Except under extremely deviant conditions, the “standard curve” regressional relationship between optical absorbance and absorbent concentration is shown to exhibit an S-shaped profile of progressively increasing slope up to an inflexion followed by a region of progressively decreasing slope. Applied to this profile, a Gaussian regression procedure is shown to deliver i) a versatile, accurate representation of these sigmoid deviations from the Beer’s Law proportional relationship between absorbance (A) and absorbent concentration (C), namely Ar = a exp – (C – b) / c2 + d in which a, b, c and d are experimentally determined constants ii) an explicit inversion of this regressional relationship C = b + c sqrt[ ln a / (Ar – d) which is required to determine the absorbent concentration corresponding to an observed value of absorbance, and iii) an expression for the slope of the regressional relationship dAr / dC = – 2 (a / c2) (C – b) exp – (C – b) 2 / c2 as a measure of the sensitivity of the experimental procedure.
John C. O’C.Young (Sun,) studied this question.