- Email: [email protected]
The locations of inflection points on titration curves for symmetrical redox reactions
ELECTROASALYTICXL
EIsevier
THE
CHEhlISTRY
Publishing
Company,
LOCATIONS
JAMES
A
IXTERF_%CIAL
- Printed
OF~INFLECITON
SYMMETRICAL
Department
AND
A4msterdam
REDOX
373
ELECTROCHEXISTRY
in The Netherlands
POINTS
ON
TITRATION
CURVES
FOR
REACTIONS
GOLDX-IN ofChemistr_~,
(Receiwzoi October
PoZyytechnic
3Ist.
of Brooklytr,
T?zstitrrle
Brooklyx,
Xew
York
(U.S._4.)
1966)
IXTRODUCTION
The question of whetherornotthelocation of the equivalence point coincides with the point of m&mum slope on redox titration cnrves hadbeen previously investigated by KOLTHOFF XVD FURXXX~. A numerical evaluation was made of the variation
of
cuprous
ion.
Fotential The
in
slope
the
titration
of
of the
titration
curye
a solution was
of
ferric
e\-aluated
ion
with
at 0.10;,
a solution
prior,
and
of
0.1%
point_ Frcm this, it was concluded that the location of just at the equivalence point" and that the “curve is symmetrical on both sides of this point". However, in retrospection, it should be recognizedthatthis numerical example couldhardlybe expected to provide evidence subsequent,
the
to the
maximum
equivalence
slope
“occurs
of any small deviations from symmetry because the difference between the values ofthetwoformalpotentialsisfairlylarge,viz., O.-V = AZ?"'_ Indeed. these authors later present a mathematical formulation of the variationin the value of the slope of the titration curve applicable to this same example, i-e_, Fez+ + Cu+ = Fe’+ + Cu’+. The conclusion was, that for any symmetrical (a1 =n3, i-e_,both reversible electron transfer couples involve the sauce number of electrons sothatthe general representation of the titration reaction is: Osl+Redz = [of the slope of the titration Re&+ 0x2) redox reaction, the "theoretical maximum curve]
occurs
before the
equivalence
point"_
No
approximations
were
required
in
the mathematical derivation used to arrive at this conclusion. It is further stated that "in cases wherethetitrationis practical, ___ the difference is so smallthatit can hardly be estimated"_ By nse of an appro-tiation, it was. however, possible to
that the magnitude of the slope just prior to the equivalence point is than at the equivalence point, but that the “difference is actually so small that it,WC&L not be observed”_ It was emphasized that the appro ximationsused demonstrate
aZways grsafev
are “only
of the same
order"
as the
difference
between
the
value
at the
point
of
maximum slope and that at the equivalence point. 01zZy for titrations where LIE"'>, 591 mV, was it possible, by use of further appro.ximations, to numerically evaluate the fraction titrated at which the point of maximum slope occurs_ The general conclusion was that "even in the simple case __. we get equations which are impracticable au~difficulttosolve". The next major theoretical investigation of the properties of redox &ration curves was made by MURGULESCU _QJD DR~GULESCU~ who deconstrated that, rigorou+y, the inflection point (corresponding to the point of mtimnrn slope) can
-. 374
J_ X.
.-: -~..~
GOLDMiX
i
never coincide with the equivalence point-even when the redo,u titration reaction is before any approxim.ations were made, from the symmetrical. This was evident, equ&ion presented for d2E/dfZ (frepresents introduction of appro_xirnations,~ the general the
location
of the
inflection
point
predicts
the fraction t&-ated): Hoxvever, after the equation presented for the evaluation ,of that
the
inflection
point
does coincide
with ~the equivalence point in a s_vmmetrical redox reaction_ Nevertheless, .it uras proposed-that the value of this equation was its utility for characterization of asyrrmetrical redox reactions. However, it should be noted that it was indeed a more general equation
than
any presented
by KOLTHOFF -*ND FURXAN l_ been presented equation 3m4has recently
for the description Because a general of titration curves for homogeneous and symmetrical redox reactions, it is now approptiate to re-investigate the location of infiection points on these curves_ It will be demonstrated that no approximations are required treatment of curves for s_v~~ntet~icaZ reactions and that
iu the following theoretical the location of the point of
~.G&zzrnz slope, as well as that of the maximum slope, may be readily determined. It is confirmed that the point of ma_&~zzc~~ slope always $~ecedes the ~equivalence point, whereas it is newly shown that the point of mirtiwzztpn slope aZways OCC-m-s sztbseqzcenf to the point at which f = +_ However, there are valueCbf AE"' for which the titration curve possesses neither a point of maximurn nor rninirnum slope. In addition, the equations presented are rigorously valid for the evaluation tions of the inflection points on curves for LZ+Z~value of AE”’ where these THE
E_XISTEXCE
_%XD LOCATIONS
For homogeneous
OF
THE
IXFLECTION
and symmetrical
redox
of the locapoints exist.
POLVTS
reactions
of the type
122Ox1 + nl Red-z. = ~2~RedI + x1 0x2 where a solution f&&ion titrated f
=
(1)
cant air-ring Red? is being titrated U@Z a solution at any potential, E, may be calculated from
{I+-
exp(nnlv))/(r+k
where ~,CJ = (F/122-) AE”’ = (RT/nl~c-F)
(E--E*), ln K.
containing
0x1,
the
=p(---nf:v)}
(2)
k = exp (--)~a), The
6 = (F/RT) AE”‘, -tz= nl~~/(m~+~z~)), and equik.lence-point potential is designated by E*, K
represents the equilibrium constant for the reaction defined by emqn. (I), and R, T and F have their customary significance_ Because only synimetrical (where ?zr = nz = p) reactions are to beg considered, , eqn(2) may be rewritten as f from .’
=
which &@ L=df.
(3)
e&P CP7p)i/G + k exF (:Pvl).}
C’.+e
one obtains .:
@ F
jr+~~p(i&J)]”
~k~~k+elcpCp~)i-esp(-pip)].
.~ _~
-.-.
..
_.
(41 .~
:_
obt+ined frdm the. .Wi&& restrictionthat IZ; =~z = P,-eqn_(~Q may. also be' readily --.rijncr~.-gene~aie;~~~~o~.previodsly.present.~5_ Equation(G), $hich pemits ~the- evalu--. : .- ation..atl&y:point,pf the_ slope of ., the titratiorx curve,.-may-conveniently~be rewritten
.. ---a& I:.‘;; -1~ -:~-_.__.~.~-2:. _. .:~. -.-'.;. : .~:__ -.
~-1J-:~~ed;ak~~l_-c~~*~, -& &$j
373-353
,_ :_~--L:
:~. ,- .‘. -:_
:
.: .-. .~-.~_.~. ~,._~ -~ ~, ~.,
._..-_:~_~
I -,-ji~ -:..
:
1~.__;,~:~ -__.~~ _
.I1_-:-._~
dl
--_.
=
U’f’
_ _.
I;
df which sfiould
[r +A t?XI’( __-jy,)]”__ .-----. zpk [ k + cash (py~) j
fume is somewhat 111orc’ convcnicr1t fcjr tl1c nun1cric:;ll evaluation be noted tlIat the slope is al7~zys fmsiliw :1ncl ncvcr ctqr1z-dto zero. At
tl1c cciuiv;llcr1cc
I )iffc:rentiatiori
+
puiIIt,
~ff_x:~~~)(-~1/,)]~~S~,(~‘VJ)-~(‘”P(-~J~J)] ---.----
:1ncl (1 fiI:1y I1tt convctnictn
-
nplmrcnt
tll:It
-
t ly rcwrit
---. - -
ttlus
0, >v tlIiIt>
-
_-
+csp(
2k I'eXp ( - fiy)]
is
It
of c?cln, (-4) yields
t zk-texy(plp)
It
w -
of dE/clj.
_
___
ten ‘2~;
+c:osil
.--
tlic
_.
--&h&l
---
j+
2[/z-+&&~l(@jJj~"
sccon(1
detrivat
r + k cusp ( --.-fiy~) ] sin11 (,hy!) -_ -
ivc: is always
rlc-t;ativt:
at
tile
cquivdcncc
tl!:1t ttie cc~uivdrlcricc point rnllst always lx: located sztbscqr4enCto tllc: infIcctic~r1 pc,irit corrc5l>oriding to tl1e location of tilt: point of maximum slope. Tl1i5 i5 in ag1ccrrie11t wit11 ttie earlier conclusions in tile literaf urt:*.“. At the inflcctio11 IH)ints it is ncc:cssary ti1at cl”E/rl/’ = o. ‘l’hen. because I)c)irit,
arlcl tl1crttfcjre
I t k c-up(-py) r11erely r,equirttci rc;kx I ;irqqcrlieIit k
_
-
and U{df that
UC dwiiys firlitc, psitivc.
