Derivation equations

Langmuir adsorption isotherm A theoretical equation, derived from the kinetic theory of gases, which relates the amount of gas adsorbed at a plane solid surface to the pressure of gas in equilibrium with the surface. In the derivation it is assumed that the adsorption is restricted to a monolayer at the surface, which is considered to be energetically uniform. It is also assumed that there is no interaction between the adsorbed species. The equation shows that at a gas pressure, p, the fraction, 0, of the surface covered by the adsorbate is given by ... 

The Cpg of real gas is calculated using the equation derived from the Lee and Kesler model ... 

In making certain mathematical approximations to the Schrodinger equation, we can equate derived terms directly to experiment and replace dilTiciilL-to-calculate mathematical expressions with experimental values. In other situation s, we introduce a parameter for a mathematical expression and derive values for that parameter by fitting the results of globally calculated results to experiment. Quantum chemistry has developed two groups of researchers ... 

Phan-Thien, N. and Tanner, R.T., 1977. A new constitutive equation derived from network theory, Non-Newtonian Fluid Mech. 2, 353-365. 

The rate of chemical diffusion in a nonfiowing medium can be predicted. This is usually done with an equation, derived from the diffusion equation, that incorporates an empirical correction parameter. These correction factors are often based on molar volume. Molecular dynamics simulations can also be used. 

The application of the Taft equation, derived from the Hammett equation (262)... 

Our approach in this chapter is to alternate between experimental results and theoretical models to acquire familiarity with both the phenomena and the theories proposed to explain them. We shall consider a model for viscous flow due to Eyring which is based on the migration of vacancies or holes in the liquid. A theory developed by Debye will give a first view of the molecular weight dependence of viscosity an equation derived by Bueche will extend that view. Finally, a model for the snakelike wiggling of a polymer chain through an array of other molecules, due to deGennes, Doi, and Edwards, will be taken up. 

The equations derived in Sec. 6.7 are based on the assumption that termination occurs exclusively by either disproportionation or combination. This is usually not the case Some proportion of each is the more common case. If A equals the fraction of termination occurring by disproportionation, we can write n = A[ 1/1 - p] + (1 - A)[2/(l - p)] and n /n = A(1 + p) + (1 - A)[(2 + p)/2]. From measurements of n and n /n it is possible in principle to evaluate A and p. May and Smith have done this for a number of polystyrene samples. A selection of their data for which this approach seems feasible is presented ... 

The Hydrate and Enol Form. In aqueous solutions, acetaldehyde exists in equihbrium with the acetaldehyde hydrate [4433-56-17, (CH2CH(0H)2). The degree of hydration can be computed from an equation derived by BeU and Clunie (31). Hydration, the mean heat of which is —21.34 kJ/mol (—89.29 kcal/mol), has been attributed to hyperconjugation (32). The enol form, vinyl alcohol [557-75-5] (CH2=CHOH) exists in equihbrium with acetaldehyde to the extent of approximately 1 molecule per 30,000. Acetaldehyde enol has been acetylated with ketene [463-51-4] to form vinyl acetate [108-05-4] (33). 

Many industrial crystallizers operate in a weU-mixed or nearly weU-mixed manner, and the equations derived above can be used to describe their performance. Furthermore, the simplicity of the equations describing an MSMPR crystallizer make experimental equipment configured to meet the assumptions lea ding to equation 44 useful in determining nucleation and growth kinetics in systems of interest. 

The analogy between equations derived from the fundamental residual- and excess-propeily relations is apparent. Whereas the fundamental lesidanl-pL-opeRy relation derives its usefulness from its direct relation to equations of state, the ci cc.s.s-property formulation is useful because V, and y are all experimentally accessible. Activity coefficients are found from vapor/liquid equilibrium data, and and values come from mixing experiments. 

Although Eq. (14-31) is convenient for computing the composition of the exit gas as a function of the number of theoretical stages, an alternative equation derived by Colburn [Tran.s. Am. Jn.st. Chem. Eng., 35, 211 (1939)] is more useful when the number of theoretical plates is the unknown ... 

