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Some theoretical considerations in analytical chemistry
AXALYTICA
SOME THEORETICAL PART
VI*.
THE
PRECISE
CHIBI ICA ACTA
397
CONSIDERATIONS CHEMISTRY
CALCULATION
IN ANALYTICAL
OF DATA
FOR
REDOS
TITRATION
CURVI;3S r:. BISI~IOI’
In the simple calculation of rcclos potentials for titration curves1 from the Nernst equation and reactant concentrations, the potentials are a logarithmic function of the ratio of the concentrations of tlic active species and appear to be inclcpcnclent of absolute concentration. ‘This also applies to the equilibrium constant of tllc overall titrimctric rcnction~ and, when the reaction coefficients are l~on~ogencous, to certain other quantitative factors”, but does not apply when the reaction cocfficicnts iLtX! In tht: latter circumstance, such factors as the cquivalcncc point inhomogcneouG3. potentials and the quantitutivcncss of the rcnction” become depcnclent on the absolute conccntrntions of the reactants. As will be shown, in cvcn the simplest rcactions having unit coefficients and involving the transfer of a single electron, the cnlculation of the potcntinls bccomcs concentration-tlcpcndcnt when illly attempt is mndc to take account of incompleteness of renction. The efIect 0f such reaction cleficicncy is to Ilattcn the horizontal part of the curve and to dccrcnse the rate of ClliLllgC of buffering is potential through the equivalence point. This enhanced potcntiostatic similar to the effect of hydrolysis of salts in acid-base titrations. The analogy with buffer solutiorPs and titrations involving such media l..l~bextends to the situation that whcrcas the simple calculation employs concentration ratios, precise calculation invokes absolute concentrations, and reveals that buffering is more efficient than the simple calculation predicts 6. The cstcnt of this potentiostasis may be judged from the esample of a customary titration with a difference in normal potential of osidant and reductant of 0.2 V and the transfer of a single electron, wherein at 0.1% from the equivalence point the alteration in potential due to incompleteness of reaction is no less than SI mV. For a reaction involving the transfer of two electrons and having a diffcrcnce in normal potential of 0.4 V, the corresponding effect of incompletcncss of reaction is diminished to o.G mV. The effect of increasing difference in normal potcntial in reclox titrations finds an analogy in the effect of increasing ionisation constant in acid-base titrations, so that reaction deficiency corresponds to solvolysis. Analogies can also be discovered with other types of ion combination reactions. The usual procedure in the preparation of a redox curve1 is to calculate the potcn* Part V see And.
C/rim. A da,
22
(rgG0)
205, Am~l. Chirn. Ada,
2G (IgG2)
3g7-.+05
398
E.
RISHOP
tial of the rcductant system prior to the equivalence point and the potential of the osiclant system after the equivalence point, and to combine the two half curves with the cquivalcnce point potential. The cffcct of neglecting the incompleteness of reaction is immediately revealed in examples such as the one first mentioned (CT].Fig. I). The two halves of the curve refuse to mate, although a continuous curve can be recorded espcrimentally and the interval between the normal potentials is generally consiclcred acceptahlc.
Vol. of O.lMoxldont
added
I;&. I. ‘I’itrrrticm CJf 100 nil of 0.1 lW rcclucta~lt with 0.1 hi oxiclant at 30~. *IIT,, - 0.3 V; u - 0 I G=d=,,,=,,.,=: I. A, A’, cA2ulatctI by ttw simple tncthrxl tlircctly from nppnrcnt conccntrntions nncl the Ncrnst equation. L3, cnlculatcil 1)~ tlic clcscribctl nicthocl.
No account of this situation, other than at the cquivalcncc point*ea, appears to be record4 in the standarc tests. It leas proved possible to evolve a mathematical treatment for the completely general cnsc, and mcthocls of calculating the reaction deficiencies and the points for titration curves arc prcscntcd here. Some observations on the consequences of the incompleteness of rcclox reactions will be offered in a subscclucnt Part.
The incompleteness of reaction at any sclectecl stage in the titration can be derived from the equilibrium constant, which in turn can bc derived from the normal potcntinls of the reactants. The apparent concentrations of the reactants can then be corrected for reaction deficiency and the true values used in the appropriate Ncrnst equation to calculate the potcntinl at the selected stage in the titration. It is first necessary to clarify the interpretation of TN,the number of electrons transferred, which appears in the expression for the equilibrium constantn. Let the osiclant system, of normal potential EON, be represented by eqn. (I), and the reductant system, of normal potential EON,be represented by eqn. (2). MultiplicaA #Ical. Chim.
A c/a, at3 (rgG2)
397-405
CALCULATION
tion of eqn. the balanced
(I)
OF DATA
FOR
REDOX
TITRATION
CURVES
by x and of eqn. (2) by y where 9tl.x = stay, followed eqn. (3) for the overall titrimetric reaction, uO.u, + 111~;i ORed anOxl
The equilibrium
constant
*
+
by addition gives
cRcdl
712E+
+ byRed
399
(1)
dOxs
cxRcdl
(2) + cZyOxz
for the overall reaction
(3)
is then (4)
At any selected point in the titration, when the reaction has reached equilibrium, potentials of the two systems,E will be the same. E = fut f
0.00021’
log
[0x13”
nI
Since
En
1
-
1%
Jl IX
,fzlLx = nny, ant1 writing A-11 I /.I 1:‘~ ---
0.0002’1’
&
=
80~
0.0002’l
=+--
[Rctl~ J”
0.0002’1 =
-
[lZcd,]e
l’JJ2 L-m
=
Eo, /t 1%
0.0002T
=
[Oxr]“~
4-
log
JJ??