Q = o. Equating
Itanh (fiWd1 bv -------
(Pwdl
2 -t tan11 (PVC)
fc,r tllc
rc:I;~t-icmsi~i~~ existirig
t~c:twc~ri
the
-
numerator
ar~cl II~VU of Q to zero,
ecl~l yields
2k2/cosh (fly&) - --. 2 + iaI-&~fiVJr jyJi-tlie
value
of ty at the
to zero. it is after
some
(13) iriflcctiori
point--and
the corresponding value of k. It is evident that far k
that
satisfy
eqn.(13)
to have physical significwxc (k 20). the values of yj~ must lx negative. Thus, again it is verified that the inflection f.
fi_Vlrclrounal. Ch‘*,n., 1.1 (1967) 373-.3B3
376
.:-
.I’
_
J _ ~A_. GOLDMAN
~.
..~
point
~5~ecedes~the equivahkce
point. As
k= ---ky.‘The valUes of k thatsatisfythik no physical si@fiGncc b&the former
pyi-+o,
-it
is seen that
eqn(r3)
becomes :
equality are o.andI-1: Thelatterpossesses corresponds to an infinitely large difference
betieenthe- tivo formal potentials (R = exp( - 0_5p) (F/RT)AZ?) _ Therefore, it -is to be~expected that as the potential difference (4iZ”‘) increases, the difference between the location of the equivalence point and that of the inflection point will decrease. Equation(r3) has been solve;& using the method of successive approximations; to obtain values of &YS for various values of k,, and the results are presented in Table I. For each iralue of pr,~i, the corresponding value offf was calculated from eqn. (3), and the slope, psi, was calculated from eqn(5). For purposes of comparison, the value of the slope, $S* atf = I is also given_ TABLE THE
I-
LOCATIONS
9x1 =
7aLq=
pilEo’
p.
OF THE
50
OF I\IASI?.IUBI
K
--p(-Es-E*)
fs
PS,
PSZ
(ml') -
0.378 o-143
no inflection
11-04
o-897
o-0539
3.03
300
0.020+
I-IO
250 300
o-00775 o.oo~gI
0.403 0.150
150
SLOPE
“9
ffnT1 IO0
POIKTS
The first column
o.ga7s 0.9983 0.9998 o-99998
lists values of ~A??‘_
16-o 1OS.I
252.5 64"_7 1670 4425-Z
101.7
252.1 Gq2.2
1669.7 4411-7
The values in all succeeding
columns
-have been calculated for zsO_ The second column lists the values of k calculated fromk =exp(-$AE"'F/zRT). Inthethird~olumnarelistedthevaluesofp(Ei-E*), values of k by use of eqn(r3). Et viz., (RT/F)p?yi obt ained from the corresponding represents the potential at the poiut of maximum slope and E*, the equivalence-
point potential_ The fourth column lists the ~values of f at this inflection point, calculated by use of eqn.(3), and the figures in the fifth and sixth columns were calculated by the use of eqns_(s) and (6),respectively_ Some of the values in the sixth column
differ from
those
previously
presented5 which were not~directly calculated @th no~special caution exercised-+th ~respecttosign;ficar~tfigures)from(dJ?/4f)~- f lexpressedasafnnctionoftheequilibrium 50 mV, ,the s&o~z~ entry in the first constant for eqn.(I)_ If p=3_, then for AE"'=
from the presenteqn.(6).butratherevaluated
column &stbeusgd,and Et-E* will be only -5-3" mV; f~will occurato:Sg_i,but Sswilln~~vbe54ascompare;dto5~_4=Ss*. It is -evi$entm~from Table I that as the .difference between ;the,tko formal potentials increases, the.value of-fiapproakhek unity, &thattithpAE"' 2300 mV,_ the point of maximum slope is indeed hardlym&tm@shable from the equivalence -.pbint,~h~._~~~ys6e~~pr~umed_Foranyp~ticularvalue of$&Y”‘.the magnitude equivalence -Of the-'sIo~peat the infl&ctiop poin~t,~55 +ji_salwaysgreaterthanthatatthe ~PoiTlt~whfe_~e~slope~_is &S~, -but- as ffre:yaIue of.$wQ' irkreases .the .&ffere_& ~.b;etw~~~~lthe-values~of-the two slopesbecornes~in_~~~ngly ~cgligible. -For anY_partic-- : ; +I&-- &_&& .&._;-~~~c~ ,~_the, ~differenc+. be-t&en Ei:g&d.:,i??*,‘: kd bee& St: and_ S+;.I ~. ~.. ~. . _ . :. I’: 1. 1;::._.. _;.__.-, _~‘__ J-. -. 1_.: ~.~ ~ -. _I_ ,. -. . _ ~-..- _._-_._..._-2 -._ , -_Ii;_~~~~~~~ai~-:~~lrr;,:rq .(f9673:~_~~13$3~~~~.~-. .-.- f_; ;I:.- .: r-1 1. _,l_ ;: -~. .~ ~~I.:: __: _;~: I_._-._-..-_~: -;m-‘_.~_~
INFLECTIO?J
POINTS
decrease
ON
as the value
TITRA-I-ION
CURVES
of p-increases_
377
When
PAZ?'=
50_mV,no
pointofmaximum
slopeexists Therefore, it is concluded that in titration-curves for symmetrical reactions, the pointofmaxinnum slope can never coincide with the equivalence
redox point
but must always precede it. As the difference between the two formal potentials the locations of the two becomes increasingly positive, the difference between points becomes less, so that for any practical titration (where the formal potential difference is sufficiently positive) the difference betweenthelocations of the equivalence point and the point of maximum slope will be too small to be experimentally observed. Investigation of the properties k there are actually two values of@+
of eqn(13) shows that for a particular whicharephysicallymeaningfulmathematical
value of
solutions. The values ofp(Ei-E'*) in Table I correspond to values ofPr+ such that I~lyrlI, the inflection whereas
whenever
meaningful mathematical solutions of eqn_(Ix)_ point corresponds to the point of mi~t~~tu~lt slope,
Ipr,uil
correspondstothe
pointof~zaximz~m
slope.
THE
LOCATIOSS
OF
THE
POIKTS
OF
3IINIMUM
--p(Ei-
E’)
SLOPE
fr
Psi
PS
(nzV) 50
I00 =50
0.37s O-143 o-0539
44-68 74-47
200
0.020~
99-90
250
0.00775 0.002g1
300
124.8 I$Q-95
atf=+
-_P(E-E*) atf=$ 30-4 50.97 75-=4 100.02 125.01
150.005
no inflection o-593 O-507 0.500s 0-5000s
102.57 102.67s
78.4 97-I 101.8 102.59 102.70
0.50003
~02.680
103.7oG
96.1 X01.4
-4tthepo~tofmill;mumslope,valuesoffr,andSiIllaybecatculatedbyuseof eqns.(3) and (5), respectively. For purposes of comparison, the values
_f=&
are also presentedin
Table
z alongwiththe
of the slope at
values of+@-E*)
corresponding
presented values of p(E--_a"') at f=$_ At f=&, and the values ofplp so calculateclweresubstituted pr+~= --qps+P(E-E~"')(FIRT). intoeqn.(5)toobtainthevaluesoftheslopeatthesepoi~tsonthecurve_ It is evident from Table 2 that the location of the point of minimum slope on the titration curve is a&ays- szrbsepemt to the point at which f=_$, although as O'increases, the location of this inflection point (corresponding to the minimum PM slope) approachesf=& so that for AE"' 2300 mV the differences in location are negligible because even at AE"'= zoo_mV. tfie difference is conside+ably 1~ than calculated
from
the
previously5
theexperimentaler-rorinlocatingthes~~ints.
alwayslargerthanitii at the value for AE"'~zoo~ mV.-JWhenpAZ?'=~o-mV,
Themagnitude
offs, although no
oftheslopeatf=+is
the differencebecomes
point
of ininimum
slope
ne&ible exists.
Ai
for this saluc of +E”‘, there is no point ‘of maximum sloyc. The rw&ed, first coIumn of Table 3, fists values of p&E”‘; the values & all succeeding columns have been calculated at 25O. Columrr. z lists the corrcspoxxding ~-alue~ of k, and in the third column are listed the ‘values of $(EI - E*) corresponding to the point of minimtzrn slope and c&x.&ted from cqxx.(x3). The fow.+h column lists the values of ill obmay
lx
tainedvia eqn.(s). at the inflection point where the slope has its minimum value which is listed in the fifth column and is calculated from eqn_(s). The sixth and seventh at columns, respectively, list the value of j6.S at f= 3, and the value of p(E -E*) f= 4. If p = 2, then for AE”’ = 50 ml’, the second entv in the first column must be mV. fg wiU occur at 0-593, and $6, witt be used, and Ei- E* will be only -zz.~ equal to 4s.
