the film follows from the derivative of the concentration profile at that position. The latter follows from a balance for A over an infinitesimal volume element, between y and y + dy, of the fictitious film ... 

Cell models constitute a second major class of empirical developments. Among these, only two will be mentioned here as constituting the most successful and widely used. The first, due to Happel (1957,1958), is useful for estimating the effective viscosity and settling velocity of suspensions. Here, the suspension is envisioned as being composed of fictitious identical cells, each containing a single spherical particle of radius a surrounded by a concentric spherical envelope of fluid. The radius b of the cell is chosen to reproduce the suspension s volume fraction

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Designating w as the pounds of dry-cake solids per unit volume of filtrate, pc as the cake density expressed as pounds of dry-cake solids per unit volume of wet filter cake, and VF as the fictitious volume of filtrate per unit of filtering area necessary to lay down a cake thickness lF, the actual cake thickness plus the fictitious cake thickness is... 

A surface contains no volume or mass, and thus no energy. Thereore, a surface can be viewed as a fictitious system whose energy content remains constant during a process (just like a steady-slate or steady-flow system). Then the energy balance for a surface can be expressed as... 

Vp = fictitious volume of filtrate per unit of filtering area necessary to lay down a cake of thickness Ip,... 

Because of the analogy between simulated and true counter-current flow, TMB models are also used to design SMB processes. As an example, the transport dispersive model for batch columns can be extended to a TM B model by adding an adsorbent volume flow Vad (Fig. 6.38), which results in a convection term in the mass balance with the velocity uads. Dispersion in the adsorbent phase is neglected because the goal here is to describe a fictitious process and transfer the results to SMB operation. For the same reason, the mass transfer coefficient feeff as well as the fluid dispersion Dax are set equal to values that are valid for fixed beds. 

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