The method is apphcable for unflocculated pulps or those in which the ionic characteristics of the solution produce a flocculent structure. If polymeric flocculants are used, the floccule size will be highly dependent on the feed concentration, and an approach based on the Kynch theoiy is preferred. In this method, the test is carried out at the expec ted feed solids concentration and is continued until underflow concentration is achieved in the cyhnder. To determine the unit area, Talmage and Fitch (op. cit.) proposed an equation derived from a relationship equivalent to that shown in Eq. (18-45) ... 

According to Ktiudsen if a small circular orifice of diameter less than the mean free path of the molecules in a container, is opened in the wall of the container to make a connection to a high vacuum sunounding the container, the mass of gas effusing tlnough the orifice, of area A, is given by an equation derived from the kinetic theoty, where tire pressure is in amiospheres. 

The critical size of the stable nucleus at any degree of under cooling can be calculated widr an equation derived similarly to that obtained earlier for the concentration of defects in a solid. The configurational entropy of a mixture of nuclei containing n atoms widr o atoms of the liquid per unit volume, is given by the Boltzmann equation... 

For a cylinder diameter, d, the following equations derived from Fig. 24-4 together with Eq. (24-46) to (24-48) are relevant ... 

In a simulation it is not convenient to work with fluctuating time intervals. The real-variable formulation is therefore recommended. Hoover [26] showed that the equations derived by Nose can be further simplified. He derived a slightly different set of equations that dispense with the time-scaling parameter s. To simplify the equations, we can introduce the thermodynamic friction coefficient, = pJQ. The equations of motion then become... 

After substituting (2.3) and (2.4) for (2.1), the integral can be evaluated with the method of steepest descents. The stationary point E = E is given by the equation derived by Miller and George [1972]... 

In X-Ray Fluorescence (XRF), an X-ray beam is used to irradiate a specimen, and the emitted fluorescent X rays are analyzed with a crystal spectrometer and scintillation or proportional counter. The fluorescent radiation normally is diffracted by a crystal at different angles to separate the X-ray wavelengths and therefore to identify the elements concentrations are determined from the peak intensities. For thin films XRF intensity-composition-thickness equations derived from first principles are used for the precision determination of composition and thickness. This can be done also for each individual layer of multiple-layer films. 

Vacancy chromatography has some quite unique properties and a number of potentially useful applications. Vacancy chromatography can be theoretically investigated using the equations derived from the plate theory for the elution of... 

Employing the conditions defined in the three data bases and the appropriate equations derived from the Plate and Rate Theories, the optimum physical... 

The different optimization equations derived in chapter 12 will then be used with these realistic chromatographic conditions in a simple optimization procedure. The conditions chosen are typical and might represent the average LC analysis. The values for (X) and (yp) are those estimated by Giddings [1] for a well-packed... 

In this optimization procedure those equations derived in chapter 13 will then be used with the data provided in Table 2. 

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

Consider a cake of moulding resin between the compression platens as shown in Fig. 4.63. When a constant force, F, is applied to the upper platen the resin flows as a result of a pressure gradient. If the flow is assumed Newtonian then the pressure flow equation derived in Section 4.2.3 may be used... 

When Re > 10 , the following equation, derived by means of the iogarith-mic velocity distribution by Prandtl and the empirical research results of Nile uradse, is valid ... 

This equation is analogous to the eontinuity equation derived for the eonserva-tion of mass in a eontinuum... 

Filters generally achieve a lower final moisture content than obtained by gravity sedimentation and are often fed from thickeners, as indicated in the schematic particulate process shown in Figure 9.2. In this chapter the principles of slurry filtration will be described and certain simplified filter design equations derived. For more complex derivations the reader is referred to specialist texts e.g. Coulson and Richardson (1991), Wakeman (1990a) and Purchas (1981). 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Bubble cap contactors are the most common The minimum diameter can be determined using the equation derived for gas separation in vertical separators (Volume 1, Chapter 4). This is ... 

Equation derived from / SS ASHRAE Handbook of Fundamentals, Chapter 22, assiim-ing an air change every five (5) minutes. Refer to the ASHRAE Handbook, Chapter 22, for additional information on naturally ventilated buildings. 

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