Poised Oxidation -Reduction Systems A Quantitative Evaluation of Redox Poising Capacity and Its Relation to the Feasibility of Redox Titrations €. R. NIGHTINGALE, JI. Department o f Chemistry and Chemical hgineering, Universify o f Nebraska, Lincoln 8, Neb. ,The capacity o f an oxidationreduction couple to resist a change in the potential o f the solution upon addition o f a reducing agent i s defined b y the differential ratio, KR/dE, where CR i s the concentration o f reductant necessary to effect unit change in the potential, E. The redox poising capacity index, p = dCR/d€, i s evaluated as a function o f the concentration of the redox couple, the number of electrons required for its oxidation or reduction, and the difference between the potential of the solution and the formal potential of the couple. The significance of maxima and minima in p vs. E curves i s discussed, and the minimum difference between the formal potential o f the redox couple and that of the reducing agent for which a titration end point can b e observed i s evaluated for 1 - and 2-electron couples. The sharpness o f a titration end point is measured b y the sharpness index, 7 = C , / p , where C, i s the stoichiometric concentration of reductant. The feasibility of a redox titration i s defined as the change in potential a t the end point of the caused b y addition o f p stoichiometric concentration o f reductant.
5
a rcducing agent, R e d , is atltl~d t o an aqueous solution an ouidation-reduction containing cou~ilo,31” ‘Mred,the ouidized form of yx.cics 11 nil1 hr reduced according to thc, rwction HES
E
o\\i t
T
red,
in the potential. as reducing
agent is added to the solution, is a function of the stoichiometric concentration of the couple, the number of electrons involved in its osidation or reduction, and the difference between the potential of the solution and the formal electrode potential of the couple. This paper discusses the efficiency of
redox couples in minimizing the chango in potential when a reductant is added to the solution, and evaluates quantitatively t h r capacity of the couples in this behavior. Acid-base buffers have long brcn used to maintain constant pH in solution. Tan Slyke (12) d t f ~ n e dacid-base buffers as “substances n hich by thcir presence in solution incrcase the amount of acid or alkali t h a t must be added to cause unit change in pH.” An) rwersible oxidation-reduction couple 11liicli dors not oxidize or rcduce the solvent behaves as a n oxidation-reduction “buffer” to resist change in t h r potential of the solution, and its behavior is analogous to a ~veakacid or a weab liaw as an acid-base buffcr. Clark (3) recognized the similarity in behavior antl suggested the term “poising action” to distinguish the behavior of redox coupl(Ls from the buffering action of acid-basr sj-stmis. Poised solutions and tho concept of poising action have receivc.tl only limited application, and n o quantitative ex aluation of poising capacity has bren puhlished. The earl). studics of Biilniann ( I ) , Clark ( 2 ) ,antl Grangcr nnd Xelson (6) utilized poised solution. t o nieasurc the effrcts of hydrogen ion concentration on organic ouidationreduction systems. Furman. Bricker, and 31cDuffie ( 5 ) recognized the eapabilitics of poiied systems in controlling the potential in electrorcductions. and Cooke, Reilley, antl Furnian ( $ ) usctl solutions 11 ith lon poising capacitj- to measure submicrogram quantitieq of ferrous iron. The most promising applications of these concepts appear to bc associated with coulonwtric procrdurcs, especially thcse a t constant current. where the primary electrode reaction does not occur with 100% current efficiency and intermediate couples are necessary t o facilitate the reactions; these procedures will be discussed in a subsequent paper. I n acid-base systems, the equilibria are established rapidly, and the pH limits are determined b y the autoprotolysis of the solvent. Acids stronger than the solvated proton are leveled b y the solvent. There is no euact redox analog of a “strong acid” or “strong
h s c ” in aqueous solution, although the solvatcd clcctrons in liquid amnionia might be a suitable example of a “strong reductant” in that solvent. In rcdos systems, tlic potcntial limits are not the theoretical (t,herniotlynaniic) onw iniposed by the solvent. but are, in practicc, dctcrniined primarily hy the kinc+cs-i.c., rat(,- of tlic solvent oxidation or rduction. I n a practical sense, :i strong rcdurtant is oiic d i o s e oxidation potential is suficiclntly positivc t o ( w ~ b l cthe reduction of watw. Siniilnrly, a strong osidant is one which oxidizes thtl solvent. Consider an aqueous solution containing a redox couplc characterized by species 11 L a ( S - ) , and b1-t3(X-)i,, in which a and h arc the charges associated n-it11 the osidizctl and reduccd forms of tlir couplr and S - is, for convenimcc, :I iuiinegative anion. K h c n a rcduct:int, Rfr(Z-),. is adtlcd tCJ the solution. the oxidizcd forni of I1 will bc reduced :wording t o Equation l a . which may be written in tcrms of t h r ionic charges as (0
- T)\L+ (D
+
((I.
