August, 1963
REACTION OF H ATOMS WITH OH-
The effect of water is large enough that the end point becomes quite indistinct above 6 M mater in titrations with 0.43 M sodium acetate. Since water loi?-ers the potentials of acid solutions more than it raises those of basic solutions, the potential a t the equivalence point is lowered somewhat. Several of the titrations with 0.43 M sodium aceta,te are shown in Fig. 5 to illustrate these points. Figures 3 and 4 illustrate the increased experimental error found on working a t very low water concentrations, and in the driest solutions the acid error is more persistent. Soaking the electrodes in moist acetic acid solutions, as was done in this work, is quite satisfactory; however, the water content of this solution should not be much greater than in the solutions to be measured. Soaking for short periods in water should probably be avoided, but prolonged soaking in water appears3 to be the best treatment whenever any acid error has been produced. To avoid prolonged exposure of the glass electrode to acids it is probably best to add acid to base in potentiometric titrations, but since HCI is easily lost from solution this was not practicable in this work. For analytical purposes some compromise must be reached between the problems due to acid error which are worst in drier solutions and the effect of
IN
RADIATION CHEMISTRY
1609
water in reducing the sharpness of the end point. Some value in the range 0.4 to 1.5 M water appears to be best for routine work. Summary.-Potential measurements with the glass electrode in glacial acetic acid can be readily obtained accurate to within A 5 mv. Still higher accuracy is presumably obtainable in “basic” sodium acetate solutions, but when an electrode has been exposed to dry HC1 solutions it will show some “acid error” until all the Si-C1 bonds have been hydrolyzed. These errors are negative in sign in dry acid solutions, but they become positive at higher concentrations of water and in sodium acetate solutions. This type of error is presumably responsible for the measurably low slope of E us. pH found by Izmailov and Aleksandrova.2 The errors are ordinarily small, under 20 mv., but by prolonged standing in dry acid solutions and appropriate manipulation of the water concentration much larger errors can be introduced. These effects are understandable in terms of a diffusion potential added to the normal hydrogen electrode response. The possibility of such acid errors should be considered along with the effect of water on the activities of acids and bases in choosing appropriate titration conditions as described above.
THE REACTION OF H ATOMS WITH OH- I X THE RADIATION CHEMISTRY OF AQUEOUS SOLUTIOIVS BY SHLOMO NEHARIAKD JOSEPH RABANI~ Department of Physical Chemistry, The Hebrew University, Jerusalem, Israel Received Recember 21, 1962 Using 200-kvp. unfiltered X-rays, the effect of pH on reduction and dehydrogenation reactions of H atoms was studied. It is concluded that the reducing radicals in water radiolysis include H atoms with a yield of about 0.48 & 0.05. This yield is independent, of pH between 2 and 13. In the alkaline solutions, H atoms are converted into a species which reacts with scavengers in the same manner as radiation produced eaq-. The reactivity of H atoms with nitrate, acetate, acetone, OH-, and bicarbonate was studied.
Introduction Hydrogen atoms generated in the gas phase2 by an electric discharge and introduced into basic aqueous solutions of chl~roacetate~ appear to react with OH-, according to reaction
H
+ OH- +HzO- (or eaq-)
(1) The product of reaction 1 reacts in basic solutions with chloroacetate ions yielding chloride, while H atoms in neutral solutions react mainly via dehydrogenation. 3 a , 4 , 6 The purpose of this work is to provide further evidence that hydrogen atoms as well as electrons are formed in the radiation chemistry of neutral and alkaline water and to yield independent support for reaction 1. This was accomplished by studying the effects of OH- ions in the reactions of H atonis formed in irradiated aqueous solutions. The primary action of X-rays on aqueous solutions (1) Chemistry Division, Arsonne National Laboratory, Argonne, Illinois. (2) G. Crapski and G. Stein, J . Phys. Chem., 68, 850 (1959). (3) (a) J. Jortner and J. Rabani, ibzd., 66, 2081 (1962); (b) J . A m . Chem. Soc., 83,4868 (1981). (4) E. Hayon and A. 0. Allen, J . Phys. Chem., 65, 2181 (1961). ( 5 ) E. Hayon and 3. Weiss, Proc. Intern. Conf. Peaceful Vses At. Energy, as, 80 (1958).
