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Chemistry, Department of
1-1-1978
The Effect of Slow Two‐Electron Transfers and
Disproportionation on Cyclic Voltammograms
Michael D. Ryan
Marquette University, [email protected]
Published version. Journal of the Electrochemical Society, Vol. 125, No. 4 (1978): 547-555. DOI. ©
Electrochemical Society 1978. Used with permission.
The Effect of Slow Two-Electron Transfers and
Disproportionation on Cyclic Voltammograms
Michael D. Ryan*
Marquette University, Department of Chemistry, Mffwaukee, Wisconsin 53233
ABSTRACT
The EE m e c h a n i s m (two-electron transfer) for cyclic v o l t a m m e t r y was
investigated in considerable detail along with the effect of disproportionation.
The theory was developed for e i t h e r the first or second electron transfer being
slow while the other one was reversible. It was possible to develop generalized
w o r k i n g curves for the height and shape of the wave regardless of the
difference i n E~ and the values of ~ and ks. This theory was then applied to
the analysis of the reduction of benzil in the presence of alkaline earth ions in
dimethylformamide.
The theory for reversible and irreversible electron
transfer has been e x a m i n e d i n some detail previously
for cyclic v o l t a m m e t r y (CV). The p r o b l e m of m u l t i electron transfers i n CV was studied first by Polcyn
and S h a i n (1). The m a i n q u a n t i t a t i v e emphasis was
placed on the deconvolution of separated waves and
only qualitative results were given for m u l t i - e l e c t r o n
waves. Myers and Shain (2) were able to q u a n t i t a tively correlate the shape and peak c u r r e n t function of
r e v e r s i b l e v o l t a m m e t r i c waves to their difference i n
Eo's if the Eo's were close together. It was found that
the shape of the wave was i n d e p e n d e n t of the relative
values of the Eo's only if the difference i n Eo's was
greater t h a n 180 mV. Otherwise, the wave will be
broader t h a n expected, even though it is still reversible. I n addition, its shape and position are i n d e p e n d e n t
of scan rate. No correlations were made for quasireversible or irreversible electron transfers.
Multi-electron transfers can put a much more stringent d e m a n d on the rate of electron transfers if the
wave is shifted significantly from its E o because of the
second electron transfer. Ruzic (3) has shown that, for
polarography, the a p p a r e n t heterogeneous electron
transfer rate constant, ks', for the o v e r - a l l two-electron transfer is less t h a n either of the i n d i v i d u a l rate
constants, k~.l or ks.2, if the second E o, E2, is more positive than the first E o, El. In particular, the rate constant that is calculated using conventional theory is not
the i n d i v i d u a l rate constant but a n a p p a r e n t one. Thus,
the theory for quasireversible electron transfers i n CV
(4) yields the value of ks'.
Competing with the m u l t i e l e c t r o n transfer mechnism is the disproportionation mechanism. If the second electron transfer is so slow as to not be i m p o r t a n t
and if E~ is positive of El, the classical disproportionation mechanism can then occur. The theory for this
mechanism has b e e n developed and verified by Saveant (5). But very little work has been done in t r y i n g
to assess the relative importance of the electron t r a n s fer mechanism (EE) as compared to the disproportionation (DISP) mechanism. For reversible electron
transfers the cyclic voltammograms will not distinguish
b e t w e e n these two mechanisms.
It is the purpose of this work to develop the cyclic
voltammetric theory more generally for the EE mechanism, including the possibility of disproportionation,
and to verify it experimentally. The emphasis will be
placed on those cases where only one wave is observed,
and either the first or second electron t r a n s f e r is slow.
Recent studies have shown several reductions that are
now k n o w n to follow the EE mechanism (6). In this
study, the electrochemical reduction of benzil in dim e t h y l f o r m a m i d e (DMF) in the presence of alkaline
* Electrochemical Society Active Member.
Key words: EE mechanism, cyclic voltammetry, disproportionation.
earth salts, w h i c h is k n o w n to be a quasireversible
two-electron transfer (6), is investigated in more detail in this work using the theory that is presented.
Experimental
Spectroquality N , N - d i m e t h y l f o r m a m i d e was obtained
from Matheson, Coleman, and Bell. The solvents were
dried as required using activated molecular sieves.
S t r o n t i u m and b a r i u m perchlorate were obtained from
G. F. S m i t h Chemical C o m p a n y and were v a c u u m dried
at 250~ for 6 hr (7). Benzil was recrystallized from
ethanol. The reference electrode is the same as was
described i n Ref. (6). Positive feedback resistance
compensation was used i n all of the fast scan rate experiments. A h a n g i n g m e r c u r y drop electrode was
used as the working electrode. All solutions were deaerated with dry nitrogen. The diffusion equations were
solved n u m e r i c a l l y using the technique of digital s i m u lation (8). The flux equations for a t w o - e l e c t r o n t r a n s fer that were derived by Feldberg (9) were used to
calculate the current.