- h)R’-’=
- r ) l r * ’ f (.
-
b)R+
(11))
\vhercx o is the ionic charge of the oxitlized form of the rcduciiig ngcsnt. Comparing Equations 1:i and 111, tlict number of electrons associatctl with tlic oxidation or rrduction of specic,s 11 anti R :UY z = ( a - b ) :111tl z = (0 rrspectirely. Equation Ib reprcwnts only on(’ of n ~ ~ large r yvariety of rrdox systems, anti in a chemical senst’, thi appear sufficiently gcmml to warrant discussion. Xot only are p H rffccte (including hydroxy- and oxygen-containing cations anti anions) omitted, but complex ionic spccics, solid p h a ~react,ants,and gaseous phase reacbants are escluded also. Thus, this initial trcatnient is designed tjo present the hasic concepts as simply as possible without loss of exactness, and such additions do not add apprwiablj- t o the theory. Hon-ever, the ubiquitous charactcrs of pH and complex ion effects are sufficiently important to warrant their consideration, and the procedure necessary for their treatment is presented in a later section. I n a purely mathematical VOL. 30, NO. 2, FEBRUARY 1958
267
sense, the notation used herein is completely general if a, b, 0,T , etc., represent the ovidation states of the species and if the formal potentials are defined t o include the concentrations of any other reactants, including ligands and their relevant equilibrium constants, as well as the activity coefficients. The expression of electroneutrality for the system defined above is
+ b{.lI+] + o[R+o] + + [ H + ] = [OH-] + [X-] +
a[M+a] r[R+r]
IZ-I
Equations 9 and 7 are equivalent to Equation 2 for the electroneutrality of the solution. The poising capacity of redox couples to resist change in the potential as reductant is added t o solution is defined as the differential ratio dCR/dE, and is measured by the redox poising capacity index, p = dCR/dE, in moles of reductant per liter of solution necessary to effect a change of 1 volt in the potential of the solution. From Equation 9,
(2)
NOW
cu
=
(0
+
~~
[11+0]
CR
sech2 8/2
1
1 - (0 - r)(tanh e/2 - 1)) (10)
and
CR
- r)* 5 RT
[X+b]
[R-’Iadded
=
=
+
[R+”] [R+?]= [ Z - ] / T (3)
where CM and CR represent the total concentrations of species M and R, respectively. Substituting Equation 3 in 2 -(o -
[R’o]
T)
+
- b) [lILa] [OH-] - [X-I
(a
+ [H+] -
bCM
The Kernst expression for couple
(4)
XI is:
where E M ” is the standard oxidation potential and E d ’ is the formal oxidation potential (11). From Equation 5 and a similar expression for couple R. &/(I
[r\l+a] =
[R+o]= C R / ( 1
+ exp + expel 01)
(6)
where a =
(o--)(tanh8/2-1) - b ) (tanh a/2 - I ) X -~dEC R
d C\I ldE = l a
- Ex’’)( a - b)S/RT
(E
and
- ER”) (0 - r)5/RT
B = (E
Substituting Equation 6 in 4
Equation i and its derivatives are more easily evaluated b y use of the following simplifications: alb
+ exp a
[ 1-1+ exp
01
I=
- [ ( a - b ) tanh 01/2 2b (a b)l
-
+
[L = -] 51 [tanh 6/2
- 11
(8)
1 +expo
Substituting Equations 8 and 7 ,
(O
2 r,
cR
(tanh 0/2
-
1) =
C M [(a - b ) tanh a / 2 - ( a + b ) ] 7
E+]+ [OH-] 268
+ [X-I
ANALYTICAL CHEMISTRY
