Reliability of Photoelectric Photometry N. T. GRIDGEhIiK Food Research Department, Lever Brothers & C’nilever, L t d . . Sharnbrook. Hedford, England quite anothei , :inti the elperiinental approach is here indispensable.
The continuous growth of photoelectric photometry in analysis has currently revived discussion on the reliability of the instruments and techniques employed, and it seems timely to separate and crystallize the main facts. The theory of the precision of the commoner instruments, as influenced by the concentration of the measured solution, is here developed to yield a general expression of which all valid expressions in the literature are shown to be special cases. Maximum precision is found normally to be associated with about 4070 transmittance, but i t is sufficient to describe the 30 to 50% transmittance range as optimum. Any more exact statement about a particular instrument can only be reached experimentally. An empiricized “error curve” is shown. The influence of solution strength on accuracy is also discussed. Finally, the precision of the transmittance-ratio technique is interpreted in the light of the general theory of observational errors.
THE TWYiMAN-LOTHIAN CURVE
To I each the equation to the Twyman-Lothian curve n-e need only call to mind the fact that optical density is, by definition, colog transmittance-i.e., D = -log 7‘. If D is differentiated with respect to T mid the coefficient is replaced by finite quantities, the relation
emerges. Dividing i ) y D,Lve obtain
which is the iequired equation. I t is important to realize that the equation is merely a mathematical corollary of the defined
0
S E of the differences between photographic and photoelec-
tric photometry lies in the fact that in the former the observed plate density is a linear function of the required optical density (absorbancy), D, whereas in the latter the observed galvanometer scale reading approximates to a linear function of the tiansmittance, T . This implies an important difference between the error distributions in the two techniques. Tmyman and Lothian (16) seem to have been the fiist to draw attention to this fact and to derive a curve relating small deviations of the galvanometer to those of the optical density estimate. Theii curve (Figure 1, A ) minimizes a t D = log e, from which the authors concluded that the precision of measurement on photoelectric instruments is a maximum when the cuvetted solution has an optical density of 0.4343-i.e., when it transmits 36.78% of the incident light. Several writers ( 1 , 3, 6, 8, I d , IS, 15) have since discussed the practical application and the conditions of validity of this curve; and some have prescribed alternative curves, minimizing at different points, for certain conditions. The over-all situation is nebulous in parts, and the present paper is an attempt a t clarification. Three preliminary remarks may be useful. Firstly, a distinction must be made between random and systematic errors; the former affect precision (reproducibility for results); the latter affect accuracy (proximity to truth). The w-ord “reliability” may be used to cover both precision and accuracy. The main interest of the Twyman-Lothian curve and its congeners lies in the information they yield about precision. Secondly, the discussion covers both spectrophotometers (operating in the ultraviolet, visible, or infrared regions) and “abridged” instruments employing color filters or interference filters and/or standard solutions. Some abridged photoelectric instruments may be less accurate than spectrophotometers, but there is no optical or electronic reason why they should be less precise, and their error distributions will in any case be comparable in form. Thirdly, while it is proper to get the theoretical background in focus, noneof our conclusionscan lessen the desirability of a practical estimate of the error distribution of a given instrument-r perhaps of a given type of instrument. To derive a likely relation between reading errors and analytical precision is one thing; to quantify the precision is
