Theoretical 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 ... 

Calculated from various theoretical equations some of the sources are themselves compilations rather than original. 

Figure 5,16. It is assumed that by using an exactly symmetric cone a shear rate distribution, which is very nearly uniform, within the equilibrium (i.e. steady state) flow held can be generated (Tanner, 1985). Therefore in this type of viscometry the applied torque required for the steady rotation of the cone is related to the uniform shearing stress on its surface by a simplihed theoretical equation given as...

Theoretical equation forms may be derived from either kinetic theory or statistical mechanics. However, empirical and semitheoretical equations of state have had the greatest success in representing data with high precision over a wide range of conditions (1). At present, theoretical equations are more limited in range of appHcation than empirical equations. There are several excellent references available on the appHcation and development of equations of state (2,3,18,21). 

Mathematical Consistency Requirements. Theoretical equations provide a method by which a data set s internal consistency can be tested or missing data can be derived from known values of related properties. The abiUty of data to fit a proven model may also provide insight into whether that data behaves correctiy and follows expected trends. For example, poor fit of vapor pressure versus temperature data to a generally accepted correlating equation could indicate systematic data error or bias. A simple sermlogarithmic form, (eg, the Antoine equation, eq. 8), has been shown to apply to most organic Hquids, so substantial deviation from this model might indicate a problem. Many other simple thermodynamics relations can provide useful data tests (1—5,18,21). 

The critical pressure ratio r can be obtained from the following theoretical equation, which assumes a perfect gas and a frictionless nozzle ... 

Forveiy thin hquids, Eqs. (14-206) and (14-207) are expected to be vahd up to a gas-flow Reynolds number of 200 (Valentin, op. cit., p. 8). For liquid viscosities up to 100 cP, Datta, Napier, and Newitt [Trans. In.st. Chem. Eng., 28, 14 (1950)] and Siems and Kauffman [Chem. Eng. Sci, 5, 127 (1956)] have shown that liquid viscosity has veiy little effec t on the bubble volume, but Davidson and Schuler [Trans. Instn. Chem. Eng., 38, 144 (I960)] and Krishnamiirthi et al. [Ind. Eng. Chem. Fundam., 7, 549 (1968)] have shown that bubble size increases considerably over that predic ted by Eq. (14-206) for hquid viscosities above 1000 cP. In fac t, Davidson et al. (op. cit.) found that their data agreed veiy well with a theoretical equation obtained by equating the buoyant force to drag based on Stokes law and the velocity of the bubble equator at break-off ... 

The second reason for modification of the displaced volume is that in real world application, the cylinder will not achieve the volumetric performance predicted by Equation 3.4. It is modified, therefore, to include empirical data. The equation used here is the one recommended by the Compressed Air and Gas Institute [1], but it is somewhat arbitrary as there is no universal equation. Practically speaking, however, there is enough flexibility in guidelines for the equation to produce reasonable results. The 1.00 in the theoretical equation is replaced with. 97 to reflect that even with zero clearance the cylinder will not fill perfectly. Term L is added at the end to allow for gas slippage past the piston rings in the various types of construction. If, in the course of making an estimate, a specific value is desired, use, 03 for lubricated compressors and. 07 for nonlubricated machines. These are approximations, and the exact value may vary by as much as an additional. 02 to. 03... 

Another error can arise when two partially resolved peaks are asymmetrical, e.g., the rear half of the peak is broader the front half. In such a situation, it is clear that there can be two sources of error, which are depicted in Figure 4. Firstly, the retention times, as measured from the peak envelope, will not be accurate. Secondly, because the peaks are asymmetrical (and most LC peaks tend to be asymmetrical to the extent shown in the Figure 4), the second peak appears higher. This can incorrectly imply that the second solute is present at a higher concentration in the mixture than the first. It follows that it is important to know the value of the specific separation ratio above which accurate measurements can still be made on the peak maxima of the individual peaks. The apparent peak separation ratio, relative to the actual peak separation ratio for columns of different efficiency, are shown in Figure 5. The data has been obtained from theoretical equations. 