Equation 10 is derived b y assuming that there is no change in either the volume of the solution or the activity coefficients and formal potentials as reductant is added to the solution. Even if the formal potentials are defined to include the concentration(s) of ligand(s), the latter assumption is not permissible if the ligand concentration changes upon the addition of reductant. I n such cases, the ligand concentration cannot be collected into the formal potential, but must he treated as shown below. If the con\.ention introduced by G. ?;. Lewis is used. the poising capacity is always positive, because addition of reductant increases the potential of the solution. While the definition of poising capacity is arbitrary. it is not necessary to use a special equation to represent the effect of oxidant on redox couples, since from Equation 9,
(9)
Clark and coworkers (8)used the reciprocal of a quantity related to poising capacity t o develop a graphical method for determining the titration end point of a system in which the potential was unsteady in the vicinity of the end point. The quantity dE/da, where is the “degree-Le., fractionof oxidation,” was evaluated for a = 0.5 where the solution is well poised, and the end point estimated by extrapolating t o (Y = 1. This procedure 178s limited, however, t o the consideration of only one redox couple a t a time, for, as the authors pointed out, ‘[there must not be superimposed upon the conditions defined. . .a change of potential of different source. .” Oxidation-reduction couples are analogous t o weak acids and weak bases and redox analogies t o strong acids and bases are not possible in aqueous solutions. When Equation 10 is compared with the analogous expression derived b y Van Slyke (IS) for the buffer capacity of acid-base systems, the major difference is the presence of the de-
.
[N+‘] =
CN/(
1
+ exp
couples a t 25' C. The maximum value of p ~ 'occurs when ('
-
(O
-
seph2
RT r)25CR
p p
+
812
I/
RT - r ) (tanh8/2
[ -(o
-
[&I+']= [JI+bJ, ( E = EM"), and sech2a/2 = 1.
1 ) ) (16)
I n order t o evaluate the maxima and minima in the p us. E curves, i t is necessary t o consider t h e second derivative of Equation 15 with respect t o potential. From 16 d2CR/dE2 = { ( a
dp/dE (c (0
-
b)3
19.46 ( a PR'
- b)23 -
- b ) 2 / ( ~- r ) v.-l
(21)
approaches its maximum value as
-
- 4 3 (&)'CN sech2 p/2 tanh p/2 + (0 r ) s (&)2C~sechzB/2 tanh8/2 (' - d ) 2 CM sech? 0.112 + R Tr ) 3 (tanh e:2 + 1) [(u CN sech2 p / 2 +
ic ' cR
RT DISCUSSION
P
= PM
+ PS + P R
where the partial poising capacities are given b y (a 2
- b)2 5 cv sechz 012
2 (0 - T ) R T
mole/liter/volt
- d)' 5 Cs sech2p/2 PN = 2 (0 - T ) RT (C
PR
RT
'
cR
(tanh e/2
The partial molar poising capacity of a n individual couple is defined b y PL' = P l / C ,
Hence, from Equation 18,
- b)2 3 sechz 0/2 v.-1 2 (0 - i r ) R T (C - d ) ' d sech2 p / 2 ps'= 2 (0 - r ) R T
y
(19)
and pa'
Y
- r ) 5 (tanh 8/2 ____
(0
RT
+ 1)
1/ 3 'q
(tanh 8/2 - I) ii (17)
( E - ERO')becomes large, and from Equation 20, ( P R ' ) ~=~ 77.84 ~ (o - T ) v.-1 Table I1 compares the relative magnitudes of ( P M ' ) ~and ~ ~ ( P R ' ) ~for ~ ~1- and 2-electron couples at 25' C. Curve 2 in Figure l , b , illustrates p~ for a 2-electron couple. Since ( p ~ and ( p ~ ' are ) proportional ~ ~ ~ to ( a - b)*, the peak height of curve 2 is four times that of curve 1. For 1-electron couples, P M decreases until for ( E - E d ) = h0.213 volt, the value is 0.1% of ( p ~ ) For ~ ~ ~ 2-electron . couples, the decrease in p~ with potential is greater than for 1-electron couples since 01 is proportional to (a - b ) . The integral of PY X dE is the concentration of reductant necessary to reduce couple &I from the ovidized form, AT-, to the reduced form, ML*.Hence the area under the p~ os. E curve for 2-electron couples must be twice that for 1electron couples. The behayior of poising capacity with potential is independent of the initial ratios [M-a]/[l\l+b] and [K++E]/
