Degree of accuracy

The potential fiinctions for the mteractions between pairs of rare-gas atoms are known to a high degree of accuracy [125]. Flowever, many of them use ad hoc fiinctional fonns parametrized to give the best possible fit to a wide range of experimental data. They will not be considered because it is more instmctive to consider representations that are more finnly rooted in theory and could be used for a wide range of interactions with confidence. 

With the advent of quantum mechanics, quite early attempts were made to obtain methods to predict chemical reactivity quantitatively. This endeavor has now matured to a point where details of the geometric and energetic changes in the course of a reaction can be calculated to a high degree of accuracy, albeit still with quite some demand on computational resources. 

Th e geoinetries obtain ed from opt irn i.ration s with sein i-ern pirical calculations describe the shapes of niolcctiles. I he calculations have varying degrees of accuracy and take more time than molecular mechan ics methods. The accuracy of the results depends on th e m olecti le. 

A chemist must realize that theories, models, and approximations are powerful tools for understanding and achieving research goals. The price of having such powerful tools is that not all of them are perfect. This may not be an ideal situation, but it is the best that the scientihc community has to offer. Chemists are advised to develop an understanding of the nature of computational chemistry approximations and what results can be trusted with any given degree of accuracy. 

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ah initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very-high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned. 

The parameters in the original parameterization are adjusted in order to reproduce the correct results. These results are generally molecular geometries and energy differences. They may be obtained from various types of experimental results or ah initio calculations. The sources of these correct results can also be a source of error. Ah initio results are only correct to some degree of accuracy. Likewise, crystal structures are influenced by crystal-packing forces. 

The first stage in the interpretation of a physisorption isotherm is to identify the isotherm type and hence the nature of the adsorption process(es) monolayer-multilayer adsorption, capillary condensation or micropore filling. If the isotherm exhibits low-pressure hysteresis (i.e. at p/p° < 0 4, with nitrogen at 77 K) the technique should be checked to establish the degree of accuracy and reproducibility of the measurements. In certain cases it is possible to relate the hysteresis loop to the morphology of the adsorbent (e.g. a Type B loop can be associated with slit-shaped pores or platey particles). 

The degree of accuracy can be assessed from the error in measurement. If Rg is the true ratio (64.5) and R is the mean, then... 

Accurate, precise isotope ratio measurements are important in a wide variety of applications, including dating, examination of environmental samples, and studies on drug metabolism. The degree of accuracy and precision required necessitates the use of special isotope mass spectrometers, which mostly use thermal ionization or inductively coupled plasma ionization, often together with multiple ion collectors. 

Part of the answer is that, unless we require a high degree of accuracy, it does not matter very much. This is illustrated by the values for Tq and for and N2, given in Table... 

This last definition should be carefully appHed as either an interpolation or an extrapolation, particularly for empirical computational methods based on diverse observations. It is critical that users of molecular modeling tools understand where it is appropriate to apply a technique and where it is not, and what degree of accuracy can be expected. 

The thermographic sensor is used as a remote sensing radiometer when a reference target is imaged. It is usually necessary to correct for emissivity and atmospheric transmission to determine surface temperature with a reasonable degree of accuracy. 

If the pump manufacturer uses motors for pump power measurement, these motors ate caUbrated to determine the horsepower from the electric power reading and caUbration curves. Such test motors ate recaUbrated periodically, ensuring the same degree of accuracy as shown by the torque meters. 

To solve a flow problem or characterize a given fluid, an instmment must be carefully selected. Many commercial viscometers are available with a variety of geometries for wide viscosity ranges and shear rates (10,21,49). Rarely is it necessary to constmct an instmment. However, in choosing a commercial viscometer a number of criteria must be considered. Of great importance is the nature of the material to be tested, its viscosity, its elasticity, the temperature dependence of its viscosity, and other variables. The degree of accuracy and precision required, and whether the measurements are for quaUty control or research, must be considered. The viscometer must be matched to the materials and processes of interest otherwise, the results may be misleading. 

Because of the diversity of available information and the continuing attempts to delineate the various world oil sands deposits, it is virtually impossible to reflect the extent of the reserves in terms of barrel units with a great degree of accuracy. The potential reserves of hydrocarbon Hquids that occur in tar sand deposits have, however, variously been estimated on a world basis to be in excess of 477 x 10 (3 x 10 bbl). Reserves that have... 

The equations given predict vapor behavior to high degrees of accuracy but tend to give poor results near and within the Hquid region. The compressibihty factor can be used to accurately determine gas volumes when used in conjunction with a virial expansion or an equation such as equation 53 (77). However, the prediction of saturated Hquid volume and density requires another technique. A correlation was found in 1958 between the critical compressibihty factor and reduced density, based on inert gases. From this correlation an equation for normal and polar substances was developed (78) ... 

When a process is continuous, nucleation frequently occurs in the presence of a seeded solution by the combined effec ts of mechanical stimulus and nucleation caused by supersaturation (heterogeneous nucleation). If such a system is completely and uniformly mixed (i.e., the product stream represents the typical magma circulated within the system) and if the system is operating at steady state, the particle-size distribution has definite hmits which can be predic ted mathematically with a high degree of accuracy, as will be shown later in this section. 

Thus, to third order in strain, the entropy along the Hugoniot is constant, and weak shock waves are nearly isentropic. For small strains, the Hugoniot can be replaced with the isentrope to a high degree of accuracy. At the initial state, the Hugoniot and isentrope have the same slope and curvature in the P-V plane. 

Computational methods have played an exceedingly important role in understanding the fundamental aspects of shock compression and in solving complex shock-wave problems. Major advances in the numerical algorithms used for solving dynamic problems, coupled with the tremendous increase in computational capabilities, have made many problems tractable that only a few years ago could not have been solved. It is now possible to perform two-dimensional molecular dynamics simulations with a high degree of accuracy, and three-dimensional problems can also be solved with moderate accuracy. 

To determine the pipeline potentials, the resultant induced field strengths have to be included in the equations in Section 23.3.2. Such calculations can be carried out with computers that allow detailed subdivision of the sections subject to interference. A high degree of accuracy is thus achieved because in the calculation with complex numbers, the phase angle will be exactly allowed for. Such calculations usually lead to lower field strengths than simplified calculations. Computer programs for these calculations are to be found in Ref. 16. 

Solving Newton s equation of motion requires a numerical procedure for integrating the differential equation. A standard method for solving ordinary differential equations, such as Newton s equation of motion, is the finite-difference approach. In this approach, the molecular coordinates and velocities at a time it + Ait are obtained (to a sufficient degree of accuracy) from the molecular coordinates and velocities at an earlier time t. The equations are solved on a step-by-step basis. The choice of time interval Ait depends on the properties of the molecular system simulated, and Ait must be significantly smaller than the characteristic time of the motion studied (Section V.B). 

X-ray structures are determined at different levels of resolution. At low resolution only the shape of the molecule is obtained, whereas at high resolution most atomic positions can be determined to a high degree of accuracy. At medium resolution the fold of the polypeptide chain is usually correctly revealed as well as the approximate positions of the side chains, including those at the active site. The quality of the final three-dimensional model of the protein depends on the resolution of the x-ray data and on the degree of refinement. In a highly refined structure, with an R value less than 0.20 at a resolution around 2.0 A, the estimated errors in atomic positions are around 0.1 A to 0.2 A, provided the amino acid sequence is known. 

The PTC 10 defines three classes of testing. The code attempts to categorize testing and thereby establish an inherent degree of accuracy. These categories are based on methods of test and methods of analysis. 

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