CHARACTERISTICS 721 =
nz
=
p.
OF
TITRATIOS
CURVES
OF SYXHETRICAL
29O --_-.
.--_
E
pAEO’ fm vl
.-
Ezo’
REDO-X
--_--
RE.tCfIOSS .--
_vitt. slope
---
_
.xrax. slope
--
Eqtri-
_~
--
vatencd fioin:
50 / -p(E-E*)
PS
4
0.5;2
30. -?
25,OO
78.4
7X.9
no infiettion
points
I 0
46.9
100
f
SO.00
0.593 4.1.66
0.897
SO.97 97.‘:
96.6
96.07
10s.
0.5
f
-P/E-
EC*)
PS
X07,
I
I f-01) I
0 x03.7
1.50
I -P(E-
E*/
PS
0.507 SJ.47
0.987s 3.08
10X.+
257.5
I
0 252.
I
200 I
0.
f -$t(E
-
E’l
PS
yx3z
X00.02
IOO.oO
101.589
102.597
O‘y3OS 99.90 10-1.57
o-9983 x.10+
0
643.7
642.3
I
250
f
0.50003
O.pOO8
0.94jij8
f
I X5.01
Iz~.oo
124.6
0
X02.70
102.689
102.678
O-403 X670.0
3
-pp(s-E”) PS
“f569.7
(;oo ,I
i
. 0.5-s
0.50003
0.99998
I
150.005
x50.00
x49.95
0.150
0
fO2.706
102.&C).+
ro2.68
-%.)=.S=
z
-_P(E--,“*f BS
44~4.7
kior cd& vdue of pAZP*, successive values of j arc given; reading from left to right. Unddrncath each value offis Listedtic corresponding value of p(E--E*) and below that the value of fls at that Iwaticm. r. Ekdroanal.
Gficn,~,
14
(IQ*)
‘3 j3-383
XhT;LECTIOh
FOISTS
There
point
TITK\TIOh’
is ~.cr arlothcr
at which
species
OS
Z.xi,rg titrated.
point
of
interest
tri;. . the Jjotenrial
E = E?“‘. T11c
volucs
379
CC’K\‘ES
of /
ut tllis
on
tiicsc
is equal point
redos
to the
titration
formal
11clx.c .xlrctiJy
curves:
potential
been
the
of the It
prcscntcd3.
may be easily verified that w = - &ri at E = Ez”’ so that bv use of eqn.(s). the v;llues In Table 3 is of tJle slope at this locntion on the cur\-e ma\ be readilk calculated. presented a comparison of these wJues with the values p~c~~iou~l~ listccl in Tab& I and 2 It is apparent from a consideration of Table 3 that a number of general statements mav be made about titration curves for symmetrical rcdos reactions. =\t f= i. P(E-k-q is never esactly equal to p&“/2. or as previouA!.5 stated, at 50?,, dccrcrtscs as the value titrated the value of E is n/ways Zzss than E?“. The disparit?, increaser. ‘It f = 4, the slope increases with increasmg values of !,4C’. and of +E”’ this value indicates that there is onl?- a 23:; approaches a limiting \.aluc of 102.~ -I; increase in the magnitude of the slope accompanying a 5009,, lncreasc in the value
for p4E”.
In contrast.
hundred times this point dots qrre#,l
the equivalence-point
slope
(at jAE’*
= 300 rnk’) is about
as large as it iq at fi4E”’ = 50 mV. and the mmitude not approach a limiting value as fiAE+” increases.
The \*aluc of _/ at \vhich E = Eze’ is ne\‘cr equ.zl to ifs, the disparity dccre;Ging as p4E” increases
of tJ\e r;lopc at
to 3. but Tht: ma,vnitude alXsa_\*S
one
occurs
SIilSC-
of the sltipe
at E = E?O is alwa_\*s less than it is at_/= 5. but becomes less so as PIE‘“’ increases. The general nature of a rcdos titration cut-x-e is readil!. discernible from Table 3. \\‘?lcn pAEn’ = 50 rnv. there are no inflection i)ointj and llcnce no locations on the cun’e
at which
the slope
is either
a masimum
or a minimum
so that
as the titration
values of _/) the slope continuously dexreases throughout the -reatcr), tlrc slope dccreasto a entire titration. kIo\vcver, \vhetr pAE” = 100 I~I\* (or h nri>:i>)llc#>t which allva1.s CCCMIS srrlscqrcz)lt to / = 3. and then rises to a )uaxinlu~rl _ als*aFs prcccdes the \vhich is located prior to tl:e equivalence point. Tlre minimum proceeds
(increving
mx.imum, whereas the minimum nf~~~vs OCCI(YSs;tbscqrtc~tl to tile point at which E =I E2“‘, \vhich irl turn is olwavs locrrtcdsubscquant to the point at \vhichfJ. It is apparent fr,?rn Table 3 that whkn p4E” 3 300 m\‘. the differences are negligible (and considcrablv lcw rigorou-; equations
th.an cspcrimcntal error) bct\\*ecn the \*alue.s predicted from the and those predicted from tile customary and simpler equations.
It i;, c\*ident that there must be some minimum vaJue of p4E” in order that the titration has at least one inflection point. This minimum vnluc of l>4E”’ is Ss.6 rn\- and corresponds to a value of k equal to o.rSSS. For k -C c. xS8S, i.e., ,+AE3’ > S5.6 mV, the redos titration tune possesses t\vo inflection points; the one corresponding to the minimum slope being located prior to tllc one conejponding to the ma_Cmum slope. When k = o.xS68, ON/_V 0112 inflection point eskts, that at / = 0.70% where P(E, -EL) = - 25.GS mV (at K = o.xSSS, thcrc exists only one X*~UC of pv, that satisfies cqn.(r3), tk., pyc = -I). For values of k such that k >0.1:888 (i.e.. pAE”< 65.6 my). tJ)e titration cume Jlas no inflection points. NIGI~TI~GALE concluded from a mathematical treatment of the poising capacity6 that in the titration oi a sdlution initially containing two redos coupIes, the second end-point (corresponding curve has to a minimum in the poising capacity, i.e., wllcrc ttlc slope of the titration its maximum value) would not be observed if the difference between the formal potcntizls of the titrant redox couple and that of the couple initially present (containing the less readily oxidized species) was less than 85.6 mV.
/. Eiecfroa~rcl. Chcm., 14 (1967) 373-333
curties &ere_pAE” 2 85.6 and S6.0 -my, .&spect_iGel~r_-From -Table .4 it ti_ apparent that even ‘yhen up.&!? is on@ 0-4 mV &s&r thanthe. mirknqti value (necessary for the existence of an inflection point} of S5_6 XIV, there is a considerable difference bet\+een-the locations of the points-~of minimnrnandmaxirn &nsl6pealthoughthemagnitudes ofthesetwFoslopesdiffer by 1t.k
ins~ktive
to
the titration
compare
only O-I. It islseen that the redox titration curve even at this small value of$A.??"' (=S6_o mV) possesses the same general nature that is observed for much greater
Dr =
85.6
mV;
R =
o.ISSS
PS
I
0.707 25.68 go.0
0.5173 42.s 92.8
3 44-4 93.6
--p(E-Ef)
s:.sg
iwirt. slope -&QEO'=
86.cmX';
R =
_wax.
szupe
0.187
f --p(E-E*)
0.518r
f 44-S 93-9
?S
o-751
o-677 d3.5 90-z
43-o 9'.9
21.9
I 0
90.3
81.5
-
T4BLE
5
THEREDOXTITRATIO~
f-
m.
0.500 o.=joq 0.5178 -0.536 0.564 --5Q3~ 0.621 0_650 0_672 0.678 0.69~
..
.O.7O7
CURVEFOR
pAE”‘=
85-6
mV_4T25°
-fJ(E--E*)
-Pw
PS
44-4 43-7 47-A +*-I
1-729
9x-6
0.y12
25-2
I.700 1.667
93-2 92.8
24-4 Z=J_I
I.600
g2.I
33-5
I.500
qr.z
36.0
I_qcm
0.721 o-735 o-767 O-770
9= 7
a_SzI
33-4 =J'_I 38.8 28.3 27.0 ~3.65
I-300 I.=00
90-z go_0
I.12
89-g
I_IOO I.050 *-cPoo
39-9 59.9 S9-9
o-go8 o-936 0.969 I .r_o32 1_06=j
f
--p(E-E=)
--PY
PS
o.gso
89.9
89.9
0.950 .0.900
20.3
89.9 89-9 89.8
0.790 0.780 0.600 O-300 0.200 0.100
ao-o =5-q 7-71 5.=4 2.57 Q -2-57 -5_r4
89-5 86.5 85-2
83.2 80.8 78-I
0
-----o-I --o-z
75-o
v&z_, the slope of the titration curve decreases,. BS the idue iucreases,untfithe:point of minimum slope is reached--'which occms subsequent
.. ;values&pAE"',
’
at Which~ E = &Or .which, in- t&n,
the- &mt
Thereafter,
k subsequent to the point._at whichf = &.~ value.? of f; the sIope~:rikes to a. maxim25 Iocation~ prior to the equivalence &oint. -From tien on,-ffiti_slope
with
progressively
increasing
oCcurrLn&~~t .~ ~~ -_oo~ntinu_ally d&creases. -: ~~ :._ _. : : :I‘$‘& only at-one pkrticular valuelof&~thzitthe~~lop~e~eshibits'neithera -no=
a"
of f to
-&xir&iti__valk~~ _ :~_~.:.:
Rather,:the -~- :
slope decreases :
C+ the titration
~~@$&urn
prodee& -until a :.