(18)
+ 1)
(a
RT
sechz 8/21
Curve 1 in Figure l , b , illustrates the manner in which the poising capacity changes with potential for the solution defined by Equation 13 ivhere CM = CN = 0.1M; ( a - b ) = (c - d ) = (0 - T ) = 1; and E u ~ 'Eso', , and ERO'are f0.2, 0.6, and 1.0 volt, respectively. K h e n the formal potentials for 1electron redox couples differ from that of the reductant by more than about 0.4 volt, the contributions of the individual couples to the total poising capacity of the solution are additive and independent of that of the reductant. From Equation 16. for (E& ERO') less than -0.4,'(0 - r ) volt, the additive nature of the partial poising capacities for the individual couples may be represented b y
Pu'
(a
=2(0--)RT-
(P'd'jmsx
(&)'CM sechz a/2 tanh a/2 +
-
PM
From Equation 19,
(20)
Table I evaluates the partial molar poising capacity for 1- and 2-electron
Table I. Partial Molar Poising Capacitiesfor 1 and 2-ElectronCouples
-
( T = 25.0' C.) Par'[2RT/(o - ?')'I & ( E - EMO'), Volt (a - b) = 1 (a - b ) = 2 0.000 0,004 0.008 0,012 0.016 0.020 0.030 0,040 0.050 0.060 0.080 0.100
1,000 0.994 0.976 0.948 0,909 0.863
0.724 0.575
0.438
4.000 3.905 3.636 3.241 2.654 2.300 1.327 0.652
n
)
~
0.200
~
,
,
I
I
08
04
E , volts
Figure
T.
Variation in CR, p, and
dp/dE with potential CM = C S = 0.1X. ( ~ 7 - b ) = ( c - d ) = 1. Et,fo' = f0.2, EsO' = 0.6, and ER"'= 1.0 volt. In b, .4,C, and E indicate poising capacity due t o species &I, N, and R, respectively. For curve 2, (a-b) = 2 (0--T) =
INfd]. If, however, one wishes to consider the titration of and N+c with R+r, Figure l,b, is the derivative curve for Figure l,a, in which the concentration of reductant, added t o a solution containing initially O.1M [&I+.] and 0.1M [N+c],is plotted as a function of potential. Figure l,n, is the inverse of the common titration curve plotting E (ordinate) vs. CR (abscissa). Maxima and Minima in p. T h e maxima and minima in the p us. E curve are evaluated analytically by equating d p / d E t o zero (Equation 17). Figure l,c, plots the entire derivative curve
Table II. Maximum Partial Molar Poising Capacities for 1- and 2Electron Couples
( T = 25.0' C.) ( PM')msxr
( PR')maxt
- r) = 2
19.46 9.73
77.84 155.68
- b) = 2 - r) = 1 - r) = 2
77.84 38.92
77.84 155.68
V-'
312
(a
- b) = 1 (0 - r ) = (0
0.120
0.140 0.160
~ -0 51 00
(a
(0
(a
1
VOL. 30, NO. 2, FEBRUARY 1958
T7 -1
269
of curve 1 in Figure 1,b. As indicated by Equation 21, the maxima in Figure l,b, a t 0.2 and 0.6 volt correspond t o the potentials of equiconcentration of the oxidized and reduced forms of species h l and Y, respectively. The minima a t 0.4 and 0.8 volt correspond, respectively, to the end points for the titration of h1+5 and X+Cn-ith R+.. The portion of the curve labeled DE indicates the increase in poising capacity as excess reductant is added t o the solution. This behavior is interesting because the concentration CR, unlike CM and CY,is variant. From Equation 20,
(0
-
T)
RT
3
(tanh el2
2
2
(0
-
A
$/RT
F
,e-
(-/ __ -0677-
T)
‘._
r’
-0856
+ 1)
As excess reductant is added t o the ) large solution, (E - E R ~ ’becomes and tanh 012 approaches unity. Hence: In cR dE
f
Figure 2.