=
KwJ [IZc&]b
0.0002?
[OXI]“’
Eo,-
,og
n2
the
fRcdz]*”
.A&,
log-‘--
I ICcdJ~~~
[.OX2]JY
[.OXI]~= (Red+
=
log
fi
(5)
The number of electrons in the espression for the equilibrium constant is therefore the least common multiple of the number of electrons transferred in each of the two half reactions. Reaction
deficieucy
In the simple calculation of curve data assuming the reaction to be complete, the ratio of [Ox] to [Red] is alone rcquircd, and is derived from the fraction of the equivalence volume of titrnnt added before the cquivalencc point, and thereafter from the excess of titrant : concentrations and dilution factors cancel out. Concentrations, when needed, can be calculated from the initial concentration and volume of titrand, the initial concentration and volume of titrant and the overall reaction stoichiomctry. Let the apparent concentrations so calculated bc Coxl, Credl, Cox, and CrcdZ at any selected point in the titration. These concentrations are inter-related through the reaction coefficients of eqn. (3). &Jove the eqzrivalence poi&, with the osidant as titrant, let the reaction dcficicncy &,~a, which has be denoted by the amount of Red 2, espressed as a concentration failed to react with 0x1 clue to incompleteness of reaction. Then, [,Os21
&a
= cm2 -;
2
[Retlz] = C,o,l a + Rrct12
[Red I] =
CIca I
CX
-
hv
(7)
CX
La 2
=
-
dy
LX
-” =
fire* l!J.v
2
A?Ju~. Chiwr. Ada,
26
(xgCi2)
397-405
E. BISHOP
400 Substitution
in (5) in terms of the reclnctant
system
gives dU
)
a
vn2A EIJ
L-__
(10)
0.0002T
n.lx = 122~~(IO) Ixcomes,
From the relationship
71*11a
A IL-0 =r
0.0002-r
in terms
of 921 and
722,
log
([I)
Since cqn. (IO) involves the L.C.M. of 121and n2, it may be of a lower order and therefort more readily solul~lc than cqn. (Ix). It is sclclom necessary to invoke the complctcly general case covet-cd by the preceding equations. Very commonly, n = b = c = d = I, but nl # TZ’L(i.e. x # y), when cqn. (10) simpliIies to (12) ancl eqn. (1x) simplifies
to Jl,?JaA 1:”
_--
(Gxa -_
QrwI,,)‘“L+“a’ .,
log
(13)
0.0002’l’
may again be of a lower order than eqn. (13). Application of the assumption that drct12becomes ncgligiblc with respect to Cox2 close to the equivalence point does not reduce the order of the equation, and it is only when the approach to equivnlcnce is so close that CTOd2becomes negligible with respect to &+ that the order diminishes and the equation collapses to first orclcr, a situation which clots not fall to be esamined in practice. For symmetrical reactions where nl = n2, which arc not uncommon, the equations resolve in to a simple quaclratic which is susceptible of formula solution : 33~111.(12)
Setting the antilog of the quantity (A -
x)&cc1
as
-i-
on the left hand side equal to A, (14) yields, (.4Crecl~ -I-2Cm
p&y--c"x3a
=
0
([5)
FoZZozvi~zg the equivalence fioint, again with the osidant as titrant, let the reaction deficiency be denotecl by the amount of Osl esprcssed as a concentration Soxl, which
CALCULATION
OF
DATA
FOR
REDOX
TITRATION
has failed to react with Redz. due to incompleteness cients of reaction (3), ~OXI] = c,,x*
+
CURVES
401
of reaction. Then, using the coeffi-
B,,x
(16)
L C
Substitution
in (3) in terms of the osidant
system gives
(20)
0.00027’
From the relationship )I, --~
n1.t: =
(20) hecomcs, in terms of PHI,It:!
(
112 A Ii,, E
my,
log
(21)
0.0002’f’
Once again in the more common (20) simplifies to
case whcrc n = b = c = d =
Xll, , 1EC, _-_--_-_~og o.ooc,2'I'
and cqn.
(21)
simplifies
I, hut nr # w, eqn.
(C wr,,-- O,,x,)‘=‘~~’ .____-__-_ (ht,,,)" Gx,
(22)
-I-fhx*P
to (23)
I:inally, for symmctricnl reactions whcrc ~11= nz, both quadratic (24), A having the same meaning as before
(20)
(A - I)r~,,x*~-I- (AC”“, -I- LCr,.#l(p,,xl-crtq
and --
(21)
rcsolvc into the
0
(=*I)
Solartion of the eqtratio?ts. Eclns. (IS) and (24) for symmetrical reactions are simply and easily soluble by manual methods. For totally unsymmetrical reactions, however, cqns. (II) and (21) may run up to the IGth order, though by using the form of (IO) and (20) an equation of the 8th order results. Even reactions with homogeneous coefficients may run cqns. (12), (13). (22) and (23) up to the 7th order. With the aid of the physical meaning of the quantities involved, manual solution is still possible by the method of successive approximations, but the operations are lengthy and tedious and a poor first guess occasions many circuits of the loop. The equations can be A trnl.
Chirn.
AC/n,
26
(1962)
397-405
E.
