counting technique. The present procedure is of particular value in studies requiring the eunrnination of large nunibers of samples over a wide range of radioactive concentrations.
(2) Comar, C. L., Wasserman, R. H., in
LITERATURE CITED
(4) Funt, B. L., flucleonics 14, N o . 8,
(1) Bauer, G C. H., Carlsson, hrvid, Lindquist, Bertil, Acta M e d . Scand. 158, 143 (1357).
“Atomic Energy and Agriculture,” pp. 249-304, Am. Assoc. Advance. of Sci., Washington, 1957. (3) Davidson, J. D., Feigelson, Philip, Inter. J. A p p l . Radiation and Isotopes 2,1(1957).
83 (1986). (5) Funt, B. L., Hetherington, rlrlene, Science 125, 986 (1957). (6) Heaney, R. P., Whedon, G. D.,
J . Clin. Endocrinol. and Metabolism 18,
1246 11958). (i)Kalimann, Hartmut, Furst, Milton, Phys. Rev. 79, 857 (1950). (8) Reynolds, G. T., Harrison, F. B., Salvini, G . ,Zbid. 78, 488 (1950). (9) White. C. G.. $elf. Samuel. Xucleonics 14, No.’lO, 46 (1956). I
.
RECEIVED for review September 16, 1958. Accepted November 12, 1958.
Precision Absorptiometry CRAMON M. CRAWFORD’ University o f California, 10s Alamos Scientific laboratory, 10s Alamos,
b To assess the significance of a recent statistical finding which seemed to invalidate differential methods, the rationale of instrumental precision was explored and applied to absorptiometry. Although these statistics were not found to b e germane, pertinent data are too few to prove the validity of published precision methods. The theory here developed stresses the need to recognize the linear scale reading without physical interpretation, and yields the important result that the Ringbom plot is insufficient to specify the range of good precision. The paper describes a new cell correction method, shows that differential methods reduce the effect of scale setting errors, corrects and extends an earlier treatment of ways to attain optimum analyte concentration, and points out the unique ability of the general differential method to use this optimum.
M
attmipts have been made to improxre the ordinary precision of absorptionietry with its 1% error or worse, all based 011 hypotheses about the principal source error. This paper reports a critical exusmination of the ideas involved, prompted partly by Cahn’s finding (16) that the statistics of actual analyses do 1106 wpport the hypothesis (predominariw of scale reading error) on which wid(.ly a w l differential methods depend. ANY
N.M .
the property to be found from the transmittancy-e.g., analyte [the substance determined (22, 49) ] concentration, identity, purity, or structure; reaction end points; photochemical quantum yields; temperature; or pH. There are several methods of improving precision known to precision absorptiometry. Both logic and convenience reserve the name differential for that method which uses reference solutions or their photometric equivalent such as filters. Precision is usually given qualitative definition (BO,22) only. I n this paper, it means also the reciprocal of the standard deviation of the result. Such a measure has the qualitative properties generally associated with precision, which the standard deviation itself does not. The quantity q , which RIandel and Stiehler (38) call sensitivity, is identical with precision as here defined. However, the discussion of the Committee on Balances and Weights (19) concerning the proper meaning of sensitivity can be extended to cover the present case, and provides ground for objection to calling \k a sensitivity.
31i‘
DEFINITIONS
Absorptiometry is here defined as the measurement of transniittancy or some function thereof. Because transmittancy is almost always used to calculate something else, precision absorptiometry is taken to mean precise measurement of Present address. Mississippi State University, State College, Miss.