may be presented schematically
HzO --+- eaq-, H a s + , H, OH, HzOZ,HZ (2) The small primary yield of hydrogen atoms (GH E 0.5)6-8is responsible for the dehydrogenation of many organic solutes in neutral solutions according to
H
+ R H +Hz + R
(3) In the presence of both R H and a solute which scavenges all the eaq- without yielding Hz,the hydrogen yield in neutral aqueous solutions will be G H ~ GH. It is to be noted that RH also scavenges all OH. In this work the effects of OH- on the yield of hydrogen were investigated using acetone or NO3- as electron scavengers and formate, acetate, or acetone as H atom scavengers.
+
Experimental The experimental procedure was similar t o that described previou~ly.~The hydrogen produced was collected and its pressure was determined by a thermal conductivity method. ( 6 ) J. T. Allan and G. Scholes, Nature, 187, 218 (1960). (7) J. Rabani and G. Stein, J . Chem. Phys., 97, 1865 (1962). (8) J. Rabani, J . A m . Chem. Soc.. 84, 868 (1962). (9) G. Czapski, J . Rabani, and G. Stein, Trans. Faradag SOC.,58, 2160 (1962).
1610 I
I
I .oo
I
I
I3
14
A
-
N
I, 0.75 c3
0.50 12
II
PH.
E'ig. l.-TJie effect of pH on G ( H a )in solutions containing sodium formate Af) and acetone (2.6 X 1 0 F AI).
0.9
v
0.0
-.
0.7
N
3 0
0.6
0.5
0.4
I
0
I
I
I
I
IO
20
30
40
1
x io3 M .
[HCO;]
Results Solutions Containing Formate and Acetone.--'l'he radiation chemistry of formic acid and formate has been investigated e x t e n s i ~ e l y . ~ The . ~ ~ formate anion reacts with H atonislOdforming H2 witaha rate c o n s t a ~ i of t~~~~ (ICH + H C O , - / ~ I I + 0 , ) 1 ~1.3 X 10-2. The rate constant of eaq- with formate is lower by a factor of a t least lo3than that of ea,,- with a c ~ t o n e . ~At the conccntrations of formate and acetone employed by us, all the ens- react with the acetone to reduce it. It will be shown later that the reaction of H a t o m with acetone may be neglected. Using M sodium formate and 2.6 X M acetone (without buffer) the experimental yield of hydrogen, G(H2), is 0.95 a t pH 7. This value is in fair agreement with that obtained previously7 in the isopropyl alcohol-acetone aqueous system. As the p1-I increases above 11, the hydrogen yield decreases rapidly (Fig. 1) to a value of G(&) 0.50 a t 0.1-0.5 dl hydroxide. (LiOH was used a t the concentration of 0.5 il/ to minimize possible changes in the absorption oi radiation.) This change in C(I1,) with pH is due to the competition of reaction 1 with reaction 3. The species produced in reaction 1 reacts with acetone without hydrogen evolution. Tlie limiting value of C(H2) 0.50 which is reached at the higher pH values is the molecular yield of hydrogen, GH*. The difference between the two extreme values of G(H2) should equal GH in agreement with the value of 0.5 h 0.1 reported previously.*-8 In Fig. 2 it is seen that a t pH 13 G(H2) iricrcases with the formate concentration from G(H2) = G1j2 = 0.40 in the absence of formate to G(I12) = 0.87 with 4.1 X M formate. The competition between reactions 1 and 3 leads to the expression (all electrons being sca\-cnged by acetone)
Fig. 2.-The effect of sodium formate concentration on G( 111) in solutions containing acetone (2.6 x M )at p€I 13.0. li,
[RH]
+
If G(H2) is plotted us. ([HC02-]/ [OH-]) [GI-I, GH G(H.I)],a straight line should be obtaincd with GII*as intercept and slope k3/kl. The analysis of the results in Fig. 1 and 2 taking GII, Gl1 = 0.95 gives (Fig. 3) h / k 1 = 11, GH, = 0.47 and GH = 0.48. Competitionfor H Atoms between Acetone and OH-. a. The Reaction of Acetone with H Atoms.--In previous worklodit was shown that acetone is dehydrogenated by H atoms
+
H 0
0.01
0.02
0 03 (GH,
0.04
0.05
t GH-G(H*)),
Vig. 3.--G(Hz) as a function of [ H C 0 2 - ] / [ O € I - in ] 2.6 X acetone: A, results of Pig. 1; 0 , results of Fig. 2.
