Curve Fitting Strategies

Last updated
Save as PDF

Page ID: 76369

Alexander Scheeline & Thomas M. Spudich
Analytical Sciences Digital Library

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Linear Working Curves (External Standards I)

For many emission sources, the intensity emitted by a particular element at a specified wavelength is proportional to the concentration of the element in a sample. Thus, saying I(λ) = I₀ + k(λ) C is a reasonable approximation to the instrument behavior. One simply makes a set of standards with various concentrations C, measures the emitted intensity in any internally-consistent set of units (photocurrent, CCD accumulated charge in fixed time, ADC counts) and uses a convenient statistics package to obtain a least-squares fit of a relationship of the form shown. An unknown concentration is then found by measuring the emission intensity and solving, C = (I(λ) -I₀)/k. Assumptions in this model include: the matrix for sample and standards is the same. Interelement effects for samples and standards is the same. Sample uptake for samples and standards is the same. For dilute solutions and simple matrices, the assumptions are reasonable and the method is widely employed.

Nonlinear Working Curves (External Standards II)

If one tries to use any method over too wide a concentration range, nonlinear response is inevitable. In atomic emission, high concentrations lead to self-absorption or self-reversal. It is common to fit the working curve to a non-linear function, at least when the degree of nonlinearity is small. One must check that the least-squares fitted working curve passes suitably close to observed data rather than being a numerical artifact (one can, after all, mechanically perform a least squares regression fit of a curve to any set of numbers. Fitting a line to data describing a circle makes no sense, but it can be done!). The fitting equation is typically I(λ) = I₀ + k₁(λ) C + k₂(λ) C². Once the constants have been fit using a series of standard solutions, the concentration of an unknown is found by using the quadratic formula: C = (-k₁(λ) + (k₁²(λ)+4k₂(λ)(I(λ) -I₀))^1/2)/(2k₂(λ)). One reason extension of this method to cubics and beyond is uncommon is the awkwardness of using the fitted curve to determine unknown concentrations. Successive approximation calculations are usually more efficient than attempts at symbolic solution.