CM
-0856
E volts
-
Variation in p and dp/dE with potential
0.1M. ( a - b ) = ( c - d ) = (0-7) = 1 1. ( E Q ‘ - E M ~= ’ ) 0.0677 volt 2. (ERO’-ENO’) = 0.0856 volt
= CN =
or, on integrating, CR 2 exp [2(0 - T ) 3 E / R T ]
+ constant
and CR increases exponentially with potential as required by the Kernst expression. The positions on the potential axis of the partial poising capacity curves depend entirely on the respective formal potentials, and the nature of the total poising capacity curve is determined by shifting the partial curves for the individual couples to the proper formal potentials. For example, if the formal potentials of couples M and N in Figure l,b, differ by only 0.3 volt, the partial curves for couples RI and 1\’ overlap and the minimum corresponding to the theoretical end point for the titration of AI+. in the presence of NtCis obtained before the stoichiometric concentration of reductant has been added. It is obvious from the manner in which the partial poising capacity curves for the individual couples overlap that the theoretical end point for the titration of &I+.will not approximate the equivalence or “stoichiometric” point (7, IO), unless, for ( a - b ) = (c - d ) and CM = CN, EU.’and EN^' differ by more than about 0.40/(a - b ) volt. Similarly the end point for the titration of K + with R+r will approximate the equivalence point only if, for (c - d ) = (o - r ) , ENO’ and E K ~ differ ’ b y more than 0.40/(0- r ) volt. If the partial poising capacity curves for couples R I and ?; overlap sufficiently, the minimum in total poising capacity will disappear. From Equation 17, for CM = CN and ( a - b) = (c - d ) , the minimum and hence the titration and ENO’ end point will disappear if EUO’ differ by less than 0.0677/(a - b ) volt,
270
ANALYTICAL CHEMISTRY
corresponding to an equilibrium constant for Equation 12 equal or less than 13.9. Curve 1 in Figure 2,a, illustrates this example for ( a - b ) = 1. The total poising capacity is the sum of the partial curves, and pmsg= 1.334 ( P M ) ~ * = . As is seen from the derivative curves in Figure 2,b, 0.0677/(a - b ) volt is the difference in formal potentials a t which the three roots of Equation 17, points A , B, and C in Figure l,c, become identical. At this point only 78.9% of CM is in the reduced form, M +b. In a similar manner, for (c - d) = (0 - r ) , a minimum in the poising capacity curve for K and R will not be obtained and an end point for the titration of K+c cannot be observed if ENO’ and ERO’differ b y less than 0.0856/ (o - r ) volt, or if the equilibrium constant for Equation 11 is less than 27.9. This is illustrated by curve 2 in Figure 2,a, for (c - d ) = 1. I n this case, however, the total poising capacity is not exactly the sum of the partial curves, since the denominator of Equation 18 is not constant. The minimum in poising capacity disappears when CR = 5 / 7 CN; a t this point, 66.7% of CN is = in the reduced form and ( E - EN”’) 0.0179/(0 - r ) volt and ( E - ERO’) = -0.0677/(0 - r ) volt. Because of the nature in which the reductant enters into Equation 16, it is theoretically possible to titrate the oxidized form of one couple in the presence of that of a second couple with more positive formal potential, if their formal potentials differ by more than 0.0677/ ( a - b) volt, whereas, if the oxidized form of a given couple is titrated directly with reductant, the difference
in formal potentials must be greater than 0.0856/(0 - r ) volt. Reduction of Solvent by Strong Reductant. T h e previous treatment has assumed t h a t t h e solvent water is neither oxidized nor reduced b u t is inert. When n-ater is reduced-i.e., Tvlien t h e potential of t h e reductant couple is more positive t h a n t h e overpotential a t which water is reduced-the equation for t h e reaction is : (o -