402
BISHOP
programmed quite easily for computational is both possible nncl fast. Culculatim
of cwve
analysis and solution by digital computer
data
Since the tlesirccl plot is of potential VCYSUStitrant volume, the basic variahlcs arc titrant volume and reaction deficiency. These may bc incorpor;Ltecl in one or more comprchcnsivc equations, but the equations then become cumbcrsomc xnd cause rcpetitional operations. It is therefore ‘more convcnicnt to use sequential operations, as follows. ’ (i) Calculate A from supplictt values of It and AE0. (ii) Calculntc COxzant1 Crc,l, (Iwforc equivalence) or C Ox1and CrcdI (after cquivalcnce) for the selected value of the iitrant volume (vide inf..a). (iii) Calculate f&l2 (before equivalence) or cjoxl (after ccluivalcncc) from the appropliate equations above, and the data of (i) and (ii). (iv) Calculate the true concentrations as rcquirccl from cclns. (G)-(g) or (rG)-(rg). (v) Substitute the concentrations from (iv) in tilt appropriate Ncrnst equation and calculate E. In operation (ii) let the molarity of the titrant (osidant) be MO, tlic molarity of the titrancl (rccluctnnt) be MII and the initial volumi: of the titrantl bc VIL ml. 13cforc the equivalcncc point, from the stoichiomctry of reaction (3), after the xlclition of v ml of titrant, the apparent conccntrutions arc given, in the alternative forms, by C ,,,,, C ,,,,(,, -
I
C’v Mu ;-_ ---. u(v -t- vn)
(25)
rlIl2vlL c1.vv I,AI ,I -- hy v Ilrlc, --.-_: -_----._ ----
”
U.Y(l/
-(-
Ivfll-
611,
IV,, (26)
rtrr,(v
vn)
v
-t
?.@a)
If-V 11 IV/, , tlur v rvro ------_ ~-m(v -f- VII) f&i12 (1~ -+- 2110
C (,x,, = .
(“7)
After the equivalence point, grcntcr precision is more rexlily acccssihlc by working in terms of the volume 7)’ of the titrant ad&xl in csccss of the equivalence volume. The cquivalcnce volume of titrant is
The total volume
V of the titration
ancl the apparent
concentrations
solution
*
V’ MC, ----.
ZY
1
( I’
-I-
CM2
----
7-_
I,\, ( I’ -}- v’)
c
(30)
v’)
cx M*,v,, :-_ I
point is then
in terms of V’ and V arc c,,,
C t#.,,
at the cquivalcncc
_-
6PlI (
M‘l
VI1
(31)
v -t_v’)
ttMnvll ,,x
-=
2
----
b(
v -t
(32 )
v’) A ~ICLI.C/rim. Ackb,
2G
(19G2)
397-405
CALCULATION
OF
DATA
FOR
REDOX
TITRATION
CURVES
403
Eqns. (25) to (32) simplify in an obvious way when the reaction coefficients are homogeneous and further when the reaction is symmetrical. Apart from the enhanced precision gained by the use of ‘u’ there is no need to change from one system to the other on passing the equivalence point. RESULTS
The method may bc illustrated by the csample of a symmetrical monoclectronic reaction, where a = b = c = d = nl = nn = I, in the titration of IOO ml of 0.1 M rcductant with 0.1 M oxidant. The significant results are presented in tabular form (Table I) for a reaction in which the difference in normal potential between osidant and reductant, AEo, is 0.2 V; the final column shows the value of the error in potential r5E arising from neglecting the incompleteness of reaction. The titration curve calculated for a reaction in which AEo is 0.3 1’ is given in Fig. I, togcthcr with the two llalf curves calculated without allowing for incompleteness of reaction. The proposed method gives a smooth curve passing accurately through the correct equivalence point potential, whereas the simple method gives two non-mating curves with a large gap in the middle despite the fairly high vnluc of AEo.
TITRATION OF 100 Illi 01’ 0. I d’f WHICH _-_
-..-_.-.----
N
=
0
-.-.-
HICDUCTANT =
c
=
d
WIT11
=
)I1
=
--.---__---
0. I 122
.-.---
=
1x1 OSIDANT I
,\NI~
----
.ilR”
AT =
3”” FOR A 100
RlThCTlON
FOR
IIlV
.._. _..._. --.^_----
--
hreda G.OI
98
* lo-..’
HH
39 99.5
8.31. IO-.! 9.3.) * I o-.’
93.75 97
99.8
I .oo.!
98
99.9
99.98 99.99 100
* 10-a
1.03*10-3 I .049 * 10-S 1.051’ 10-S 1.053*10-n
101 120
..I
13.4 LG.2
13s IO.!
4’ 6.)
IHO
99 39.5 100
81
I22 I .)O
2.22
240 -
100
Box3 100.0
I.o5I~Io-~
IOO.O‘?
I .O.)‘)’
10-a
100.1
r.o31* I .005
IO-3 * I o-3
100
--.#O
I .)O
100.5 101
--12 -+ 20
I.!2 XI
3x
6.4
101
-t-
103
-I- (J.?
4*
101
9.38’ IO-, 8.3.) * 10-J
I
-t- Ho
26.
IO.!
cl.OG*
I I I
.9
-t_9H
13.9
100.2 IOO.fi
._.-
I
.._ --_
lo-’
-_.-..--__---_._--__
106.
_.-- ..____ -- __._. --.-.-.--.-.----
I
-__-.-
It may be noted that in the simple method the calculation before equivalence bnsecl on the reductant system is entirely unconnected with the calculation after equivalence based on the osidant system, ant1 a separate calculation is required for the equivalence point potential. In the proposed method, the whole series of calculations, including that of the ecluivalencc point potential, can be made on tither rccluctant or osidant system: there is no need to change over on passing the equivalence point. Anal. C/rim. Aclo, zG (rgbz) 397-405
E.