Jf
The conductivity tube wm kept at constant temperature ( i c e watc,r) in order to in(-rrase the tirrursvy of the measurements. 17nfiltered X-rays (200 kvp.) were used. l'he close rate, 1000 ritds nih-1, w,w detmmined using the Fricke dosimeter: d l FcS04 i n 0.1 V I-trSOr,takingGF,I- = 14.5. For most of thcl cvperuiients SiLOII \yas used to adjust ttic 1115, here dcfincd by 10p" = 10" [OH-]. The espcrinrcntttl G values reported are the averitge of :tt lcwst two determinations. Tlie gaa volume above the solution (30 cc.) was about 100 cc.
+ CH3COCHs +Hz + CHaCOCH3
(5)
reaction 5 being relatively s ~ o w - .In~ ~order to determine the value of kg, competition experiments with CuSO4were carried out in 0.1 N HzS04. Cu++was added as it was foundL3to react with H atoms, being reduced to the Cu+ ion, with a rate constant of'l ~ I - I + c ~ ~ s o , / ~ H + o ~ = 3.2 x 10-3. (10) (a) I). Srnithirs and IC. .J. Hart, J. A m . Chem. Soc., 84, 4775 (19GO); (1)) E.J. Hart, i b i d . , 73,68 (1931); (c) 7 6 , 4198 (1054); (d) H. Fricke, E. .J. Ilart, and €1. P. Sniitli. J. Chem. Z'hye., 6,229 (1938); (e) T.J. IIardwick, Radiation Res., l a , B (1960). (11) J. Rebani, .I. I'hj~s. Chem., 66, 361 (1962). (12) I n this pnprr rate constitntr Iirc givcn relativc to kit + 0 2 t o e n a l h
iiireot comparison witti previous rcsults.a.7.11 (13) J. 11. Bavendale and 1). H. Sniithics, %. physik. Chem. ( I h n k f u r t ) , 7,242 (1956).
REACTION OF H ATOMS WITH OH-
August, 1963 cu++
+ €1 + c u + + I-I+
(6)
Using 2.6 X i l l acetone in 0.1 N &So4 (total dose, 1000 rads) a value of G(H2) = 2.8 was obtained. This value is considerably lower than the sum7 G,GH G1j2. This is due to the competition of reaction 7 and, as will be shown later, of reaction 8 with reactions 5 and 9
+
+
+ CII3COCH3 +(CH3COCH3)H + C€I+2OCH3 +CH3COHCH3 e-- + €Iaq++H
enQ-
(7)
IN
RADIATION CHEMISTRY
1611
By the use of cq. 11 me get kl/k6 = 12. Using this value it is possible to calculatc kH+IICOn-/kl = (kS/kl X ks/ke X ~ I I + I I C O ~ - / ~l 1G which agrees with the ratio krT+IIr~Ol-/kl mcasured by direct conipetition of formate and OH- for the H atoms. Competition for H Atoms between Acetate, Nitrate, and OH-.-Expcriments in which acetate and nitrate werc used as €I atom and e,,,- scavcngers, respectively, were carried out. I n these experiments, all eSq- in the bulk probably react with NO3-
(8) (9)
In the presence of both acetone and CuS04, neglecting the reaction of CuSO4 with earl- we obtain
G, -
without forming Hz. H atoms may rcact with acetate to form HS according to rcaction 3 and with nitrate to form reduction products. TABLE I IIYI,RO~JCN YIEI.I)S I N SOLUTIONS O F ACETATE-NITRATEAT VARIOUSpH VALUES Total doso 4000 rads [ S a wetate] = 0.1 M ; [NrrNOJ =
1
