Volume derivation, expressions

The above example gives us an idea of the difficulties in stating a rigorous kinetic model for the free-radical polymerization of formulations containing polyfunctional monomers. An example of efforts to introduce a mechanistic analysis for this kind of reaction, is the case of (meth)acrylate polymerizations, where Bowman and Peppas (1991) coupled free-volume derived expressions for diffusion-controlled kp and kt values to expressions describing the time-dependent evolution of the free volume. Further work expanded this initial analysis to take into account different possible elemental steps of the kinetic scheme (Anseth and Bowman, 1992/93 Kurdikar and Peppas, 1994 Scott and Peppas, 1999). The analysis of these mechanistic models is beyond our scope. Instead, one example of models that capture the main concepts of a rigorous description, but include phenomenological equations to account for the variation of specific rate constants with conversion, will be discussed. 

In developing these ideas quantitatively, we shall derive expressions for the light scattered by a volume element in the scattering medium. The symbol i is used to represent this quantity its physical significance is also shown in Fig. 10.1. [Our problem with notation in this chapter is too many i s ] Before actually deriving this, let us examine the relationship between i and 1 or, more exactly, between I /Iq and IJIq. 

Derive expressions for the velocity profile, shear stress, shear rate and volume flow rale during the isothermal flow of a power law fluid in a rectangular section slit of width W, depth H and length L. During tests on such a section the following data was obtained. 

Units may be combined together into derived units to express a property more complicated than mass, length, or time. For example, volume, V, the amount of space occupied by a substance, is the product of three lengths therefore, the derived unit of volume is (meter)3, denoted m3. Similarly, density, the mass of a sample divided by its volume, is expressed in terms of the base unit for mass divided by the derived unit for volume—namely, kilogram/(meter)3, denoted kg/m3 or, equivalently, kg-m-3. The SI convention is that a power, such as the 3 in cm3, refers to the unit and its multiple. That is, cm3 should be interpreted as (cm)3 or 10-6 m3 not as c(m3), or 10 2 m3. Many of the more common derived units have names and abbreviations of their own. 

Flashover occurs in a room causing 200 g/s of fuel to be generated. It is empirically known that the yield of CO is 0.08. Yield is defined as the mass of product generated per mass of fuel supplied. The room has a volume of 30 m3 and is connected to a closed corridor that has a volume of 200 m3. The temperatures of the room and corridor are assumed uniform and constant at 800 and 80 °C respectively. An equation of state for these gases can be taken as pT = 360 kg K/m3. Steady mass flow rates prevail at the window of the room (to the outside air) and at the doorway to the corridor. These flow rates are 600 and 900 g/s respectively. Derive expressions for the mass fraction of CO in the room and corridor as a function of time assuming uniform concentrations in each region. 

The articles and discussion comments contained in this volume derive from a Conference on Inorganic Reaction Mechanisms. The organization of the conference was stimulated both by an interest of the National Science Foundation in surveying the current status of this area and by a need expressed by some 100 researchers active in mechanistic investigations whose opinions were canvassed by the organizers. 

Squire63 used a model of the gel phase in which the volume elements available to solvent within the gel were regarded as a combination of cones, cylinders, and crevices, and derived expressions for the volumes available to a solute of Stokes radius a in these three types of pore. Certain arbitrary assumptions regarding the distribution of solute among the different types of pore gave the following equation ... 

Outline the logic used in deriving expressions for the osmotic pressure and second virial coefficient due to excluded-volume interactions ... 

For the differential control volume shown (assume a 2n circumferential extent), derive expressions for the heat-transfer rate, including internal heat generation and thermal conduction. State the net conduction heat-transfer rate in terms of the temperature distribution that is, evaluate f q ndA. Take care when evaluating the differential areas. Identify and neglect higher-order terms. 

Equation (8) constitutes the basic thermodynamic equation for the calculation of the radius of the globules. Of course, explicit expressions, in terms of the radius of the globules and volume fraction, are needed fort, C and af before such a calculation can be carried out. Expressions for Af will be provided in another section of the paper, but it is difficult to derive expressions fory and C. One may, however, note that y (and also C) depends on the radius for the following two reasons (1) its value depends upon the concentrations of surfactant and cosurfactant in the bulk phases, which, because the system is closed, depend upon the amounts adsorbed on the area of the internal interface of the microemulsion (2) in addition to the above mass balance effect, there is a curvature effect on y (this point is examined later in the paper). 

A mathematical description of several mutually connected chemical reactions usually requires some simplification. According to Bohm, all reactive intermediates are assumed to exist in a stationary state, and all rate constants are assumed to be independent of the length of the growing macromolecules. As the derived expressions should also describe heterogeneous polymerizations, the author used the numbers of particles in unit volume of reacting medium instead of concentrations. Some expressions can be simplified in this... 

The value of the partition coefficient in this derivation, AT, is the ratio of the concentration of solute in the coating to the concentration of solute in the ambient (vapor) phase, with all concentrations being expressed in units of mass of solute per unit volume. Alternative expressions for the partition coefficient can be derived for concentration units of (moles of solute/coating volume) or (mass of solute/coating mass), or on a mole fraction basis. The value of K will be dependent on the concentration units used. For our purposes in the remainder of discussion, will refer specifically to the partition coefficient using the concentration units of mass per unit volume as described above. 