-_;. .
-~.~ .~
I_NI;LECTIOS
I’OISTS
I;c,r 5t1 C>llh’ ix1.5cr nc)
iliflcctic~n
ON TITI~hT1ON
csxmy>lc:, arc ti1crc: Imints
wticn tlircc:
ASI
CURVES
:I inf~riohn.sic*lxlrt.ntly cliffcrant
xcicl tyyat5
is of
titr;itiBcl
wit 11 a
titr:ttic)ri
(-Iii
vt*5
mc,rioac-iclic: I)t)sGblc-:
(3)
; (t)) only
ttit: inflcctioli I)oilit cc)rrc:s~mndirlg 10 tlic. ln:isirll~ill~ *;ICBIP vsj:;tr.. c>,(c) t xv<) infbct ion I)oints c,riri IxGrit:, Jcn t:si:.t 011 tilt titl.kticBri ~:lix~~t, tllc* onts ‘.o~~(~C;I~W)II(~~I~(; to the point of minimum slop is Iocatcd virt1i:ilIy at / = 0, ;~ntl tlicrttfort: only tilt: exist
to tile maxiniuni slope exists at 3 non-x::10 val~c c~f f, sulutior!s ttie cunccntiation of which esc-cccls I0 -4 1;. ~1s concentratic)n of tilt: acid is clcc:rc‘;~s~fcf,the ioc-.?tiori of Clic: point of minimum -but 11c’vcr c:sc:c:c:c\s /’ ; SIOIIC IIIOVt!S a\ViIy frCl111 zt:ro tc~\V;lItI~ I;ilC;Cr v;llUcS c>f f 1)IoViclc~~l ttiat tile c~jIlr;crltI-ntivll ui tile ba~c is not f;rc:ltcr than that of tlit? :ac*id 41c)pe rnov6ls Cownrrk sm:ilhtr irritiirllyancl tl2C ltr<:aCi<)n cif tht: point c,I m.3simurn iIlfltx:tinn wllic-tl is thct initial
I)oint.
corrtxpontiiri~
virtually
unity
for
values c:f /_ As LVCcoritinuc tion value is rctachcci where paint
c*f n1axiriiun~
sloptt,
to
clccrasc:
the point Icx-:ltc:cl
at
the!
init
ial
cwnwntr;ttion
01 minimum a v.allltf
c,f
f
of
r&cl,
slop2 ~2.2scs to exist cc,nsidcrnl>ly
less
than
a concentradtliouC;li tiic:
Ilnity,
still
When the initial acicl cunccntr;ition is allowed to dccrcasc cvc:n further. tllc ln,cntinn of the maximum slops mnves tnward CWII smaller values of / urllil firidly it ti1c slcq_x: of tllc CUNC cll?crcnscS too ccz~scs tc>exist, am1 tlitm tlii cm~h>ut. tiic titration.
ftxists.
continuc,us!y
fIonl
it:, initial
valut:
at / -
0. Thus,
it is .?pprcnt
that
there
is a si~milnr-
ity betwt:en these acid-base titration curves and the titration curves for symmetrical redox reactions in the cxistcnctt of three- cliffctrcnt types elf conclitions which proviclc: three apprclntly tliffprent types of c~-~e_s. I~~lrtlwrmorc:. wlwn the primary indcpcn dent variable: (the initial acid concentration in the :icid-base titration; ,+A??” in the rcclox titr;ltic>n) h;Ls CIvalue which is kss than some minimal orle, no inflectron IMnts exist. In the titrntion of :a monol>asic: wt:;& acid of initial c.onc:t:ntration C, with ;L niononcidic strong hasca of concentration C, the value of f, at which I:ii t ] = Kn. is /_ Eleclroawal.
(ihem..
14
(1907)
373- $33
~-..382 i----.
y-,
-. :;_. :‘- ;,_‘. :~ :.
;;
-~
_
_-_.-
:.~:
‘.
-.-I _:_:_:~:~~.“~-.__~‘-
-_
.J;
A-_.GOLD+
_.~
: the-s&es
oif
ak:~l&'~thA
g-when
PKks~
In contradistidktion;~fora symme&al exceeds 41 Perhaps this dif&en&i6is not a point ofi~the redox titration ~~ :-‘~~~k.(~~theregio~~~d~f~z) &here thevalue ofK, and~onIy.K,_d&ectly deter&n= The variation of-the location of. -. the- pctential, -viz_, .E--_E *I (R~/F)mk=&AE"'_ minimum slope ontheseacid-base titration .the infkcti~&@oint corr+po&ingtothe ~~ c&ves;'is sim&rtothe behavior of values offat which [H+]~=K,.-because it may bc&r at vahres of~greaterorIessthan~_~enpX,<.~~pK~~,the pointqfminimum slope (when it e&ists) always occurs prior to the point at which [H+]=Kn, wmch in for PK,~>~PKw, ~. tafn precedes the "halfway point"(where f=$-2); whereas po;ltatwhich[H+]=K,. pointof rninim~slo~e(~vheaitesists)alavaysfollows&e
latter 2dways ~&curs tier
which
the half-way
point_ The inflection
the
point ckresporid-
mgto therni&num slope always occurs before theinflection pointcorrespondingto the location of rhein4he.m axiruuruslope. In other words, the disparity between and that of the half-way point is flection poink corrtzspondkg to them inknuti~slope +lways.greater than that-between the point at which [H+]=K, and the half-way ~~point.‘I%owever, in a:symmet&cal redox.reaction,.the location of the point of mim~rrkrn slope always. occurs after the point at which E = EzO’, which in turn always this sequence resembles the hollows Ithe half-xvay point, so that in a formal sense,
acid-base titratjon where$Ka> and for small values,
Ka >o:I,
to.:exist,
but
it is o&y
for
the
Q e.g.,
pKxv_
Both
for relatively large values of K,, e-g-,
the point of minimum slope. ceases Ks< 5 -IO-~',
srnd
value&of~K,
that the point
of maximum
slope
to-the fact that the .lo~ation'oftheminimumslcpemayoc+ron either~sideofthehalf-waypointbutthe maximum slope can never occur before the half-way point, and-~indeed,_ceases to may
also cease
to etista_ Esserkxlly.
this may
be
attributed
.ersst hefore it ever reaches the~half-way point. In contrast, for symmetrical redox reactions,thelocation oftheminimum slope canonlyfollowthehalf-waypoint-When m.nel,ther a maximum nor-a minimum slope exists (Kn<( -?o-12), the slope of the
-titration curve thronghout the entire titration, continuously. decreases from initial-valueatf=o, asitdoes inthe redostitration where pAE”r
its
..
FI& titration curves of symmetrical (2~1=ns~=p) redok reactions, it has been denioktrated thkthere are-always two inflecti& points whenever p&E"' > 85.6 CV equivalenceat 25O-;the location ofthepoint~of' maxinmmslopeis~alwayspriortothe ruin&slope is always:located subsequent~tntkhalf_. point, where23 the pqint of 300 mV, Way point: F~rpractic~~-tItra~ions.where~~AE O'S customtiygreater~thari
INFLECTION
POINTS
OK
TITRATION
CURVES
383
REFERENCES
III_ z I-G_ MITRGULESCUXXDCDRAG~LESCU,~. 3 + 5 6 7 8 g
PikpzR.Chenz., 185A(rg4o) J. A.-GOLD~UN,]. ElectroanaLGhem.. 11~(rg66) 355. E.BISHOP,AIZ~Z.C~~~.~C~~~, 27(1962)253. J. A. GOLDB~AW.J_ EZec#roanuZ_ Chem., II (1966) 416~ E. R. NIGHTINGALE, JR., Anal_ Chem.. 30 (1958) 267. L_ X~EITES AND J_ _X. GOLDMSW, AtraZ. Chi+~z. Acta, 30 (1964 203. J_ A. Go~D~IAE;A~DL.MEITEs,A~~uZ.C~~~Z.~~G#~~, 30 (Ig6q)zS. L. MEITES AWD J_ A_ GOLDM_KN, AmzL Chim Acta, ag (1963) 472. J.