INSTRUMENTAL PRECISION
Notation. Instrumental measurement consists of calculating a result, y, from a n instrumental reading, R , here considered t o be on a linear but otherwise arbitrary scale. I n turn, R is a function of several experimental variables, or conditions, typified by 5 , . Because y is intended to measure one of the zl, say $1, the experiment is usually arranged so that R is sensitive only to zl. The explicit recognition of R, divorced from any interpretation of its physical meaning, is important, because the physical meaning frequently changes with a change in the manner of using the instrument, and because the significance of any conclusion is more apparent when the quantity of practical interest-the
dial reading-is unambiguously identified as such. For instance, in ordinary absorptiometry, the dial reading measures transmittancy, but in differential absorptiometry the dial reading measures either a ratio or a more complicated function of two or three transmittancies. Assignment of a single symbol for the dial reading in the latter case provides warning that contact with experiment has been lost if these transmittancies are separated algebraically. Improving Precision. The result, y, is calculated from a theoretical or empirical relation y = y(R) whose differential is d g = y’(R) dR. The latter can be interpreted as providing a conversion of the uncertainty, dR (as measured, perhaps, by U R ) , into the corresponding uncertainty, dy, of the result. The curve slope, y’, thus acts as a weight factor, or error coefficient, which by becoming small can make the final uncertainty small. The quantities entering y’ which are susceptible to experimental variations are therefore given values which make y’ small. I n particular, R will be such a quantity. By selecting R so that y’ has its least value, a given error in R will cause least error in y. If it can be shown experimentally that dR does not depend on R-Le., dR is constant-the precision has been optimized with respect to R. The investigation of the dR us. R relation is usually neglected in absorptiometry, but conditions exist in which constancy mould not be expected and cannot be assumed. If dR does depend on R, the least value of d y implies the least value of y’dR, and will not necessarily coincide with least values of either y’ or of dR alone. Whether or not dR depends on R, a decrease of dR may be possible. The uncertainty, dy, may be predicted, for changes whose magnitudes (including sign) are known, by differentiating the assumed relation y = y(R) taking R = R(z,): VOL. 31, NO. 3, MARCH 1959
343
Minimum Error. Minimizing uu or 21 be identified, their independence proved, their distributions measured, and their functional relations to R evaluated either theoretically or empirically. If skewed distributions are present, appropriate cross terms must be added to Equation 2. Predicted and measured data must agree within the random error for an adequate theoretical function. An empirical function should be fitted well enough so that the terms of Equation 2 which represent the uncertainties contributed by the adjustable parameters can be neglected in comparison with the other uncertainties. The zi cannot be identified once for all, because a change in experimental arrangement might introduce new z j or discard some of the old. Hoxever, the zi reported for one experimental arrangement are useful as trial identities for another. When uy has been correctly predicted with Equation 2, the identification of the z, may be deemed complete. Snesarev (64) claims to have done this for one experimental arrangement in absorptiometry, but gives few details about the arrangement. Formal mathematical minimization of Equation 2 is difficult and probably impractical in general. To examine the conditions which minimize each harmful sensitivity in turn is feasible, however, and these conditions may be found identical or a t least compatible. When they are incompatible, minimizing the harmful sensitivity associated with the largest error is also feasible. A validated Equation 2 has other uses too: to find out how precisely the experimental conditions must be controlled to obtain a specified precision in y , or when some terms are unknown to set upper limits to the precision attainable by using the terms which are known. u v / y requires t h a t the
Real changes in the z, may take place either during or between measurements, but changes which are slow compared to the speed of instrumental response may be considered to take place between measurements. Fictitious changes may be postulated to take into account the uncertainty in z i values. Changes in an experimental condition sufficiently slow or infrequent to permit correction or recalibration, as with phototube replacement, can be removed as sources of variance by performing such recalibration. For highest precision this is mandatory, but it is nearly so for less precise work if the precision must be known, because the accumulation of the needed data is much slower than the changes themselves unless the latter are artificially induced, and the induction can alter the kind of distribution obtained. The average error is obtained through squaring Equation 1 and summing over the measurements to be averaged. The result can be very complicated, but if the distributions of the z, are symmetrical about their means, as in the normal distribution, positive and negative cross terms of any given magnitude should occur with equal frequency and hence cancel (63). For a large number, X, of measurements, therefore, N
Snesarev (64) writes this in fractional form, which is more symmetrical in R , and calls dR/dy the “useful sensitivity,,’ while dR/dz, he calls the “harmful sensitivity.” Probably he would not include the measured property z1 among the xi. Certainly variations in z1 which are to be measured should be excluded from url, but extraneous variations are included, for instance, as caused by loss or contamination during sample preparation when z1 is analyte concentration. The quantity (dR/dy) is really an inverse sensitivity intended to equal the direct sensitivity (dR/dzl). Therefore, one might prefer to call (dR/bzl) the helpful sensitivity, even though as a coefficient of u , ~in the above sense it is also a harmful sensitivity. Snesarev apparently overlooked the distinction made here, that a tendency in the desired direction is helpful, but only the resultant of helpful and harmful tendencies is useful. An equation referring to relative error instead of absolute error is obtained by dividing Equation 2 by y2. 344
ANALYTICAL CHEMISTRY
APPLICATION TO CONCENTRATION MEASUREMENT
The calculation of analyte concentration can combine the theoretical and empirical approaches to some extent, for Beer’s law has theoretical justification (65)but contains in effect one adjustable parameter, u, whose proper selection may (or may not) provide adequate fit to the data. Beer’s law may be mitten: 1
c = -In T ZL
n-here
u
= (In 10) ab
T
3
P/P,
(3) (4) (5)
and where a is absorptivity, b is the path length of the light, and P and P o are the radiant powers leaving and entering the sampk, respectively.
