Absolute equivalence

This kind of representation for a single nuclear spin is absolutely equivalent to the classic vector model. 

Because part of the anomalous dispersion component is jt/2 out of phase with the isomorphous, real component, the net observable effect is a breakdown of Friedel s law regarding the perfect equality of the magnitudes of and If-h-k-i- That is, the two need not be absolutely equivalent but can demonstrate some slight difference A I anom = fhki — f-h-k-i- This difference will normally be imperceptible and within the expected statistical error of most X-ray diffraction intensity measurements, but with care in data collection, and judicious choice of X-ray wavelength, it can be measured and used to obtain phase information in conjunction with isomorphous replacement phase determination, or even independently, as described in Chapter 8. 

Molar mass determination methods can be classed as absolute, equivalent, or relative. Absolute methods allow the molar mass to be directly calculated from the measured quantities without the need for assumptions concerning the physical and/or chemical structure of the polymers. In contrast, equivalent methods require a knowledge of the chemical structure of the macromolecules. Relative methods depend on the chemical and physical structure of the solute as well as on the solute-solvent interaction these methods require calibration against another molecular mass determination method. 

Fig. 3.7 Response S of the SnO nanowires operated in self-heating mode and with extemai microheater to [NOJ = 0.5 ppm. The maximum response to this gas with and without heater (/ =10 nA) is the absolute equivalent (inset) (Reprinted with permission from Prades et al. (2008), Copyright 2008 American Institute of Physics)...

In our example, measurement efforts are minimized to a maximum of two calibration series. If the recovery function is not explicitly required by guidelines that apply to one s practice, one may skip the preparation and independent measurement of series of control samples. An instant comparison of calibrations in the matrix and in pure solvent will suffice. The objective evidence of this statistically sound comparison proves absolutely equivalent to evaluating the recovery function. 

The calculation tells us that this view of things also yields expression [4.24] for the isotherm. In fact, this means that a zone of divariance with localized water molecules can be viewed either as a substoichiometric hydrate in relation to a superior hydrate, or as an overstoichiometric hydrate in relation to an inferior hydrate (or an anhydrous salt). The two models are absolutely equivalent. 

The BSSE is difficult to calculate accurately. We know there will be a BSSE for any finite basis set, but most of the time we do not know its magnitude. Later in this chapter we will discuss the counterpoise method invented by Boys and Bernardi, a common method to estimate the error using basis sets from both centers to calculate both the atom and the diatomic. The method often works well, but it does have an inherent error. The error is the following what we desire is really the energy of, say, the diatomic and the energy of the separate atoms calculated with the same functions as used by the diatomic. In our hydrogen molecule example, the occupancy of each atomic function from its own center is not exactly one because the functions overlap. So we should use only part of the atomic function from the other center in our calculations of the atomic energy to get an absolutely equivalent basis for both atom and molecule. That proves to be almost impossible to do except in certain cases that are explored later. 

In order to test the measuring system and verify the measuring principle, measurements were initially conducted using steel balls and then with non-spherical, but regularly-shaped particles. The measurements with the steel balls allow an assessment of whether the three optical sub-systems operate to an absolutely equivalent standard. From the results, one can also assess the highest attainable measuring reproducibility, the accuracy and resolution of such a system (i.e. in the absence of any shape influence). 

The differential flatness systems concept was introduced by Michel Fliess, and his teamwork through the concepts ofdifferential algebra (Fliess, 1994). They conceive a system as a differential field, which is generated by a set of variables (states and inputs). Later, Martin (1997) redefined this concept in amore geometric context, in which flatness systems could be described in terms of absolute equivalence. 

Scale of Operation In an acid-base titration the volume of titrant needed to reach the equivalence point is proportional to the absolute amount of analyte present in the analytical solution. Nevertheless, the change in pH at the equivalence point, and thus the utility of an acid-base titration, is a function of the analyte s concentration in the solution being titrated. 

One particular case of multicomponent diffusion that has been examined is the dilute diffusion of a solute in a homogeneous mixture (e.g., of A in B -h C). Umesi and Danner compared the three equations given below for 49 ternaiy systems. All three equations were equivalent, giving average absolute deviations of 25 percent. 

The mercury barometer (Fig. 10-11) indicates directly the absolute pressure of the atmosphere in terms of height of the mercuiy column. Normal (standard) barometric pressure is 101.325 kPa by definition. Equivalents of this pressure in other units are 760 mm mercury (at 0°C), 29.921 iuHg (at 0°C), 14.696 IbFin, and 1 atm. For cases in which barometer readings, when expressed by the height of a mercuiy column, must be corrected to standard temperature (usually 0°C), appropriate temperature correction factors are given in ASME PTC, op. cit., pp. 23-26 and Weast, Handbook of Chemistty and Physics, 59th ed., Chemical Rubber, Cleveland, 1978-1979, pp. E39-E41. 

Suction Limitations of a Pump Whenever the pressure in a liquid drops below the vapor pressure corresponding to its temperature, the liquid will vaporize. When this happens within an operating pump, the vapor bubbles will be carried along to a point of higher pressure, where they suddenly collapse. This phenomenon is known as cavitation. Cavitation in a pump should be avoided, as it is accompanied by metal removal, vibration, reduced flow, loss in efficiency, and noise. When the absolute suction pressure is low, cavitation may occur in the pump inlet and damage result in the pump suction and on the impeller vanes near the inlet edges. To avoid this phenomenon, it is necessary to maintain a required net positive suction head (NPSH)r, which is the equivalent total head of liquid at the pump centerline less the vapor pressure p. Each pump manufacturer publishes curves relating (NPSH)r to capacity and speed for each pump. 

For practical engineering purposes, relative density at 15°C (288 K), widely used in countries outside the United States, is considered equivalent to specific gravity at 60°F (288.6 K). With the adoption of SI units, the American Petroleum Institute favors absolute density at 288 K instead of degrees API. 

A stream of 100 gpm (379 1/min) liquid expanding from 1,000 psi (6,900 kN/m ) to 100 psi (690 kN/m ) if it does not flash will develop approximately 40 hp (29.9 kW). Flashing usually more than doubles the power. Although this is only a small fraetion of the total required refrigeration, it is near the eoldest point in the system and is valuable. A hundred hp is the equivalent to a quarter of a million Btu per hour (73.3 kW) of heat. This would vaporize 2,900 lb (1,315 kg) per hour of liquid nitrogen at an absolute pressure of 20 psi (138 kN/m ). 

A. ot-Chloroelhyl ethyl ether. A mixture of 200 g. (201 ml.) of redistilled paraldehyde, b.p. 121-122.5° (equivalent to 4.54 moles of acetaldehyde), and 200 g. (254 ml., 4.34 moles) of absolute ethanol is placed in a 1-1. three-necked flask fitted with a mechanical stirrer and a gas inlet tube reaching to the bottom of the flask. The mixture is cooled to —5° in a mixture of Dry Ice and acetone, and dry hydrogen chloride (Note 1) is passed into the stirred reaction mixture maintained at about —5° until 200 g. (5.48 moles) has been absorbed. During this operation, which requires about 2 hours, the reaction mixture separates into two layers. The upper layer of crude a-chloroethyl ethyl ether is re-... 

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