T)
+
+ R+r = w) 2 Hz (o - r ) OH- + R+”
H,O
(22)
The expression of electroneutrality is o[R+”]
+
r[Rtr1
and since CR =
@+’]added
[Z-lh-, (0
-
T)
+
[H+l = [OH-] [Z-I
+ [R+’I +
[RSo] = [OH-]
[€?+’I
- [H’]
(23) =
(24)
The Nernst expression for the reduction of water is E = EWO’ R 5 T In [OH-]fi~,*/n
+
where E,”’ includes the proper overpotential term for the electrode surface in question. For unit partial pressure of hydrogen gas, [OH-] = exp 6
(25)
where 6 =
(E
- Ewo’) S / R T
Substituting Equations 25 and 8 in 24 - (’
-2T)cR (tanh e/2 - 1) = exp 6 - KWexp - 6
(26)
where KW is the ion concentration product of water. The poising capacity for the solution defined by Equation 23 is given by
Wax
Cs
PI\ z ievp 6
+ Kw eup
-
6) 3 / ( 0 -
T)
mole/liter/volt
RT
(28)
If ( E - Ew"') 2 -0.355 or 5 -0.473 volt, corresponding t o [OH-]/[H+] 2 100 or s 0.1, pn- increases approximately exponentially with potential. PR offers negligible contribution to the total poising capacity, as the concentration [R+'] doc^ not increase, because of the continuous reduction of nater. Sharpness of End Point and Feasibility of Titration. I t is desirable to chai ncterizr titration end points by a function t h a t exhibits a masiniuni rate of changr about t h e end point. Tlie reciprocal of t h e poising index is not :done sufficicnt (Q), as it represents the. rstcx of change of potential relative to thc c.onc.entration of added reductant instead of relative t o the fraction of the stoichiomctric concentration, Cs, of wductant nrcrssary to obtain the t>quiwlence point. The necessary corrcction may be ninde by multiplying the reciprocal by Cs,and the sharpness of an t.nd point i5 defined by the sharp1 1 ( w index = Cs/p. From Equation 1 G, ? s C s l p =
2 CS(O- r ) R T / 3 (a - b ) 2 CMsech2 a/2 (c - d ) 2 CK sech2 p/2
+
If (a - b ) = ( 6 - d ) and Equation 30 becomes (0
e
?ma.
For (ER" - Ew") greater than about 0.20/(0 - T ) volt, the partial poising capacity due t o water is given by
z
CM
=
CS
- T ) RT cosh2cu/2 = CN ( a - b ) 2 5
Thus for (c - d ) = (0 - T ) = 1, the addition of 0.1% of the stoichiometric concentration of reductant about the exp - a/2)2 (31) end point will produce, a t 25' C., a For ( E - EW"')/(U- b ) greater than change of (ER" - E>,"' - 0.390) volt 0.118/(a - b ) volt, cosh* 4 2 , is, within about the end point irrespective of the I%, equal to exp a/4, concentration of the reactants. Conversely, for any given AE, the necesCs(o - r)RT In vmnY z In 01 (32) sary difference between formal poten4 Cif ( a - b)* 3 tials of the redox couple and that of and In qmax incrcascs approuiniatcly the reducing agent may be evaluated. linearly with ( a - b ) and ( E - EX"'). From Equation 33, the relation between Curve 2 in Figure 3 indicates that a the feasibility and the sharp'ness index much sharper end point is obtained is given by n-hen ( a - b ) = (c - d ) = 2 . I n a 2 R T In 4 (0 - T ) 5 p x x similar manner, if E$' and ER" differ AEe(37) 200 R T (0 - r ) 3 sufficiently and (c - d ) = (o - T ) , the magnitude of q m a x at the end point for or, a t 25" C., ' may be approxithe titration of S rnated b y
+
pH AND COMPLEXING EFFECTS
Chemists have for years attempted to describe the feasibility of a potentiometric titration b y the magnitude of the potential change "about the end point." The feasibility, AE, may be defined quantitatively as the change in potential caused by the addition of p % of the stoichiometric concentration of reductant between p / 2 % short and p / 2 70 beyond the theoretical end point, E+,, - E - - p ?
