Characterization of the Color Quality of Indicator Transition Co m ple me nta ry Trist im uIus Color i metry CHARLES N. REILLEY, H. A. FLASCHKA,I SIF LAURENT,’ and BERTEL LAURENT2 Deparfment o f Chemistry, University of North Carolina, Chapel Hill, N. C.
,The over-all quality of an indicator end point i s a composite function of the sharpness and color quality of the transition. While the former i s related to the net effect of the various competitive equilibria and can be calculated rigorously, the latter involves the subjective perception of color and i s readily handled b y combining the principles of tristimulus colorimetry with Beer’s law. This derivation leads to a most useful equation given in abbreviated form as P, = G, J(Q, - G,) = G, - JV,, where P, corresponds to the chromaticity coordinates of the observed color, G, to the coordinates of the illuminant color, Q, to the coordinates of the absorbed color (complementary color point), and J to a “color concentration.” Q, i s a concentration-independent parameter characteristic of the colorant V7species, and its position-i.e., relative to the illuminant point describes the relative grayness of the indicator color-e.g., its dirtiness. An indicator’s color change quality i s related numerically b y these quantities and account i s made of the number of chromaticity and memory steps crossed in the transition. The proposed theory permits calculation of the relative quantities of inert dyestuffs necessary for screening an indicator, thus eliminating trial and error procedures. The calculation i s based on the simple i
V,,,I, = 0. An extended
equation, 0
equation describes the dichromatistic tendency of a colorant, a factor negligible for practical indicators at proper concentration levels.
I
visual end points of a titration, terms such as “good, bad, >harp, dragging, indistinct, abrupt” :ire frequently used. Despite the fact that such terms are vague and always -ubjectivc, they can also be misleading because the over-all quality of the oiid point response is a composite function of a t least two important and entirely distinct factors. N DESCRIRIXG
1218
ANALYTICAL CHEMISTRY
The first factor involves the chemistry of the reaction which governs the color change-, Le., rate of reaction and equilibrium constants of the various reaction systems. The only limitation which may arise here will be a practical one caused by numerous competing equilibria and lack of available data. Theories dealing with these aspects have been presented in detail for acidbase and redox titrations and more recently for chelometric titrations (1-8, 13, 16). The second factor consists of the color phenomena in the solution and their relation to the observing eye. This problem is much more complex and complicated for treatment in a rational manner because it involves physiological and psychological parameters whose treatment, if possible, enforces some rather deliberate assumptions. Some widely accepted theories for describing colors exist, but for application to the present problem they had to be modifid and considerably extended. The development of formulas to permit comFlete characterization of the color factor, including mathematical treatment of screening, dichromatism, and related phenomena, not possible previously, constitutes the purpose of the present paper. MATHEMATICAL TREATMENT FOR SUBTRACTIVE COLORS
T o specify the color transition of a visual indicator, a logical choice is to employ the chromaticity values of the International Commission on Illumination (official abbreviation, CIE, 6). The specification method based on tristimuli values is highly developed and widely applied for characterization of additive colors. Two studies in this direction have been made by Van K y k and Clark (15) and King (11). Honever, in the case of a n indicator subtractive colors are involved and the concentration of the indicator and any other colorant present influences the final color in a rather complicated way. The essential difficulty arises from the
fact that the intensity of the light leaving the titration vessel is related to the concentration in an exponential manner (Beer’s law). Therefore, the simple treatment of additive colors is no longer applicable and a combination of the known tristimulus scheme iyith Beer’s law was necessary. Because more than one colorant may be present in a titration system, the following general form of Beer’s law is employed :
- In T =
2.303 log
T
=
+2.30,3 -qbibs
=
J$here T denotes the transmittance a t wave length x, c, the concentration of substance i; and Abs the absorbance. k, denotes the product of the absorptivity of substance i and length of the light path. Expressing the concentration of a particular colorant in mole fraction terniP, the usual definition is introduced pi =
ci
;
n
ci, the total concentra-
where c = i
tion. Defining the total absorptivity times path length for all the colored coinponents together, taking into account their relative concentrations, as
and combining Equation 1 with 2 and 3 yirlds cK Abs -(4) 2.303
I n accord with the usual procedure for specifying color by the tristimulub scheme, +h(x), $2(x), and &(A) are introduced to denote the weighting functions (6). 1 Present address, Georgia Institute of Technology, Atlanta, Ga. 2 Present address, Ormangsgatan 61 1, Vallingsby, Stockholm, Sweden. I
The location of any point in the chromaticity diagram. IT hich allows specification of a color, is obtained by calculating its coordinates, P,, from the following formula
This iorniula is actually equivalent to three expressions usunlly written n ith other s j mbols and 1vhoi.e rclationqhip is denoted below.