Differentiating Equation 3 and dividing by c, dc/c =
dT/(T In T )
(6)
Here dc/e is the relative concentration error, the quantity which one ordinarily wishes to minimize. The relative error coefficient is 1 / ( T In T), which minimizes at T = l/e = 36.8%, or A (absorbancy) = -log T = 0.4343, a fact which has been known at least since 1927 (68). Since Twyman and Lothian’s paper (67) came later, the graph of this equation is not appropriately called the Twyman-Lothian curve as does Gridgeman ($8). For constant dT, T = 36.8% represents a minimum error condition which does not depend on u. RINGBOM-AYRES PLOT
The negative slope of the RingbomAyres plot (2, 61) of log c us. (1 - T ) , or the positive slope of its more convenient form, log e us. T (4), will always give valid empirical relative error coefficients, whereas the theoretical Equation 6 or its graph will give correct results only when the theory is applicable -i.e., when Beer’s law is a reasonable approximation. I n either case, the information thus obtained must be combined with the dT us. T relation before conditions of maximum precision or the range of good precision can be found. Taking for granted that dT is independent of T is unjustified, because several instances of known dependence are given below, and more may exist. Difficulty was caused by one such instance (nonlinear scales) (1, $8, 29). The literature frequently states that tlir range of good precision was found from the Ringbom plot alone; such statements should no longer appear. LlcBryde (36) has criticized the Ringbom plot on different grounds. Theoretically, three principal kinds of dT us. T behavior are expected: proportionality, (rough) proportionality to the square root of T , and constancy. 1. dT Proportional to T. This relation implies t h a t the error in absorbancy is independent of absorbancy. Cahn (16) lists cell errors, sample preparation errors, wave length errors, and source change errors under this heading, but does not discuss what he means by these names. One may expect that an\- error directly affecting the absorptivity, n, or the optical path length, 6, will be capable of causing dT to be proportional t o T , as both a and 6 are logarithmically related to transmittancy. An error in analyte concentration caused b r loss or contamination during sample preparation is in the same class-that is, even
unskewed distributions of a, b, or c cause skewed distributions of T . Changes of temperature directly affect absorptivity ( I S ) but may be made to cause negligible error by sufficiently good thermostating. Wave length changes also affect absorptivity. When absorptirity changes significantly with wave length, neither wave length nor slit width controls should be disturbed after initial setting, during or between calibration and measurement. The filter of filter photonieters must remain in unchanged condition. The probable importance of these precautions is a clear deduction from Cahn (16). Of other possible sources of such errors, changes of source, cuvette, or photocell arc sufficiently infrequent that correctioiis should be made or the calibration repeated. The potential error can thus be made negligible. CELL CORRECTIOKS. Banks, Grimes, and Bystroff ( 7 ) have treated cell corrections, showing that if a is the full scale setting and p is the zero setting,
Table 1.