AE
(34)
From the Nernst expressions
./! - C ~ ( o - - i j ( t a n h 8 / 2 -
and substituting Equation 34 in 35
For the titration of more than one couple, Cs, is multivalued, and it is usunlly desirable to determine the correct value for each end point separately; for the first end point, Cs = (o - r ) / ( a - b ) CV and for the second, Cs = (o - T ) / ( c - d ) CZ. Curve 1 in Figure 3 plots the sharpness index as a function of potential for the system previously defined by Equation 13 and characterized in Figure 1. If the formal potentials of redox couples hI and N differ sufficiently-Le., by more than that about O.4O/(a - b) volt-from of the reductant, the maxiniuni in the sharpness index, f m z x , corresponding to the minimum in p a t the end point for the titration of M+" is giI-en by
I
0
'
\ I
00
04
\ '
E, volts
08
Figure 3. Variation in sharpness index, p, with potential
cs = 0.l'lf.
Ehp' = $0.2, EhO' = 0.6, and ERG' = 1.0 volt 1. ( a - b ) = ( c - d ) = ( 0 - T ) = 1 2. ( a - b ) = ( c - d ) = 2
CM
=
The systems defined b y Equations 2 and 13 are not sufficiently general, and unless the formal potentials are defined t o include the concentrations of any reactants, including ligands and their. relevant equilibrium constants, other than the species being oxidized or reduced, these equatiow do not represent systems which contain p H or other complexing effects. The formal potentials may be defined to include the reactant and ligand concentrations only if the solution is "buffered" with respect t o the reactants and ligands; otherwise, the formal potential will change if the ligand concentratio11 changes upon the addition of reductant. The mathematical treatment of an oxidation-reduction systeni in which the concentrations of reactants or ligands change with the addition of reducing agent is somewhat more coniplicated than the simple procedure indicated for the systems defined b y Equations 2 and 13. The asymmetry introduced into the Nernst expression for such systems requires a more laborious procedure for evaluating numerically the poising capacity or any of its related quantities. The presentation iyhich follons indicates a method by which the poising capacity of redox systems containing reactants or ligands other than the species being oxidized or reduced may be evaluated. As a simple example of a n osidationreduction system involving a reactant other than the species being oxidized or reduced, consider a redos couple characterized by the half-reaction Q+'
+ H+ + e-
=
HQLk+'
VOL. 30, NO. 2, FEBRUARY 1958
(38)
271
in which (j - k ) = 1. The expression of electroneutrality for a system containing the couple Q + ’ / Q + k + + ’ and t o which reducing agent R+r is being added is j[Q+I
[H+]
+ (k +
+-
o[R+O]
+
1) [H&+k+ll
+ r[R+r] =
+
[OH-] (39)
+ [Z-I
[X-1
in which X- is the anion associated with species Q. The stoichiometric concentration CQis given b y CQ =
+ [HQ+k+’]
[&-’I
(40)
Substituting Equation 40 into 39, and collecting terms, (k
+ 1) CQ + ( j - k - 1) [Q+j]+ + o[R+”]+ ~ [ R ’ F ]= [OH-] +
[H+]
iX-I
+ [Z-I
introduction of additional reactants necescitates additional known relationships, usually expressions of stoichiometry, By virtue of Equation 38, the total hydrogen ion in the system is given by CE
=
:H-j
+
[HQ+k+’]
142)
Solving for (H’) and substituting Equations 40 and 42 into 41,
+
~ C Q i j - k)[Q+] r[R+r] = [OH-]
+ o[R+’]+
+ [X-] + [Z-J
(43)
-4 similar substitution for [OH-] in the above equation has not been indicated because, for simplicity, it mill be assumed that [OH-] is negligibly small compared with [H’] in this system. Such an assumption simplifies greatly the mathematical treatment of the problem with only insignifiqant loss of rigor. The Xernst expression for couple Q is