In this way Equation 9 becomes
To change the quotient form of Equation 12 into a form consisting of sums of concentration tcrniq, a Taylor espansion u p to the second order in c is applied : 1
so - c s a + C2SP = + c -Sa- + s1 a 11 i Sa ’/2
P, =
P?
Placing Equation 13 in 12 and again neglecting tcrms higher in c than second order gives
=
P =
P , denotes a color point in the chromaticity diagram and, according to whether T becomes 1, 2, or 3, i t is the familiar poordinate 5 , y, or P, respectively.
This equation may be rewritten in the following form:
Combining Equation 1 with 4 yields T(X) =
(7)
e-cK(X)
Putting this into 3 gives
Equation 14a can be written in the follon-ing simple form,
Be( a u Equation ~ 7 contains the concentration variable in exponential form a n d is therefore difficult to manipulate, a Ta>lor elpansion is applied to give €’,
=
f+il--
2
CK +
‘/z
f p,(l - C K +
‘/z c2K2-
’To simplify the handling of the formula. all members higher than second order are neglected. This is also justified from a practical point of view, as the cmcentration of the indicator in an actual titration system, and hence the value of cK, is sufficiently low. For the qake of simplicity in further derivations the follon ing definitions are introduced :
s +,dX
(loa)
ffr
=
f +.,KdX
(lob)
rSr
=
S+X2dX
(10c)
sa
=
IT,
r =
1
3
r = l
- c’E~ - G, + Q9)
(15)
where the following definitions have been used
)dX
(9)
=
((27
c Z K 2- . . .)dX
r = 1
U T
Pp = G, - cE(Q, - G,)
(17)
It is now possible to attribute phSsical significance to most of these terms. I n definition 16, both numerator and denominator represent constants which are independent of the colored system and coordinates G, define the illuminant point ( 7 ) ,on the chromaticity diagram. Because the same coordinates are obtained in any sjstem where I( is independent of the wai-e length and such systems show only different depths of gray, this point has been designated as the graj point. G,. The gray point Coordinates may also rcsult for systems where K is dependent on the wave length in a special n-a?. Such systems will appear gray only with a particular illuminant and nil1 exhibit color when illuminated by a source exhibiting a different spectral distribution.
The quotient, Q,. in Equation 17 is a constant which specifies, in a concentration-independent ~ - n y the , color of a subtractive color system. According to the definition, this constant is related to the light absorbed b j the solution. Consequently point Q r will be located in the chromaticity diagram a t a place corresponding to the complementary color of the true color point, P,. The true color point, P,. is concentration-dependent and m o w s from the gray point, G, (at zero concentration) toward the periphery of the chromaticity diagram as the concentration increases. At lorn concentration, the path of the color point can be approximated by a straight line. At higher concentrations i t deviates from the straight line and this is equivalent to a color change. This phenomenon of dichromatism is well known. Because of this dichromatimi, the constant Q r allows complete specifications of the true color only a t low “color concentration”-i.e , a t small absoihance. The parameter Qf (together u i t h Q,) determines this drviation appro\imatelj and hence iq a measurr for the dichromatistic tendency of the color system. Because terms higher than second order in c were neglected in the derivation (Equation la), this parameter is able to correct for the dichromatistic deviation only u p through intermediate concmtration ranges. Because indicators are allvayq used in low concentrations, this limitation is not an aggravating factor Parameter E is defined in Equation 19. The numerator, Sa, contains Ii trrms, ~3 hereas the denominator, Sa,does not. Therefore the dimension of E is that of an “absorptivity” times sample path length and consequently thi. quantity may be called “effective” absorptivity times path length; the adjective “effective” denotes that the a( tual absorptivities have been n eightetl according to their effect on visual obserration of color. One must not confuse this t r r m with the luminosity in the CIE system. The luminosity is related only to t h r Y terms. whcrc:is S contains the sun1 X I’ Z of all tbree chromaticity intcgrals. An ins1,cction of Equation 13 s h o ~ s that the roordinatw of the true color point. P,. are quadratic functions of thc product, cE. this product being given the e> mho1 J according to the definition
+ +
J
= ch’
(20)
Becauce the Froduct of absorptivity, Fath length, and concentration is absorbanre, J may be considered as an effective absorbance, taking into account the absolute absorbance and the eye’s ability to detect it. Equation 15 can be greatly simplified by introducing the following definitions: VOL. 32, NO. 10, SEPTEMBER 1960
1219
V,
- G, v, + Q:‘
= Qr
w,=
(21)
I
I
I
I
I
I
(22)
‘
I
Putting these and 20 into 15 gives P , = G,
- JV, - J2W.
