VOL. 7 (1952) .ASALSTICA CIII;\lICA ACTA =5

THE JIATHE1IATICAL TREAThIENT OF REDOX REACTIONS IN

VOLUMI~TRIC AN.4LYSIS

The conditions pertaining to the cquivalencc point of a rcdos rcnction are of particular interest in determining whether a reaction is quantitative and uscfnl in volumetric analysis. The criteria required are, I the equilibrium constant of the reaction; II the redox potential at the equivalence point; III the ratio of the concentrations of reactant to product at the end point (i.e. the “quantitativc- ncss” of the reaction); and IV the difference bctwccn end and equivalence points.

It is important to realist that the end point of a titration as shown by operation of the indicating mechanism may be significantly different from the cquivalcncc point of the reaction, at which the stoichiometric amount of reactant has been added; for example in the titration of a weak acid with a strong base using an indicator changing at pri 5, or the potentiomctric titration of ferrous iron with perrnanganatc.

Setting aside considerations of compatibility, a rcdox titration is based upon the interaction of two reversible rcdox systems, whose normal potentials differ by a sufficient amount. oxidising agent u Ox, + ne P c Red, normal potential E,, reducing agent b Red, + ne + d Ox, normal potential E,,

aOx, + b Red, + c Red, + d Ox, . . . . . I

The summation equation may be taken to rcprcsent the generaliscd form of redox reaction.

The criteria mentioned above may readily be formulated in the simple case where a = b = c = d = I, and criteria I, II, and III in the case where n = c and 6 = d. A full treatment thereof may be found in standard texts, and of criterion IV by KOLTHO~, andneed not be repeated in detail here. No attempt to formulate the completely general case appears to be recorded in the literature, probably for the reason that such attempts have not been successful. Although this attempt is in no way unique in this respect, it is believed that during its course proof

Rcfcrc9rcc p. rg.

16 E. BISHOP VOL. 7 (1952)

has been obtained that such formulation is a mathematical impossibility. The mathematical operations have been both lengthy and complicated, so that, since the result is largely negative, only a brief summary and the conclusions need be stated here.

I. Eqzrilibrium conslanl. At equilibrium, the Law of Mass Action can be applied to equation r, and assuming that the active concentrations are represented,

Phil” [Rcd*lb = K [RW” KW”

By the NERNST equation,

0.0002 T E, = E(-Jl + -~ Px,l”

9% log [Red,]”

0.0002 T 6, = j& - ---

11 log W-%1”

co4” At equilibrium, the potential of the system is uniform, and

E, = E, = 6. 0.0002 T LOx,l” 0.0002 T .

. . E = Eol + ---- 91

log [Red,]c = &k? - --- n.

0.0002 T .

{

[Ox, I” -- . . tRcd21” = Eoz - EOl 96 log [Rc-J,]C + log

Px21” I

. ‘Z(&), - J&l) [Ox,]” [Red21D . . 0.0002 T = log [R&jJ [Ox2]‘l = log10 I<.

[Red,]* log [Ox,]J

Whence a value for I< may bc calculated from the normal potentials of the two systems. This, then, is satisfactory and without complication.

II. Eqzrivalence-+oinl +otertliaZ. If there be imposed upon the system the conditions pertaining to the equivalence point, viz., that an exactly equivalent amount of 6x, has been added to the solution of Red,, we can obtain valence-point potential.

In the less gcncral cast whcrc c = a ant1 d = 6, which esomplcs in prncticc, WC have at the cquivalcncc point :

log [OS,]”

[Reel,]” =

Eo2 + 0.0002 7’ 31

a value for the equi-

co\-crs nearly all the

iI0x,1* Ob’ [Reel,]’

n 0.0002 7’ [Ox,] = q), + 92 log cfi;;tT = -02

F _t- 6 0.0002 7’ 91 l”6’

CO%?1 -- -(RCCl,]

when cc

DE,, = q + c”pG-l,, [Ox,1

[Rcd,l

VOL. 7 (1952) REDOX REACTIONS 17

13ut, \II~CC the substances react LII the rutlu a,‘O.

LO%1 [Red,] ---- --L-- at the cqui\-alcncc polnt. Cl~.cd,l - [OS,]

COs,l cod . . [X&J-[-= I

+ DC * Eep = _5___2_ . . (a i- 0)

1Yhere u, b, c, d arc all different, it is felt form;

and a.5 log:,, = 0

that the espression should take the

7? fl nbcd E,,3 + f, abed Ear . AL@=

-f,abcd__--_- _ .

possibly ECP = (a+~) Eo2 + (b+d) Eel

, a+b+c+d

which gives reasonable values in practical examples, but attempts to check this with the turning point of the calculated titration curves have been incon- clusive. Attempts to derive values of fdcd and to general& on the above opera- tions have resulted in insoluble equations.