EZectroa~d.
375.
Chem..
14 (1967)
373-383
EIsevier
THE
CHEhlISTRY
Publishing
Company,
LOCATIONS
JAMES
A
IXTERF_%CIAL
- Printed
OF~INFLECITON
SYMMETRICAL
Department
AND
A4msterdam
REDOX
373
ELECTROCHEXISTRY
in The Netherlands
POINTS
ON
TITRATION
CURVES
FOR
REACTIONS
GOLDX-IN ofChemistr_~,
(Receiwzoi October
PoZyytechnic
3Ist.
of Brooklytr,
T?zstitrrle
Brooklyx,
Xew
York
(U.S._4.)
1966)
IXTRODUCTION
The question of whetherornotthelocation of the equivalence point coincides with the point of m&mum slope on redox titration cnrves hadbeen previously investigated by KOLTHOFF XVD FURXXX~. A numerical evaluation was made of the variation
of
cuprous
ion.
Fotential The
in
slope
the
titration
of
of the
titration
curye
a solution was
of
ferric
e\-aluated
ion
with
at 0.10;,
a solution
prior,
and
of
0.1%
point_ Frcm this, it was concluded that the location of just at the equivalence point" and that the “curve is symmetrical on both sides of this point". However, in retrospection, it should be recognizedthatthis numerical example couldhardlybe expected to provide evidence subsequent,
the
to the
maximum
equivalence
slope
“occurs
of any small deviations from symmetry because the difference between the values ofthetwoformalpotentialsisfairlylarge,viz., O.-V = AZ?"'_ Indeed. these authors later present a mathematical formulation of the variationin the value of the slope of the titration curve applicable to this same example, i-e_, Fez+ + Cu+ = Fe’+ + Cu’+. The conclusion was, that for any symmetrical (a1 =n3, i-e_,both reversible electron transfer couples involve the sauce number of electrons sothatthe general representation of the titration reaction is: Osl+Redz = [of the slope of the titration Re&+ 0x2) redox reaction, the "theoretical maximum curve]
occurs
before the
equivalence
point"_
No
approximations
were
required
in
the mathematical derivation used to arrive at this conclusion. It is further stated that "in cases wherethetitrationis practical, ___ the difference is so smallthatit can hardly be estimated"_ By nse of an appro-tiation, it was. however, possible to
that the magnitude of the slope just prior to the equivalence point is than at the equivalence point, but that the “difference is actually so small that it,WC&L not be observed”_ It was emphasized that the appro ximationsused demonstrate
aZways grsafev
are “only
of the same
order"
as the
difference
between
the
value
at the
point
of
maximum slope and that at the equivalence point. 01zZy for titrations where LIE"'>, 591 mV, was it possible, by use of further appro.ximations, to numerically evaluate the fraction titrated at which the point of maximum slope occurs_ The general conclusion was that "even in the simple case __. we get equations which are impracticable au~difficulttosolve". The next major theoretical investigation of the properties of redox &ration curves was made by MURGULESCU _QJD DR~GULESCU~ who deconstrated that, rigorou+y, the inflection point (corresponding to the point of mtimnrn slope) can
-. 374
J_ X.
.-: -~..~
GOLDMiX
i
never coincide with the equivalence point-even when the redo,u titration reaction is before any approxim.ations were made, from the symmetrical. This was evident, equ&ion presented for d2E/dfZ (frepresents introduction of appro_xirnations,~ the general the
location
of the
inflection
point
predicts
the fraction t&-ated): Hoxvever, after the equation presented for the evaluation ,of that
the
inflection
point
does coincide
with ~the equivalence point in a s_vmmetrical redox reaction_ Nevertheless, .it uras proposed-that the value of this equation was its utility for characterization of asyrrmetrical redox reactions. However, it should be noted that it was indeed a more general equation
than
any presented
by KOLTHOFF -*ND FURXAN l_ been presented equation 3m4has recently
for the description Because a general of titration curves for homogeneous and symmetrical redox reactions, it is now approptiate to re-investigate the location of infiection points on these curves_ It will be demonstrated that no approximations are required treatment of curves for s_v~~ntet~icaZ reactions and that
iu the following theoretical the location of the point of
~.G&zzrnz slope, as well as that of the maximum slope, may be readily determined. It is confirmed that the point of ma_&~zzc~~ slope always $~ecedes the ~equivalence point, whereas it is newly shown that the point of mirtiwzztpn slope aZways OCC-m-s sztbseqzcenf to the point at which f = +_ However, there are valueCbf AE"' for which the titration curve possesses neither a point of maximurn nor rninirnum slope. In addition, the equations presented are rigorously valid for the evaluation tions of the inflection points on curves for LZ+Z~value of AE”’ where these THE
E_XISTEXCE
_%XD LOCATIONS
For homogeneous
OF
THE
IXFLECTION
and symmetrical
redox
of the locapoints exist.
POLVTS
reactions
of the type
122Ox1 + nl Red-z. = ~2~RedI + x1 0x2 where a solution f&&ion titrated f
=
(1)
cant air-ring Red? is being titrated U@Z a solution at any potential, E, may be calculated from
{I+-
exp(nnlv))/(r+k
where ~,CJ = (F/122-) AE”’ = (RT/nl~c-F)
(E--E*), ln K.
containing
0x1,
the
=p(---nf:v)}
(2)
k = exp (--)~a), The
6 = (F/RT) AE”‘, -tz= nl~~/(m~+~z~)), and equik.lence-point potential is designated by E*, K
represents the equilibrium constant for the reaction defined by emqn. (I), and R, T and F have their customary significance_ Because only synimetrical (where ?zr = nz = p) reactions are to beg considered, , eqn(2) may be rewritten as f from .’
=
which &@ L=df.
(3)
e&P CP7p)i/G + k exF (:Pvl).}
C’.+e
one obtains .:
@ F
jr+~~p(i&J)]”
~k~~k+elcpCp~)i-esp(-pip)].
.~ _~
-.-.
..
_.
(41 .~
:_
obt+ined frdm the. .Wi&& restrictionthat IZ; =~z = P,-eqn_(~Q may. also be' readily --.rijncr~.-gene~aie;~~~~o~.previodsly.present.~5_ Equation(G), $hich pemits ~the- evalu--. : .- ation..atl&y:point,pf the_ slope of ., the titratiorx curve,.-may-conveniently~be rewritten
.. ---a& I:.‘;; -1~ -:~-_.__.~.~-2:. _. .:~. -.-'.;. : .~:__ -.
~-1J-:~~ed;ak~~l_-c~~*~, -& &$j
373-353
,_ :_~--L:
:~. ,- .‘. -:_
:
.: .-. .~-.~_.~. ~,._~ -~ ~, ~.,
._..-_:~_~
I -,-ji~ -:..
:
1~.__;,~:~ -__.~~ _
.I1_-:-._~
dl
--_.
=
U’f’
_ _.
I;
df which sfiould
[r +A t?XI’( __-jy,)]”__ .-----. zpk [ k + cash (py~) j
fume is somewhat 111orc’ convcnicr1t fcjr tl1c nun1cric:;ll evaluation be noted tlIat the slope is al7~zys fmsiliw :1ncl ncvcr ctqr1z-dto zero. At
tl1c cciuiv;llcr1cc
I )iffc:rentiatiori
+
puiIIt,
~ff_x:~~~)(-~1/,)]~~S~,(~‘VJ)-~(‘”P(-~J~J)] ---.----
:1ncl (1 fiI:1y I1tt convctnictn
-
nplmrcnt
tll:It
-
t ly rcwrit
---. - -
ttlus
0, >v tlIiIt>
-
_-
+csp(
2k I'eXp ( - fiy)]
is
It
of c?cln, (-4) yields
t zk-texy(plp)
It
w -
of dE/clj.
_
___
ten ‘2~;
+c:osil
.--
tlic
_.
--&h&l
---
j+
2[/z-+&&~l(@jJj~"
sccon(1
detrivat
r + k cusp ( --.-fiy~) ] sin11 (,hy!) -_ -
ivc: is always
rlc-t;ativt:
at
tile
cquivdcncc
tl!:1t ttie cc~uivdrlcricc point rnllst always lx: located sztbscqr4enCto tllc: infIcctic~r1 pc,irit corrc5l>oriding to tl1e location of tilt: point of maximum slope. Tl1i5 i5 in ag1ccrrie11t wit11 ttie earlier conclusions in tile literaf urt:*.“. At the inflcctio11 IH)ints it is ncc:cssary ti1at cl”E/rl/’ = o. ‘l’hen. because I)c)irit,
arlcl tl1crttfcjre
I t k c-up(-py) r11erely r,equirttci rc;kx I ;irqqcrlieIit k
_
-
and U{df that
UC dwiiys firlitc, psitivc.