The kinetic analysis of the sigmoid pH-rate profile will yield numerical estimates of the pH-independent parameters K, k, and k". With these estimates the apparent constant k is calculated using the theoretical equation over the pH range that was explored experimentally. Quantitative agreement between the calculated line and the experimental points indicates that the model is a good one. A further easy, and very pertinent, test is a comparison of the kinetically determined value with the value obtained by conventional methods under the same conditions. 

The pH-independent plateau from about pH 5 to 9 represents reaction of the acylsalicylate anion. It is obvious from the pH-rate profile that k" is much larger than k. The theoretical equation for k, the observed first-order rate constant, is derived in the usual way from Eq. (6-71). 

What do we leam from this comparison of the general theoretical equations (5.24) and (5.25) with the specific experimental equations (5.19) and (5.18) of solid state electrochemistry The answer is mathematically obvious and physically important. In solid state electrochemistry one has ... 

It must first be noted that the experimental Equation (7.11), in conjunction with the general theoretical Equation (7.13) implies directly21 32 33... 

This does not imply necessarily T y = T r = 0 or T s = 0, where is the Volta potential on the electrolyte surface. But for the subsequent analysis it is useful to notice that equation (7.15) is mathematically equivalent (in view of the general theoretical Equation (7.13) with the key experimental equation (7.11). 

It is also of interest to mention here that Stevenson s (10) theoretical equation, k = 27r(e2a /ju)1/2 predicts a value for i5 = 3.5 X 10 9. This is close to the experimental value 1.2 X 10 9 discussed in the text and suggests that Stevenson s theoretical equation has some validity for Reactions 15 and 16. Stevenson s equation predicts that i5 and i6 differ... 

Steady State Population Density Distributions. Representative experimental population density distri-butions are presented by Figure 1 for two different levels of media viscosity. An excellent degree of theoretical (Equation 8) / experimental correlation is observed. Inasmuch as the slope of population density distribution at a specific degree of polymerization is proportional to the rate of propagation for that size macroanion, propagation rates are also observed to be independent of molecular weight. 

TABLE 4 Theoretical Equations for the Transport of a Single Spherical Particle in a Constant Electric Eield... 

Equations (44), (45), and (45 ) are alternative expressions for the theoretical equation of state for an ideal rubber. Equations of state for swollen networks, derived in Appendix B, have the same form. Only the magnitude predicted for r at a given elongation is affected by swelling. 

The theoretical equation of state for an ideal rubber in tension, Eq. (44) or (45), equates the tension r to the product of three factors RT, a structure factor (or re/Eo, the volume of the rubber being assumed constant), and a deformation factor a—l/a ) analogous to the bulk compression factor Eo/E for the gas. The equation of state for an ideal gas, which for the purpose of emphasizing the analogy may be written P = RT v/Vq) Vq/V), consists of three corresponding factors. Proportionality between r and T follows necessarily from the condition dE/dL)Ty=0 for an ideal rubber. Results already cited for real rubbers indicate this condition usually is fulfilled almost within experimental error. Hence the propriety of the temperature factor... 

Anthony, Caston, and Guth obtained considerably better agreement between the experimental stress-strain curve for natural rubber similarly vulcanized and the theoretical equation over the range a = 1 to 4. KinelP found that the retractive force for vulcanized poly-chloroprene increased linearly with a — l/a up to a = 3.5. 

These experimental results show conclusively that the deformation factor occurring in the theoretical equation of state offers only a crude approximation to the form of the actual equilibrium stress-strain curve. The reasons behind the observed deviation are not known. It does appear, however, from observations on other rubberlike systems that the type of deviation observed is general. Similar deviations are indicated in TutyP rubber (essentially a cross-linked polyisobutylene) and even in polyamides having network structures and exhibiting rubberlike behavior at high temperatures (see Sec. 4b). 

Perhaps the most convincing test of the theoretical equation of state is provided by the work of Schaefgen on so-called multilinked polyamides, which would not ordinarily be classified as rubberlike. 

Much attention has been directed since olden times towards ion solvation, which is a key concept for understanding various chemical processes with electrolyte solutions. In 1920, a theoretical equation of ion solvation energy (AG ) was first proposed by Born [1], who considered the ion as a hard sphere of a given radius (r) immersed in a continuous medium of constant permittivity (e), and then defined AG as the electrostatic energy for charging the ion up to ze (z, the charge number of the ion e, the elementary charge) ... 