III. Qzrantitalivertess of reaction. A measure of the degree of completeness of the reaction may be obtained from the ratio of [Ox,] to [Red,] at the equivalence point.

Again, in the simple case where c = n and d = 0, this ratio is readily obtained in terms of the equilibrium constant. As in II. at the equivalence point :

~O~,l [~ed,3

[OX,]~ [RcdJb -____

lTrorn I’ I’ = [Red,]” [Ox,]~

Substituting ; {*..}a+b={~~}“+~~ I(

COx,l a+6 I--- .

[Red,] = V Ic’ . . Application of equivalence point conditions to the general case where n, b, C, d

are all different gives:

18 E. BISHOP VOL. 7 (1952)

I) [Ox,]” [RcdJb = “-r;

[Red,1 -- 3) [Ox,] = : ;

COx,l CRe41 from which an cxprcssion for -.---- or ___-

[Red11 LOX*1 in terms of I< is required.

Application of the above operations yields equations always containing an unknown in the answer.

If c+d= n + b, then the equations arc homogeneous for the required ratios;

Thus, for example,

FWJ ----- = [Ox,1

If a-t-b is not equal to c+d, then I) is not homogeneous and thercforc I), z)

and can be solved

and 3) cannot bc solved for simultaneous homogeneous roots. It is therefore mathematically impossible to obtain the desired expression

unless a fourth condition can be evolved from the equivalence point relationships. The expression for the equivalence point potential may supply this condition, but as the proposed expression is incapable of proof, and the reasonable assump- tion of a functional form in Ear and Eoa yields equations of excessive fluidity, no useful result can be extracted therefrom.

By using approximations, some useful data may be obtained in particular cases where numerical values of the constants arc kno\vn.

Thus, it can bc shown that:

[Ox,] {w-v - (c+d)} [Red,] = - K {(G)“(G)‘}

[Ox,] {w-w - (c+k} =- K ((4)’ (G)“}:

whence it can be seen that:

[Red,] R+6 +)” (5)’ h’ [Ox,]c+J

c Px,l and [Red,] = d l

VOL. 7 (1952) REDOX REACTIONS 19

Although these expressions alI involve [0x2], they are useful, since a close approsimation to [OxJ can usually be obtained in practice.

IV. Coincidence of end and equivalence fioa’nts. The end point will occur when dsE/dVs = o, the equivalence point being determined by the stoichiometric considerations concerned.

It is possible to obtain an cspression of this difference when n = b = c = d = I,

by using certain approximations, though the mathematical operations are lengthy and difficult and KOLTHOFF AND PunMAN who show this calculation state that even in the case where c = a and d = b the equations arc not practicable. Esperi- mentally, the difference can be shown to be considerable in unsymmetrical reactions such as ferrous iron with dichromate or permanganate.

From this brief summary it can be seen that the mathematical trcatmcnt of redox reactions is incomplete and unsatisfactory, the more so when it is remcm- bered that such treatment as is possible cannot properly be applied to non- reversible reactions, to which class many reactions involving hydrogen ion belong.

The kindness of 11fr. P. MACNAUGIITON-Sn~IITlI in checking the mathematics and confirming the conclusions in II and III above is gratefully acknowledged.

The mathematical trcatmcnt of reclox rcactlons has been cxanunccl. Except for the cquillbrlum constant. the general cast 15 not susccptlblc of strict mathcm,lticd formulation, and cvidcncc has been aclvancccl to show that such treatment lci Imposslblc.

Nous ~~011s cffcctu6 unc Ctuclc mathtimaticluc clcs rdactions d’oxydo-r6cluction. A part la con&ante cl’dqulllbrc, 11 11’1 IL pas dc p~ssibilitd clc formulatloii mathd- maticluc exactc.

ZUSAIcIMENBASSUNG

DE mathcmatlschc Behandlung des Rcdoxreaktions ist gccxaminicrt wordcn. hllt Ausnnhmc von dcm Equlllbrlumkonstantcn 1Qsst sich cler Gcneralsatz nicht gcnau mathematisch formulieren. und Bcwels ist gcgcbcn worden urn zu zcigcn class solchc Behancllung unmogllch ist.

REFERENCE

1 I. XI. KOLTILOFF AND ?S. H. ~?URMAN, Potc~~tzotmt~~c T~tvnt~ows, pp 55-60. New York (1931).

Reccivccl March 5th. 1952