3 TRANSMITTANCE(1).
GALvANOMETERREADINP
100
Figure 1. Ordinates of TwymanLothian Curve, A, Are Sums of Those of Curves B and C Applicable to Imperfections (in the Same Direction) of Setting, Respectively, the Zero End, and the Unit End of the Transmittance Scale These curves have no necessary connection with random errors
relation between D and T that was our starting point. Given the relation, we see that a small absolute change in T is necessarily associated with a fractional change in D that is greater by the factor reciprocal T In 2‘. As T is customarily measured on a zerounity scale, In !/’ and, in consequence, the factor itself are always negative; this means that the associated displacements have opposite signs-as indeed is obviously bound to be. This is practically all there is to be said about the equation and the curve per se; in particular, there is no connection with the LambertBouguer law, with Beer’s law, with instrumentation, with the distribution of errors in T,or even with the statistical theory of errors. 445
446
ANALYTICAL CHEMISTRY
There are few experimental plots of the curve recorded in the literature, and none that demonstrates its correspondence to an observed random error distribution. Hamilton ( 5 )has plotted it from observations on graded standard solutions in a photoelectric colorimeter whose galvanometer (transmittance) scale was in effect shifted + I % throughout its length. His plot substantiates two points: (1) the validity of Beer’s law for the chosen wave bands in his solutions; and (2) the linearity of the instrumental “reaction” to changes in optical density. But it contributes nothing to knowledge of the precision or accuracy of the instrument. (Incidentally, when 6T is as high as 1% of a galvanometer scale, the Twyman-Lothian curve, based as it is on a differential coefficient, is not accurately applicable a t the transmittance extremes-for example, a shift of T = 0.05 to 0.06 is accompanied by a 6.1% fall in D , and a shift of T = 0.05 to 0.04 by a rise of 7.5% in D, whereas & l / T In T is &6.7%.) In the same paper Hamilton ( 5 ) shows how the TwymanLothian curve can be regarded as a summation of two curves, one relating 8D/D6T to T with an error in the zero setting of the galvanometer, the other with the same error (in the same direction) in the 100% setting. To visualize the effect of these setting errors it is helpful to imagine an elastic scale laid along, and being normally coincident with, a true rigid scale, both ranging linearly from zero to unity. If the two unit ends are clamped and we stretch or compress the elastic scale a linear amount, z, at the zero end, the displacement of any point T from truth will be inversely proportional to its distance from the zero end-i.e., it will be (1 - T ) x . If, similarly, we clamp the two zero ends and consider a play of x a t the unit end of the elastic scale, the displacement of T will be Tx. And if, thirdly, we use no clamps but displace the whole elastic scale, without distortion, by an amount z, the displacement of T will be I in all positions. Note that (1 - 2‘)s T z = z. Now if we consider T in Equation 1 replaceable by ( 1 - T ) z , T z and x, we can write down two equations that add up to a third, and plot from them two error functions whose curves add up to the Tvjyman-Lothian curve as shown in Figure 1, which operation, to quote Hamilton (6),shows “that if an error in the same direction is made a t both ends of the scale, the additive effect is to shift the whole scale, and to produce the same error in every reading.”
+
passes through the receiver-amplifier system” for assessment of the energy transmittance through the sample solution, as an esample of the practical validity of the Twyman-Lothian analysis, but the reasoning seems to contain a non sequitur. Equation 1needs remolding if it is to express precision based on random errors. On the assumption that the galvanometer readings perfectly reflect transmittance changes, we may replace aT by UT, the standard deviation of transmittance (range 0 to l), and which will thus be equivalent to one hundredth of UG, the standard deviation of the galvanometer reading (0 to 100 scale). This permits a similar change in the numerator of the left-hand side: 6D/D can be replaced by oD/D, which is one hundredth of the coefficient of variation (percentage standard deviation) of D and, therefore, of the assay itself. So
expresses the coefficient of variation of the assay as a multiple of the standard deviation of the transmittance estimate on which it is based. What is of practical importance is the part of the curve (Figure 1, A ) where F (which, after other authors, may be called the error function) is reasonably small. In other words, its actual value is less important than its relative value along the galvanometer scale. APPLICATION TO PRECISION OF ESTIMATlON
As a general rule a photoelectric photometric estimation consists of three operations: (1) setting the zero (dark current) on the galvanometer scale, (2) setting the unit (100%) transmittance point on the scale, and (3) reading transmittance T through the sample solution. Settings 1 and 2 may involve errors: let us call them ZUT and UUT, respectively, where z and u are numerical factors, and UT is the error of T. Now the setting errors will contribute to the uncertainty of T over and above that covered by UT alone, and the contributions will depend on the value of 7‘ as shown in an earlier section. The position is summarized below: Source of Error Reading of test solution only Setting of zero transmittance
DISTRIBUTION OF RANDOM ERRORS
Effect on Estimate, T. of T e s t Transmit tance 7!’ f U T
*