.4. Forward B. Reverse
P’
= ( a - P ) (““‘ p,’ --pz ‘:’)
+ fi
(8b)
wlicreupon substitution shows that
That is, one will obtain the same reading for a given P as before, if the boundary conditions are set as R = a’ when P = P I r ’ ,and R = p’when P = P,”’.This reciprocity is suggested by Table I. One is free to select a and p, or else a’ and p’, but not all four, because given txo, the remaining two have values defined by Equation 8. Because to every forward choice of a and p there corresponds a unique a’ and p’, and conversely, it follows that if the forward boundary conditions have been established, there is a reverse set of boundary conditions TT-hich is automatically satisfied, and conversely. If the cells are matched, Pl’ = PI” and P,‘ = P1”,so that a = CY’ and B = p’. HoJvever, if the cells are not matched, the points ( a, cl) and (p, Q) are the appropriate ends of the calibration curve, and not the points (a’,cl)
Boundary Conditions Definitions of CY and fi When Read R = a P = Pl’ R=p P = P2‘
Definitions of a’ and p’ Boundary Conditions When Read R = a’ P = PI“ R = p’ p = p2t“
considered of dubious validity with regard to time effects. Neglecting time effects, then, one may wish to use several sample cuvettes to speed the work. The foregoing scheme may be supplemented with corrections to allow this. For constant boundary conditions, Equation 7 may be written in the form R1 = kTIT IC’, where T is the sample transmittance and T1is the transmittance of the first cuvette. For a second cuvette, Rz = kT2T IC’, and so on. The readings R1,Rz,Ra,etc., for a given sample (constant T ) may be easily measured, and two such samples, not necessarily standard (dashed lines, Figure I ) , are sufficient to plot the curves shown, as the modified Equation 7 predicts straight lines for linear photocells, and the results to be obtained are independent of the assigned values of T, which can then be arbitran.. Any reading on any cuvette can now be corrected to the reading that would be observed on the first or any cuvette by vertical projection. Rigorously speaking, this procedure compares cuvettes a t the same concentration rather than the same transmittance. Only if the cuvettes have the same optical path lengths will the same concentration imply the same solution transmittance. n’evertheless, the procedure makes, in effect, a linear approximation correction for the exponential error caused by path differences, as well as a “perfect” correction for constant differences in cuvette transparency. The residual error should be small, if the cuvettes are not too badly matched in path length. Only by individually calibrating each cuvette could one hope to do better. This graphical “multiple-sample cuvette” procedure has a more complicated algebraic equivalent; it is applicable t o general differential absorptiometry and to any special case thereof, including the transmittance-ratio and ordinary methods. COMPARISON. Method I11 of Banks, Grimes, and Bystroff ( 7 ) is in principle identical with this single-sample cuvette procedure. hlethod I1 is a t best an approximation, requiring that all information (matched pairs) necessary to apply the more exact (for constant matching) and no more difficult Method I11 be a t hand before it can be used. Method I1 should therefore not be used.
+
+
L
where subscripts denote reference solutions and primes denote cuvettes. Beyond this equation, the present author prefers a different analysis. From Equation 7 , one can see that R = CY when P = PI’, and that R = p when P = P2’,These are the boundary conditions. One may now define a’ and 8’ by the eqaations
Operational Analysis of Cell Corrections
I
Transrnittancy
Figure plot
1.
Schematic cell correction
and (p‘, c2). Boundary conditions, on the other hand, are most easily set with a’ and 8’. One aspect of the problem of cell corrections is therefore to find corresponding pairs of (a,p) and (a’,8’). Corresponding pairs can be readily measured: the algebraic expression (Equation 8) of the experimental relation was given only for explicitness, and need not be used in practice. When these pairs do not change with time, no corrections are necessary. The procedure of setting boundary conditions with a’ and B’, while plotting a and p on the calibration curve, permits the use of individual reference cuvettes without error. Banks, Spooner, and O’Laughlin (9) have reported that, experimentally, corresponding pairs do change with time. These changes are usually in one direction during a series of measurements, but the direction is random (8). Unfortunately, the measurements which show that a change has occurred do not show the source of the change. Freeland and Fritz (24) have described methods for measuring cuvette transparency and path length ratios, but neither these factors nor deposition of solute mould be likely to produce changes as described above. Electrical instabilities are a plausible source. The applicability of cell corrections can be examined only when the source of change is known, and existing methods niust be
VOL. 31, NO. 3, MARCH 1959
345
