Square-law

The reason why lanthanides of high atomic number emerge first is that the stability of a lanthanide ion-citrate ion complex increases with the atomic number. Since these complexes are formed by ions, this must mean that the ion-ligand attraction also increases with atomic number, i.e. that the ionic radius decreases (inverse square law). It is a characteristic of the lanthanides that the ionic radius... 

Burning times for coal particles are obtained from integrated reaction rates. For larger particles (>100 fim) and at practical combustion temperatures, there is a good correlation between theory and experiment for char burnout. Experimental data are found to obey the Nusselt "square law" which states that the burning time varies with the square of the initial particle diameter (t ). However, for particle sizes smaller than 100 p.m, the Nusselt... 

Equation set (9.104) approximates to an inverse square law, and increases the controller gains at low speeds, where the control surfaces are at their most insensitive. 

Figure 7.8 The structure of InCls " showing square-pyramidal (C4 ) geometry. The In-Clapex distance is significantly shorter than the In-Clbase distances and In is 59 pm above the basal plane this leads to a Clapex-In-Clbase angle of 103.9° which is very close to the theoretical value required to minimize Cl Cl repulsions whilst still retaining C4, symmetry (103.6°) calculated on the basis of a simple inverse square law for repulsion between ligands. [NEt4]2[TlCl5] is isomoiphous with [NEt4]2[InCl5] and presumably has a similar structure for the ardon.

The square law relationship also implies that the instrument measures RMS values. It can be used on either A.C. (up to the lower audio range if special compensating circuits are employed) or D.C. The instrument reading can be... 

If ri is double Fq it will be seen that SPLq — SPLj will be approximately equal to 6dB (20 log 2). This gives us the principle of a decrease in level of 6 dB per doubling of distance (inverse square law). For the line source the same calculation produces a difference of only 3 dB per doubling of distance. 

The B/V intensity ratio is an excellent relative measure of magnitude and it is possible to derive a B/V magnitude and, using Equation 2.7, derive a calibration curve for the temperature of a star (Figure 2.4) so that the temperature of the star can be measured directly by telescopes. Now, with a measure of the luminosity of a star the radius can be determined, but there is a problem the luminosity of a star as measured on Earth depends on how far away the star is - the Inverse Square Law - so the distance to the star must also be known to understand the absolute luminosity of the star. 

The solar flux can be calculated via Stefan s law from the observed surface temperature of the Sun, and the level of radiation at a known distance is calculated via the inverse square law (Figure 7.6). 

Consider the amount of radiation arriving on the surface of the Earth at a distance of 1 AU or 1.5 x 1011 m. The total flux of the Sun is distributed evenly over a sphere of radius at the distance of the planet, d. From the luminosity calculation of the Sun, F, the solar flux at the surface of Earth, FEarth, is F/47t(1.5 x 1011)2 = 1370 Wm-2 from the least-square law of radiation discussed in Example 2.4 (Equation 2.4). Substituting this into Equation 7.6 with the estimate of the albedo listed in Table 7.2 gives a surface temperature for Earth of 256 K. 

Inverse Square Law The drop of intensity of radiation (for example) with the square of the distance 1/(4tt d2). 

The ionic model, developed by Bom, Lande, and Lennard-Jones, enables lattice energies (U) to be summed from inverse square law interactions between spherically symmetrical charge distributions and interactions following higher inverse power laws. Formation enthalpies are related to calculated lattice energies in the familiar Bom-Haber cycle. For an alkali fluoride... 

Fig. 20. Square-law functional dependence of spin-lattice relaxation rates R z on segmental order parameters for a homologous series of disaturated phosphat-...

The chain ion-radical mechanism of ter Meer reaction has been supported by a thorough kinetic analysis. The reaction is well-described by a standard equation of chain-radical processes (with square-law chain termination) (Shugalei et al. 1981). This mechanism also explains the nature of side products—aldehydes (see steps 13 and 14) as well as vicinal dinitroethylenes. Scheme 4.37 explains formation of vic-dinitroethylenes. 

In Figure 8, the growth rates have been plotted against supersaturation. For each temperature, a square law dependence was found. This is shown directly in Figure 9. Accepting a square law, the corresponding growth rate constants, k, were evaluated, where,... 

Gypsum growth rates show a square law dependence on supersaturation with an activation energy of 64 kJ/mol. Growth appears to be surface kinetics controlled. 

According to the type of scale division, a distinction is made between two forms of compression vacuum gauges those with a linear scale (see Fig. 3.7) and those with a square-law scale (see Fig. 3.8). In the case of the compression vacuum gauges of the McLeod linear-scale type, the ratio of the enclosed residual volume Vc to the total volume V must be knovm for each height of the mercury level in the measurement capillary this ratio is shown on the scale provided with the instrument. In the case of compression vacuum gauges with a square-law scale, the total volume and the capillary diameter d must be known. 

Fig. 3.8 McLeod compression vacuum gauge with square-law scale (equation 3.11)...

As can be seen from Eq. (25), (nc)-p is proportional to the square of the photon flux nc. It should also be proportional to the product of the maximum absorption cross section ffmax and the cross section oi at wavelength . This relation has been checked experimentally in 48>. The relation between the fluorescence output (wc)F and the excitation power nc for an aqueous solution of rhodamine resulted in a straight line in a double-logarithmic plot with a slope of 2.05 0.1, thus verifying the square law of two-photon absorption. 

For a broadband r.f. amplifier of bandwidth Afi sending a signal to a square-law diode detector and thence to a low-frequency video amplifier of bandwidth A/j the noise power is (Dicke 1946 Robinson 1974)... 

Throughout this section all V are functions of z they have been printed bold in (8.20) to emphasize that they contain phase information. They are summed as complex quantities, i.e. with respect to both their amplitude and phase. But in the usual experimental implementation, where V is measured through a diode detector, the phase information of V is not available. Therefore, if the system has been calibrated to give square law detection, the measured signal may be represented as... 

Next we note that the induced field varies inversely with r. It makes sense that the field should decrease as we get farther from the antenna, but the inverse first-power dependence may be unexpected since we are more familiar with inverse-square laws. However, it is the energy or intensity of the light that varies according to an inverse-square law. In the next section we convert this expression for E to an expression for energy the more familiar r 2 functionality appears then. 

To determine the maximum possible efficiency of square-law delayed fluorescence, we set a limit to Iai such that the deviation from eq. (72) is just becoming appreciable. We choose as this arbitrary limit the conditions where the rate of disappearance of triplet by self-quenching has risen to a value one-fifth of that for the radiationless decay. Thus, at this limit ... 

Although at high rates of light absorption the square law dependence on rate of light absorption is no longer obeyed, the intensity of P-type delayed fluorescence is still proportional to the square of the intensity of triplet-singlet emission because the latter is always proportional to... 

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