The ratio k7/kg = 0.5 is known7 for 0.1 N HzS04. , GH = 0.5 and Ge- = 3.0 for 0.1 N HzS04, Taking G ~ I = k8/k6 = 0.3 is obtained from the experiments in the abscnce of CuS04 according to eq. 10. Values of G(Hz) = 2.3, 1.8, and 1.25 wcrc obtained in solutions 3 X and M CuSO4, containing 1 X rcspectively. Defining A = 1 167[CH3COCH3]/ kg[H+] plot of (GI< Ge-/A)/(G(HZ) - GHJ US. [CuS04]/[CH3COCH3]should give a straight linc with (1 4- k8/lc6)as an intercept and k6/k6 as a slope. Thc analysis of these results shows an agrcemcnt with k8/k6 = 0.3 and k 6 / k 6 = 70. Reaction 8 is also supported by Ab isopropyl alcohol to a the result that addition of solution of 2.6 X A I acctonc in 0.1 N H2S04 increases G(H2) from 2.8 in the absence of isopropyl alcohol to 3.5. In these experiments H atoms react with isopropyl alcohol6 according to reaction 3, but do riot react with acctonc, while thc earl- are shared betwccn Hag+and acetonc. Thus the increase in G(H2) due to isopropyl alcohol should be equal to G, the yield of rcaction 8 when acetone only is present. From this result the value k8/k6 = 0.3 is obtained again. b. The Effect of Varying pH on G(H2) in Acetone M Solutions.-In neutral solutions of 2.6 X acetonc, S(H2) = 0.87 (total dosc 3000 rads). This value is lower by 0.08 than that obtained in the presence of M formate, because of reaction 8. Assuming GH, = 0.47 and GII = 0.48 as in the formatc experiments, ks/ks = 0.2 in agreemcnt, within experimental error, with the value found in thc acid solutions. Adding and lo-' 111 NaOH, G(Hz) decreased to 0.79 and 0.49, respectively, showing again competition of reaction 1 with reactions 5 and 8. The equation derived to express this compctition is
+
+
[5 x
Seutral 10-2nr NLlIICOIl
SO;i-
0.86 74 .73 .65 .62 .54
11 0 11.3
2 3 4 5 6
11 c, 12 0 13.0
+ 11
--f
M
G(1Iz)
PIT
1:XlJt.
SO2
+ OH-
(01'
ISOJI-)
ki/ka
..
90 50 60
50
..
(12)
The effect of pH on G(H2) in acctatc-nitrate aqucous solutions is shown in Table I. G(Hz) in experiment 1 is determined by thc compctition of reactions 3 and 12 neglecting rcaction 1. From this experiment k d k 3 = 2,; can be calculated according to the procedure employcd previously. In a solution containing Ai' NaKOa without RH, we found G(Hz) = G112 = 0.52. This value and QII = 0.48 were used in thc calculations. The calculated valucs of kl/k3 at pH 11, accordipg to cq. 13
+
/i3
[acctate]
arc givcn in thc lust column of Table I. All thc values agrce within a factor of 2. Since
are k i i ~ w n it, ~is~possible ~~ to show that k3/kl in the acetate system is consistent with the results obtained in the formate and acctone systems. The Reactivity of Bicarbonate toward H Atoms.The rate constant of the reaction of bicarbonate with H atoms (probably forming an intermediate addition product) was found by studying the competition between bicarbonate and methanol. Thc results are shown in Table 11. ( 1 4 ) J . .Jortner, M. Ottolonghi, J. Rabani, and G. Stein, J . Chem. Phya.. 37, 2488 (1962).