The multiple-sample cuvette procedure is a more convenient version of Method I, except that in Method I all readings are corrected to the properties of a hypothetical cuvette instead of a real one, as here. This hypothetical standard corresponds to none of the cuvettes actually used, and has a transmittance which is a linear function of the incident light flux rather than being constant. Time dependence of corresponding pairs caused by changes in the reference cuvettes or instrumental drift away from the initially chosen a' and p' have the effect of altering the "standard" cuvette for any of these procedures. The error so caused is minimized by use of the general differential method, as will be shown. INDETERMIXATE ERRORS. Returning to error sources which make dT/T constant, four others should be listed:
1. Photocell noise, or fluctuations of light source brightness in photoelectric instruments (60). 2. Operation of the Weber-Fechner law in visual instruments. 3. Emulsion sensitivity in photographically recording instruments. 4. Reading errors on logarithmic scales (44). The first causes the differential method to be worthless (60); this is not true for the last of these if the scale reading is a general linear function of log T. The assumed dT:T relation may be substituted in Equation 6 to show that the relative concentration error becomes zero a t T = 0, or complete absorption, a very different conclusion than would be drawn from a Ringbom plot alone. Zero error is not attainable, however, partly because for small T the d T : T relation changes to that next discussed. 2. dT Proportional to 4 F . Cahn (16) attributes this relation to statistical variation in photon arrival a t the phototube. The reasoning is not given, but seems to be as follows: Logarithms are taken and then Equation 5 is differentiated:
analogous to Equation 1. By similar reasoning, this is converted to the analog of Equation 2:
Lindsay (34) derives a formula equivalent to: up
=
4-3
(12)
A like equation can be written involving Po. Substituting these together with Equation 5 in Equation 11, u$ = ( t / P o )( T
346
+ T')
ANALYTICAL CHEMISTRY
(13)
If T is sufficiently small, T 2can be neglected, so that the standard deviation of T , averaged over time t, is proportional to 4 F . Putting this relation into Equation 6, it is easy to prove Cahn's result that the optimum value of T is 1 / e 2 or 13.5% transmittance. If T 2 is not neglected, the minimum error condition is In T 2T 2 = 0, having an approximate solution of 10.9% transmittance. Because this differs by an easily measurable amount from Cahn's result, one concludes that T is not sufficiently small. The binomial distribution is assumed in this argument, a distribution which for reasonably large numbers of photons does have skewness approaching zero as required by Equation 11. If the argument is to apply to differential absorptiometry, the point of departure must be Equation 14 instead of Equation 15. 3. dT Independent of T. Cahn (16) lists scale reading errors on linear T scales and dark current drift under this heading. Nonlinear scales can be taken into account (48)-for example, suppose T = tan*R, where R is the linear scale reading. Then dt = 2 tan R sec2 R dR, which can be substituted in Equation 6 to find minimum error conditions under the assumption that dR is independent of R. One effect of dark current drift is to alter the zero scale setting, and Gridgeman (29)has shown that the error in T so caused is (1 - T ) d p , where dp is the change in zero. The drift may also cause an error in the full scale setting equal in both sign and magnitude to the zero error; the net effect of the zero and full scale errors on T is then to make dT independent of T . It is not clear that drift will always satisfy this condition, so that it is incorrect to state categorically that dT errors due to drift are constant. Dark current drift is one cause of a more general problem of the effect of scale setting errors on the final result. This problem has been a source of confusion in the past (18, 6.2). Hamilton (3.2) showed that the zero and full scale setting error can add to yield the error curve of Equation 6, and later ( S I ) implied (perhaps inadvertently) that the setting errors actually cause the error expressed by Equation 6, and that other errors-e.g., scale reading error-are to be taken into account separately. The additivity is misleading, and the latter idea entirely wrong. Gridgeman (28) has given the best analysis of the problem, an analysis extended in the present paper to cover general differential absorptiometry . Solving Equation 7 for P and substituting in Equation 5 ,
Differentiating Equation 14 and substituting in Equation 6,
+ +
These three equations give relative sensitivities related to the coefficient of Equation 1. If da = d p in both magnitude and sign, as may happen in dark current drift, the resulting error is:
