instruments. Another limitation is the inability to determine a true absorption curve. For these reasons,
it is often desirable to use a spectrophotometer which uses a prism or grating monochromator, the
technique being known as spectrophotometry.
Random errors associated with the measurement of absorption impose a limitation on the degree of
precision attainable. At low absorbances, noise in the measuring circuit becomes the governing factor
while at high absorbances, the small amount of radiation reaching the detector necessitates high
sensitivity (amplifier gain) settings. If the error in measuring the radiation intensity is constant over the
operating range of a detector, it can be shown that the relative error in the absorbance reading passes
through a minimum value. The absorbance corresponding to the minimum error can be calculated by
differentiating the Beer-Lambert equation twice and setting the second differential to zero. Thus,
Converting to a natural logarithm and differentiating
whence by substitution and rearrangement
where dC/C represents the relative error in concentration and dT is the constant error or uncertainty in
measuring T (instrumental error).
Converting to a natural logarithm and differentiating a second time gives
Equating to zero to find the minimum in the curve
Hence, for an absorbance of 0.434, there is a minimum in the relative error of the measurement. Figure
9.4 (curve A) shows how the relative error varies with absorbance (dT = 1%) for a simple instrument
incorporating a photovoltaic detector (p. 282). Measurements outside the range 0.2–