Question 6 of 7: Spectrophotometric Measurement (Beer-Lambert Law)

Question 6: Spectrophotometric Measurement (Beer-Lambert Law) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

a) Combining the two given relations, $v=kT=k\cdot10^{-\varepsilon Cl}$ — the output voltage is an exponentially DECAYING function of concentration (Fig. 2), starting at $v=k$ when $C=0$ and falling toward zero as $C$ grows. Its slope $dv/dC=-\ln(10)\,\varepsilon l\,v$ is directly proportional to $v$ itself, so the curve is steepest (most sensitive, in absolute volts per unit concentration) at LOW concentration, where $v$ is still close to its full-scale value $k$. As concentration rises the curve flattens toward zero: $v$ becomes very small and approaches the instrument's own electrical noise floor, so at HIGH concentration a given amount of electrical noise represents an increasingly large fractional error in $v$ — and therefore in the concentration inferred from it — making the high-concentration end of the curve the region most affected by measurement noise.

b) Absorbance is computed from the RATIO (equivalently, the subtracted logs) of the sample transmittance to the blank/baseline transmittance, $A=-\log(T_{sample}/T_{blank})$. For a dilute solution, $T_{sample}$ is only marginally different from $T_{blank}$ (both close to the maximum, near-100% transmission), so the true absorbance signal is the small DIFFERENCE between two large, nearly equal readings. Any noise, drift, or imprecision in either the sample or the blank measurement is no longer small relative to the quantity being measured — it becomes a large FRACTIONAL error in the small computed difference, a classic subtraction-of-near-equal- numbers error amplification. This is why dilute-solution measurements are disproportionately sensitive to baseline drift and detector noise compared with mid-range concentrations.

c) At high concentration the transmitted light intensity ($T\to0$) falls toward the detector's own noise floor, and STRAY LIGHT — light that reaches the detector without having passed fully through the absorbing sample (leakage around the optical path, scattered light, imperfect monochromator rejection) — becomes the dominant residual signal instead of the true attenuated beam. Once the genuine transmitted signal is comparable to or smaller than the stray-light floor, further increases in concentration no longer produce a proportional decrease in the measured signal; the reading plateaus and the measured absorbance deviates from (falls below) the true Beer-Lambert value. The highest reliably measurable concentration is therefore set by the point where the true transmitted signal is no longer distinguishable from the combination of the detector noise floor and instrument stray light — not by any limit in the absorbing chemistry itself.