# IV. Scientific Background

The central relationship that covers the basics of absorption and fluorescence spectrophotometry is the Beer‐Lambert Law:

$A = \varepsilon bC = \log_{10}T = -\log_{10}\dfrac{I}{I_0}\tag{1}$

Where

• A     Absorbance
• T     Transmittance
• ε     Molar absorptivity, L mol‐1 cm‐1
• b     Light path length through specimen, cm
• C    Absorber concentration, mol L‐1
• I     Intensity detected in the presence of sample
• I0   Intensity detected absent sample but with solvent present

Greater detail is contained in a vast literature, including discussion of spectrometer design,7 detectors,8 and measurement non‐idealities.9,10 A review of spectrometer instrument design is thus omitted here, though simplified discussions do appear in the "High School Teacher Module" and "Student Module" accompanying this paper. Critical ideas include how diffraction gratings function:

$n \lambda = d (\sin \alpha - \sin \beta) \tag{2}$

with

• n     diffraction order (integer)
• λ     wavelength (nm)
• d     grating groove space (nm)
• α   incidence angle of light on grating, measured counter‐clockwise from grating normal
• β     exit angle of observed, constructively‐interfering light, measured counter‐clockwise from grating normal

Low‐granularity optical detectors (8‐10 bit intensity digitization per pixel or channel) for chemical analysis have previously been employed by, e.g., Suslick,11,12 Whitesides,13‐16 and others.17‐21 Accepted practice is that charge‐coupled arrays are employed for low‐light‐level applications (fluorescence, Raman, atomic emission), while diode arrays are used for absorbance measurements (including commercial instruments by Agilent, Ocean Optics, Stellarnet, and others). In the limiting case that readout granularity is the limiting noise source, the signal‐to‐noise ratio for concentration measurement is given by:

$\dfrac{C}{\delta C} = \dfrac{-I_0 T\ln T}{\delta I\left(I + T^2 \right)^{1/2}}\tag{3}$

For an 8 bit digitizer, I0 / δI = 255 or less. A plot appears in Figure 1.

Figure 1. Signal‐to‐noise Ratio for an 8 Bit Digitized Photosignal. Assumptions include: I0 is full‐scale exactly, at A=∞, I = 0 (no stray light), and intensity is high enough and stable enough that no noise source other than digitizer resolution contributes to measurement uncertainty.

What is not obvious is that instrument alignment, actual use of digitizer range, and positional stability of the detector all overwhelm the influence of digitizer resolution for the open geometry employed here. One may thus ignore all the usual theory and mathematics prior to having students build instruments; they will discover measurement flaws without the burden of algebra in advance.