Large numbers, multiplying

The rate is thus the number of collisions between A and B - a very large number - multiplied by the reaction probability, which may be a very small number. For example, if the energy barrier corresponds to 100 kj mol , the reaction probability is only 3.5 x lO l at 500 K. Hence, only a very small fraction of all collisions leads to product formation. In a way, a reaction is a rare event For examples of the application of collision theory see K.J. Laidler, Chemical Kinetics 3 Ed. (1987), Harper Row, New York. 

G is a multiplier which is zero at locations where slip condition does not apply and is a sufficiently large number at the nodes where slip may occur. It is important to note that, when the shear stress at a wall exceeds the threshold of slip and the fluid slides over the solid surface, this may reduce the shearing to below the critical value resulting in a renewed stick. Therefore imposition of wall slip introduces a form of non-linearity into the flow model which should be handled via an iterative loop. The slip coefficient (i.e. /I in the Navier s slip condition given as Equation (3.59) is defined as... 

An array ion collector (detector) consists of a large number of miniature electron multiplier elements arranged side by side along a plane. Point ion collectors gather and detect ions sequentially (all ions are focused at one point one after another), but array collectors gather and detect all ions simultaneously (all ions are focused onto the array elements at the same time). Array detectors are particularly useful for situations in which ionization occurs within a very short space of time, as with some ionization sources, or in which only trace quantities of a substance are available. For these very short time scales, only the array collector can measure a whole spectrum or part of a spectrum satisfactorily in the time available. 

A surprisiagly large number of important iadustrial-scale separations can be accompHshed with the relatively small number of zeoHtes that are commercially available. The discovery, characterization, and commercial availabiHty of new zeoHtes and molecular sieves are likely to multiply the number of potential solutions to separation problems. A wider variety of pore diameters, pore geometries, and hydrophobicity ia new zeoHtes and molecular sieves as weU as more precise control of composition and crystallinity ia existing zeoHtes will help to broaden the appHcations for adsorptive separations and likely lead to improvements ia separations that are currently ia commercial practice. 

Lagrange Multiplier Method for programming problems, 289 for weapon allocation, 291 Lamb and Rutherford, 641 Lamb shift, 486,641 Lanczos form, 73 Landau, L. D., 726,759, 768 Landau-Lifshitz theory applied to magnetic structure, 762 Large numbers, weak law of, 199 Law of large numbers, weak, 199 Lawson, J. L., 170,176 Le Cone, Y., 726... 

To find the power series expansion of Eq. (30) in ub, ojc, u>d we can thus replace the first-order responses of the cluster amplitudes and Lagrangian multipliers and the second-order responses of the cluster amplitudes by the expansions in Eqs. (37), (39) and (44) and express OJA as —ojb ojc — ojd- However, doing so starting from Eq. (30) leads to expressions which involve an unneccessary large number of second-order Cauchy vectors C m,n). To keep the number of second-order... 

A brief study of the available data related to limits of inflammability in Part Two shows that these parameters are subject to high experimental uncertainty. For a large number of substances, the experimental values are widely dispersed. When they are submitted to quality estimation using statistical tools, in many cases they reveal that it is impossible to use them with confidence. The examples of difficulties raised by the statistical analysis of the LEL data can be multiplied. 

Dissemination. A second goal is the dissemination of the results of demonstration projects to a large number of plants in the industrial sector, in order to obtain a multiplier effect. 

The generation of large numbers of complex, poly-atomic and multiply charged ions in the sputtering process [15,16,17] creates potentially severe problems for low resolution secondary ion isotopic analysis. To minimize the formation of hydride and... 

Multiplying a concentration profile, or column of C, with a factor is equivalent to multiplication of the corresponding column of T with the same factor. Any one element of each column vector of T can be chosen freely while the other elements in that column define the shape of the concentration profile. In order to avoid numerical problems with very small or very large numbers in each column of T, we choose the largest absolute element of each column of the matrix of initial guesses TgUess and keep it... 

For example, to express 6,403,500,000 in scientific notation, first change the number to a decimal between 1 and 10, that is 6.4035. Now, multiply this decimal by a power of 10, determined by the number of placeholders the decimal was moved. This is a large number, so the power of 10 will be positive. 

See how much shorter the number in scientific notation is than the number written out the long way Also, comparing two large numbers is easier when they re written in scientific notation. You look at the power of 10, first, and then compare the multiplier. 

The exact solution of the problem leads to the same expression with a proportionality constant between 3 and 5, depending on the definition of the thickness of the boundary layer. In the following sections, the preceding evaluation procedure is applied to a large number of problems, particularly to complex cases for which limiting solutions can be obtained. As already noted in the introduction, the terms in the transport equations will be replaced by their evaluating expressions multiplied by constants. The undetermined constants will then be determined from solutions available for some asymptotic cases. 

Thermal-ionization mass spectrometers (TIMS) combine a hot-filament source with a magnetic-sector mass spectrometer. The mass spectrometers are operated at low to moderate mass-resolving power. A large number of elements can be measured with thermal ionization mass spectrometry. Special care is taken to purify the samples using ion exchange columns. Samples are loaded onto the filaments along with an emitter, and a typical run may take several hours. Modem systems have multiple collectors so that several isotopes can be measured simultaneously. High-precision measurements are done with Faraday cup detectors, but low-abundance isotopes can be measured on electron multipliers. Modem machines are capable of precisions of 0.1 to 0.01 permit. 

We can see that this equation involves e twice. On its first occurrence it is multiplied by the dimensionless initial concentration of the reactant n0. It has already been mentioned that n0 will generally be a very large number, 7t0 1. The first term in eqn (3.15) therefore involves the multiple of a small number e and a large number jt0 as well as the exponential term involving e. If it0 is of the same order of magnitude as the inverse of e (we say if ti0 is of the order of -1, or write n0 0(e-1)) then their product in (3.15) will be neither large nor small. We can thus express their product as another dimensionless group pQ which will then be defined by... 

To understand how this shorthand notation works, consider the large number 50,000,000. Mathematically this number is equal to 5 multiplied by 10 X 10X 10X 10X 10 X 10 X 10 (check this out on your calculator). We can abbreviate this chain of numbers by writing all the 10s in exponential form, which gives us the scientific notation 5 X 107. (Note that 107 is the same as lOx lOx 10x lOx 10 X 10 X 10. Table A. 1 shows the exponential form of some other large and small numbers.) Likewise, the small number 0.0005 is mathematically equal to 5 divided by 10 X 10 x 10 X 10, which is 5/104. Because dividing by a number is exactly equivalent to multiplying by the reciprocal of that number, 5/104 can be written in the form 5 X 10-4, and so in scientific notation 0.0005 becomes 5 X 10-4 (note the negative exponent). 

W hen a formula contains subscripts — the small numerals that indicate how many of a kind — be certain to multiply the atomic weight by the number indicated by the subscript. In cases where, the formula is preceded by a large number, be sure to multiply the molecular weight by this number.)... 

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