Root-mean-square ratio

The eoneept of an Amplifieation Faetor (AF) is introdueed in the new API 616 standard. Amplifieation faetor is defined as the ratio of the eritieal speed to the speed ehange at the root mean square of the eritieal amplitudes. 

A bottle contains 1.0 mol He(g) and a second bottle contains 1.0 mol Ar(g) at the same temperature. At that temperature, the root mean square speed of He is 1477 m-s 1 and that of Ar is 467 nvs-1. What is the ratio of the number of He atoms in the first bottle to the number of Ar atoms in the second bottle having these speeds Assume that both gases behave ideally. 

Determine the ratio of the number of molecules in a gas having a speed ten times as great as the root mean square speed to the number having a speed equal to the root mean square speed. Is this ratio independent of temperature Why ... 

Turbulent mass burning rate versus the turbulent root-mean-square velocity by Karpov and Severin [18]. Here, nis the air excess coefficient that is the inverse of the equivalence ratio. (Reprinted from Abdel-Gayed, R., Bradley, D., and Lung, F.K.-K., Combustion regimes and the straining of turbulent premixed flames. Combust. Flame, 76, 213, 1989. With permission. Figure 2, p. 215, copyright Elsevier editions.)... 

Signal-to-noise ratio Ratio of signal intensity to the root-mean-square noise level. 

The root-mean-square distance Vr separating the ends of the polymer chain is a convenient measure of its linear dimensions. The dissymmetry coefficient will be unity for (VrV 0< l and will increase as this ratio increases. 

Apart from their utility in determining the correction factor 1/P( ), light-scattering dissymmetry measurements afford a measure of the dimensions of the randomly coiled polymer molecule in dilute solution. Thus the above analysis of measurements made at different angles yields the important ratio from which the root-mean-square... 

Eq. (16 ) reverts to (8) for sufficiently small values of the ratio, r/r, of the displacement length to the maximum length. The bracketed quantity in the exponential of Eq. (16 ) may be looked upon as a correction factor on the value of which should have been used in the simpler Eq. (8). For r/vm, less than about one-half, this correction is negligible. The ratio of the root-mean-square length to the maximum length varies inversely as the square root of the maximum length, according to Eq. (13), i.e., / / m = from which it... 

In order to verify the conditions of this averaging process, one has to relate the displacements during the encoding time - the interval A between two gradient pulses, set to typically 250 ms in these experiments - with the characteristic sizes of the system. Even in the bulk state with a diffusion coefficient D0, the root mean square (rms) displacement of n-heptane or, indeed, any liquid does not exceed several 10 5 m (given that = 2D0 A). This is much smaller than the smallest pellet diameter of 1.5 mm, so that intraparticle diffusion determines the measured diffusion coefficient (see Chapter 3.1). This intrapartide diffusion is hindered by the obstades of the pore structure and is thus reduced relative to D0 the ratio between the measured and the bulk diffusion coeffident is called the tortuosity x. More predsely, the tortuosity r is defined as the ratio of the mean-squared displacements in the bulk and inside the pore space over identical times ... 

Number of test sections, 22 Number of data points, 638 Average ratio, 0.997 Root-mean-square error, 6.13%... 

Here, a is the elongation ratio of the polymer chains in any direction and (r/j1/2 is the root-mean-square, unperturbed, end-to-end distance of the polymer chains between two neighboring crosslinks (Canal and Peppas, 1989). For isotropically swollen hydrogel, the elongation ratio, a, can be related to the swollen polymer volume fraction, u2,j> using Eq. (11). 

S° Defined as the ratio of Sest to the root-mean-square of the data. It is a measure of the goodness of fit. The smaller the value of S°, the better the fit. 

The ratio of the root mean square lengths is called the chain expansion factor ... 

Fig. 21. Ratio between the correct molecular weight (M) and that calculated from dissymmetry (Md), as a function of the root mean square end-to-end distance calculated from dissymmetry (< h2 >y2) for different values of e 30 ...

Fig. 12.1 (continued) (c) Isotope effects on mean square amplitudes (upper curve) and root mean square amplitudes (lower curve) as a function of temperature for hypothetical nondissociating molecules. At low temperatures the molecules are in the ground state and the amplitude is nearly independent of temperature. At higher temperature the vibrational amplitudes increase due to excitation into upper levels (Fig. 12.1) but the ratios drop smoothly to the classical value of unity at very high temperature (Fig. 12.1)... 

Because sensitivity depends on so many different experimental factors, NMR spectroscopists generally use the signal-to-noise ratio, SIN, as a figure of merit for sensitivity comparisons. For example, in a comparison between NMR probes or spectrometers from two vendors, the spectral SIN measured for a standard sample acquired with specified acquisition parameters and probe geometry would provide a direct indication of relative sensitivity. The SIN is calculated for an NMR experiment as the peak signal divided by the root mean square (RMS) noise, given by Equation 7.6, and is directly related to the performance of the radiofrequency coil [3,6]... 

Secondly come methods depending on W asastjema s Criterion (36) that the distance D should be split into structural radii in the ratio of the Electron Cloud radii y at approximately their root-mean-square values yrms-... 

Hindered rotation is studied for the disaccharides composed of basic p-glucopyranose units. The van der Waals Interactions are calculated for the Lennard-Jones, Buckingham, and Kitaygorodsky interatomic potential functions. Values of the ratio of unperturbed to free-rotation root-mean-square end-to-end distance are calculated for chains composed of the unsolvated disaccharide repeating units. 

In the limit as ftact the rate of reaction of encounter pairs is very fast. The Collins and Kimball [4] expression, eqn. (25), reduces to the Smoluchowski rate coefficient, eqn. (19). Naqvi et al. [38a] have pointed out that this is not strictly correct within the limits of the classical picture of a random walk with finite jump size and times. They note the first jump of the random walk occurs at a finite rate, so that both diffusion and crossing of the encounter surface leads to finite rate of reaction. Consequently, they imply that the ratio kactj TxRD cannot be much larger than 10 (when the mean jump distance is comparable with the root mean square jump distance and both are approximately 0.05 nm). Practically, this means that the Reii of eqn. (27) is within 10% of R, which will be experimentally undetectable. A more severe criticism notes that the diffusion equation is not valid for times when only several jumps have occurred, as Naqvi et al. [38b] have acknowledged (typically several picoseconds in mobile solvents). This is discussed in Sect. 6.8, Chap. 8 Sect 2.1 and Chaps. 11 and 12. Their comments, though interesting, are hardly pertinent, because chemical reactions cannot occur at infinite rates (see Chap. 8 Sect. 2.4). The limit kact °°is usually taken for operational convenience. 

Ratio of standard deviation to root mean square of the data. b Values for PhCO approximated using those for MeCO. r CONHj approximated as C02R. 

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