only T (1 - QZOT The applicability of the Twyman-Lothian curve to the distribuSetting of unit transmittance tion of random photometric errors necessitates three assumptions only T i TXCT of instrumentation: (1) that the standard deviation, UQ, of a galvanometer reading is 20 uniform a t all points on the scale; (2) that I ERROR FACTOR A( ZERO ,RANSMITTANCE SEnlNG the 0 to 100 galvanometer scale is an exact linear reflection of 0 to 1 transmittance and, in particular consequence, that UQ = 1 o o U T ; and (3) that the trim of the galvanometer scale is perfect when the test solution is being assayed. This third item means that we must be able to assume that the standard deviation of the estimation of the transmittanee through the sample is the only operative error-Le,, that there are no “setting” errors to be taken into account, Factors affecting accuracy, such as stray light, reflectance losses, fluorescence, slitwidth variation, electrical nonlinearity, and divagations from the Beer-Lambert-Bouguer laws, do not impinge on the validity of the error curve. They may falsify mean values but will not modify the reproducibility of 0 4 . . . , , , . , . I, . , , . . . , ,1060 2 those means. 2 0 0 The f i s t two of the three assumptions can ERROR FACTOR(U) AT 1002 TRANSMITTANCE SETTING probably be safely adopted for most instruFigure 2. Influence of Presumptive Random Errors of Scale-End Settings ments; the third is more questionable. (as Multiples 5ut and uut of the Error, ut, of the Reading of the Test TransCole (s), some of whose terms differ from the mittance 2‘) on Optimum Precision usage in this paper, mentions double-heam When I = u 0 (the Twyman-Lothian condition) the optimum point has an error function of 2.718 ( = unity on the left-hand ordinate) with T o p t . = 0.3678 and Dopt. = 0.4343 spectroscopy, in which “only one signal I
l-----l
-
V O L U M E 2 4 , NO. 3, M A R C H 1 9 5 2
447 -
equivalent to ours, but he juxtaposes with it an alternative form, in which the d T 2 1 is replaced by ( T l ) , based on a so-called ' ' r a n g e- o f - e r r o r method." This method involves the simple addition of errors and is statistically abhorrent. The same equation also a p pears (in a slightly inaccurate, and presumably misprinted, form) in Steams's discussion (15)of the optimum transmittance point in an instrument employing a Martens photometer and electrical balancing. His finding is quoted elsewhere in the same publication (p. 112) and the derived optimum transmittance points (0.2784) are marked on calibration curves whose flow visibly belies the significance of these points. Condition 111 involves a double I . . . . * . . . . I 0 0.5 0 0.5 I.o error (twice the variance) in the setTRA NSMlllANCE ting of 100% as in the reading of R A Ns M ~TTANC E ,GA LVANOM ET E R RE AD ING Figure 4. Observed Error DistribuT , and no error in the zero settmg. 100 tion on a Beckman SpectrophoThere are theoretical reasons for Figure 3. Three Illustrative Error tometer debiting the 100% setting with Functions a higher error in some types of s e i and u b i being the errors of the scale setinstrument. t i n g of, respectively, the zero and full transCondition IV is of interest in that it yields the same error mittance termini, as multiples of the error, ct, function as that of another situation discussed by Cole (3)-vis., of the reading of the test transmittance, T a techni ue involving four operations: setting for (1) a free (air) light pa&, (2) transmittance through a cuvette of pure solvent, (3) the air path again, and (4) transmittance through the cuvette I€we assume that the three error sources are independent-i.e., filled with the test solution. If the four operations are assumed zero correlation-the joint contribution to the over-all error of T to have equal errors, the final effect on precision is the same as if will be the square root of the sums of the squares of the component there were a single 100yo transmittance setting with a variance errors. ( I n dealing with uncorrelated random errors, we cannot triple that of the test-transmittance setting. make simple additions as in the discussion of the simple correlt~ted Condition V is the complement, as it were, of condition 11: no displacements centering around Figure 1.) Therefore the estierror in the 100% setting but an error in the zero setting equal to mate of 7' will be that of the 2' reading. Condition VI covers errors of the same magnitude in all three T i C T d T L ( Z 2 u2) - 2TzZ 22 1 oDerations. Robinson and Cole (14) give (in effect)this eauation. b i t again an unacceptable "range-of-erroP" derivation is given as and the error function corresponding to the Twyman-Lothian an alternative. In so far as any one condition is selectable, this is curve (Equation 2) can be adapted to fit this estimate by the subthe most likely to cover null-point instruments of the Beckman stitution of this function of UT for u p itself. So type. Perhaps the most significant feature of the set of theoretical error functions whose characteristics are given in Figure 2 is their compactness. The range covered, from zero scale-setting errors is the required equation. \I?iatever values are given to z and ZL to errors a t both setting points twice the size of the actual Treadthe value of F will be infinity at T = 0 and T = 1 and will miniing, is fairly comprehensive; yet the error function minimum is mize unsharply somewhere between. A11 the corresponding curves will resemble, and lie within, the Twyman-Lothian curve, which always between 2.7 and 4.8, and the optimum transmittance itself follows Equation 3 with the limiting values z = u = 0. The point within 0.27 and 0.52 (corresponding