SHLOMO XEHARI AND JOSEPH RABANI
1612
"01. 67
TABLE I1 assuming that the scavenging of eaq- by acetone or biG( H2) IS SOLUTIOKS CONTAIXING BICARBOXATE AKD METHASOL carbonate is the factor determining G(H2) in these solutions. Dose 4000 rads, [NaHCOa] = 5 X Af ~ + C H ~ O H
[CHaOHl, M
kH+NsHCOa
........
3.0 x 1.0 x 3.0 x 1.0 x 1.0 x
0.67 .78 .81 .91 .99 1.03
10-4 10-3 10-3 10-2 10-1
... 150 60 50 60
...
Discussion GH. -The present investigation provides further evidence for a small primary product yield of hydrogen (GH &% 0.5). kl is independent of the source of hydrogen atoms (X-rays or gas phase generated hydrogen atoms).15 This supports the idea that GH is the yield for primarily produced H atoms. The value GH G 0.5 was previously found7 constant from pH E 2 to pH G 8. This range is now extended up to pH 13. On the Possible Source of GH.-Theoretical consideration@ as well as experimental e v i d e n ~ e ~ - ~show ~'~~~' that the reducing radicals formed primarily by the action of X-rays on water and dilute aqueous solutions are mainly electrons. Ge,,- is about 85% of the total yield of the reducing radicals.6-s It was found previouslyiss and confirmed here that the measured GH is not due to any reaction of eaq- in the bulk, since with electron scavengers the limiting GE is found. In the tracks, two possible reactions which may yield hydrogen atoms are
The value of G(H2) = 0.67 in 0.05 M XaHCO3 mithout methanol seems to be higher than the molecular hydrogen yield, GHz. This is probably due to the low rate of the reaction of bicarbonate with H atoms, so t,hat recombination of H atoms may occur to some extent. I n 0.1 X methanol G(H2) is interpreted as equal to (GH* GH). Assuming as previously that GH = 0.48 and neglecting H atom recombination in the methanol solutions, kH+methanol/JCH+bicalonate could be calculated (last column of Table 11). The higher value found in 3.0 X lo-* 114 methanol may be due to the neglect of H atom recombination, which probably occurs a t this low scavenger concentratrack tion. (14)18 cas- H30+ -+ H From the results of Table 11, taking ~ H + C H ~ O H / ~ H + O ~ = 8.5 X lop5 l1 it is found that kH+bicarbonate/kH+Oz = and6 1.5 X This value is also confirmed by experiments tvit'h 5 X M sodium acetahe a'nd 5 X HzO* +H OH (15) lon2 f W sodium bicarbonate. At 4000 rads, G(H2) = If GH is G14, it should depend on the concentration of 0.87 was found. From this one may obtain kII+bicarbonate/ eaq- scavengers as well as on the [OH-]. In our experik H + o , = 1.0 x lo-'. ments no such effect has been observed. Further work Reactions with eaq-,-It is knownlOd that the hydrowith pulsed radiolysis techniques is in progress on the gen yield in neutral solutions of RH is in many cases rates of homogeneous reactions of eaq- and the possible GK. This excess yield higher than the sum of Gsz species which are originally formed in the tracks. seems to be due to reactions of eaq-, since electron The Reaction of H with OH-.-At the higher pH scavengers such as acetone,'j ferri~yanide,',~ and bicarvalues, when [OH-] is sufficient to compete with other bonate8 decrease G(H2) in such solutions. The actual solutes for the hydrogen at'onis, these atoms are conmechanism by which eaq- is converted to HZis not verted to a species which reacts with electron scavengers known. This may be by recombination of eaq- with and does not dehydrogenate the organic solutes RH. itself or by its reaction with water. The determined k1 is independent of the system used TABLE I11 (acetone, formate-acetone, acetate-nitrate, chloroG(H2) IN SODIUMFORMATE SOLUTIONS.EFFECTOF ACETONE acetate3), and therefore we conclude that the effect of A N D BICARBONATE pH on G(H2) is due to reaction 1 and not to any specific Dose 1000 rads, [formate] = 1.0 X 10-3 M in lo-* M Na3P0 pH dependent secondary reactions of the solutes. buffer The radical species formed by reaction 1 behave in a [Acetone], 'iA [Bicarbonate], M G(Hz) Initial pH similar manner to radiat,ion produced eaq-. Both species ........ ........ 1.58 8.5 react with acetone and with nitrate without forming HZ ........ 1 . 0 x 10-4 1.62 Kot determined and with chloroacet,ate forming C1- rather than H2.3-6 1 . 0 x 10-6 . 1.30 Not determined I n bot'li cases no evidence has been found for direct de........ 1.0 x 10-3 1.27 7.6 hydrogenation of RH.i9 It is tempting to suggest that 5 . 0 x 10-5 ........ 1.18 8.2
+
+
+
+
, , ,
........
