Polynomial Equations based on Bouguer–Lambert and Beer Laws for Deviations from Linearity and Absorption Flattening

Polynomial through the origin equations based on the Bouguer—Lambert and Beer laws were proposed for the accurate representation of positive and negative deviations from linearity and absorption flattening. Cubic and higher order equations of absorbance on concentration, thickness, and molar absorptivity do not provide explicit inverse equations which are required to determine the concentration, thickness, and molar absorptivity. Quadratic equations provide explicit inverse equations. The proposed quadratic equations are $$A = ad + b{{d}^{2}}$$ and $$A = ad - b{{d}^{2}}$$ for positive and negative deviations from linearity of the Bouguer—Lambert law, $$A = \varepsilon dc + {{\xi }}d{{c}^{2}}$$ and $$A = \varepsilon dc - {{\xi }}d{{c}^{2}}$$ for positive and negative deviations from linearity of the Beer law, and $$A = f\varepsilon - g{{\varepsilon }^{2}}$$ for absorption flattening, where $$A$$ is absorbance, $$a$$ and $$b$$ are linear and nonlinear absorption coefficients, $$c$$ is molar concentration, $$d$$ is thickness, $$\varepsilon $$ and $${{\xi }}$$ are linear and nonlinear molar absorptivities, $$f$$ and $$g$$ are linear and nonlinear coefficients. Concentration, thickness, and molar absorptivity are determined from inverse equations.