Decay with distance squared

The higher the barrier, the larger the exponential coefficient p and the more dramatically the electron transfer rate decays with distance. By a fortunate coincidence of units, the P in " is approximated by the square root of the barrier height in eV. Thus for typical biological redox centers that must overcome a barrier of about 8eV to be ionized in a vacuum, we can estimate the P for exponential decay of electron transfer in vacuum to be about 2.8 Much less of a barrier is presented by a surrounding organic... 

Analysis of sound travel will help when planning noise control in a room. Noise travels away from a source. The energy level in a sound field will decay with the square of the distance from a source. However, most rooms have reflective surfaces, such as floors, walls, and ceilings. Placement of noise sources in a layout can help with noise control. For example, placement of noise sources in a comer near highly reflective floor, ceiling and walls will concentrate noise energy in one location and one direction. Conversely, placement of noise sources away from reflective surfaces may give noise a chance to dissipate to an acceptable level. 

Three-pulse ESEEM spectrum of perdeuterated P-carotene imbedded in Cu-MCM-41 exhibits an echo decay with an echo modulation due to deuterons. The three-pulse ESEEM is plotted as a function of time, and curves are drawn through the maximum and minima. From ratio analysis of these curves, a best nonlinear least-squares lit determines the number of interacting deuterons, the distance (3.3 0.2A), and the isotopic coupling (0.06 0.2MHz). This analysis made it possible to explain the observed reversible forward and backward electron transfer between the carotenoid and Cu2+ as the temperature was cycled (77-300 K). 

So the free energy of attraction decays very slowly, viz. reciprocally with the distance, and even slower than it does in the case of flat plates, where it goes reciprocally with the square of the distance, (eq. 48c of Part II). At larger separations of the two spheres the decay is of course faster, as for very great distances the attraction must die out as 1/if , but nevertheless the decay remains slower than 1/H until s exceeds 2.4, as may be read from a double logarithmic plot of eq. (85). 

Point-symmetry, also referred to as spherical or unconfined geometry, has the lowest degree of flame confinement. The flame is free to expand spherically from a point ignition source. The overall flame surface increases with the square of the distance from the point ignition source. The flame-induced flow field can decay freely in three directions. Therefore, flow velocities are low, and the flow field disturbances by obstacles are small. 

The spherical Bessel function of first order j f) decays as 1/r and therefore 0 °(r) decays as 1/r. Thus also the DAFH orbital demonstrates inverse square decay with the distance. The delocalization indices being proportional to the squared amplitude of 0(r) exhibit 1/r decay. Sub-... 

It should be noted that when there is no jet reinforcement of the flow, i.e., the exhaust hood is used in its conventional mode, then in the two-dimensional form of the Aaberg principle the fluid flow velocity due to the exhaust decays approximately inversely proportionally to the distance from the exhaust opening. However, for three-dimensional exhaust hoods the fluid velocity outside the hood decays approximately inversely as the square of the distance from the exhaust hood. Thus in the three-dimensional conventional hood operating conditions the hood has to be placed much closer to the contaminant in order to exhaust the contaminant than is the situation for the two-dimensional hood (see section on Basic Exhaust Openings). Thus for ease of operation it is even more vital to develop hoods with a larger range of operation in the three-dimensional situation in comparison with two-dimensional hoods. 

A similar treatment applies for the unstable regime of the phase diagram (v / < v /sp), where the mixture decays via spinodal decomposition.For the linearized theory of spinodal decomposition to hold, we must require that the mean square amplitude of the growing concentration waves is small in comparison with the distance from the spinodal curve. 

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