(23)
V , and V, can now be interpreted as vectors and mathematically handled as such. Equation 23 serves as simple and unified basis for the interpretation and calculation of color phenomena in rolutions (such as designation of color change, quality of end points, dichromatism, screening, and related factors). The important goals which were reached in that equation are: Only a single concentration-dependent term, J , is present. One quantity is completely independent of the color system, G,. Only two other quantities, V , and W,, remain and these “color constants” are sufficient to describe a dilute color system fully. EXPERIMENTAL DETERMINATION OF COLOR CONSTANTS
Remarkably, the only experimental data necessary to calculate all the parameters are a n absorption spectrum taken in the visible range. The analytical concentration of the colorant need not be known but in practice should be chosen to allow accurate absorption readings. Parameter J defined in Formula 20 can be calculated from a b s o r p tion data without knowledge of the analytical concentration. J , the “effective absorbance,” can be considered also as the “optical or color concentration” and the true color points, P,, can be calculated by Equation 23 in terms of J units. To calculate the color constants of Equation 23 several convenient quantities are introduced m-ith the help of Equation 4, so that the constants can be directly evaluated from absorption data. These quantities are presented below in the order in which they are needed for calculation of the color constants: 0 7
=
f *&A
(24)
Figure 1. Chromaticity diagram representing parameters describing dichromatism
02
0
tristimulus data (4,s). The 10 selected ordinates method (10) has proved to be sufficiently accurate for describing color properties of indicators. Xomographs sold by Bausch & Lomb (Catalog S o . 33-29-12) also allow rapid and sufficiently accurate calculation of this type of data. The numerical mlues of the three integrals given by Equation 24 are independent of the color system and are constants for any given method of tristimulus calculation and hence the magnitude needs to be determined only once. Equations 25 and 26 are obtained through combining 4 with 10b and lOc, respectively, and allow calculation of A, and B, directly from absorption spectra. The mode of practical calculation is identical with that of the CIE method, except that absorbance (or absorbance squared for Equation 26) readings are used instead of transmittance. The derivation of Formulas 27 to 29 is a simple summation of the 2, y, and i terms of the corresponding formula in the series 24 to 26. The numerical calculation proceeds accordingly. With these quantities and their numerical values at hand, the constants are easily obtained by Equation 30 to 33 : G
-
2’
- Sa
a
S a r x A , = -2.303 r=l r-1
(28)
Q:
=
J
The integrals in Formulas 24 to 26 may be calculated in a way analogous to any of the standard methods for obtaining
1220
06
ANALYTICAL CHEMISTRY
08
X
r=1
3
04
1 -2(S~~r - SUB,) 2s*
E cE =
S 2.303 A
SO
(32) (33)
From the numerical values of G,, QI,and Qf, the magnitude of the vectors, Ti,
and JVr, can be evaluated by using Equations 21 and 22. These data in combination with J values permit calculation of the color function (Equation 23). DICHROMATISM
The ~imp1c.t edimatr of the importance of the third term in Equation (23) can bc deriwd as follow^. Equation 23 ma: be writtrn as: P , = G , - ( I-,
+ WJ)J
The third term ( a n be neglected n-hen 1r-J
. availahle intlicators varies considci.ably and. for the indicators listcd ill Tablil 11. this value was obtaincti tlwough photometric titration with ruitable nwtal ions and ’or chelons. Onc particular batch of an indicator, foi, cxmiple, \va+s found to lie only 40% puw. Svvertheless. even this rather liniitc-ti tallle shows clearly the value of this somewhat obji,rativc approach to classifying the color quality of an indicator transition. SCREENING