404
BISHOP
The author wishes to cxprcss thanks to Mr. D. J. %ONE for valuable discussion on computer programming and to Dr. J. WILLIAMS and Mr. R. G. DHANESHWAR for checking the mathematics. APPENDIX
of the
kIxanafh
ntctl~od
of calc~ulation
for
synwaetvical
reactior~s
For instance, let nl = 712= z; u = 0 = c = d = I; LIE,, = 0.2 V; tcmpcraturc 3o”, Ii-lo = M,L = 5.1o-rild. = antilog 6.G67 == 4.045 * 10”. (‘I’llis is the cq\lilihrium (i) A = antilog (2.0.2/o.&) constant for tllc reaction). (ii) Aftei adding 99.9 ml (v) of OSiClilIlt to I00 nil (zl,() of rccluctnnt, from eclns. (2G) and (27), Crcda = 2.501. IO-+ md Coxz = 2.499. x0-2. (iii) From eqn. (15) tllc rcclction clcficicncy is given by r’iWll2 L: +-_-j
-
I
( ~(/lC”.‘l*
-I- 2c,,x,,)2 -I- .I (A -- I )c”“zq .
’ -- (‘K”.,lz
-I- 1C”X2) 1
(33)
Substitution of tile vducs of A, Crcti,,_ ancl Cox, frown (i) arid (ii) xml solution of (33) gives &,Q = 4.56 *Io-~~. (iv) From eclns. (6) and (7), [OSZ:~ = 2.499*10-z - 4.56. Io--~~= 2.499. IO-“; [Ii&] = Z.~OI- IO--B -I-4.56. IO-” ==1z.951. IO-~_
(VI Exnr~jde of the method of s~tcccssivc nfifimvinu~clions for steb (iii) for Itormgemorrs ccnd i~ihontoge~teoi~s rcacliom 2,3’ = 3; (4 = b = c = d = I; temperature For instiLllCc, Ict 911 --_ 3, jlz = 2; N = 3cP; d1.0 = 0.1 v; ICI<,= 3.33’IW-“, A111= 5*10-2. From step (i), A = x01”. After adding 99.9 ml of osiclnnt to IOO ml of rccluctant, C rctl, = 2.5'10-4, CJX,, = 2.5'10~2. (iili) From cqn. (12j, )b (3.5* IO -2- h,,.,l 3 I[>10-z?-----_------* to-0 -I-orrp 0;r,ul,,yr.5 .
or (2.5 * IO--~-*
On-II )3 2
..____--
h,p.s
I--.-* 10-b -t_ orP,p
._---.---
IO’”
--_ *
Inspection cluickly shows that to rcmovc the powers of IO, &(I., is of the or&r Try 2 x0-4. l
r..IXa. 10-111 --_-_ ----_---_> _)'~~-S.p,"*~3.~0-11.,010 A
trial at 2.5 * 10-'1 gives a value of less than I.
1
(34) of x0-4.
CALCULATION At =
2.3-10-4, o.g6. (iv)
=
2.58.
so [Os2]
OF DATA
FOR
2.4775~10-‘o/2.32~2.55~~IO-1~ brcdg = =
2.33.
REDOS =
TITRATION
X.06.
At
CUHVES
2.35*10-J,
405
2.4765s/2.35”s2.63
IO-Is.
2.5.1~~
-
2.33.10-d
=
2.48*10-2;
[I’
=
2.5.10-G
f
2.33.
IO-J
IO+. L?.‘@.Jo-~
0.06
I:2
=
IT,,*, _+ ___,_
}(Jg
-
2,5x.
,o-.I
-_
#y”
.I
:! ’
0.0305
\‘.
‘I’tic clcl~cndcncc
of ttic potentials during a rcdox titration tltmn tlic nholutc conccntmticms of tllc rcnctants in the gcncrrrt ciisc is dcmonstratcd and illi analogy is drawn Iwtwccn ttw cffccts of incomplctcncss of reaction in rctlos titrations and salt hydrolysis in mid-lmsc rc;uztions. ‘I’tlc interprch~iOn CJf dlc cc~tlilibriIlni consbnt of die O\I?ratt rcdOx rcactiOl1 is cl;lrifiCd, mitt ri~OrOllS C’Xprcssions For the calculation of rcnction clcficicncics arc’ clcvclqxxt. ~lcttiocls of calculating thtn for titration cur\93 Jnaking ;dhvancc for rcirction dcficicncy arc offcrcd togctlicr with cxamplcs of tticir
x.l~t~ticirtion.
I.‘;iutcur cffcctuc 11m2Ctudc thhriftuc sur In rclilticlJY cntrc Its twtcnticls cl Its conc62nlration~ ctcs rhctifs, au ccmrs d’lrn titragc r&lox, tnontmnt t’analo~ic cntrc Its clfcts tl’unc rhction inccmq~lbtc Des IlrCttIortcs dc c;11culs ttc tow tic titragcs rctlox ct t’hydrcalysc clans Its rchctions aciclc-tmsc. coclrbcs dc titragc sent prr>pusf?cs ct clcs cxcrnptcs scjnt tlonnds. iL
En wird rtic 2\thiingi~kcit dcr t’otcntiatc tIci l~cttox-‘l’itr;ltirln~J~ ~011 clcr I
SOME THEORETICAL PART
VI*.