to optical density 0.57 minima of these curves can be obtained by differentiation of F with to 0.28). Bearing in mind that the minima of the curves are on respect to T and equalization of the differential coefficient to zero. very flat portions, we shall be pretty safe if we prescribe the I t will be found that F is a minimum when transmittance region 0.3 to 0.5 as optimum. Finer specificity is impossible without material evidence, and it is interesting to reflect that direct experimental estimation of the precision of a given instrument would be much simpler (as well as more satisfactory) The influence of the magnitude of z and u on the minimum error function, Fmin.,on the optimum transmittance point, TOpt. than an investigation of the nature and magnitude of the actual (and the corresponding Do,$.)is shown in Figure 2, Equations 3 and 4 being the bases of calculation. The maximum values explored of z and u are 2, and the curve of the error function for this circumstance is compared in Figure 3 t o the simplest circumstance Table I. Influence of Magnitude of Errors of Zero Setting z = u = 0 and to an intermediate one, z = u 1. ( Z O T ) and 100% Setting ( U O T ) on Error Function F, and on Optimum Scale Points All the other curves covered by the characteristics shown in Condition z u Numerator of F Fmin. Topt. Dapt. Figure 2 fall between the innermost and outermost curves of I 0 0 1 2.718 0.3678 0.4343 Figure 3. Further details of some special cases are given in Table I1 0 1 4Fq-l 2.878 0.3299 0.4816 I and commented on below. I11 0 4% 4 2-1 3.007 0.3063 0.5153
+
+
+
+ +
=i
Condition I. This, as already indicated, is the TwymanLothian case; an error in T only is assumed. Fmin.is the lowest possible; if z and/or u have any values other than zero Fmin. becomes greater than 2.718. It is improbable that condition I ever obtains. Condition 11. Here the assumption is perfect zero setting and an error in the 100% setting equal in magnitude to the error of the test reading. Cole (3) gives an equation that is mathematically
IV
v
0 1 1
4%
1-34
z/T2-2T+2 VI 42Tp 2T+2 Denominator of F = T In T . transmittance recorded. T D = corresponding optical density. UT standard deviation of T
-
0 1
-
3.117 0.2872 0.5418 3.207 0.4300 0.3665 3.358 0.3882 0.4110
.
448
ANALYTICAL CHEMISTRY can be denoted by hv;therefore, the estimate DS will have a coefficient of &v(Ds - DR)/Ds, and it is plain that as the difference Ds - DR decreases and/or the magnitude of Ds increases so the error of the estimate of Ds dwindles. At first sight it appears that if D s and DR are identical the error of the estimate vanishes, but this is absurd and to show what really happens we must take into account the dependence of v on Ds - DR according to the information set out in the previous sections. At this stage, it should be noted that in practice it does not really matter whether D s or DR is the greater; the weaker solution, whichever it may be, will be the one to use for the 100% setting. The difference in optical densities may therefore be mitten as I D s - DE I. The test transmittance can, as usual, be denoted by T-Le., I D s - DR 1 = log T . Now we have already demonstrated that v will minimize according to Equation 4, from which it follows that, for-any value of I D s - DR 1,
setting and determinative errors with the object of constructing the curve of the error function from Equation 3. Figure 4 shows an error-transmittance curve empiricized from replicated data on the absorption of aqueous potassium chromate given in one of the reports compiled for the 1945 U.S.P. vitamin A study by the Division of Physics and Electrical Engineering of the National Research Council of Canada (11) and analyzed by the present writer. The instrument was a Beckman. ACCURACY
According to the work of Vandenbelt et al. (17) and of Glover
( 4 ) ,a t least some instruments of the Beckman type begin slightly to overestimate a t optical densities below 0.5 ( T = 0.316) and to underestimate beyond 1.5 ( T = 0.032). It is conceivable that the former results from reflection losses and the latter from stray light in the optical system. Typical courses of these departures from linearity (taken from Vandenbelt, 17) have been used to construct the right-hand curves of Figure 5 which illustrate, by contrast to the left-hand curves, the effect of the nonlinearity on the limits of error of measurements along the galvanometer transmittance scale. Noninstrumental sources of inaccuracy such as fluorescence and Beer’s law deviations are in special case and lie outside the scope of the present discussion. These effects can be countered by calibration curves. In so far as the accuracy of an estimate of absorption may instrumentally depend on the optical density actually measured, a photoelectric photometer designed to embody a telescopic cuvette would have its attractions. All work could then be done a t one (the optimum, obviously) transmittance point, the solution light path being the variable metric. The principle of what might v , elongate) is of course be called “teinophotometry” ( ~ ~ i v e i to integral to the Duboscq-type visual color comparator. (J. R. Edisbury has suggested the alternative “trombophotometry” which, pace the etymological purists, is certainly evocative. ) The technical problem of producing an optical plunger device of sufficiently fine accuracy to be worthy of incorporation in, and in fact refine the performance of, a high-precision photoelectric instrument, is, admittedly, not negligible, but might be worth consideration. And a point to weigh in the consideration is the link with the subject of the next section : the transmittance-ratio technique.