Q = o. Equating
Itanh (fiWd1 bv -------
(Pwdl
2 -t tan11 (PVC)
fc,r tllc
rc:I;~t-icmsi~i~~ existirig
t~c:twc~ri
the
-
numerator
ar~cl II~VU of Q to zero,
ecl~l yields
2k2/cosh (fly&) - --. 2 + iaI-&~fiVJr jyJi-tlie
value
of ty at the
to zero. it is after
some
(13) iriflcctiori
point--and
the corresponding value of k. It is evident that far k
that
satisfy
eqn.(13)
to have physical significwxc (k 20). the values of yj~ must lx negative. Thus, again it is verified that the inflection f.
fi_Vlrclrounal. Ch‘*,n., 1.1 (1967) 373-.3B3
376
.:-
.I’
_
J _ ~A_. GOLDMAN
~.
..~
point
~5~ecedes~the equivahkce
point. As
k= ---ky.‘The valUes of k thatsatisfythik no physical si@fiGncc b&the former
pyi-+o,
-it
is seen that
eqn(r3)
becomes :
equality are o.andI-1: Thelatterpossesses corresponds to an infinitely large difference
betieenthe- tivo formal potentials (R = exp( - 0_5p) (F/RT)AZ?) _ Therefore, it -is to be~expected that as the potential difference (4iZ”‘) increases, the difference between the location of the equivalence point and that of the inflection point will decrease. Equation(r3) has been solve;& using the method of successive approximations; to obtain values of &YS for various values of k,, and the results are presented in Table I. For each iralue of pr,~i, the corresponding value offf was calculated from eqn. (3), and the slope, psi, was calculated from eqn(5). For purposes of comparison, the value of the slope, $S* atf = I is also given_ TABLE THE
I-
LOCATIONS
9x1 =
7aLq=
pilEo’
p.
OF THE
50
OF I\IASI?.IUBI
K
--p(-Es-E*)
fs
PS,
PSZ
(ml') -
0.378 o-143
no inflection
11-04
o-897
o-0539
3.03
300
0.020+
I-IO
250 300
o-00775 o.oo~gI
0.403 0.150
150
SLOPE
“9
ffnT1 IO0
POIKTS
The first column
o.ga7s 0.9983 0.9998 o-99998
lists values of ~A??‘_
16-o 1OS.I
252.5 64"_7 1670 4425-Z
101.7
252.1 Gq2.2
1669.7 4411-7
The values in all succeeding
columns
-have been calculated for zsO_ The second column lists the values of k calculated fromk =exp(-$AE"'F/zRT). Inthethird~olumnarelistedthevaluesofp(Ei-E*), values of k by use of eqn(r3). Et viz., (RT/F)p?yi obt ained from the corresponding represents the potential at the poiut of maximum slope and E*, the equivalence-
point potential_ The fourth column lists the ~values of f at this inflection point, calculated by use of eqn.(3), and the figures in the fifth and sixth columns were calculated by the use of eqns_(s) and (6),respectively_ Some of the values in the sixth column
differ from
those
previously
presented5 which were not~directly calculated @th no~special caution exercised-+th ~respecttosign;ficar~tfigures)from(dJ?/4f)~- f lexpressedasafnnctionoftheequilibrium 50 mV, ,the s&o~z~ entry in the first constant for eqn.(I)_ If p=3_, then for AE"'=
from the presenteqn.(6).butratherevaluated
column &stbeusgd,and Et-E* will be only -5-3" mV; f~will occurato:Sg_i,but Sswilln~~vbe54ascompare;dto5~_4=Ss*. It is -evi$entm~from Table I that as the .difference between ;the,tko formal potentials increases, the.value of-fiapproakhek unity, &thattithpAE"' 2300 mV,_ the point of maximum slope is indeed hardlym&tm@shable from the equivalence -.pbint,~h~._~~~ys6e~~pr~umed_Foranyp~ticularvalue of$&Y”‘.the magnitude equivalence -Of the-'sIo~peat the infl&ctiop poin~t,~55 +ji_salwaysgreaterthanthatatthe ~PoiTlt~whfe_~e~slope~_is &S~, -but- as ffre:yaIue of.$wQ' irkreases .the .&ffere_& ~.b;etw~~~~lthe-values~of-the two slopesbecornes~in_~~~ngly ~cgligible. -For anY_partic-- : ; +I&-- &_&& .&._;-~~~c~ ,~_the, ~differenc+. be-t&en Ei:g&d.:,i??*,‘: kd bee& St: and_ S+;.I ~. ~.. ~. . _ . :. I’: 1. 1;::._.. _;.__.-, _~‘__ J-. -. 1_.: ~.~ ~ -. _I_ ,. -. . _ ~-..- _._-_._..._-2 -._ , -_Ii;_~~~~~~~ai~-:~~lrr;,:rq .(f9673:~_~~13$3~~~~.~-. .-.- f_; ;I:.- .: r-1 1. _,l_ ;: -~. .~ ~~I.:: __: _;~: I_._-._-..-_~: -;m-‘_.~_~
INFLECTIO?J
POINTS
decrease
ON
as the value
TITRA-I-ION
CURVES
of p-increases_
377
When
PAZ?'=
50_mV,no
pointofmaximum
slopeexists Therefore, it is concluded that in titration-curves for symmetrical reactions, the pointofmaxinnum slope can never coincide with the equivalence
redox point
but must always precede it. As the difference between the two formal potentials the locations of the two becomes increasingly positive, the difference between points becomes less, so that for any practical titration (where the formal potential difference is sufficiently positive) the difference betweenthelocations of the equivalence point and the point of maximum slope will be too small to be experimentally observed. Investigation of the properties k there are actually two values of@+
of eqn(13) shows that for a particular whicharephysicallymeaningfulmathematical
value of
solutions. The values ofp(Ei-E'*) in Table I correspond to values ofPr+ such that I~lyrl
whenever
meaningful mathematical solutions of eqn_(Ix)_ point corresponds to the point of mi~t~~tu~lt slope,
Ipr,uil
correspondstothe
pointof~zaximz~m
slope.
THE
LOCATIOSS
OF
THE
POIKTS
OF
3IINIMUM
--p(Ei-
E’)
SLOPE
fr
Psi
PS
(nzV) 50
I00 =50
0.37s O-143 o-0539
44-68 74-47
200
0.020~
99-90
250
0.00775 0.002g1
300
124.8 I$Q-95
atf=+
-_P(E-E*) atf=$ 30-4 50.97 75-=4 100.02 125.01
150.005
no inflection o-593 O-507 0.500s 0-5000s
102.57 102.67s
78.4 97-I 101.8 102.59 102.70
0.50003
~02.680
103.7oG
96.1 X01.4
-4tthepo~tofmill;mumslope,valuesoffr,andSiIllaybecatculatedbyuseof eqns.(3) and (5), respectively. For purposes of comparison, the values
_f=&
are also presentedin
Table
z alongwiththe
of the slope at
values of+@-E*)
corresponding
presented values of p(E--_a"') at f=$_ At f=&, and the values ofplp so calculateclweresubstituted pr+~= --qps+P(E-E~"')(FIRT). intoeqn.(5)toobtainthevaluesoftheslopeatthesepoi~tsonthecurve_ It is evident from Table 2 that the location of the point of minimum slope on the titration curve is a&ays- szrbsepemt to the point at which f=_$, although as O'increases, the location of this inflection point (corresponding to the minimum PM slope) approachesf=& so that for AE"' 2300 mV the differences in location are negligible because even at AE"'= zoo_mV. tfie difference is conside+ably 1~ than calculated
from
the
previously5
theexperimentaler-rorinlocatingthes~~ints.
alwayslargerthanitii at the value for AE"'~zoo~ mV.-JWhenpAZ?'=~o-mV,
Themagnitude
offs, although no
oftheslopeatf=+is
the differencebecomes
point
of ininimum
slope
ne&ible exists.
Ai
for this saluc of +E”‘, there is no point ‘of maximum sloyc. The rw&ed, first coIumn of Table 3, fists values of p&E”‘; the values & all succeeding columns have been calculated at 25O. Columrr. z lists the corrcspoxxding ~-alue~ of k, and in the third column are listed the ‘values of $(EI - E*) corresponding to the point of minimtzrn slope and c&x.&ted from cqxx.(x3). The fow.+h column lists the values of ill obmay
lx
tainedvia eqn.(s). at the inflection point where the slope has its minimum value which is listed in the fifth column and is calculated from eqn_(s). The sixth and seventh at columns, respectively, list the value of j6.S at f= 3, and the value of p(E -E*) f= 4. If p = 2, then for AE”’ = 50 ml’, the second entv in the first column must be mV. fg wiU occur at 0-593, and $6, witt be used, and Ei- E* will be only -zz.~ equal to 4s.
