Semiempirical mass equation

von Weizsacker developed a crude theory of nuclear masses in 1935. The theory takes as its basis the idea that nuclei behave like incompressible uniformly charged liquid drops. How can we account for the variation of nuclear masses We begin by stating that 

Weizsacker s mass equation has evolved into what is called the semiempirical mass equation, which begins by parameterizing the total binding energy of species Z, 4 as 

The justification of this representation of the total binding energy of the nucleus is as follows  

Since there are A nucleons in the nucleus and the nuclear force saturates, we expect each nucleon to contribute to the total binding energy. This term is known as the volume term. The coefficient av is the energy by which a nucleon in the interior of the nucleus is bound to its nearest neighbors and is a parameter to be determined experimentally. 

The third term reflects the decrease in binding due to the Coulomb repulsion between the protons. The Coulomb energy of a uniform sphere can be written as 

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Figure 2.5 Relative contributions of the various terms in the semiempirical mass equation to the average binding energy per nucleon from W. E. Meyerhof, Elements of Nuclear Physics. Copyright 1967 by McGraw-Hill Book Company, Inc. Reprinted by permission of McGraw-Hill Book Company, Inc.

Myers and Swiatecki (1966) have proposed a modification of the semiempirical mass equation that gives a better description of the experimental masses. This modification can be summarized in the following equation ... 

We will now look at some of the predictions of the semiempirical mass equation. The first question we pose is what happens if we hold A constant and vary Z (neglecting for a moment the pairing term). We can write... 

Use the semiempirical mass equation to derive an expression for the energy released in a decay. For fixed Z, how should the energy released depend on A ... 

For very neutron-deficient (i.e., proton-rich) nuclei, the Q value for proton emission, Qp, becomes positive. One estimate, based on the semiempirical mass equation, of the line that describes the locus of the nuclei where Qp becomes positive for ground-state decay is shown in Figure 7.11. This line is known as the proton-drip line. Our ability to know the position of this line is a measure of our ability to describe the forces holding nuclei together. Nuclei to the left of the proton dripline in Figure 7.11 can decay by proton emission. 

Using the semiempirical mass equation, verify that Qa becomes positive for A > 150. 

With the semiempirical mass equation (3.8) estimate the binding energy per nucleon for 6, Al, Co, and Compare the results with the observed values in Table 3.1. 

In 4.4 we found that fission of heavy nuclei like is exoergic and in 4.6 that it is a common decay mode for the heaviest nuclei. From the semiempirical mass equation (3.8) it can be found that fission of nuclei with A S 100 have positive 0-values. Why is the decay by spontaneous fission only observed for nuclei with A 230 ... 

J. Water Pollut. Control Fed., V. 51, N. 11, Nov. 1979, pp. 2602-2614 A preliminary design methodology is presented for the sidestream softeners in zero-discharge cooling water systems. The methodology consists of semiempirical chemical equations which were calibrated by experiments and which are linked to the mass balances of the important chemical species. The only inputs required are the makeup water quality and certain cooling water system parameters. 20 refs, cited. 

Vacuum pyrolysis of 12 has been recently studied by high-resolution mass spectrometry200. The ionization potential (IP = 9.17 0.10 eV) for silole 2 was found to be in excellent agreement with the IP (9.26 eV) calculated by the semiempirical PM3 method. Thermodynamic calculations on possible decomposition mechanisms of 12 have shown that 2 is probably formed from the primary intermediate silylene 9 via two consecutive hydrogen shifts (equation 3). 

The substance-specific kinetic constants, kx and k2, and partition coefficient Ksw (see Equations 3.1 and 3.2) can be determined in two ways. In theory, kinetic parameters characterizing the uptake of analytes can be estimated using semiempirical correlations employing mass transfer coefficients, physicochemical properties (mainly diffusivities and permeabilities in various media), and hydro-dynamic parameters.38 39 However, because of the complexity of the flow of water around passive sampling devices (usually nonstreamlined objects) during field exposures, it is difficult to estimate uptake parameters from first principles. In most cases, laboratory experiments are needed for the calibration of both equilibrium and kinetic samplers. 

The kinetic parameters reported in Table 1 was applied to calculate the mean value of activity of the catalyst in two industrial reactors to simulate their performance. The mass-transfer parameters of the bubble columnns were calculated using semiempirical equations [5,6], The physical constant values are reported in previous works [1,2]. 

Parameters k and k2 can be easily related to the hydrodynamic conditions (flow rate, stirring rates) and to the current density by empirical equations. The influence of the current density can also be related to the reagent dose for parameter k and to the bubble generation for parameter k2 (the flow rate of cathodically generated hydrogen is proportional to the current density). Thus, this semiempirical model considers easily and simultaneously the gas-liquid mass transfer, the collections of solid particles in electroflotation processes, and the effect of the current density. 

Experimental Mass Transfer Coefficients. Hundreds of papers have been published reporting mass transfer coefficients in packed columns. For some simple systems which have been studied quite extensively, mass transfer data may be obtained directly from the literature (6). The situation with respect to the prediction of mass transfer coefficients for new systems is still poor. Despite the wealth of experimental and theoretical studies, no comprehensive theory has been developed, and most generalizations are based on empirical or semiempirical equations. 

The recommended approach to modeling is to create models based on fundamental balances (of mass, species, energy, population) and basic kinetics and use them to build a complete model of the precipitator, as shown in earlier sections. Such a set of equations is known as a physical or a mechanistic model. Complete physical models are difficult to create and solve because they require identification in advance of all physical and chemical subprocesses, properties, and parameters. That is why the semiempirical models of a form similar to the complete physical models (but usually simpler) and with fewer equations are often used for scaling up. Parameters of such models are often given in lumped form, some of them fitted to available experimental data obtained from the small-scale system. Such a model can be useful for scaling up, but one cannot be sure that the scale-up will be completely correct because there is no guarantee that the model contains the complete mechanism (88). However, scale-up errors should be smaller than in the case of purely empirical models. CFD codes that are based on reasonable simplifications (closures) regarding their accuracy can be placed between the physical and semiempirical models their application was demonstrated earlier. 

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