which except for sign has exactly the form of Equation 17. This is what Hamilton (8.2) showed experimentally. However, Equation 17 is both mathematically and conceptually a separate equation, and does not lose its signscance as showing the effect of reading error even if Equation 18 is also valid. If da and d p represent scale reading errors made by the operator while setting LY and p, they will presumably be uncorrelated and random, so that a rootmean-square combination of errors in the manner of Equation 2 is appropriate. Setting errors actually encountered will be partly due to reading errors and partly to instrumental drift, in varying proportions. I n either case, and also for the error represented by Equation 17, clearly the error is minimized under the same conditions: making the relative error coefficient 1 TT )small ' by using reference solutions closely spaced around 37% transmittance-Le., by using general differential absorptiometry. The same could be readily shown with respect t o slide-wire nonlinearities. One important feature of general differential absorptiometry not previously pointed out is that it can provide the kind of advantage (decrease of error coefficient numerator) gained in the transmittance-ratio and trace analysis methods while still using the optimum denominator value (37% T ) , whereas both transmittance-ratio and trace analysis techniques must accept a penalty in the denominator to obtain profit in the numerator. For trace analysis, the penalty is unavoidable, but this is not true of the transmittance-ratio method. A comparison made by measuring a given sample by all four methods uses the numerator alone, and does not do justice to the general method, except for reflectance measurements of the sort made by Nimeroff (60). The absolute error in reflectance can be obtained by solving Equation 7 for P and differentiating:
( In
The P’s may represent either transmitted or reflected light, of course; evidently there is no optimum P for either, and hence no optimum reflectance. This would cease to be true if a nonlinear function of P were sought, which is the reason that there is an optimum transmittance for concentration measurement. The device used by Banks, Spooner, and O’Laughlin (9) to avoid the necessity for rerunning calibration curves for each change of reference solution-plotting 1 O - a b c os. R to obtain a straight line-is very convenient, but depends on photocell linearity and on adherence to Beer’s law, both of which should be checked. The logical extreme of general differential absorptiometry makes the scale correspond to as small a concentration range as possible. This can be done usefully in two cases without even using reference solutions, and thus not meeting the definition of differential, even though the principle is the same: photometric titrations (26) and teinophotometry (88). I n these cases one does not need to know just what concentration range the scale corresponds to.
incurring an error. The extent to which this is true will now be examined for general differential absorptiometry, assuming dT constant. Four methods of changing concentration-i.e., transmittancy-towards the optimum suggest themselves: addition or removal of m moles of analyte, or addition or removal of solvent (m = 0). The first (addition of analyte) was apparently first proposed in 1917 (SS), given honorable mention in 1950 (do), and used as recently as 1953 (15). McBryde ($6) challenged but made an error in his analysis. Addition of solvent was discussed in 1957 (W). The remaining two have been neither challenged nor justified. Clearly, in all four cases the mass of analyte is conserved: V,ci
+ m = Vfc/
(20)
where i = initial and f = final. N c Bryde’s definition of r is convenient : r =cr/c,
=
VJVj
+ ml(V/c,)
(21)
Direct measurement on solutions of concentrations ci and cf is associated with errors Ei and El predicted bv Equation 6. The ratios are:
EXPERIMENTAL EVALUATION OF dT vs. T
Cahn’s finding (16) that the standard deviation of absorbance is experimentally independent of absorbance is the onlv test of the dT us. T relation known to the author. This result is based on only a few points whose self-consistency does not inspire confidence that the data they represent were taken under conditions of statistical control. Afore iniportant, these data were taken on many instruments, which is a procedure not used in practical analysis. Gridgeman (SO) and others have felt that collaborative experimental teste of the value of differential techniques should be undertaken by some responsible professional group. This proposal is here warmly endorsed, with the notation that differential techniques can be profitably studied only jn relation to the general problem of attaining precision. Many studies of photometric variability have been made, but apparent1)none intended t o measure the dT 2s. T relation under practical conditions: measurements and calibrations both on a single instrument v-ith a standard technique. 4 t present, use of methods intended to produce precision appears to be based primarily on an act of faith that the conditions which justify them can he and are satisfied. EFFECTING TRANSMITTANCY CHANGE
The preceding discussion has tacitly assumed that it is possible to change transmittance or transmittancy without
where the last transformation was made by applying Equations 3 and 21. To find the error, Ed’,associated with calculating ci from measurement on cf, Equation 20 is differentiated:
Combining equations 22 and 23, finally, E,’ = ( V,t/V,)T,(l-r)E,
(24)
