Relative error of the absorbance

Our first chapter in this set [4] was an overview the next six examined the effects of noise when the noise was due to constant detector noise, and the last one on the list is the first of the chapters dealing with the effects of noise when the noise is due to detectors, such as photomultipliers, that are shot-noise-limited, so that the detector noise is Poisson-distributed and therefore the standard deviation of the noise equals the square root of the signal level. We continue along this line in the same manner we did previously by finding the proper expression to describe the relative error of the absorbance, which by virtue of Beer s law also describes the relative error of the concentration as determined by the spectrometric readings, and from that determine the... 

So let us continue. We now wish to generate the expression for the relative error of the absorbance, A A/A, which we again obtain by using the expression in equation 48-25... 

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 (d T= 1%) for a simple instrument incorporating a photovoltaic detector (p. 282). Measurements outside the range 0.2-... 

Accuracy Under normal conditions relative errors of 1-5% are easily obtained with UV/Vis absorption. Accuracy is usually limited by the quality of the blank. Examples of the type of problems that may be encountered include the presence of particulates in a sample that scatter radiation and interferents that react with analytical reagents. In the latter case the interferant may react to form an absorbing species, giving rise to a positive determinate error. Interferents also may prevent the analyte from reacting, leading to a negative determinate error. With care, it maybe possible to improve the accuracy of an analysis by as much as an order of magnitude. 

One cm3 of the reactant/product/catalyst mixture was sampled periodically during the reaction for the transmission infrared analysis (Nicolet Magna 550 Series II infrared spectrometer with a MCT detector). The concentrations of reactants and products were obtained by multiplying integrated absorbance of each species by its molar extinction coefficient. The molar extinction coefficient was determined from the slope of a calibration curve, a plot of the peak area versus the number of moles of the reagent in the IR cell. The reaction on each catalyst was repeated and the relative error for the carbamate yield measured by IR is within 5%. 

B. The true absorbance of a sample is 1.000, but the monochromator passes 1.0% stray light. Add the light coming through the sample to the stray light to find the apparent transmittance of the sample. Convert this back into absorbance and find the relative error in the calculated concentration of the sample. 

Within the limits of experimental error, the relative efficiencies of photons absorbed by the monomeric form of dyes 3, 5, and 6 were the same for exposures made in air and under vacuum. This was the case also for the H-aggregated forms of dyes 5 and 6, but photons absorbed by the H-aggregate of dye 7 were relatively more efficient for exposure under vacuum. The relative efficiency of photons absorbed by the J-aggregated dye 3 was much larger for exposure under vacuum than in air, but the reverse was the case for dye A. 

Differential spectrophotometry [22,23] consists in the measurement of the absorbance of a solution of the given element, not with reference to the solvent used, but with reference to a solution of this element (in the form of a coloured complex) of known concentration, slightly lower than the concentration of the solution studied. In this technique, the measuring error is in proportion to the difference of concentrations (and not to the concentration of the analyte in the solution under test), which enables one to reduce the relative error. Grey filters of appropriate absorbance have been proposed as references [24]. 

The theoretical bases of differential spectrophotometry have been presented [22]. The relative error of absorbance measurement is 0.2-0.5%, and is less in differential spectrophotometry [25-28] than in the regular method. Hence, the precision of differential spectrophotometry is comparable with that of gravimetric and titrimetric methods. This fact enables the technique to be applied in the determination of higher contents of the analytes. 

Ideally, the transmittance scale on a linear detector is fixed by a 0% T measurement (dark current measurement on the detector) and a 100% T measurement (total illumination of the detector by Iq). A sample attenuates the Iq intensity signal, and the sample transmittance and hence the absorbance are obtained. All these individual measurements are subject to noise and drift errors and combine to give an overall measurement standard deviation, This standard deviation is related to the relative error of measurement, rearranging Equation (1.7) and obtaining the partial derivative. Note that the molarity, M, in Equation (1.5) has been replaced by C the concentration in g L-1. The relative error function [9] is given by Equation (1.8) ... 

Figure 16.27 illustrates the dependence of the relative error on the transmittance, calculated for a constant error of 0.01 T in reading the scale. It is evident from the figure that, while the minimum occurs at 36.8% T, a nearly constant minimum error occurs over the range of 20 to 65% T (0.7 to 0.2 A). The percent transmittance should fall within 10 to 80% T (A = 1 to 0.1) in order to prevent large errors in spectrophotometric readings. Hence, samples should be diluted (or concentrated), and standard solutions prepared, so that the absorbance falls within the optimal range. 

The firebox model has a heat loss term that can be calculated as a parameter, however the loss term is essentially a small difference between large numbers (total heat fired minus total heat absorbed in radiant plus convection sections), so it is subject to large relative errors. Additionally, the heat loss may be calculated as a heat gain depending on the accuracy of the fired and absorbed heat duties. To keep the heat loss from the firebox reasonable, it is typically set to a small value, on the order of 1% of the heat fired. 

Optical analysis can be done relatively fast. NIR analysis can be performed within a few seconds, depending on the magnitude of the analytical signal, sample absorbance, and overall error of the measurement. Simple analytes like moisture require only two to three wavelengths. On the other hand, more complex analytes may be calibrated best using up to six wavelengths. Multiple linear regression is the best calibration technique [125]. Near-infrared analysis is typically a secondary analytical method, i.e. it has to be calibrated with several samples of known concentrations. 

---