CHARACTERISTICS 721 =
nz
=
p.
OF
TITRATIOS
CURVES
OF SYXHETRICAL
29O --_-.
.--_
E
pAEO’ fm vl
.-
Ezo’
REDO-X
--_--
RE.tCfIOSS .--
_vitt. slope
---
_
.xrax. slope
--
Eqtri-
_~
--
vatencd fioin:
50 / -p(E-E*)
PS
4
0.5;2
30. -?
25,OO
78.4
7X.9
no infiettion
points
I 0
46.9
100
f
SO.00
0.593 4.1.66
0.897
SO.97 97.‘:
96.6
96.07
10s.
0.5
f
-P/E-
EC*)
PS
X07,
I
I f-01) I
0 x03.7
1.50
I -P(E-
E*/
PS
0.507 SJ.47
0.987s 3.08
10X.+
257.5
I
0 252.
I
200 I
0.
f -$t(E
-
E’l
PS
yx3z
X00.02
IOO.oO
101.589
102.597
O‘y3OS 99.90 10-1.57
o-9983 x.10+
0
643.7
642.3
I
250
f
0.50003
O.pOO8
0.94jij8
f
I X5.01
Iz~.oo
124.6
0
X02.70
102.689
102.678
O-403 X670.0
3
-pp(s-E”) PS
“f569.7
(;oo ,I
i
. 0.5-s
0.50003
0.99998
I
150.005
x50.00
x49.95
0.150
0
fO2.706
102.&C).+
ro2.68
-%.)=.S=
z
-_P(E--,“*f BS
44~4.7
kior cd& vdue of pAZP*, successive values of j arc given; reading from left to right. Unddrncath each value offis Listedtic corresponding value of p(E--E*) and below that the value of fls at that Iwaticm. r. Ekdroanal.
Gficn,~,
14
(IQ*)
‘3 j3-383
XhT;LECTIOh
FOISTS
There
point
TITK\TIOh’
is ~.cr arlothcr
at which
species
OS
Z.xi,rg titrated.
point
of
interest
tri;. . the Jjotenrial
E = E?“‘. T11c
volucs
379
CC’K\‘ES
of /
ut tllis
on
tiicsc
is equal point
redos
to the
titration
formal
11clx.c .xlrctiJy
curves:
potential
been
the
of the It
prcscntcd3.
may be easily verified that w = - &ri at E = Ez”’ so that bv use of eqn.(s). the v;llues In Table 3 is of tJle slope at this locntion on the cur\-e ma\ be readilk calculated. presented a comparison of these wJues with the values p~c~~iou~l~ listccl in Tab& I and 2 It is apparent from a consideration of Table 3 that a number of general statements mav be made about titration curves for symmetrical rcdos reactions. =\t f= i. P(E-k-q is never esactly equal to p&“/2. or as previouA!.5 stated, at 50?,, dccrcrtscs as the value titrated the value of E is n/ways Zzss than E?“. The disparit?, increaser. ‘It f = 4, the slope increases with increasmg values of !,4C’. and of +E”’ this value indicates that there is onl?- a 23:; approaches a limiting \.aluc of 102.~ -I; increase in the magnitude of the slope accompanying a 5009,, lncreasc in the value
for p4E”.
In contrast.
hundred times this point dots qrre#,l
the equivalence-point
slope
(at jAE’*
= 300 rnk’) is about
as large as it iq at fi4E”’ = 50 mV. and the mmitude not approach a limiting value as fiAE+” increases.
The \*aluc of _/ at \vhich E = Eze’ is ne\‘cr equ.zl to ifs, the disparity dccre;Ging as p4E” increases
of tJ\e r;lopc at
to 3. but Tht: ma,vnitude alXsa_\*S
one
occurs
SIilSC-
of the sltipe
at E = E?O is alwa_\*s less than it is at_/= 5. but becomes less so as PIE‘“’ increases. The general nature of a rcdos titration cut-x-e is readil!. discernible from Table 3. \\‘?lcn pAEn’ = 50 rnv. there are no inflection i)ointj and llcnce no locations on the cun’e
at which
the slope
is either
a masimum
or a minimum
so that
as the titration
values of _/) the slope continuously dexreases throughout the -reatcr), tlrc slope dccreasto a entire titration. kIo\vcver, \vhetr pAE” = 100 I~I\* (or h nri>:i>)llc#>t which allva1.s CCCMIS srrlscqrcz)lt to / = 3. and then rises to a )uaxinlu~rl _ als*aFs prcccdes the \vhich is located prior to tl:e equivalence point. Tlre minimum proceeds
(increving
mx.imum, whereas the minimum nf~~~vs OCCI(YSs;tbscqrtc~tl to tile point at which E =I E2“‘, \vhich irl turn is olwavs locrrtcdsubscquant to the point at \vhichfJ. It is apparent fr,?rn Table 3 that whkn p4E” 3 300 m\‘. the differences are negligible (and considcrablv lcw rigorou-; equations
th.an cspcrimcntal error) bct\\*ecn the \*alue.s predicted from the and those predicted from tile customary and simpler equations.
It i;, c\*ident that there must be some minimum vaJue of p4E” in order that the titration has at least one inflection point. This minimum vnluc of l>4E”’ is Ss.6 rn\- and corresponds to a value of k equal to o.rSSS. For k -C c. xS8S, i.e., ,+AE3’ > S5.6 mV, the redos titration tune possesses t\vo inflection points; the one corresponding to the minimum slope being located prior to tllc one conejponding to the ma_Cmum slope. When k = o.xS68, ON/_V 0112 inflection point eskts, that at / = 0.70% where P(E, -EL) = - 25.GS mV (at K = o.xSSS, thcrc exists only one X*~UC of pv, that satisfies cqn.(r3), tk., pyc = -I). For values of k such that k >0.1:888 (i.e.. pAE”< 65.6 my). tJ)e titration cume Jlas no inflection points. NIGI~TI~GALE concluded from a mathematical treatment of the poising capacity6 that in the titration oi a sdlution initially containing two redos coupIes, the second end-point (corresponding curve has to a minimum in the poising capacity, i.e., wllcrc ttlc slope of the titration its maximum value) would not be observed if the difference between the formal potcntizls of the titrant redox couple and that of the couple initially present (containing the less readily oxidized species) was less than 85.6 mV.
/. Eiecfroa~rcl. Chcm., 14 (1967) 373-333
curties &ere_pAE” 2 85.6 and S6.0 -my, .&spect_iGel~r_-From -Table .4 it ti_ apparent that even ‘yhen up.&!? is on@ 0-4 mV &s&r thanthe. mirknqti value (necessary for the existence of an inflection point} of S5_6 XIV, there is a considerable difference bet\+een-the locations of the points-~of minimnrnandmaxirn &nsl6pealthoughthemagnitudes ofthesetwFoslopesdiffer by 1t.k
ins~ktive
to
the titration
compare
only O-I. It islseen that the redox titration curve even at this small value of$A.??"' (=S6_o mV) possesses the same general nature that is observed for much greater
Dr =
85.6
mV;
R =
o.ISSS
PS
I
0.707 25.68 go.0
0.5173 42.s 92.8
3 44-4 93.6
--p(E-Ef)
s:.sg
iwirt. slope -&QEO'=
86.cmX';
R =
_wax.
szupe
0.187
f --p(E-E*)
0.518r
f 44-S 93-9
?S
o-751
o-677 d3.5 90-z
43-o 9'.9
21.9
I 0
90.3
81.5
-
T4BLE
5
THEREDOXTITRATIO~
f-
m.
0.500 o.=joq 0.5178 -0.536 0.564 --5Q3~ 0.621 0_650 0_672 0.678 0.69~
..
