Theoretical and Experimental Investigation of Factors Affecting Precision in Molecular Absorption Spectrophotometry L. D. Rothman' and S. R. Crouch Department of Chemistry, Michipn State University, East Lansing, MI 48824
J. D. Ingle, Jr.2 Department of Chemistry, Oregon State University, Corvallis, OR 9733 1
An expression which indicates the dependence of the relative precision of molecular absorption measurements on the transmittance and experimental variables is presented. The theory takes into account imprecision due to amplifier noise, readout quantization noise, dark current shot noise, photocurrent shot noise, source flicker noise, and cell positioning. Procedures for experimental evaluation of the instrumental variables in the theoretical expressions are outlined. The standard deviation for measurement of absorbances from zero to three under different instrumental conditions was determined from repetitive measurements of the absorbance of dichromate solutions of different concentrations. The data indicate that precision depends to a large extent upon the choice of instrumental variables and can be limited by cell positioning imprecision. Comparison of theoretical and experimental standard deviations indicates the validlty of the theoretical expression.
A recent treatment ( I ) of the precision to be expected in molecular absorption spectrophotometric measurements has revealed that the assumption that optimum measurement precision occurs near 37% T may be grossly in error under certain conditions. In this treatment, theoretical expressions were derived which indicated how the relative precision of an absorbance measurement ( U A / A )was dependent on the transmittance, on instrumental variables, on three different types of instrumental noise, on the mode of readout, and on the type of current measurement technique. Limiting forms of the theoretical expressions were identified and used t o predict how the relative precision varies with transmittance, and a t what transmittance measurement precision is optimal if specific types of instrumental noise are dominant. Many modern instruments permit the operator sufficient latitude in the choice 'of instrumental operating conditions such that for a particular analysis any of the noise sources previously described could become predominant. Since both the magnitude of the relative measurement precision a t a given transmittance and the dependence of the relative measurement precision on transmittance depend on the limiting types of noise and, hence, instrumental operating conditions, it is important to understand how to use the theory. In this study, a single beam spectrophotometer with a readout linear in transmittance was used to determine the relative measurement standard deviation as a function of transmittance under different experimental conditions. The results of the experimental study are compared to the relative measurement standard deviation as a function of Present address, Department of Chemistry, University of Georgia, Athens, GA 30602. Author to whom correspondence should be addressed. 1226
ANALYTICAL CHEMISTRY, VOL. 47, NO. 8, JULY 1975
absorbance as predicted by theory. This comparison indicates under what conditions the theory correctly predicts experimental behavior and suggests a modification to the previous theory that is needed under certain conditions. In the first section, a theoretical expression for the dependence of u*/A on instrumental variables is developed which treats the noise sources previously considered ( 1 ) and, in addition, considers the contribution of imprecision in cell positioning to the overall relative measurement precision. The second section describes the measurement system and techniques used in the experimental study. The third section contains a comparison of the experimentally measured relative measurement standard deviation of the spectrophotometric measurements to the theoretically predicted values. The contributions of various individual noise sources to the total measurement precision are discussed. Studies were performed with measurement precision limited by 1) amplifier-readout noise, dark current shot noise, and excess noise, 2) photocurrent shot noise, 3) source flicker noise, and 4) imprecision in sample cell positioning. The final section compares the uncertainty introduced by limited readout resolution to the uncertainty from other instrumental noise sources. This comparison clearly demonstrates the constraints which limited readout resolution may place upon the overall measurement precision.
THEORY Measurement precision in molecular absorption spectrophotometry is dependent upon a number of factors. T o understand how these factors affect precision, it is first necessary to understand what these factors are and how they relate to the measurement process. The most commonly followed procedure for a spectrophotometric measurement involves: 1) introduction of the reference solution into the optical path, 2) the measurement of the reference photocurrent (100% T ) ,3) measurement of the dark current (0% T ) ,4) introduction of the sample into the optical path, and 5 ) measurement of the sample photocurrent (962' of the sample). Overall measurement precision is influenced by imprecision in each step. The measurements in steps 2, 3 and 5 are imprecise due to noise from the amplifier-readout system and to shot and excess noise in the dark current. The excess dark current noise is non-fundamental noise in the dark current which is present in addition to the theoretically predictable shot noise. The measurements in steps 2 and 5 are also imprecise due to shot and secondary emission noise in the photocurrent, fluctuations in light source intensity, and irreproducibility in performing steps 1 and 4. The contribution of each source of imprecision, except cell positioning imprecision has been previously treated ( I 1. Hence, the effects of noise sources not related to the cell positioning step will be treated briefly followed by detailed treatment of uncertainty due to sample cell positioning. Precision in the Absence of Cell Positioning Uncer-
~
tainty. Classical treatments ((see for example (2-4) and references in these articles)) of measurement precision in spectrophotometry have considered the effect of reading errors on the relative precision of an absorbance measurement. When such treatments are applied to a spectrophotometer with a linear transmittance readout, the result is the familiar prediction that absorbance measurement precision is optimized a t 36.8% T. Many modern spectrophotometers either are or may be equipped with low noise amplifiers and high resolution readout devices or have scale expansion capabilities such that the precision of absorbance measurements may not be limited by the amplifierreadout system. In such cases, the measurement precision will rarely be optimized a t 36.8% T ( I , 3, 4 ) but rather in the O-ll% T range (I), and careful consideration must therefore be given to the effect of the predominant noise source. The actual goal of any theoretical treatment of precision in spectrophotometry is to predict the magnitude of the relative uncertainty in a measurement of the concentration of some chemical species. The relationship between the relative concentration standard deviation ( u c / c ) and the relative absorbance standard deviation (uA/A) may he easily established by applying propagation of uncertainty mathematics to Beer’s law. If Beer’s law holds, then uc/c = UA/A. T h e theory presented here will express the relative measurement precision as the relative absorbance standard deviation, u.dA. Equations 1-4 have been applied to a description of the relative precision of absorbance measurements performed on an instrument with a linear transmittance readout ( I ) where T is the transmittance, UT is the standard deviation in the transmittance, and UA/Ais the relative absorbance standard deviation, and all other terms are defined in Table I. Equations 2 and 4 take into account the variance in the following three measured voltages: Ert, corresponding to 100% transmission; &, corresponding to 0% T; and, Est,corresponding to the sample transmittance.
~
~~~
= = = = ti ( u , ) ~ +=~
(u,)~
(us),
This derivation assumes t h a t all noise sources are statistically independent, that there is negligible unidirectional instrumental drift between the measurements of 100% T , 0% T and the sample transmittance, ust