Poor color changes in the end pointdesignated I)? low values of ICh and I.,r-liave for a long time been the subject of improvemerit b y screening. Thi-: ;)rocedure, however, has been a highly empirical undertaking, and often not entirelJ- successful, because i t is rather improbable that a single inert dye r i l l be fouiid having the necessary properties. Ailthoughthis prohlein can be overcome by the lice of two inert volorants, the trial-and-error procedure becomes even more involved. Therefore, it seemed worthwhile to eliminate the empirical nature of screening by establisliiiig a theoretical basis for the entire process. This is easily achieved through application of the color const.ants and the formulas derived above. The basis of esact screening consists simply in the addition OF selected inert colorants in amounts such as to obtain a “hueless” end point. The ideal situation is achicved \Then the solution at the end point coritains 110 color a t all. I n practice, lion-ever, only chromatic rolors can be cancelled coiiipletely and a certain aniount of achromatic color (gray) remains. Hence, a properly screened system has to appear gray a t tlie end point. End points occurring in certain color fields are distinctive, Y S they allow sensitive detection of appearance or disappearance of neighboring hues. To these special (chromatic) colors, the (achromatic) d o r gray must be appended. K h m m any chromatic colors neighbor only tn.o hues, gray neighbors all liuea, tliua offwing the possibility to detect seiisitively the disappearance or appcwancc of any hue. Furthermore, not only does a chromatic color disappear when the screened end point is reached, hut the color appearing beyond tlie end point is perfectly complc~mentaryto the first, thus enhancing the mcmory factor tremendously. As previously shown, tlie concentration-iiicl~pendrnt coordinates (normalized to unit concentration) of a color niirture are obtained b y summation of vectorsT. E for the individual species involved For mixtures coni-
-
posed of individual colorants in concentrations other than unity, the color coordinates of the mixtures are obtained b y summation of vectors V J of the individual species. Be+ cause vectors V J originate from the gray point and because in this scheme the coordinates of the gray point are (O,O), a misture of i colorants must fulfill the following condition to yield a perfect gray
-
n
20
-
Consider the screening of a n end point, which occurs when the mole fractions of the indicator form before and after i t are 1- p and q, respectively, and two screening dyes are used, designated by subscripts I and I1 [qVroJa
+ (1 -
+
+
VT,IJI V,IIJIII = 0 (64)
q)V,bJb
Equation 64 can be set up as two equations for the X and the Y term (1. = 1 and T = 2), and solved for J , and JII, yie1ding
line connecting the color point of the form before the end point (VJ and after the end point (V7,,J. It is important to realize that end point Ti, I does not lie exactly halfway between these two points because of the difference in the E values of the limiting forms. The figure contains also the complementary color points of five selected screening dyes. For choosing the screening dye pair, a straight line is drawn from V , , through the gray point. The color point of the screening dye or dye mixture must be located on this line (dashed part). None of the five screening dyes lies on this line and consequently a mixture of a suitable pair is required. Since the V , points of a mixture of two dyes must be located on a straight line connecting their individual T’, points, only pairs 5-2 or 1-2 could be used to screen this methjl red end point. -4 closer inrpection of the situation shows immediately that pair 1-2 14 superior to $2. Intersection A is further from the gray point than intersection B and hence the gray content of the mixture methyl red-1-2 is less than methyl red-5-2. Consequently, although the end point for both cases is.in the gray point (relative grayness in-
(65b)
If a pair from several dyes, thought to be suitable for a given screening purpose. is selected and their color constant inserted in Formulas 65, and 65b, either J r or JI1may be negative. This indicates conclusively that this particular pair will not give a gray end point. I n this n a y Formulas 65a and 65b can decide JThether or not a given pair of screening dyes can be used. T h e n J , or JIlis zero, the dye yielding the zero value need not be used, for the other dye alone is sufficient for complete screening. Although the formulas distinguish between possible and impossible pairs. they do not indicate which of several possible pairs is best. To approach this goal, the problem has to be divided into steps. First the color coordinates for the unscreened end point are calculated by the following formulas:
finite), the end point mith the latter system nil1 exhibit less absolute grayness and the limiting colors of thP screened indicator will therefore bc bnghter. This latter feature can bo demonstrated by connecting A and B n i t h T’, and Vb, respectively, thushowing that the lines leading to B are closer to the gray point. Now that the proper combination ichosen, Formulas 65, and 65b can btx applied to determine the re1a t’ive concentrations rrquired. Because the V, , coordinates have already bccn calculated, it is simpler to use the following equations, which are equivalent to, but qhorter than, 65a and 65b: Ji =
