Frequently Used Equations

The most frequently used equations of state are summarized in Table I. Many of the equations used are based on the Tait (133) equation... 

Many different types of equations and inequalities are used to solve math word problems. Each type has its own methods for solution, and many have some quirks to watch out for when solving. Here are some more frequently used equations and inequalities ... 

One most frequently used equation is called the Exponential-Modified-Gauss (EMG) function (Jeansonne and Foley, 1991 and 1992 Foley and Dorsey 1983 and 1984). It is defined as a Gaussian peak (Eq. 6.143) superimposed by an exponential decay function hexp... 

A modified Nielsen modef is another frequently used equation, cially in the form proposed by Nicolais and Narkis ... 

The most frequently used equations that relate the vapor pressure of a material to temperature are as follows. In moderate temperature ranges, the Antoine equation provides a reasonable empirical prediction, given by... 

In most cases, n is an integer or a half-integer (see Chapter 4). Therefore, a more complete but less frequently used equation representing the rate constant as a function of temperature is... 

The majority of modifications of the cubic equations is based on the Van der Waals equation. Up to the present time many cubic equations of state [8, 9] have been proposed. Below are given some, most frequently used, equations. 

Activities and activity coefficients are commonly used in order to rewrite some frequently used equations into more pleasing form. In this manner, some relations can be written in especially short form. There is a certain difficulty, though, in the fact that these quantities are introduced and applied in many variations. To introduce the subject, we will choose the most general form that is less often used but can be most easily understood. 

Simulation of powder ESEEM spectra is usually performed by a numerical integration over the magnetic field directions. Frequently used Equations [54, 57, 61, 82-85] are reproduced below. Angular selection can be taken into account in a manner analogous to that used in powder ENDOR simulations. 

The literature contains numerous formulations for this dependence on temperature one frequently used equation [2.23] is... 

Equation (44) and in Eq. (43) can be evaluated analytically if a mathematical expression is available for the specific heat. Many of these have been published, and two frequently used equations are... 

The precise manner in which material reduces its volume in response to applied pressure is described by the equation of state, which relates volume reduction with the bulk modulus of the material and its first derivative with respect to pressure Bo. The most frequently used equation is universal the Birch-Mumaghan equation of state (EOS) [3-5] which can be expressed as follows ... 

Several theoretical and empirical equations have been used to describe the Tj,-composition dependence of miscible blends. The two frequently used equations are the Fox equation [equation (4.44)] and the Gordon-Taylor equation [equation (4.45)] ... 

For powder reactions (Fig. 2.15), a complete description of the reaction kinetics must take into account several parameters, thereby making the analysis very complicated. Simplified assumptions are commonly made in the derivation of kinetic equations. For isothermal reaction conditions, a frequently used equation has been derived by Jander (36). In the derivation, it is assumed that equalsized spheres of reactant A are embedded in a quasi-continuous medium of reactant B and that the reaction product forms coherently and uniformly on the A particles. The volume of unreacted material at time t is... 

A large number of thermodynamic relations may be derived from the above equations by conventional manipulations. Table 2.1 summarizes the most frequently used equations. 

In stress analysis, designers frequently use equations originally derived for materials... 

Another less frequently used equation which can represent gas-phase properties is a reduced Gibbs function. This was successfully used by Craven etal. (1989) to correlate the gas-phase data for methanol. 

The other class of isotherms follows from the theory of mobile or partially mobile adsorption [8,33]. One of the most frequently used equations of this type is the Hill-de Boer isotherm [34,35]. Moreover, an important attribute of gas adsorption is the formation of multilayer surface films. One partieularly popular and successful equation describing multilayer adsorption is the Brunauer-Emmet-TeUer isotherm [36]. Studies of many authors have been focused on an extension of various modifieations of the BET equation to heterogeneous surfaces [5,6]. This problem will be discussed in Seetion II1.B.2.C. 

Another frequently used equation for binary gas diffusion coefficients is derived based on kinetic theory and experimental data (Bird et al., 2002 Slattery and Bird, 1958) as... 

Many equations approximate the friction losses that can be expected with the flow of liquid through a pressure pipe. The two most frequently used equations in plastic piping systems are ... 

The most frequently used equations of state of ionic surfactant adsorption layers were discussed in [19, 21]. Among these the Frumkin adsorption model describes the adsorption behaviour of ionic surfactants very well... 

We must note here that different forms for y arise from the models because of the different nature of the models, and that there is a very different physical significance also for the quantities P/. Pi and (P ). It must also be clear that the rate coefficient of the adiabatic channel model does not depend on any of these model assumptions rather, it can be derived in its general form from the very general considerations discussed in Section 2.4.1. However, the simple models for deriving transmission coefficients in Section 2.4.2 and 2.4.3 provide some physical insight and could also be made the basis for simple numerical calculations of transmission coefficients beyond the step function behavior, which leads to the most frequently used equation (54), i.e., equation (46) with (y) = 1. See also our calculation of transmission coefficients. ... 

Within this ehapter we are mostly concerned with physical adsorption. Our first task is to explore some simple and frequently used equations for adsorption isotherms. We will foeus on the acidity/basicity of the adsorbate and adsorbent as a eriterion for the adsorbate-adsorbent interaction energy, and make a special case of the adsorption of aleohol molecules from apolar solvents. The alcohol molecules self-assoeiate and form molecular clusters in apolar and inert solvents. This aggregation proeess may influence the adsorption process and will be encountered when diseussing and analyzing the alcohol adsorption isotherms. 

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