Approximate calculation

Make the following approximate calculations for the surface energy per square centimeter of solid krypton (nearest-neighbor distance 3.97 A), and compare your results with those of Table VII-1. (a) Make the calculations for (100), (110), and (111) planes, considering only nearest-neighbor interactions, (b) Make the calculation for (100) planes, considering all interactions within a radius defined by the sum... 

The group contribution method allows the approximate calculation of solubility by summing up fragmental values associated with substmctural units of the compounds (see Section 7.1). In a group contribution model, the aqueous solubility values are computed by Eq. (12), where log S is the logarithm of solubility, C is the number of occurrences of a substmctural group, i, in a molecule, and is the relative contribution of the fragment i. 

The annual processing cost Apo for a similar plant of a different size designed for an annual produc tion rate Ro can be approximately calculated from an equation of the form... 

The processing cost per unit of production for a plant with an annual production Ro can be approximately calculated from... 

In either equation, /c is given by Eq. (16-84) for parallel pore and surface diffusion or by Eq. (16-85) for a bidispersed particle. For nearly linear isotherms (0.7 < R < 1.5), the same linear addition of resistance can be used as a good approximation to predict the adsorption behavior of packed beds, since solutions for all mechanisms are nearly identical. With a highly favorable isotherm (R 0), however, the rate at each point is controlled by the resistance that is locally greater, and the principle of additivity of resistances breaks down. For approximate calculations with intermediate values of R, an overall transport parameter for use with the LDF approximation can be calculated from the following relationship for sohd diffusion and film resistance in series... 

The Sherwood number, Sh, is estimated from Table 16-9, and the dispersion parameters Yi and Jo ffom Table 16-10 for well-packed columns. Typical values are a 1-4 and b 0.5-1. Since HETP -2HTU, Fig. 16-13 can also be used for approximate calculations. 

The availability of energy from an explosion can be approximately calculated in most cases but the method used depends upon the nature of the explosion. 

Approximate calculations of this activation energy have been made in a number of examples using the quanmm theory of molecular binding, by making assumptions concerning the stmcture and paitition functions of the Uansition state molecule. 

The approximate calculation of the surface energies of metals as a function of crystal structure described earlier uses the enthalpy of sublimation, s, and the co-ordination number to calculate the energy as a function of the atomic concentration on the surface. The atomic areas of the principal configurations are as follows ... 

Example Approximate calculation of the hardness of solids. This concept of shear yielding - where we ignore the details of the grains in our polycrystal and treat the material as a continuum - is useful in many respects. For example, we can use it to calculate the loads that would make our material yield for all sorts of quite complicated geometries. 

The stress in the crank shaft is calculated approximately from the power and speed as follows. Bear in mind that approximate calculations of this sort may be in error by up to a factor of 2 - but this makes no difference to the conclusions reached below. Referring to Fig. 16.9 ... 

Constants C and K can be determined from several measurements of filtrate volumes taken at different time intervals. There are some doubts as to the actual constancy of C and K during constant pressure filtration. Constants C and K depend on r (specific volumetric cake resistance), which, in turn, depends on the pressure drop across the cake. This AP causes some changes in the cake, especially during the initial stages of filtration. When the cake is very thin, the main portion of the total pressure drop is exerted on the filter medium. As the cake becomes thicker, the pressure drop through the cake increases rapidly but then levels off to a constant value. Isobaric filtration shows insignificant deviation from the expressions developed. For approximate calculations, it is possible to neglect the resistance of the filter plate, provided the cake is not too thin. Then the filter plate resistance, Rf, is equal to zero, C = 0, and r = 0. Hence, a simplified equation is = Kr. 

Often in plant operations condensate at high pressures are let down to lower pressures. In such situations some low-pressure flash steam is produced, and the low-pressure condensate is either sent to a power plant or is cascaded to a lower pressure level. The following analysis solves the mass and heat balances that describe such a system, and can be used as an approximate calculation procedure. Refer to Figure 2 for a simplified view of the system and the basis for developing the mass and energy balances. We consider the condensate to be at pressure Pj and temperature tj, from whence it is let down to pressure 2. The saturation temperature at pressure Pj is tj. The vapor flow is defined as V Ibs/hr, and the condensate quality is defined as L Ibs/hr. The mass balance derived from Figure 2 is ... 

We can end this discussion of ATP and the other important high-energy compounds in biology by discussing the daily metabolic consumption of ATP by humans. An approximate calculation gives a somewhat surprising and impressive result. Assume that the average adult human consumes approximately... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

Figure 2.Virtual crystal approximation calculations (solid line) compared with coherent potential approximation calculations for Fe-Co (longdashed line), Fe-Ni (dot-dashed line) and Fe-Cu (dashed line). The fcc-bcc energy difference is shown as a function of the atomic number.

These equations can also be used for approximate calculations of F c . The simplest approximation is MFA that neglects all statistical fluctuations and correlations. Therefore, the MFA expression Fmfa given by Eq. (13) corresponds to the omission of the last term in Eq. (16). 

It would be interesting to compare these results with those of a CPA (coherent potential approximation) calculation. For a real test of both methods a system should be found which shows a more pronounced change when going from one phase to the other, than found for the system presented in this paper. 

Read, F, for the approximate calculated Cs, ft/sec from Table 9-37 at calculated liquid rate, Ib/hr-ft. Then read Y from IMTP Packing Pressure Drop chart. Figure 9-2IG and then read curves showing pressure drop, (may require interpolation). 

Note The actual length under tension in the hole can be obtained from pertinent graphs or approximately calculated from... 

Variamine blue (C.I. 37255). The end point in an EDTA titration may sometimes be detected by changes in redox potential, and hence by the use of appropriate redox indicators. An excellent example is variamine blue (4-methoxy-4 -aminodiphenylamine), which may be employed in the complexometric titration of iron(III). When a mixture of iron(II) and (III) is titrated with EDTA the latter disappears first. As soon as an amount of the complexing agent equivalent to the concentration of iron(III) has been added, pFe(III) increases abruptly and consequently there is a sudden decrease in the redox potential (compare Section 2.33) the end point can therefore be detected either potentiometrically or with a redox indicator (10.91). The stability constant of the iron(III) complex FeY- (EDTA = Na2H2Y) is about 1025 and that of the iron(II) complex FeY2 - is 1014 approximate calculations show that the change of redox potential is about 600 millivolts at pH = 2 and that this will be almost independent of the concentration of iron(II) present. The jump in redox potential will also be obtained if no iron(II) salt is actually added, since the extremely minute amount of iron(II) necessary is always present in any pure iron(III) salt. 

Kessler, P., Comfit, rend. 240, 1058, Calcul approch du terme du second ordre en th<5orie des perturbations." Approximate calculation of the second-order term in perturbation theory. 

The potential usefulness of x-ray emission spectrography for trace analysis is implicit in the results of approximate calculations presented in Chapter 4. Thus, it was estimated that the intensity of cobalt Ka generated under practicable conditions in a monolayer (area, 1 sq cm) of cobalt atoms might give 133 counts per second (4.16). Such a sample weighs 0.2 pg. 

Connor J. N. L., Sun H., Hutson J. M. Exact and approximate calculations for the effect of potential anisotropy on integral and differential cross-sections Ar-N2 rotationally inelastic scattering, J. 

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