J,
J’l.CV2 11 Vl.IIV2.1
-
V*..Vl.II Vl.IVZ.11
(67aj
Placing these coordinates as well as V , points of the screening dyes in the chromaticity diagram simplifies the choice of a n optimum screming dye pair.
From the J values calculated by 67 a-c the proper amounts of indicator and the t n o screening dyes are then mixed to give the screened indicator system. According to Formula 33, the optical concentration of a solution can be directly obtained from absorption data and expressed in J units. This is a very suitable way and stock solutions of indicators and screening dyes can be labeled in these concentration units.
Consider, for example, the screening of methyl red in Figure 12. Here the desired screening was to take place at q = 0.5. Point V, e lies on the straight
Figure 13 shows the spectra of four dyes selected for screening purposes and Figure 14 shows the spectra of methyl red in its acid and base forms. VOL. 32, NO. 10, SEPTEMBER 1960
1229
Wave
length (A)
Figure 13. Absorption spectra of four selected screening dyes. Numbers in parentheses refer to color points in Figure 1 2
X
Figure 1 2. Chromaticity diagram illustrating principle of screening Absorption spectra of screening dyes (1, 2, 3, and 4) given in Figure 13
A milture of methyl red-dye 1-dye 2 in the proper ratios was prepared according to the calculation above. From the known data the absorption curve of thc mixture at the gray end point m s calculated and the result is given as the solid line in Figure 14. Kext the milture was used as indicator and a titration performed until a n end point was reached which appeared gray. An absorption curve of this solution is sho\vn as a dashed line in Figure 14. The agreement between the calculated eurvc and the experimental curve is obvious. The titration apparently was not performed to the eaact end point. This may be attributed to the use of a n illumiriarit n.hose spectral distribution 11as slightly different from that of C I E illuminant C. Also near the end point the eye has some difficulty in detccting the traces of a color change and the titration was stopped with the acid form slightly in excess of the theoretical amount. This can easily be seen by comparison n i t h the two other curves in the diagram, which represent the absorption curves of the acid and base forms of the screened indicator. The absorption curve of the gray screened indicator a t the end point iq not a horizontal line. T h a t means the obtained achromatic color is not a n illuminant-independent gray. The mixture will appear gray to the eye of the average observer only if illuminated with illuminant C (according to C I E specifications) or a n illuminant having a very similar spectral distribution. With any other illuminant a chromatic color may be exhibited according to the spectral distribution of that particular illuminant. 1230
ANALYTICAL CHEMISTRY
Alternatively the screening could be accomplished in such a n a y that the gaps in the absorption curve for the indicator a t p = 0.5 are filled, thus yielding a nearly uniform horizontal absorption line. This procedure results in an illuminant-independent gray. However, this procedure, beside being very tedious practically, n ould lead to a mixture having a higher amount of grayness, thereby obscuring considerably the chromatic colors occurring in this region. The four selected screening dyes were chosen from a study of a large number of commercially available dye samples. I n selecting screening dyes the following important points were considered. The dyes should be bright; this means that their gray content should be very low or, in other words, their V , points should be located near the periphery of the chromaticity diagram. Otherwise the screened mixture will have a high amount of gray. Further, the dyes must not take part in any chemical reaction nhieh nould result in a color change. The dyes should not fluoresce, for othern ise an incorrect absorption curve would be obtained giving false color constants for USE in the above formulas. Although the dyes sclected are the best ones found from a series of available sample$, the study was not exhaustive. Further investigations may discover other colorants superior to these. The four dyes named in Figure 13 are sufficient to screen any indicator because the “screening square’’ surrounds the end point completely and the connecting lines of the square are well removed from the gray point.
