Square root of the observation time

In infrared, the noise from the detector is random and proportional to the square root of the observation time. It is thus possible to write ... 

Another deviation from the pattern of ordinary diffusion must be expected if the reactant and product molecules are subjected to single-file conditions, i.e. if (i) the zeolite pore system consists of an array of parallel channels and if (ii) the molecules are too big to pass each other. In this case, the molecular mean-square displacement z t)) is found to be proportional to the square root of the observation time, rather than to the observation time itself. First PFG NMR studies of such systems are in agreement with this prediction [8]. By introducing a mobility factor F, in analogy to the Einstein relation for ordinary diffusion. 

For an FTIR spectrometer, all of the spectral elements are observed for time T. Thus, the time spent sampling the resolution Av is also equal to T. Since the signal-to-noise ratio improves as the square root of the observation time... 

The significance of Equation 32 is that it indicates that peak broadening is proportional to the square root of the retention time. As observed from chromatograms, the longer the retention time, the wider the peak is. 

The linear relations observed between the fraction converted and the square root of the reaction time indicate that the rate-determining step of the exchange reaction is a diffusion process. It seems reasonable to assume that only the alkali iodides located at the edge sites of the crystallites can take part in the exchange reaction. The rate controlling process is the diffusion of the alkali iodide through the interlayer spaces to the outer edges in order to react. 

Pb +, Zn +, and UOI in molten KSCN was studied polarographically and chronopotentiometrically by Yanagi and co-workers.Well-defined and reproducible chronopotentiograms for the reduction of Cd-, Pb +, and Zn + were obtained the product of current density and the square root of the transition time was proportional to concentration for each system. The reduction of Pb + and Zn + was found to be reversible. The diffusion coefficients reported were of the order of 10 cm-/sec. Two reduction steps were observed for UO + polarographically and chronopotentiometrically. The processes are diffusion controlled and were proposed to be UO2+ + U02 + UOg. The diffusion coefficient of UO ... 

The concentration [MB] constantly experiences tiny fluctuations, the duration of which can determine linewidths. It is also possible to adopt a traditional kinetic viewpoint and measure the time course of such spontaneous fluctuations directly by monitoring the time-varying concentration in an extremely small sample (6). Spontaneous fluctuations obey exactly the same kinetics of return to equiUbrium that describe relaxation of a macroscopic perturbation. Normally, fluctuations are so small they are ignored. The relative ampHtude of a fluctuation is inversely proportional to the square root of the number of AB entities being observed. Consequently, fluctuations are important when concentrations are small or, more usehiUy, when volumes are tiny. 

Ruths and Granick [95] have studied the self-adhesion of several monolayers and adsorbed polymers onto mica. For loose-packed monolayers, the adhesion, in excess of a constant value observed at low rate, increased as a power law with the square root of the separation rate. In the case of adsorbed diblocks, the excess adhesion increased linearly with logarithmic separation rate. The time effects were ascribed to interdigitation and interdiffusion between the contacting layers. 

The units for variance are, therefore, squared, which can cause difficulties. If the observations are measuring time for instance, the variance may be given in seconds squared (s2), which is meaningless. The square root of the variance is, therefore, used to return to the original units. This is the SD. 

At constant temperature, the observed widths of the spectral functions decrease with increasing mass of the collisional pair. This fact is a simple consequence of the mean translational energy of a pair, jm v = kT, which is the same for all pairs. The interaction time is roughly proportional to the reciprocal root mean square speed, and thus to the square root of the reduced mass. 

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