5.0
, , , ,
x
10-8
1.17
8.5
I n Table I11 the effect of acetone on G(H?)in loF3M formate solution is compared to the effect of sodium bicarbonate. During the irradiation the pH was increased by about 0.5 p H unit. The results show that in order to obtain the same decrease in G(Hz), [HC03-] should be about 100 times [acetone]. These experinients do not exclude the possibility t,hat eaq- is scavenged by the COZpresented in equilibrium with the HCOZ- a t this pH. (ke-+coZ/ Jc, -+HIO t G. Scholes, private communication.) A value of he -+acetone/ke-+bicarbonate 2 100 iS obtained
(15) I n the case of gas phase generated hydrogen atoms, the choice of parameters (G. Czapski, J. Jortner, and G. Stein, J . Phya. Chem., 66, €464 (1961)) leads to a value of ka+oZin agreement with the radiation chemical value. Although the "absolute" value for k ~ + is 0 ~not certain, relative rate constants for H atoms obtained with radiation can be compared t o those obtained with gas phase generated H atoms. (16) (a) H. Frohlich and R. L. Platzman, Phys. Rea., 92, 1152 (1953); (b) G. Stein, Discussions Faraday Soc., i2, 227, 289 (1952); (c) J. Weiss, Xature, 186, 751 (1960). (17) (a) N. F. Barr and A. 0. Allen, J . Phys. Chem., 68, 928 (1959); (b) J. T. Sworski, J . Am. Chem. floc., 76, 4687 (1954); (0) F. S. Dainton and D. B. Peterson, Nature, 186, 878 (1960); (d) G. Dobson and G. Hughes, Trans. Faraday Soc., 67, 1117 (1961); (e) G. Czapski and A. 0. Allen, J . P h y s . Chem.. 66, 262 (1962); ( f ) G. Czapski and H. A. Schaarz, ibid., 66, 471 (1962); (g) 8. R.Anderson and E. J. Hart, ibid., 65, 804 (1961). (18) C. Lifshitz. Can. J . Chem., 40, 1903 (1962). (19) J. H. Baxendale and G. Hughes, 2. physik. Chem. (Frankfurt), 14, 30B (1958).
SPECTROPHOTOMETRIC ANALYSIS OF REACTION A~XTURES
August, 1963
the intermediate formed in reaction 1 is identical with eaq- formed by the primary action of radiation on water. On the Origin of GH,.-H~O~,~~ NOa-, and OH- have about the same reactivity for H atoms. If the precursors of GH* were :K atoms, the effect of these three ~ be the same. However, the experisolutes on G H would mental results show that OH- has a negligible effect compared with tha,t of HzOz and NOs-, while the last (20) G. Czapski, J. Jortner, and G. Stein, J . Phys. Chem., 66, 964 (1961).