Since, in the transition of an acidbase indicator, p is directly related to the p H of the end point, a more exact location of the end point to a desired y H value can be achieved via screening. This is especially important in the titration of weak acids or bases and for dilute solutions. For maGmum benefit of screening i t is not advisable to proceed too far to one or the other limiting color or in other words to establish q or 1 - p too close to 1, where dq d nil. is small. I n such a caSe any chromatic color just before or after the end point noould be very weak and obscured b y the gray content. The usual practice in chelometric titrations is to establish the end point as the point where the indicator is present almost completely in its unmetallized form or, in case of some backtitration procedures, in its metallized form. The error introduced by this choice of end point is usually negligible. I n the titration of magnesium with EDTA using Eriochronie Black T as indicator, the ~ K of E the indicator and the p h l of the end point are identical, indicating that the correct end point R ould be actually midway in the transition of Eriochrome Black T . However, i t is practically impossible (for reasons discussed above) to establish a n exact end point as a certain shade of violet. With a screened indicator any point between the limiting colors can be chosen as end point, thus offering the possibility of terminating the titration a t the equivalence point. On the other hand, with other p K and p h l relationships i t may be perfectly correct to titrate until the indicator is nearly completely, or a t lead for the observer’s eye completely, in the unmetallized form. I n such a case, screening to the mid-point would introduce a small systematic error. Bccause the indicator concentration is much smaller than the concentration
,
0 ,4 -,
,
,
,
,
,
,
,
,
l
400
/
5 00
Figure 15. I
I
400
I
I
I
I
!
I
700
600 Wave length (h)
Screening of methyl red
. ...
I
length
,
~
l
700
(X)
Screening of copper-PAN end point
x.
500
Figure 14.
l
b-.L.ll
I
Upper. Absorption spectra of metollired ond nonmetallized indicator lower. __ calculated absorption spectrum for screened end point absorption spectrum of screened indicator actually titrated to best observable neutral color
__-
Absorption curves of acid and base forms of methyl r e d lower. . acid form of screened indicator bore farm of screened indicator -- calculated absorption curve of screened indicator at theoretital end point (half previous concentration) absorption curve of screened indicator actually titrated to best observable neutral color Upper,
'
6 00
Wave
01
I
____
of the iiietal titrated and only a part of the indicator will be left in the metallized form in such a procedure, this ('rror is usually negligible. Furthermore, this small systematic error will bc ba1:inced by a decrease in the usually 1argt.r nunsystematic error caused by difficulties in locating the end point \:-ith an unscreened indicator, Titration to complete conversion of the indicator t o one limiting form eliminate$ casy detcction of overshooting the end point :tiid evaluation of the degree of overshoot.. I n scrwning indicators for chelometric titrations, another important' fact must be considered. Unlike acidb n h c tit,rationswhere the same speciesnanic~ly. H + ions--is titrated, regardl t w of what the acid or base may be, the titrated species in a chelometric~ titration changes from metal to metal. Furthc~rniorc,the same metal may be titrwtcd under diff(,rent ronditiona; thus rquiring a specially screened indicator for each case. For example, the sliadr of the violet of the zincErioi.lrronic~ Mack T complex depends upon the amount of ammonia present; as anothcr example, the color of free pyroc.:itechol violet depends highly upon the p H of the solution. Other examples can be added a d libitum. Therefore in general screening for chelometric titration will be effective only