RT 2
2
c_Q
2
cR
+ k ) ] + [x-]
[i]- k ) tanh ~ / 2- ( j
=
C ~ ’ [ e x p (r 2.3pH)
+ I]
(45)
where [ H + ] = esp -2.3 p H and y = ( E - EM”’)( j - k)S/RT. Substituting Equations 3, 6, and 45 into 13 and neglecting [OH-],
[X-] (46)
where K = y -2.3 pH. The transform into hyperbolic functions is
-(j -
-,
+ -i o
(49)
As indicated by Equation 49, evaluation of the poising capacity for systems involving reactants or ligands will require more than the single parameter of potential. I n this example, pH is also rrquired, because h = y - 2.3 pH. I n general. an additional parameter nil1 be required for each additional reactant. From the Kernst expression, Equation 44, it is readily observed that a single potential measurement is insufficient to establish the ratio [Q+?]/[H&+k’l], and that, in this example, a measurement of p H is also required. An alternative procedure 1%ill sometimes facilitate the evaluation. especially for systems in which a specific indicator electrode (such as the glass electrode for hydrogen ion) is not readily available. Substituting Equations 40 and 42 into 44, and rearranging,
-
2
{CQ
+ (3 - k )
23
272
- 3) tanh K j 2 -
(j
+ k)]
ANALYTICAL CHEMISTRY
(47)
-
Biilmann, II.* U. S. Publ. Health Repts. 38, 443 (1923). Cooke, V. D., Reilley, C. AI., Furman. N. H.. AXAL. CIiElI. 23. 1662(1951). (5) Furman, S. H:, Bricker, C. E., McDuffie, B., J . Wash. dcad. Sci. 38, 159 (1948). (6) Granger, F. S., Nelson, J. > I . , J . iim. Chem. SOC. 43. 1401 (1921). Kolthoff, I. >I., Stenger, V. .\., “Volumetric Analysis,” 2nd ed., Vol. I, p. 1, Interscience, S e w York, 1942. Phillips, RI., Clark, 1%‘. M., Cohen, B., U . S. Publ. Health Repts., Snppl. 61 11927). Smith, T. B., “Bnalytical Processes,” 2nd ed., p. 207, Edward Arnold Co., Idondon, 1940. Swift, E. H., “Introductory Qiiantitative Analysis,” p. 39, PrenticeHall, Ken- Tork, 1950. Ibid., p. 109. Van Slyke, D. D., J . B i d . Chem. 5 2 , 525 (1922).
(51)
1) =
-
!QTI]
[Z-t (52)
The poising capacity of the system is again given by differentiating Equation 52 with respect to potential, and
p =
dCR - = dE
i
j
CR s w h 2 8/2 + (0 - r ) 9 RT
- k)dlQ-J]/dE -(o
(55).
LITERATURE CITED
S o n substituting Equations 3, 6, and 8 into 43 and neglecting [OH-] - (0 _ - r ) C_ x ~ - 8//2 (tanh
1)
The author wishes t o express his appreciation to G. -4.Gallup for numerous discussions.
+
(CH - CQ -,] - CQ eup - y = 0
RT
ACKNOWLEDGMENT
exp - -, = [Q”] {CE - CQ -k \ Q L ? ] ] / ICQ - .Q-’Il (50)
+ [Q-’]
{CQ - [Q+j]/exp -~ r)2 5 C R _ sech2 _e/2
Equation 55 does not involve the concentration of the reactant (hydrogen ion), but in comparison with Equation 49 it has in effect substituted the concentration of Q+! for that of hydrogen ion. Thus this procedure is primarily advantageous if the concentration of one species of the oxidation-reduction couple is more easily determined than the corresponding concentrations of the reactant or ligand species.
RT
e\p
5
- (0 - P) (tanh 8/2 -
io - P y 5 C R (tanh 8/2 - I ) \
k)2
RT
Differentiating Equation 48 with respect t o potential, the poising capacity of the system defined by Equation 39 is given b y
From Equation 51 -1 - [(j
and substituting 54 in 53, the alternative expression for poising capacity becomes
(48)
(Q”]’
[Q-i]
y
(tanh e/2 - 1) =
01‘
Substituting Equation 40 into 44 and rearranging,
+ CH - CQ + exp -
[&+’I
(54)
- (O - ’)
(41)
As in a11 equilibrium calculations, the
-(j - k ) 5
and substituting Equations 8 and 47 into 46, the expression of electroneutrality in its final form becomes
- T ) (tanh 8/2
(53)
- 1)
RECEIVEDfor review October 12, 1956. Accepted September 16, 1957. Division of Analvtical Chemistry, 131st Meeting, ACS, Miami, Fla., l p r i l 1957.
