Application of an integrator in the calculation of diffusion coefficients from plotted diffusion curves: Theory and practice, including an estimation of the accuracy of the method illustrated also by the calculation of a set of known Gaussian curves

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Abstract
The formulas used for calculation of diffusion coefficients in free diffusion are stated. The calculations request the 0th, first, and second moments of the curve to be calculated. This is usually made by calculation of moments of segments of the figure between the curve and the base line, followed by summation.It is proposed to replace this rather laborious procedure by the use of an integrator. A survey is given on the theoretical background and the principles for such instruments. Some equations are stated for the type of instrument used in these investigations. From these equations it is possible, by inserting the values in the formulas used for the calculation of diffusion coefficients, to express the latter for a certain curve as a function of the rotation of the three integration rolls in the instrument. The relative influence on the final result for these integration rolls is considered. For one of the rolls this can be shown to be insignificant, when the moment axis is chosen very near the central axis of the curve. Our experience has shown the standard error in a normal measurement to be 2 × 10−3 and 3 × 10−3 of a revolution for the two other integration rolls. Some curves are given which show the requests one must put on the relative number of rotations for these rolls to make certain that the relative standard error, introduced by the integrator, will not exceed certain values on the final result. The curves are drawn for values from 2 to 0.1%.From a known Gaussian distribution a set of diffusion curves is drawn. The diffusion coefficient is calculated from these curves both by the usual method and by the integrator. The results obtained by the integrator compare well with those by the usual method if the peaks are not drawn too narrow.

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