If then, we write
= log e
u2
- 2z2/T
+ zZ/TZ + l/T2
(5)
the symbol F’ will express the error reduction achieved, scil., the coefficient of variation of the transmittance-ratio method relative to the optimum coefficient of variation of the conventional method. Clearly, F‘ minimizes when D s = DR ( =, say, D )and, therefore, T becomes unity. So
To evaluate F’, or to plot F’ against D s and DR, it is necessary to decide upon the most likely equation for F . The one on which Figure 6 is built is t$t given for condition VI in Table I ; it is perhaps the fairest average” error-function of all those considered. In point of fact, Figure 6 would need only trivial modification had any of the other equations been used.
TERMINAL
COMPLETE ACCURACY
The precision of an optical density estimate made on instruments of the type we are here concerned with can be increased by the use of standard solutions. Ringbom (12) and Kortum (10)pioneered what has come to be known as the transmittance-ratio technique, and which has recently been discussed and exemplified by other workers ( 1 , 2, 6, 7 ) . Its dependence on the functional features of the instrument and in particular its limitation with a certain type of filter colorimeter are dealt with by Hiskey et al. ( 7 ) . The principle is simple.
-
+
DS Fmin.
TRANSMITTANCE-RATIO TECHNIQUE
Suppose we have a sample solution of optical density Ds and a reference standard solution of the same material of optical density DR. Let DR < Ds. Let us now measure D s on a photoelectric photometer with the modification that the 100% galvanometer setting is made with the reference solution instead of, as normally, the pure solvent, in the cuvette. This means that the sample-solution transmittance we subsequently record will correspond to the optical density DS DR. Knowing DR, we can then calculate Ds. Now the estimate of D s - DR will have a certain coefficient of variation that
1/22
I NACC URAG I E S
(€AN ALUE
I
0
-
-
05 TRANSMITTANCE
3
05 TRANS M I TTANCE
Figure 5. Distortion of Typical Limits of Error (Precision) Produced when terminal inaccuracies of the form noted by Vandenbelt occur
et al.
(17)
V O L U M E 24, NO. 3, M A R C H 1 9 5 2
449
All points below the parallel (in Figure 6) drawn from the ordinate at unity represent a gain in precision over the best possible conditions of estimating Ds solo-Le., diluted to be read a t TOpt.-but not necessarily a gain over estimating Ds at any other transmittance. The curves of Figure 6 may in fact be regarded as cut from the “best” plane of a three-dimensional figure illustrating the effect on F’of (1) Ds; ( 2 ) DR; and ( 3 ) all possible points on the transmittance scale a t which the Ds, by dilution, might be estimated solo and conventionally. To assess the practical value of the transmittance-ratio technique, the above theoretical treatment must be lined up with other items. It is probably fair to state that the precision of the conventional technique is high enough for many analyses. When estimates of absorption within especially fine limits are demanded, the transmittance-ratio technique is worth consideration. It must then be borne in mind that the reference solutions must be standardized with gre&t accuracy if the method is to have real value, for implicit in the theory is the negligibility of the error of translation of optical density into concentration. Furthermore, the extra trouble of the special technique has to be weighed against
a 5 >
G:
