Mathematics of the Variation and Mole Ratio Methods of Complex Determination

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1 Joural of the Arkasas Academy of Sciece Volume 22 Article Mathematics of the Variatio ad Mole Ratio Methods of omplex Determiatio James O. Wear Souther Research Support eter Follow this ad additioal works at: Part of the Physical hemistry ommos Recommeded itatio Wear, James O. (1968) "Mathematics of the Variatio ad Mole Ratio Methods of omplex Determiatio," Joural of the Arkasas Academy of Sciece: Vol. 22, Article 18. Available at: This article is available for use uder the reative ommos licese: Attributio-NoDerivatives 4.0 Iteratioal ( BY-ND 4.0). Users are able to read, dowload, copy, prit, distribute, search, lik to the full texts of these articles, or use them for ay other lawful purpose, without askig prior permissio from the publisher or the author. This Article is brought to you for free ad ope access by ScholarWorks@UARK. It has bee accepted for iclusio i Joural of the Arkasas Academy of Sciece by a authorized editor of ScholarWorks@UARK. For more iformatio, please cotact scholar@uark.edu, ccmiddle@uark.edu.

2 97 Joural of the Arkasas Academy of Sciece, Vol. 22 [1968], Art. 18 Arkasas Academy of Sciece Proceedigs, Vol. 22, 1968 MATHEMATIS OF THE VARIATION AND MOLE RATIO METHODS OF OMPLEX DETERMINATION J. O. Wear Souther Research Support eter Little Rock, Arkasas INTRODUTION Two well kow methods for complex determiatio have bee used for some time to obtai the ratio of the atoms, molecules, or ios i the species. These are the Job variatio method (1) as modified by Vosburgh ad ooper (2) ad the mole ratio method (3). The variatio method works for most complexes where the complex has a measurable property such as optical absorptio that is differet from the reactats. The ratio of uits i the complex is determied by a maximum or miimum i a plot of A A (differece i expected ad observed magitude of a property) versus mole fractio of oe of the complexig agets. This maximum or miimum is the itersectio of two straight lies for a strog complex. For a weaker complex there is a itersectio of a lie from the poits ear mole fractio 1.0. I the mole ratio method, the ratio of the uits i the complex species is determied by the iflectio poit i a plot of a property such as optical absorptio versus the mole ratio of the reactats. This method oly works for a strog complex. Sice both of the methods ivolve the itersectio of two straight lies i the limit of very strog complexes, oe should be able to develop equatios for these lies. These equatios would explai why the method works ad allow for further iformatio to be obtaied from slopes ad itercepts. This developmet is the object of this discussio. VARIATION METHOD For the discussio of Ohe variatio method, oly the maximum i A will be cosidered rememberig that maximum ca be replaced by miimum i all cases. A will be cosidered as the differece betwee observed optical absorptio ad expected optical absorptio at a give -wavelegth if o complexig occurs. Vosburg ad ooper (2) used the followig equatios: AA A obs " A calc b(a A c + A ab c B + a c +c c )-b [a ft M (1-x)+a B Mx]. (!) c c M(l-x)-cA (2) Published by Arkasas Academy of Sciece,

3 98 Joural of the Arkasas Academy of Sciece, Vol. 22 [1968], Art. 18 Arkasas Academy of Sciece Proceedigs for the reactio c B Mx-c c A + B (4) t AB Where b is the sample path legth; aj is the absorptivity; c^, g, ad cq are cocetratios i the moles/liter of A, B, ad AB, respectively; M is the sum of the moles/liter of A ad B iitially; ad x is the mole fractio of B. Iitially for a strog complex, B is the limitig reaget ad the followig equatios are obtaied: c 0 Mx - cc Where A Aj deotes - L MX - M(l-x) Mx: (3) " AA_ (a -a -a )bmx (5) I AB -jj- A A before the maximum. After the maximum A is limitig ad will be essetially all complexed, the followig equatios are obtaied: c 0 c Mx-M(l-x) c_ M(l-x) A B AA (a I3; c -a A -a )bm(l-x) (6) B where A Ajj deotes A A after the maximum. ad At the itersectio of Equatios (5) ad (6): Aa I Aa II x which is the result of Vosburgh ad ooper for. Usig Equatios (5) ad (6), (aq-a^-ag) ca be de- termied three the slope of AAt the slope termiea i iree ways me siupe uiuat versus x, te siope of oi Ajj versus x, ad th/e itercept of,aajj at x O. Sice b, M, ad are kow ad a^ ad ap ca be determied idepedetly, a q ca be calculated. The average of the three values for aq should give a depedable value. If < 1, the procedure are slightly differet: AAj (a -a -a B )bmx AA II (a c ~ a A~ a B ) (1 " x > works the same but the fial equatios 98

4 99 Joural of the Arkasas Academy of Sciece, Vol. 22 [1968], Art. 18 Mathematics of omplex Determiatio MOLE RATIO METHOD Yoe ad Joes (3) itroduced the mole ratio method which is applicable to strog complexes without ay theory. I the followig, the mathematics of the mole ratio method willbe developed ad it willbe show how absorptivities ca be obtaied. Assumig that the complex formatio is represeted by the followig equatio: A + B Z AB (4) where A, B, ad AB ca be atoms, molecules, or ios, the followig ca be writte for some additive property such as optical absorptio: " A b(a A A +a B B +a c) (7) I Equatio (7), b is the sample path legth, aj is the absorptivity, cj is the cocetratio i moles 1 liter ad represets the complex AB. For a strog complex where B is limitig: c A c - c c (8) B * B " c 0 (9) were the superscript represets the cocetratio assumig o complex is formed. The added amout of A is held costat while B is varied so c^ is a costat but c^ is varyig with each additio of B. Substitutig Equatios (8) ad (9) i Equatio (7) yields: - A (a aa c )bc c + a A bc A (10) Now 0 v o 1 A/ (U) (11) Whe Equatio (11) is substituted for q p i Equatio (10) oe obtais: /» A T (a -a ) bc / \ A B I A I + a D bc c o J B A (12) \ A / Whe eough B is added that A becomes limitig: A (13) A (14) c c B B " (15) Now if Equatios (13- (15) are substituted ito Equatio (7), oe obtais the followig equatio: Published by Arkasas Academy of Sciece,

5 100 Joural of the Arkasas Academy of Sciece, Vol. 22 [1968], Art. 18 Arkasas Academy of Sciece Proceedigs A II (a c" a B )bc A + 3 B bc B (16) By multiplyig the last term o the right had side of Equatio (16) by (cj^/c ) oe obtais: A II (a - a B )b + a B b A( B / A> (18) Equatios so that (12) ad (18) willitersect at the poit where c B o c - -S cfa A plot of A A versus c /cj^ will give a straight lie up to the poit where A becomes limitig, ad the a chage i slope will occur. If a secod complex is formed, the slope will deped o it. Whe o further complexig occurs the plot willlevel off at a costat value uless B absorbs at the wavelegth. From Equatios (12 ad (18) the a^ ad a^ - a R ca be determied from itercepts at a mole fractio of 0. The a@ a» ad ag ca be determied from the slopes. ombiig these fidigs or usig previous kowledge of a* ad ag two idepedet values of a c ca be obtaied. APPLIATIONS These methods have bee used to determie absorptivities of complexes of NpO^"*" with pheolthalei ad complexes of NpO 2 with oxalate (4). The absorptivities were determied for two complexes each with a precisio of better tha \ /v. Usig the absorptivities, cocetratios of all species were determied ad the equilibrium costats were calculated. SUMMARY From the slopes of the lies before ad after the maximum (or miimum) i the stadard plots of optical absorptio versus mole fractio of complexig aget used i the variatio method of complex determiatio, the absorptivity of the complex ca be determied. The mathematics of the mole ratio method have bee developed, ad with this developmet, equatios for the determiatio of absorptivities evolved. Thje slopes of the lies i the stadard absorptio versus mole ratio plots are used for the determiatio of absorptivities.. i 100

6 101 Joural of the Arkasas Academy of Sciece, Vol. 22 [1968], Art. 18 Mathematics of omplex Determiatio REFERENES 1. Job, A. him. 10 9, 113 (1928). 2. Vosburgh, W., ad ooper, G. R., J. Amer. hem. Soe. 63, (1941). 3. Yoe, J. H., ad Joes, A. L., Aal. hem. 16, (1944). 4. Wear, J. O., U. S. Atomic Eergy ommissio S-RR » Published by Arkasas Academy of Sciece,