for a particular metal and for this metal only under particular conditions. Furthermore, during the titration Lrith a screened indicator, the titration conditions have to be established and maintained more rigidly. As an examplr of an EDTA titration inr.olving a scrpened indicator, the c o p p ( ~ - P A Ksystem is presented in Figure 15. The scrcening mixture was PAS-dyc 2-dye 3 and q was established again as 0.5. The curve calculatcd from t,he concentrations of the mixture at the gray point is the solid line. The dashed line is the experimental spectral curvc of a copper solut'ion titrated to the gray end point. Comparing these two curves and taking into arcount the absorption curves of the pure limiting colors lead to the conclusion that' this end point \\-as overshot to a small extent, because the red is l o w r than t h t yellow. The agreement ehon-s clearly the effectiveness of the scrccning calculation. SAMPLE CALCULATION
T o summarize the use of the equations derived in this article, consider the example of the violet dye 3 (CI 697) whose absorption spectrum is gircn in Figure 13. For C I E illuminant C and using the 10 selected ordinates method ( I O ) , the following values result:
.ibsorbance a t Selectcd Ordinates 3
2
1
0 010 0 020 0 398 0,392 0 34.5 0 328 0 284 0 180 0 090
0.073 0 215 0.325 0.394
0 009 0 007 0 009
0 394
0 01 1
0 016 0 021 338 0 027 278 0 043 0 026 084 0_ 092_ 2 033 2 804 0 241 ~ _ _ 0,09804 0,10000 0.11812 0.1993 0.2804 0.0286 ( = AI) ( = i l l ) ( = -1,)
Sum Factors Sum X factors
(Formula
0 0 0 0 0
0.010
356
34i
25) u* =
0.9804 1.0000
=
1.1812
~1
~3
=
S, = 3.1616
G2 = !?
so
=
0.316
GB = 2
=
0.374
Check:
ZG, = 1.000
su
(Formula loa-c) (Formula 1 l a )
(Formula 1 6 )
VOL. 32, NO. 10, SEPTEMBER 1960
1231
Q1
A &
=
=
Transmittance at Selected Ordinates
0.392
A Q? = 2 = 0.552 SA A Qa = 3; = 0.056
1
(Formula 17)
0 978
1
+ +
‘‘
1
(1.5464 X 2-XT5.082 X lo-’ X 9.804 X lo-’ - 31.6144X 10-l X 0.6189 X 10-I) , = -0.085 Q$ = -0.260 (Formula 32) Q3” = S0.345 Check: ZQI = 0.000 =
0.5082 2.303 X __3.1616
F
0.302 - 0.310 = T7g = 0.552 - 0.316 = V B= 0.056 - 0.374 Check: ET,’ = 0.000 T’i
=
5
(Formula 33)
+ 0.082 + 0.236 - 0.318
(Formula 21) 61-1 = 0.082
- 0.085 = - 0.003 - 0.260 = - 0.024
0.236 FT’? -0.318 0.345 = Check: ZW, = 0.000 TT-2
= =
+
+ 0.027 (Formula 22)
Grayness = 0.590 (periphery value found by plotting Q p into chromaticity diagram, drawing a line through G, and Qr,and reading) 1 0.590 - 0.552 = = 3.i~i.16x 0.500 0.552 - 0.316 0.09 (Formula 61) 0.316 0.78 g = o,590 -~ X ---- = 0.09 (Formula 62) 4.50
QZ
True Color Point P I = GI - J V i - JzWi = 0.310 - 0.370 x 0.082 (0.370)e X 0.003 = 0.280 Pz = 0.232 (Formula 23) P , = 0.488 Check: Z P , = 1.000
+
Check of True Color Point. Convert absorbance data of Table I into transmittance to yield for the 10 selected ordinate method:
1232
ANALYTICAL CHEMISTRY
17,’
+~236 X 85 _ 82’ + 236z
--82 X~260
=
-o.020 _
_
(Formula 36) NOMENCLATURE
The formula number where the symbol is defined or used for the first time is given in parenthew.
A,
Q:
qi T
Sa Sp
S, SA
SB 5”
r-
-*
9.69 x IO-’ F X J = 9.69 X lo-’ X 3.70 X lo-’ = 0.036 (Note: 0.036