Equation (3) suggests that the membrane potential in the presence of sufficient electrolytes in Wl, W2, and LM is primarily determined by the potential differences at two interfaces which depend on charge transfer reactions at the interfaces, though the potential differences at interfaces are not apparently taken into account in theoretical equations such as Nernst-Planck, Henderson, and Goldman-Hodgkin-Katz equations which have often been adopted in the discussion of the membrane potential. 

A negative current wave appeared in the polarogram as shown in curve 1 of Fig. 5, though the wave was not observed in the absence of NADH in W or CQ in DCE. The logarithmic analysis of the current wave based on the theoretical equation for the electron transfer [42,54,55] indicated that the wave was caused by two-electron transfer at the interface and controlled by the diffusion of NADH in W. 

In this chapter, the voltammetric study of local anesthetics (procaine and related compounds) [14—16], antihistamines (doxylamine and related compounds) [17,22], and uncouplers (2,4-dinitrophenol and related compounds) [18] at nitrobenzene (NB]Uwater (W) and 1,2-dichloroethane (DCE)-water (W) interfaces is discussed. Potential step voltammetry (chronoamperometry) or normal pulse voltammetry (NPV) and potential sweep voltammetry or cyclic voltammetry (CV) have been employed. Theoretical equations of the half-wave potential vs. pH diagram are derived and applied to interpret the midpoint potential or half-wave potential vs. pH plots to evaluate physicochemical properties, including the partition coefficients and dissociation constants of the drugs. Voltammetric study of the kinetics of protonation of base (procaine) in aqueous solution is also discussed. Finally, application to structure-activity relationship and mode of action study will be discussed briefly. 

In deriving theoretical equations of the current-potential (or time) curves of ion-transfer voltammetry of a dibase we shall make the following assumptions ... 

In deriving theoretical equations of the current-potential (or time) curves of ion transfer of an acid we shall make essentially the same assumptions as the assumption 1-6 above. It is noted here that theoretical equations of the more general case, that is, of a dibasic acid, such as expressed by AH2 = AH + H, AH = A + H, can be derived [24], but are not included here, to save space. The formal formation constant, and formal dissociation constant,, in the a phase is defined by... 

For hydrophobic, (virtually) nonionizable substances [i.e., those that show no ionic species of significance in the pH range 1 to 10 (e.g., diazepam)], solubility can usually be improved by addition of nonpolar solvents. Aside from solubility, stability is also affected by solvents in either a favorable or a nonfavorable direction [6], Theoretical equations for solubility in water [7] and in binary solvents [8] have been reported in literature, but in general the approach in preformulation is pseudoempirical. Most often the solubility changes as the concentration of nonpolar solvent C2, increases. For binary systems it may simply be a monotonely changing function [9], as shown in Fig. 2. The solubility is usually tied to the dielectric constant, and in a case such as that shown by the squares, the solubility is often log-linear when plotted as a function of inverse dielectric constant, E, that is,... 

If the formulator knows a priori the theoretical equation for the formulation properties of interest, no experimentation is necessary. However, much of the work in pharmaceutics has been in the pursuit of such relationships, and to our knowledge most have not been determined. Therefore, it remains the task of the formulator to generate the relationships between the variables for the particular formulation and process. 

Surface tension, o, as shown in previous theoretical equations for "vap of boiling liquid metals or for q"nx of ordinary liquids, is included in the form (comparable effect of ct is found on q"c. However, its effect on q"ub is not so certain, as will also be discussed later under the effect of subcooling. 

Film boiling on horizontal cylinders. In his pioneering work on film boiling on a horizontal cylinder, Bromley proposed a classical theoretical equation in dimensionless form ... 

The actual response of Well A to the injection of acid cannot fit with any kind of theoretical equation for undamaged primary porosity or damaged double porosity reservoir. Therefore, the acid attack in secondary porosity formations proceeds very differently, as expected. 

In a least squares parameter estimation, it is desired to find parameters that minimize the sum of squares of the deviation between the experimental data and the theoretical equation. 

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