Empirical, equation observations

Comparison with the empirical Equation (1.4) shows that = /re /S/z eg and that n" = 2 for the Balmer series. Similarly n" = 1, 3, 4, and 5 for the Lyman, Paschen, Brackett and Pfimd series, although it is important to realize that there is an infinite number of series. Many series with high n" have been observed, by techniques of radioastronomy, in the interstellar medium, where there is a large amount of atomic hydrogen. For example, the (n = 167) — ( " = 166) transition has been observed with V = 1.425 GFIz (1 = 21.04 cm). 

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

The study of the hydrogen atom also played an important role in the development of quantum theory. The Lyman, Balmer, and Paschen series of spectral lines observed in incandescent atomic hydrogen were found to obey the empirical equation... 

The values of AG f have been calculated using the Uhlig formula [Eq. (4)] for the charge-independent part and the empirical equations [Eqs. (45), (46), (49), (50), and (51)] for the charge-dependent part. Table 5 gives the calculated values of ACrlf for 34 ions, and the values are compared in Fig. 12 with the observed values shown in Table 4. As seen... 

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

In the absence of any equations based on fundamental material behaviour, equations such as these should be used for extrapolation, although it is generally dangerous to extrapolate in this way by more that than one decade in time. Other empirical equations may be used, provided that Occam s principle is observed, i.e., the theory should be no more complicated than is necessary to fit the observations. Higher order polynomials in time should be avoided, since they have a tendency to diverge rapidly towards infinity, or minus infinity, when they are extrapolated beyond the range of measurements. 

Activity coefficients can be determined by experimental observations. Since they are functions of ionic strength, temperature and pressure, marine scientists typically estimate values at the environmental conditions of interest from semi-empirical equations. In dilute solutions, the activity coefficient of a monoatomic ion can be calculated from the Debye-Hiickel equation ... 

The pressure peaks observed in the combustion tests shown in Fig. 14.19 are computed as a function of L/D as shown in Fig. 14.20. The peak pressures computed by means of the Lenoir-Robillard empirical equation are confirmed by the measured pressure at the head-end of the motor. It is evident thatp values predicted without erosive burning are significantly lower than the measured maximum pressures. Fig. 14.21 shows the erosive ratio, 8 = r/to, as a function of the mass flow rate per unit cross-sectional area in the port, G. The erosive ratio increases with increasing Mach number in the port at constant L/D. 

If X is the fraction of A converted to B and y is the fraction of A converted to C then, for many systems, it has been observed that x and y conform to the empirical equation... 

The question that we need to ask ourselves is whether the longitudinal dispersion can be predicted accurately for these rivers. Equation (6.35), which predicts that >l/(m+/i) = constant, is shown in Table 6.4 to have a large range of constants, probably because of the variations in cross section and morphology seen in natural streams. Fisher (1973) observed that this constant seemed to depend on mean surface width, W, and substituted W for h in the numerator of equation (6.34) to develop the following empirical equation to characterize longitudinal dispersion coefficient in rivers ... 

Alvarez Gonzalez et al. (19) stated that those systems in which the value of n is less than unity move the azeotrope to richer ethanol compositions, even to breaking the azeotrope. This was also observed for the systems presently under consideration. Reference 19 proposes a correlation of the equilibrium data by the empirical equation ... 

The above empirical equation can be derived theoretically, and the deviation of the experimentally observed Ce isotope ratio data from Curve I illustrates the extent to which the tropospheric atmosphere is not instantly and uniformly mixed this enables us to follow the eastward movement of the nuclear debris around the world... 

The principal innovations that have been made in the discussion of the theory of the chemical bond in this edition are the wide application of the electroneutrality principle and the use of an empirical equation (Sec. 7-10) for the evaluation of the bond numbers of fractional bonds from the observed bond lengths. A new theory of the structure of electron-deficient substances, the resonating-valence-bond theory, is described and used in the discussion of the boranes, ferrocene, and other substances. A detailed discussion of the valence-bond theory of the electronic structure of metals and intermetallic compounds is also presented. 

PHYSICAL VARIABLES. Currently, the pressure dependence of burning velocity is in dispute. Most workers find that Ul P B, where n is between 0.1 and 0.5. Recent work at the U. S. Bureau of Mines indicates that the exponent on the pressure is actually a function of the burning velocity (49, 58), an observation that may clarify the situation. Figure 7 is a plot of the exponent, n, in the empirical equation... 

For practical purposes, within the NN frame the ionic strength dependence can be taken into account in the estimate of the free energy and of the melting temperature by using empirical equations with numerical prefactors. Such equations display a leading logarithmic dependence on salt concentration which reproduces the experimental observations well and allows simple and reliable calculations [3, 15]. 

Nevertheless, at present a distinct tendency is observed not to restrict oneself to the semi-empirical equations, but to apply kinetic models corresponding to the physico-chemical substantiated mechanisms. As a rule, modifications of the well-known adsorption (L-H) and impact (E-R) mechanisms are used. As usual, the complex mechanism always has a constituent accounting for the adsorption mechanism. 

It seems likely that the magnitude of p represents the "extent of electron demand at phosphorus , in the T. S. or in other words is a measure of "electron transfer" in the T.S., a term (=z) which appears in the semi-empirical equations describing the nucleophilic reactivity of tricoordinated phosphorus. This concept is reinforced by (i) the observation (from the data of Bokonov. and Goetz ) that p-values based on the pKg values of protonated phosphines increase with increasing pKa, i.e. increase with a shift of the equilibrium (eqn. 4) to the left and... 

Two relaxation processes, called 8 and y can be observed, the last as a shoulder of the low temperature side of the f i relaxation [37], Assuming symmetry for these two relaxations the analysis according to Fuoss-Kirkwood empirical equation (2.35) can be performed as in previous systems [36,37,69], Due to the presence of the /3 relaxation an exhaustive analysis of this systems allow to obtain important information about these relaxation processes. 

Empirical Kinetic Equation for Coke Formation The experimental observations given in section Vlb(l), (2), and (3) lead to the empirical equation for coke formation... 

Wiedensohler (1988) developed an approximation for the Fuchs model, taking advantage of the observation that for an aerosol in charge equilibrium, the fraction of particles of any size with 3 or more elementary units of charge of the same sign can be calculated from Eq. 12.40. Then, for small particles with up to 2 elementary units of charge, he proposed the empirical equation... 

As illustrated in Figure 7, the release profiles of both tablet coatings show a linear behavior only at small times, and then exponential increases in concentrations were observed. Based on this behavior, the release profiles were fitted to an empirical equation as... 

Three-dimensional Hiickel molecular orbital calculations have been performed on a series of molecules made from the atoms H, C, N, O, F, and CL It has been found that the sum of the Mulliken overlap populations is closely related to the energy of atomization. For 40 compounds not containing carbon, the observed energies of atomization can be reproduced with a mean deviation of 11.1 kcal./mole by a simple empirical equation which includes a term to account for the extra stability of polar molecules. A slightly more complicated expression is needed for compounds of carbon, and the fit to the observed data is not as good. 

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