When the dimensions of the scatterers are much smaller than the wavelength of sound simple expressions for f(9) and c are obtained in terms of the complex elastic moduli of the inclusion and host materials, and the volume fraction of the inclusions. Alternately, static self-consistent mean field models can be used to derive expressions for the complex effective moduli of the composite material in terms of the complex elastic moduli and volume fractions of the component materials [32,33,34,35]. The propagation wavenumber c can then be expressed in terms of the effective complex moduli of the composite using Eqs.9 and 10. Particularly interesting. 

Derive expressions showing how the heat capacity of a material at constant pressure and constant volume changes with application of an electric field. 

Explicit expressions for the isothermal compressibility, the partial molar volumes and the derivatives of the chemical potentials with respect to concentrations are obtained in terms of the KB integrals. These equations are employed to derive expressions at infinite dilution, which are relevant in the calculation of the solubilities in ternary mixtures in terms of those in the binary constituents. 

The Kirkwood—Buff formalism was used to derive an expression for the composition dependence of the Henry s constant in a binary solvent. A binary mixed solvent can be considered as composed of two solvents, or one solvent and a solute, such as a salt, polymer, or protein. The following simple expression for the Henry s constant in a binary solvent (H2t) was obtained when the binary solvent was assumed ideal In = [In f2,i(ln V — In V ) + In i 2,3(ln Vj — In V)]/ (In — In V ). In this expression, i 2,i and i 2,3 are the Henry s constants for the pure single solvents 1 and 3, respectively V is the molar volume of the ideal binary solvent 1—3 and and Vs are the molar volumes of the pure individual solvents 1 and 3. The comparison with experimental data for aqueous binary solvents demonstrated that the derived expression provides the best predictions among the known equations. Even though the aqueous solvents are nonideal, their degree of nonideality is much smaller than those of the solute gas in each of the constituents. For this reason, the ideality assumption for the binary solvent constitutes a most reasonable approximation even for nonideal mixtures. 

The purpose of this part is to derive expressions for the Kirkwood-Buff integrals (KBIs) in binary systems. For binary mixtures, Kirkwood and Buff obtained the following expressions for the partial molar volumes, the isothermal compressibility and the derivatives of the chemical potential with respect to concentrations... 

The purpose of this paper is to shed additional light on the cosolvent concentration dependence of OSVC in water— protein—cosolvent mixtures. The Kirkwood—Buff theory of solutions was used to derive an expression which connects OSVC to the thermodynamic properties of water—protein— cosolvent mixtures. These properties can be subdivided into two groups (1) those due to a protein-free water—cosolvent mixture, such as concentrations, isothermal compressibility, partial molar volumes, and the derivative of the water activity coefficient with respect to the water molar fraction and (2) those of infinitely dilute (with respect to the protein) water—protein—cosolvent mixtures, such as the partial molar volume of the protein at infinite dilution (VT) and the derivatives of the protein activity coefficient with respect to the protein and water molar fractions (/21 and J22). It was found that the derived expression for OSVC contains three contributions (1) ideal mixture contribution B >),... 

The power of the theory developed by Shih et al. (1) lies in the fact that it is possible to experimentally determine if a system is in the strong- or weak-link. The strain at the limit of linearity increases as a function of the volume fraction of network material for the weak-link regime while it decreases for the strong-link regime. Below we derive expressions for the relationship between the strain at the limit of linearity and network material volume fraction. 

For a reaction in which some of the coefficients in the balanced equation are not equal to 1, deriving expressions for the changes in the partial pressures of the products and reactants requires care. Consider the combustion of ethane at constant volume ... 

This is the van der Waals equation. In this equation, P,V,T, and n represent the measured values of pressure, volume, temperature (expressed on the absolute scale), and number of moles, respectively, just as in the ideal gas equation. The quantities a and b are experimentally derived constants that differ for different gases (Table 12-5). When a and b are both zero, the van der Waals equation reduces to the ideal gas equation. 

Equation (16) constitutes a physically derived expression for the binary diffusion coefficients This equation may be written in a more useful form by expressing the number of i-j collisions per unit volume per second (Vj ) in terms of more basic molecular parameters. Since there are Uj molecules of type i per unit volume, v j =, where t,j is the average... 

As an example of a set of standard parameters, Table 4.1 lists all the potential-dependent information needed to perform an energy-band calculation for (non-magnetic) chromium metal. In the following, chromium is used as an example when we discuss the physical significance of each of the four potential parameters (4.1). At the end of the chapter we derive free-electron potential parameters, give expressions for the volume derivatives of some se-... 

The question arises as to how long it will take a reactor operating in the plug flow regime to reach a steady state for a specific set of reaction kinetics, volume, and flow rate. To solve this problem we need to solve both in time and in space. If the kinetics are simple, then we can solve the problem analytically, that is, we can derive expressions for the concentrations that are functions of time and position. However, often the kinetics are not straightforward and analytical solutions must be surrendered in favor of numerical solutions. 

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