Equation 24 permits one to compute from a proposed method of changing concentration whether or not benefit [meaning (Vf/V,)T,(l-‘)< 11 mill result, if Beer’s law is valid and dT is constant, and applies to any of the four methods described or to any combination of them. It also assumes no error in m, which will he the most favorable case for these methods. Addition of Analyte. Here c / > cl, hence r > 1 and (1 - r ) is negative. Since (0 5 T , 5 l), then T,(l-r) 2 1. Also, V/ > V,, so that (Vf/V,) > 1. Plainly benefit cannot result, and the method cannot be recommended. McBryde argues correctly that T’< 1, but the relevant question is whether T’ < T. He also omits consideration of the volume changes involved. Volume changes are significant, because in the intended application to trace analysis, accuracy is sacrificed by leaving part of the analyte behind in a volumetric flask. Al-
though the volume must be known, routine “making to volume” must be eschewed unless the standard volume is that of the cuvette. Subtraction of Analyte. Here the above arguments apply in reverse, with the reverse result. Nevertheless, the method will not ordinarily be experimentally aonvenient, and the subtraction must be done in such a manner t h a t m is accurately known. Addition and Subtraction of Solvent. With m = 0, r = ?‘,/VI and Ej = E,’. Setting dEi‘/dr = 0 as found from Equation 24, the minimum error condition r In T, = -1 is obtained. But from Equations 3 and 21, r = (In T//ln Ti), so that T f = 36.8%, in agreement with the analysis which ignored the manner of effecting transmittancy change. One may consider this as justification for either adding or removing solvent to get the optimum transmittancy for measurement. CONCENTRATION RANGE FOR PRECISION
Although ideas about the extent to which absorber, path length, and concentration may be chosen seem to be well understood in some quarters, much confusion exists still. Analyses of the sort given earlier fix a range of absorbancy which will give precision. Absorbancy is the product of absorptivity (characteristic of the absorber, and determined, perhaps uniquely, by the available color-forming reactions and freedom from interferences), the optical path length ( determined by the available cuvettes), and analyte concentration (easily adjustable for major constituents, but not in trace analysis). Evidently, the precision range of absorbancy may be obtained in infinitely many ways. The contribution of du to the orer-all error (no matter how dT depends on T, since u does not appear in the error coefficient) is minimlzed by making u large (50). For this reason, because it aids in trace analysis or nith weak absorbers, and because Beer’s law is more likely to be valid, long optical paths coupled with rather dilute solutions are to be preferred except when the solvent absorbs strongly and there is trouble getting enough light. If the measurements on u (or other empirical parameters) are careful and numerous enough to make du small compared to other errors, the analysis predicts the same maximum precision for every size of cuvette, no matter what the absorptivity, provided only that the absorbancy giving maximum precision is used. This is probably not true, but the reasons are obscure. ERROR SUMMARY
Harmful sensitivities have been evalVOL. 31, NO. 3, MARCH 1959
347
uated, subject to assumptions, for da, dp, and dR in Equations 15, 16, and 17, while the previous paper (50) gave those for dP,, du, dcl, and dcz, except that for arbitrary scale ends, R/100 must be replaced with ( E - p ) / ( a - p). Means have been suggested for making errors due to temperature, wave length, source, photocell, or cuvette changes insignificant, so that these need not be considered as zi. The size of any error sets a limit on the precision attainable, whether or not the other errors are known-for example, the readability of the Beckman D U scale is about + O.lyo. The error coefficient in the ordinary method reaches a minimum value of e. Thus no analysis carried out in the ordinary way on a Beckman D U can hope to have a concentration error less than + (O.l)e or f 0.27%. Many reported analyses come close to this limit. APPLICATIONS
Differential absorptiometry has been applied to alumjnum (6), beryllium (69), chromium ( I C ) , copper ( I I ) , erbium (IO), manganese (14, 60), molybdenum ( 4 ) , neodymium (9, IO), nickel ( l a ) ,niobium (47), platinum (3), plutonium (@), praseodymium ( I O ) , samarium (IO), tantalum (4?‘), titanium (17, 43, 45), uranium (5, 23, 56), and zirconium (26, $9) among the metals, and also to cyanide (46), fluoride (85), and phosphate (27). Aspects of precision absorptiometry other than the differential technique are less easily indexed, and no search for their applications has been made. ACKNOWLEDGMENT
The author thanks Charles V. Banks, P. G. Grmes, and R. I. Bystroff for helpful correspondence, and P. E. Rouse for helpful discussion, concerning the ideas of this paper. He is grateful also to the Los Alamos Scientific Laboratory of the University of California for permission to publish the work, which was performed under the auspices of the U. S. Atomic Energy Commission. LITERATURE CITED
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