.O.7O7
CURVEFOR
pAE”‘=
85-6
mV_4T25°
-fJ(E--E*)
-Pw
PS
44-4 43-7 47-A +*-I
1-729
9x-6
0.y12
25-2
I.700 1.667
93-2 92.8
24-4 Z=J_I
I.600
g2.I
33-5
I.500
qr.z
36.0
I_qcm
0.721 o-735 o-767 O-770
9= 7
a_SzI
33-4 =J'_I 38.8 28.3 27.0 ~3.65
I-300 I.=00
90-z go_0
I.12
89-g
I_IOO I.050 *-cPoo
39-9 59.9 S9-9
o-go8 o-936 0.969 I .r_o32 1_06=j
f
--p(E-E=)
--PY
PS
o.gso
89.9
89.9
0.950 .0.900
20.3
89.9 89-9 89.8
0.790 0.780 0.600 O-300 0.200 0.100
ao-o =5-q 7-71 5.=4 2.57 Q -2-57 -5_r4
89-5 86.5 85-2
83.2 80.8 78-I
0
-----o-I --o-z
75-o
v&z_, the slope of the titration curve decreases,. BS the idue iucreases,untfithe:point of minimum slope is reached--'which occms subsequent
.. ;values&pAE"',
’
at Which~ E = &Or .which, in- t&n,
the- &mt
Thereafter,
k subsequent to the point._at whichf = &.~ value.? of f; the sIope~:rikes to a. maxim25 Iocation~ prior to the equivalence &oint. -From tien on,-ffiti_slope
with
progressively
increasing
oCcurrLn&~~t .~ ~~ -_oo~ntinu_ally d&creases. -: ~~ :._ _. : : :I‘$‘& only at-one pkrticular valuelof&~thzitthe~~lop~e~eshibits'neithera -no=
a"
of f to
-&xir&iti__valk~~ _ :~_~.:.:
Rather,:the -~- :
slope decreases :
C+ the titration
~~@$&urn
prodee& -until a :.
-_;. .
-~.~ .~
I_NI;LECTIOS
I’OISTS
I;c,r 5t1 C>llh’ ix1.5cr nc)
iliflcctic~n
ON TITI~hT1ON
csxmy>lc:, arc ti1crc: Imints
wticn tlircc:
ASI
CURVES
:I inf~riohn.sic*
xcicl tyyat5
is of
titr;itiBcl
wit 11 a
titr:ttic)ri
(-Iii
vt*5
mc,rioac-iclic: I)t)sGblc-:
(3)
; (t)) only
ttit: inflcctioli I)oilit cc)rrc:s~mndirlg 10 tlic. ln:isirll~ill~ *;ICBIP vsj:;tr.. c>,(c) t xv<) infbct ion I)oints c,ri
to tile maxiniuni slope exists at 3 non-x::10 val~c c~f f, sulutior!s ttie cunccntiation of which esc-cccls I0 -4 1;. ~1s concentratic)n of tilt: acid is clcc:rc‘;~s~fcf,the ioc-.?tiori of Clic: point of minimum -but 11c’vcr c:sc:c:c:c\s /’ ; SIOIIC IIIOVt!S a\ViIy frCl111 zt:ro tc~\V;lItI~ I;ilC;Cr v;llUcS c>f f 1)IoViclc~~l ttiat tile c~jIlr;crltI-ntivll ui tile ba~c is not f;rc:ltcr than that of tlit? :ac*id 41c)pe rnov6ls Cownrrk sm:ilhtr irritiirllyancl tl2C ltr<:aCi<)n cif tht: point c,I m.3simurn iIlfltx:tinn wllic-tl is thct initial
I)oint.
corrtxpontiiri~
virtually
unity
for
values c:f /_ As LVCcoritinuc tion value is rctachcci where paint
c*f n1axiriiun~
sloptt,
to
clccrasc:
the point Icx-:ltc:cl
at
the!
init
ial
cwnwntr;ttion
01 minimum a v.allltf
c,f
f
of
r&cl,
slop2 ~2.2scs to exist cc,nsidcrnl>ly
less
than
a concentradtliouC;li tiic:
Ilnity,
still
When the initial acicl cunccntr;ition is allowed to dccrcasc cvc:n further. tllc ln,cntinn of the maximum slops mnves tnward CWII smaller values of / urllil firidly it ti1c slcq_x: of tllc CUNC cll?crcnscS too ccz~scs tc>exist, am1 tlitm tlii cm~h>ut. tiic titration.
ftxists.
continuc,us!y
fIonl
it:, initial
valut:
at / -
0. Thus,
it is .?pprcnt
that
there
is a si~milnr-
ity betwt:en these acid-base titration curves and the titration curves for symmetrical redox reactions in the cxistcnctt of three- cliffctrcnt types elf conclitions which proviclc: three apprclntly tliffprent types of c~-~e_s. I~~lrtlwrmorc:. wlwn the primary indcpcn dent variable: (the initial acid concentration in the :icid-base titration; ,+A??” in the rcclox titr;ltic>n) h;Ls CIvalue which is kss than some minimal orle, no inflectron IMnts exist. In the titrntion of :a monol>asic: wt:;& acid of initial c.onc:t:ntration C, with ;L niononcidic strong hasca of concentration C, the value of f, at which I:ii t ] = Kn. is /_ Eleclroawal.
(ihem..
14
(1907)
373- $33
~-..382 i----.
y-,
-. :;_. :‘- ;,_‘. :~ :.
;;
-~
_
_-_.-
:.~:
‘.
-.-I _:_:_:~:~~.“~-.__~‘-
-_
.J;
A-_.GOLD+
_.~
: the-s&es
oif
ak:~l&'~thA
g-when
PKks~
In contradistidktion;~fora symme&al exceeds 41 Perhaps this dif&en&i6is not a point ofi~the redox titration ~~ :-‘~~~k.(~~theregio~~~d~f~z) &here thevalue ofK, and~onIy.K,_d&ectly deter&n= The variation of-the location of. -. the- pctential, -viz_, .E--_E *I (R~/F)mk=&AE"'_ minimum slope ontheseacid-base titration .the infkcti~&@oint corr+po&ingtothe ~~ c&ves;'is sim&rtothe behavior of values offat which [H+]~=K,.-because it may bc&r at vahres of~greaterorIessthan~_~enpX,<.~~pK~~,the pointqfminimum slope (when it e&ists) always occurs prior to the point at which [H+]=Kn, wmch in for PK,~>~PKw, ~. tafn precedes the "halfway point"(where f=$-2); whereas po;ltatwhich[H+]=K,. pointof rninim~slo~e(~vheaitesists)alavaysfollows&e
latter 2dways ~&curs tier
which
the half-way
point_ The inflection
the
point ckresporid-
mgto therni&num slope always occurs before theinflection pointcorrespondingto the location of rhein4he.m axiruuruslope. In other words, the disparity between and that of the half-way point is flection poink corrtzspondkg to them inknuti~slope +lways.greater than that-between the point at which [H+]=K, and the half-way ~~point.‘I%owever, in a:symmet&cal redox.reaction,.the location of the point of mim~rrkrn slope always. occurs after the point at which E = EzO’, which in turn always this sequence resembles the hollows Ithe half-xvay point, so that in a formal sense,
acid-base titratjon where$Ka> and for small values,
Ka >o:I,
to.:exist,
but
it is o&y
for
the
Q e.g.,
pKxv_
Both
for relatively large values of K,, e-g-,
the point of minimum slope. ceases Ks< 5 -IO-~',
srnd
value&of~K,
that the point
of maximum
slope
to-the fact that the .lo~ation'oftheminimumslcpemayoc+ron either~sideofthehalf-waypointbutthe maximum slope can never occur before the half-way point, and-~indeed,_ceases to may
also cease
to etista_ Esserkxlly.
this may
be
attributed
.ersst hefore it ever reaches the~half-way point. In contrast, for symmetrical redox reactions,thelocation oftheminimum slope canonlyfollowthehalf-waypoint-When m.nel,ther a maximum nor-a minimum slope exists (Kn<( -?o-12), the slope of the
-titration curve thronghout the entire titration, continuously. decreases from initial-valueatf=o, asitdoes inthe redostitration where pAE”r
its
..
FI& titration curves of symmetrical (2~1=ns~=p) redok reactions, it has been denioktrated thkthere are-always two inflecti& points whenever p&E"' > 85.6 CV equivalenceat 25O-;the location ofthepoint~of' maxinmmslopeis~alwayspriortothe ruin&slope is always:located subsequent~tntkhalf_. point, where23 the pqint of 300 mV, Way point: F~rpractic~~-tItra~ions.where~~AE O'S customtiygreater~thari
INFLECTION
POINTS
OK
TITRATION
CURVES
383
REFERENCES
III_ z I-G_ MITRGULESCUXXDCDRAG~LESCU,~. 3 + 5 6 7 8 g
PikpzR.Chenz., 185A(rg4o) J. A.-GOLD~UN,]. ElectroanaLGhem.. 11~(rg66) 355. E.BISHOP,AIZ~Z.C~~~.~C~~~, 27(1962)253. J. A. GOLDB~AW.J_ EZec#roanuZ_ Chem., II (1966) 416~ E. R. NIGHTINGALE, JR., Anal_ Chem.. 30 (1958) 267. L_ X~EITES AND J_ _X. GOLDMSW, AtraZ. Chi+~z. Acta, 30 (1964 203. J_ A. Go~D~IAE;A~DL.MEITEs,A~~uZ.C~~~Z.~~G#~~, 30 (Ig6q)zS. L. MEITES AWD J_ A_ GOLDM_KN, AmzL Chim Acta, ag (1963) 472. J.
EZectroa~d.
375.
Chem..
14 (1967)
373-383