THE
PRECISE
CHIBI ICA ACTA
397
CONSIDERATIONS CHEMISTRY
CALCULATION
IN ANALYTICAL
OF DATA
FOR
REDOS
TITRATION
CURVI;3S r:. BISI~IOI’
In the simple calculation of rcclos potentials for titration curves1 from the Nernst equation and reactant concentrations, the potentials are a logarithmic function of the ratio of the concentrations of tlic active species and appear to be inclcpcnclent of absolute concentration. ‘This also applies to the equilibrium constant of tllc overall titrimctric rcnction~ and, when the reaction coefficients are l~on~ogencous, to certain other quantitative factors”, but does not apply when the reaction cocfficicnts iLtX! In tht: latter circumstance, such factors as the cquivalcncc point inhomogcneouG3. potentials and the quantitutivcncss of the rcnction” become depcnclent on the absolute conccntrntions of the reactants. As will be shown, in cvcn the simplest rcactions having unit coefficients and involving the transfer of a single electron, the cnlculation of the potcntinls bccomcs concentration-tlcpcndcnt when illly attempt is mndc to take account of incompleteness of renction. The efIect 0f such reaction cleficicncy is to Ilattcn the horizontal part of the curve and to dccrcnse the rate of ClliLllgC of buffering is potential through the equivalence point. This enhanced potcntiostatic similar to the effect of hydrolysis of salts in acid-base titrations. The analogy with buffer solutiorPs and titrations involving such media l..l~bextends to the situation that whcrcas the simple calculation employs concentration ratios, precise calculation invokes absolute concentrations, and reveals that buffering is more efficient than the simple calculation predicts 6. The cstcnt of this potentiostasis may be judged from the esample of a customary titration with a difference in normal potential of osidant and reductant of 0.2 V and the transfer of a single electron, wherein at 0.1% from the equivalence point the alteration in potential due to incompleteness of reaction is no less than SI mV. For a reaction involving the transfer of two electrons and having a diffcrcnce in normal potential of 0.4 V, the corresponding effect of incompletcncss of reaction is diminished to o.G mV. The effect of increasing difference in normal potcntial in reclox titrations finds an analogy in the effect of increasing ionisation constant in acid-base titrations, so that reaction deficiency corresponds to solvolysis. Analogies can also be discovered with other types of ion combination reactions. The usual procedure in the preparation of a redox curve1 is to calculate the potcn* Part V see And.
C/rim. A da,
22
(rgG0)
205, Am~l. Chirn. Ada,
2G (IgG2)
3g7-.+05
398
E.
RISHOP
tial of the rcductant system prior to the equivalence point and the potential of the osiclant system after the equivalence point, and to combine the two half curves with the cquivalcnce point potential. The cffcct of neglecting the incompleteness of reaction is immediately revealed in examples such as the one first mentioned (CT].Fig. I). The two halves of the curve refuse to mate, although a continuous curve can be recorded espcrimentally and the interval between the normal potentials is generally consiclcred acceptahlc.
Vol. of O.lMoxldont
added
I;&. I. ‘I’itrrrticm CJf 100 nil of 0.1 lW rcclucta~lt with 0.1 hi oxiclant at 30~. *IIT,, - 0.3 V; u - 0 I G=d=,,,=,,.,=: I. A, A’, cA2ulatctI by ttw simple tncthrxl tlircctly from nppnrcnt conccntrntions nncl the Ncrnst equation. L3, cnlculatcil 1)~ tlic clcscribctl nicthocl.
No account of this situation, other than at the cquivalcncc point*ea, appears to be record4 in the standarc tests. It leas proved possible to evolve a mathematical treatment for the completely general cnsc, and mcthocls of calculating the reaction deficiencies and the points for titration curves arc prcscntcd here. Some observations on the consequences of the incompleteness of rcclox reactions will be offered in a subscclucnt Part.
The incompleteness of reaction at any sclectecl stage in the titration can be derived from the equilibrium constant, which in turn can bc derived from the normal potcntinls of the reactants. The apparent concentrations of the reactants can then be corrected for reaction deficiency and the true values used in the appropriate Ncrnst equation to calculate the potcntinl at the selected stage in the titration. It is first necessary to clarify the interpretation of TN,the number of electrons transferred, which appears in the expression for the equilibrium constantn. Let the osiclant system, of normal potential EON, be represented by eqn. (I), and the reductant system, of normal potential EON,be represented by eqn. (2). MultiplicaA #Ical. Chim.
A c/a, at3 (rgG2)
397-405
CALCULATION
tion of eqn. the balanced
(I)
OF DATA
FOR
REDOX
TITRATION
CURVES
by x and of eqn. (2) by y where 9tl.x = stay, followed eqn. (3) for the overall titrimetric reaction, uO.u, + 111~;i ORed anOxl
The equilibrium
constant
*
+
by addition gives
cRcdl
712E+
+ byRed
399
(1)
dOxs
cxRcdl
(2) + cZyOxz
for the overall reaction
(3)
is then (4)
At any selected point in the titration, when the reaction has reached equilibrium, potentials of the two systems,E will be the same. E = fut f
0.00021’
log
[0x13”
nI
Since
En
1
-
1%
Jl IX
,fzlLx = nny, ant1 writing A-11 I /.I 1:‘~ ---
0.0002’1’
&
=
80~
0.0002’l
=+--
[Rctl~ J”
0.0002’1 =
-
[lZcd,]e
l’JJ2 L-m
=
Eo, /t 1%
0.0002T
=
[Oxr]“~
4-
log
JJ??
