Rapid determination of areas of Gaussian curves and diffusion coefficients

A new and quick method for calculating the area under a Gaussian curve from its height and width ( HW) has been developed and tested for its accuracy in area and diffusion coefficient determinations. This method has been used to determine whether the peaks from diffusion experiments, which theoretically should be Gaussian actually are Gaussian; the experiments were carried out in ultracentrifuges using schlieren optics. This analysis shows that the schlieren peaks deviate slightly (as much as 4%), in a reproducible manner, from Gaussian curves. The overall average deviation curve for these data provides an empirical estimate of the factor needed to correct a given width measurement, at a certain fractional height, to its Gaussian curve value. We used this height-width-corrected method to determine the areas of peaks in 49 different photographs of schlieren peaks from seven diffusion experiments. In comparison with actual measured areas, the areas calculated with the HW method had a probable error of only 0.7%, a standard deviation of only 0.9%, and a maximum deviation of only 2.3%. The corresponding values for diffusion coefficients were also very accurate. Because of the ease and accuracy of the HW method, it is expected that it will supplant the popular, but tedious, method of numerical integration to determine areas under Gaussian peaks from synthetic boundary experiments and sedimentation velocity experiments. This will allow rapid and accurate determination of diffusion coefficients, relative concentrations, and initial concentrations. Furthermore, in principle, this new method can be applied to any type of experiment involving a Gaussian peak whose broadening is diffusion controlled.