1613
two substances have similar effects on G H ~and ~ ~ ~ - ~ similar reactivity for eaq-.14317e This is consistent with the view that ea,-, and not H atoms, is the main precursor of GH,. Acknowledgments.-The authors wish to thank Drs. E. J. Hart, M. S. Matheson, G. Scholes, and Prof. G. Stein for valuable comments. (21) (a) H. A. Mahlman and J. W. Boyle, J . Chem. Phys., 27, 1434 (1957); (b) J. A. Ghormley and C. J. Hochanadel, Radiation Res., 3, 227 (1955).
SPECTROPHOTOMETRIC ANALYSIS OF REACTIOK MIXTURES. 111 BY S. AIRTSWORTH Department of Biochemistry, University of Shefield, She ffield,England Received December 68, 1966
A method is described whereby the absorption of the kth component in a reaction mixture may be evaluated without prior knowledge of the spectra of the other components. The method requires that among the nm optical density readings that relate t o n states of the system and m wave lengths there shall be either k - 1 states where the kth component is absent or IC - 1 wave lengths where i t does not absorb light. These data are then used with optical density readings to which the kth component makes a contribution to evaluate its absorption either as a function of the state of the system or of the wave length. The results obtained may be used in the further analysis of the system.
Introduction I n the previous paper of this series2 a method was described to give the number of absorbing species in a reaction mixture .using spectrophotometric data in matrix form. The method also gives information concerning the possible interrelationships among the components of the mixture. Prior knowledge of the spect,ra of the pure componlents is not required. This paper will describe an extension of the method whereby, if the required experimental criteria are met, the absorption of one or more of the components may be calculated, both. as a function of the state of the system and of the wave length. Calculation of Absorption.-In what follows, the optical density of the reaction mixture will be represented as the sum ad products of two parameters x and y, one of which is a function of the wave length and the other a function of the state of the system associated, respectively, with i and j variations but not otherwise identified. Thus n
dij
=
k=l
(1)
XikYjk
where ZikYjk is the optical density of the kth component in the mixture defined by i and j and n is the total number of components. For ij variations, the ij optical density readings may be set out as an absorbance matrix Aij
!/dl1dzl . . . dill\
11.. .. . . . . . . . . (2) j/dij d2j . . . dijj/ which is the product, taken column by row of the matrices Aij
I or
. . . Xiill . . . %in
~ 1 ~1 2 .1
Xln X2n
=
IIy11 ylz . . . Y j l yj2
yln
. . . yjn
1
(3)
( 1 ) This work was supported in part by tlie iltornic Energy Commission. (2) s. .4insworOh, J . Phas. Chem., 66, 1968 (1961).
Aij
= XY
Assuming that the initial concentrations of reactants are arbitrary, a reaction system with n components has an absorbance matrix with rank n det A,, # 0 and it can easily be shown that det An,
=
det Y X i k X l k
+ det YXzkXzk . . . -k det
YXnkXnk
(4) where X n k is the coefficient of Xnk in det x. It is possible to suppose, however, that there exist conditions where the kth component does not contribute to the total absorption of the mixture. Such a situation might arise at the beginning or towards the end of the reaction when the concentration of k is either zero or sufficiently small to be negligible. Alternatively, there may be a region of the spectrum examined where k ceases to absorb light. I n either case, the rank of det Annis reduced to (n - 1). We will assume that this reduction in rank is associated with variations i = 1, 2 . . .(n - 1) but not with the ith variation. Equation 4 then reduces to det Ann
=
det
a
Z-1 det A,,
Y X i k X i k = %ik
(5) Thus, if variations j = 1 . . . n and i = 1 . . . (n - 1) are held unchanged, Xik multiplied by an undetermined constant may be evaluated as a function of i. This treatment is applicable to any system where a decrease in rank of 1 unit can be brought about by the choice of variations and can be applied more than once to the same reaction system if the required criterion is met. If n such %k-spectra are available, either through calculation or by independent measurement, the ratios of the yjk constants are uniquely determined for
y where
X-l
(6)
is directly proportional to the inverse of x.