=
KwJ [IZc&]b
0.0002?
[OXI]“’
Eo,-
,og
n2
the
fRcdz]*”
.A&,
log-‘--
I ICcdJ~~~
[.OX2]JY
[.OXI]~= (Red+
=
log
fi
(5)
The number of electrons in the espression for the equilibrium constant is therefore the least common multiple of the number of electrons transferred in each of the two half reactions. Reaction
deficieucy
In the simple calculation of curve data assuming the reaction to be complete, the ratio of [Ox] to [Red] is alone rcquircd, and is derived from the fraction of the equivalence volume of titrnnt added before the cquivalencc point, and thereafter from the excess of titrant : concentrations and dilution factors cancel out. Concentrations, when needed, can be calculated from the initial concentration and volume of titrand, the initial concentration and volume of titrant and the overall reaction stoichiomctry. Let the apparent concentrations so calculated bc Coxl, Credl, Cox, and CrcdZ at any selected point in the titration. These concentrations are inter-related through the reaction coefficients of eqn. (3). &Jove the eqzrivalence poi&, with the osidant as titrant, let the reaction dcficicncy &,~a, which has be denoted by the amount of Red 2, espressed as a concentration failed to react with 0x1 clue to incompleteness of reaction. Then, [,Os21
&a
= cm2 -;
2
[Retlz] = C,o,l a + Rrct12
[Red I] =
CIca I
CX
-
hv
(7)
CX
La 2
=
-
dy
LX
-” =
fire* l!J.v
2
A?Ju~. Chiwr. Ada,
26
(xgCi2)
397-405
E. BISHOP
400 Substitution
in (5) in terms of the reclnctant
system
gives dU
)
a
vn2A EIJ
L-__
(10)
0.0002T
n.lx = 122~~(IO) Ixcomes,
From the relationship
71*11a
A IL-0 =r
0.0002-r
in terms
of 921 and
722,
log
([I)
Since cqn. (IO) involves the L.C.M. of 121and n2, it may be of a lower order and therefort more readily solul~lc than cqn. (Ix). It is sclclom necessary to invoke the complctcly general case covet-cd by the preceding equations. Very commonly, n = b = c = d = I, but nl # TZ’L(i.e. x # y), when cqn. (10) simpliIies to (12) ancl eqn. (1x) simplifies
to Jl,?JaA 1:”
_--
(Gxa -_
QrwI,,)‘“L+“a’ .,
log
(13)
0.0002’l’
may again be of a lower order than eqn. (13). Application of the assumption that drct12becomes ncgligiblc with respect to Cox2 close to the equivalence point does not reduce the order of the equation, and it is only when the approach to equivnlcnce is so close that CTOd2becomes negligible with respect to &+ that the order diminishes and the equation collapses to first orclcr, a situation which clots not fall to be esamined in practice. For symmetrical reactions where nl = n2, which arc not uncommon, the equations resolve in to a simple quaclratic which is susceptible of formula solution : 33~111.(12)
Setting the antilog of the quantity (A -
x)&cc1
as
-i-
on the left hand side equal to A, (14) yields, (.4Crecl~ -I-2Cm
p&y--c"x3a
=
0
([5)
FoZZozvi~zg the equivalence fioint, again with the osidant as titrant, let the reaction deficiency be denotecl by the amount of Osl esprcssed as a concentration Soxl, which
CALCULATION
OF
DATA
FOR
REDOX
TITRATION
has failed to react with Redz. due to incompleteness cients of reaction (3), ~OXI] = c,,x*
+
CURVES
401
of reaction. Then, using the coeffi-
B,,x
(16)
L C
Substitution
in (3) in terms of the osidant
system gives
(20)
0.00027’
From the relationship )I, --~
n1.t: =
(20) hecomcs, in terms of PHI,It:!
(
112 A Ii,, E
my,
log
(21)
0.0002’f’
Once again in the more common (20) simplifies to
case whcrc n = b = c = d =
Xll, , 1EC, _-_--_-_~og o.ooc,2'I'
and cqn.
(21)
simplifies
I, hut nr # w, eqn.
(C wr,,-- O,,x,)‘=‘~~’ .____-__-_ (ht,,,)" Gx,
(22)
-I-fhx*P
to (23)
I:inally, for symmctricnl reactions whcrc ~11= nz, both quadratic (24), A having the same meaning as before
(20)
(A - I)r~,,x*~-I- (AC”“, -I- LCr,.#l(p,,xl-crtq
and --
(21)
rcsolvc into the
0
(=*I)
Solartion of the eqtratio?ts. Eclns. (IS) and (24) for symmetrical reactions are simply and easily soluble by manual methods. For totally unsymmetrical reactions, however, cqns. (II) and (21) may run up to the IGth order, though by using the form of (IO) and (20) an equation of the 8th order results. Even reactions with homogeneous coefficients may run cqns. (12), (13). (22) and (23) up to the 7th order. With the aid of the physical meaning of the quantities involved, manual solution is still possible by the method of successive approximations, but the operations are lengthy and tedious and a poor first guess occasions many circuits of the loop. The equations can be A trnl.
Chirn.
AC/n,
26
(1962)
397-405
E.
402
BISHOP
programmed quite easily for computational is both possible nncl fast. Culculatim
of cwve
analysis and solution by digital computer
data
Since the tlesirccl plot is of potential VCYSUStitrant volume, the basic variahlcs arc titrant volume and reaction deficiency. These may bc incorpor;Ltecl in one or more comprchcnsivc equations, but the equations then become cumbcrsomc xnd cause rcpetitional operations. It is therefore ‘more convcnicnt to use sequential operations, as follows. ’ (i) Calculate A from supplictt values of It and AE0. (ii) Calculntc COxzant1 Crc,l, (Iwforc equivalence) or C Ox1and CrcdI (after cquivalcnce) for the selected value of the iitrant volume (vide inf..a). (iii) Calculate f&l2 (before equivalence) or cjoxl (after ccluivalcncc) from the appropliate equations above, and the data of (i) and (ii). (iv) Calculate the true concentrations as rcquirccl from cclns. (G)-(g) or (rG)-(rg). (v) Substitute the concentrations from (iv) in tilt appropriate Ncrnst equation and calculate E. In operation (ii) let the molarity of the titrant (osidant) be MO, tlic molarity of the titrancl (rccluctnnt) be MII and the initial volumi: of the titrantl bc VIL ml. 13cforc the equivalcncc point, from the stoichiomctry of reaction (3), after the xlclition of v ml of titrant, the apparent conccntrutions arc given, in the alternative forms, by C ,,,,, C ,,,,(,, -
I
C’v Mu ;-_ ---. u(v -t- vn)
(25)
rlIl2vlL c1.vv I,AI ,I -- hy v Ilrlc, --.-_: -_----._ ----
”
U.Y(l/
-(-
Ivfll-
611,
IV,, (26)
rtrr,(v
vn)
v
-t
?.@a)
If-V 11 IV/, , tlur v rvro ------_ ~-m(v -f- VII) f&i12 (1~ -+- 2110
C (,x,, = .
(“7)
After the equivalence point, grcntcr precision is more rexlily acccssihlc by working in terms of the volume 7)’ of the titrant ad&xl in csccss of the equivalence volume. The cquivalcnce volume of titrant is
The total volume
V of the titration
ancl the apparent
concentrations
solution
*
V’ MC, ----.
ZY
1
( I’
-I-
CM2
----
7-_
I,\, ( I’ -}- v’)
c
(30)
v’)
cx M*,v,, :-_ I
point is then
in terms of V’ and V arc c,,,
C t#.,,
at the cquivalcncc
_-
6PlI (
M‘l
VI1
(31)
v -t_v’)
ttMnvll ,,x
-=
2
----
b(
v -t
(32 )
v’) A ~ICLI.C/rim. Ackb,
2G
(19G2)
397-405
CALCULATION
OF
DATA
FOR
REDOX
TITRATION
CURVES
403
Eqns. (25) to (32) simplify in an obvious way when the reaction coefficients are homogeneous and further when the reaction is symmetrical. Apart from the enhanced precision gained by the use of ‘u’ there is no need to change from one system to the other on passing the equivalence point. RESULTS
The method may bc illustrated by the csample of a symmetrical monoclectronic reaction, where a = b = c = d = nl = nn = I, in the titration of IOO ml of 0.1 M rcductant with 0.1 M oxidant. The significant results are presented in tabular form (Table I) for a reaction in which the difference in normal potential between osidant and reductant, AEo, is 0.2 V; the final column shows the value of the error in potential r5E arising from neglecting the incompleteness of reaction. The titration curve calculated for a reaction in which AEo is 0.3 1’ is given in Fig. I, togcthcr with the two llalf curves calculated without allowing for incompleteness of reaction. The proposed method gives a smooth curve passing accurately through the correct equivalence point potential, whereas the simple method gives two non-mating curves with a large gap in the middle despite the fairly high vnluc of AEo.
TITRATION OF 100 Illi 01’ 0. I d’f WHICH _-_
-..-_.-.----
N
=
0
-.-.-
HICDUCTANT =
c
=
d
WIT11
=
)I1
=
--.---__---
0. I 122
.-.---
=
1x1 OSIDANT I
,\NI~
----
.ilR”
AT =
3”” FOR A 100
RlThCTlON
FOR
IIlV
.._. _..._. --.^_----
--
hreda G.OI
98
* lo-..’
HH
39 99.5
8.31. IO-.! 9.3.) * I o-.’
93.75 97
99.8
I .oo.!
98
99.9
99.98 99.99 100
* 10-a
1.03*10-3 I .049 * 10-S 1.051’ 10-S 1.053*10-n
101 120
..I
13.4 LG.2
13s IO.!
4’ 6.)
IHO
99 39.5 100
81
I22 I .)O
2.22
240 -
100
Box3 100.0
I.o5I~Io-~
IOO.O‘?
I .O.)‘)’
10-a
100.1
r.o31* I .005
IO-3 * I o-3
100
--.#O
I .)O
100.5 101
--12 -+ 20
I.!2 XI
3x
6.4
101
-t-
103
-I- (J.?
4*
101
9.38’ IO-, 8.3.) * 10-J
I
-t- Ho
26.
IO.!
cl.OG*
I I I
.9
-t_9H
13.9
100.2 IOO.fi
._.-
I
.._ --_
lo-’
-_.-..--__---_._--__
106.
_.-- ..____ -- __._. --.-.-.--.-.----
I
-__-.-
It may be noted that in the simple method the calculation before equivalence bnsecl on the reductant system is entirely unconnected with the calculation after equivalence based on the osidant system, ant1 a separate calculation is required for the equivalence point potential. In the proposed method, the whole series of calculations, including that of the ecluivalencc point potential, can be made on tither rccluctant or osidant system: there is no need to change over on passing the equivalence point. Anal. C/rim. Aclo, zG (rgbz) 397-405
E.
404
BISHOP
The author wishes to cxprcss thanks to Mr. D. J. %ONE for valuable discussion on computer programming and to Dr. J. WILLIAMS and Mr. R. G. DHANESHWAR for checking the mathematics. APPENDIX
of the
kIxanafh
ntctl~od
of calc~ulation
for
synwaetvical
reactior~s
For instance, let nl = 712= z; u = 0 = c = d = I; LIE,, = 0.2 V; tcmpcraturc 3o”, Ii-lo = M,L = 5.1o-rild. = antilog 6.G67 == 4.045 * 10”. (‘I’llis is the cq\lilihrium (i) A = antilog (2.0.2/o.&) constant for tllc reaction). (ii) Aftei adding 99.9 ml (v) of OSiClilIlt to I00 nil (zl,() of rccluctnnt, from eclns. (2G) and (27), Crcda = 2.501. IO-+ md Coxz = 2.499. x0-2. (iii) From eqn. (15) tllc rcclction clcficicncy is given by r’iWll2 L: +-_-j
-
I
( ~(/lC”.‘l*
-I- 2c,,x,,)2 -I- .I (A -- I )c”“zq .
’ -- (‘K”.,lz
-I- 1C”X2) 1
(33)
Substitution of tile vducs of A, Crcti,,_ ancl Cox, frown (i) arid (ii) xml solution of (33) gives &,Q = 4.56 *Io-~~. (iv) From eclns. (6) and (7), [OSZ:~ = 2.499*10-z - 4.56. Io--~~= 2.499. IO-“; [Ii&] = Z.~OI- IO--B -I-4.56. IO-” ==1z.951. IO-~_
(VI Exnr~jde of the method of s~tcccssivc nfifimvinu~clions for steb (iii) for Itormgemorrs ccnd i~ihontoge~teoi~s rcacliom 2,3’ = 3; (4 = b = c = d = I; temperature For instiLllCc, Ict 911 --_ 3, jlz = 2; N = 3cP; d1.0 = 0.1 v; ICI<,= 3.33’IW-“, A111= 5*10-2. From step (i), A = x01”. After adding 99.9 ml of osiclnnt to IOO ml of rccluctant, C rctl, = 2.5'10-4, CJX,, = 2.5'10~2. (iili) From cqn. (12j, )b (3.5* IO -2- h,,.,l 3 I[>10-z?-----_------* to-0 -I-orrp 0;r,ul,,yr.5 .
or (2.5 * IO--~-*
On-II )3 2
..____--
h,p.s
I--.-* 10-b -t_ orP,p
._---.---
IO’”
--_ *
Inspection cluickly shows that to rcmovc the powers of IO, &(I., is of the or&r Try 2 x0-4. l
r..IXa. 10-111 --_-_ ----_---_> _)'~~-S.p,"*~3.~0-11.,010 A
trial at 2.5 * 10-'1 gives a value of less than I.
1
(34) of x0-4.
CALCULATION At =
2.3-10-4, o.g6. (iv)
=
2.58.
so [Os2]
OF DATA
FOR
2.4775~10-‘o/2.32~2.55~~IO-1~ brcdg = =
2.33.
REDOS =
TITRATION
X.06.
At
CUHVES
2.35*10-J,
405
2.4765s/2.35”s2.63
IO-Is.
2.5.1~~
-
2.33.10-d
=
2.48*10-2;
[I’
=
2.5.10-G
f
2.33.
IO-J
IO+. L?.‘@.Jo-~
0.06
I:2
=
IT,,*, _+ ___,_
}(Jg
-
2,5x.
,o-.I
-_
#y”
.I
:! ’
0.0305
\‘.
‘I’tic clcl~cndcncc
of ttic potentials during a rcdox titration tltmn tlic nholutc conccntmticms of tllc rcnctants in the gcncrrrt ciisc is dcmonstratcd and illi analogy is drawn Iwtwccn ttw cffccts of incomplctcncss of reaction in rctlos titrations and salt hydrolysis in mid-lmsc rc;uztions. ‘I’tlc interprch~iOn CJf dlc cc~tlilibriIlni consbnt of die O\I?ratt rcdOx rcactiOl1 is cl;lrifiCd, mitt ri~OrOllS C’Xprcssions For the calculation of rcnction clcficicncics arc’ clcvclqxxt. ~lcttiocls of calculating thtn for titration cur\93 Jnaking ;dhvancc for rcirction dcficicncy arc offcrcd togctlicr with cxamplcs of tticir
x.l~t~ticirtion.
I.‘;iutcur cffcctuc 11m2Ctudc thhriftuc sur In rclilticlJY cntrc Its twtcnticls cl Its conc62nlration~ ctcs rhctifs, au ccmrs d’lrn titragc r&lox, tnontmnt t’analo~ic cntrc Its clfcts tl’unc rhction inccmq~lbtc Des IlrCttIortcs dc c;11culs ttc tow tic titragcs rctlox ct t’hydrcalysc clans Its rchctions aciclc-tmsc. coclrbcs dc titragc sent prr>pusf?cs ct clcs cxcrnptcs scjnt tlonnds. iL
En wird rtic 2\thiingi~kcit dcr t’otcntiatc tIci l~cttox-‘l’itr;ltirln~J~ ~011 clcr I