Theory
The generalized EE mechanism that is discussed i n
this paper is given by reactions [1]-[3]
El
A - t - e ~- B
[1]
E~
B + e ~- C
[2]
kd
2B --> A + C
[3]
Each electron transfer has associated with it a ks,i and
~i value, where i : 1 or 2, for the first or second electron transfer, respectively. The analysis of the mechanism is divided into two m a j o r sections, depending
upon w h e t h e r the first or second electron transfer is
rate limiting. In both cases, the mechanism is studied
in the regions where only one wave is seen.
While each i n d i v i d u a l step has an ~i associated with
it, the over-all electron transfer coefficient that is
observed if one wave is seen has been derived by
Mohilner (10). If the first electron transfer is limiting,
then
an - - a l
[4]
If the second electron transfer step is limiting, the observed an is
an = 1 § ~
[5]
First-electron trans]er limiting.-- Reversible electron
trans]er.--Using the theory that was derived for polarography (3) and verified by digital simulation in
this work for CV, the value of ks' can be calculated as
follows
ks' : ks.1 exp[--alFaE/2RT]
[6]
547
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548
Y. Eleetrochem. Soe.: ELECTROCHEMICAL SCIENCE A N D T E C H N O L O G Y
w h e r e hE _-- E2 -- El. As stated earlier, it is the ks'
value that is e x p e r i m e n t a l l y measured. Thus, ~1 and hE
must be k n o w n in o r d e r to calculate ks,1. In order to
generalize the results, the dimensionless electron transfer rate parameter, %, will be used and is defined as
follows
-: ks/%] aaDA
[7]
and
a = nFv/RT
[8]
w h e r e v is the scan rate and t h e o t h e r terms h a v e
either been defined or have t h e ir usual electrochemical
significance. Combining Eq. [6] and [7], we obtain
[ --alFAE ]
~1' : ~1 exp
2RT
[9]
The shape of the cyclic v o l t a m m o g r a m depends on
its r e v e r s i b i l i t y (%1') and on the difference in Eo's
(hE). These are two different p a r a m e t e r s w h i c h must
be e v a l u a t e d separately. Cyclic v o l t a m m o g r a m s can
be divided into t h r e e g e n e r a l classes depending on the
va l u e of ~1'. These classes are: (i) reversible, ~1' > 7,
(ii) quasireversible, 0.1 < %s' < 7; and (iii) i r r e v e r s ible, %f' < 0.1. Within these t h r e e classes, the shape of
the w a v e is also d e p e n d e n t upon hE. In this paper, for
each of the t h r e e classes, the effect of AE on these
classes is examined. S e v e r a l l i m i t i n g cases h a v e alre ad y been described. F o r the r e v e r s i b l e and i r r e versible classes, Nicholson and Shain (11) h a v e d eriv ed the shape and b e h a v i o r of the w a v e if ~E > 180
mV. Myers and Shain (2) and Polcyn and S h ai n (1)
h a v e investigated the reversible case if ~E < 180 mV.
These papers can be consulted to d e t e r m i n e the shape
of the w a v e and the v a l u e of ~E. The i r r e v e r s i b l e case
was also studied by Polcyn and Shain (1) for AE <
180 mV but q u a n t i t a t i v e analysis was possible only if
two wav es w e r e seen.
The disproportionation reaction (reaction [3]) has
no effect on the shape of the w a v e if the first electron
t r a n s f e r is the slow step. This is because B is r a p i d l y
reduced to C at the electrode surface. This is not to
say that the disproportionation reaction does not occur
but only that the reaction has no effect on the w a v e
w h e n it occurs in this situation. This was verified by
digital simulation.
will depend upon hE in addition to r
This effect is
not large and can be corrected for in the same w a y as
shown in Eq. [32] later in this paper.
Irreversible
electron t r a n s f e r . - - I n the case w h e r e
single scan is used, the position, shape, and size of the
reduction w a v e depends on the v a l u e of ~s', as, and AE.
The w a v e does not occur at its t h e r m o d y n a m i c a l l y
defined E o b u t at a m o r e n e g a t i v e p o t en t i al depending
upon scan rate. In the t h e o r y as d e r i v e d by Nicholson
and Shain (11) for m u l t i e l e c t r o n i r r e v e r s i b l e transfer,
it was assumed that all the electron transfers except the
slow one w e r e fast, at least at the potentials w h e r e the
reduction w a v e occurs. P o l c y n and Sh ai n (1) h a v e
sh o w n that the w a v e will be affected if this assumption
does not hold true but no q u a n t i t a t i v e t h e o r y was de rived if one w a v e is observed. In o r d er to p r ovi de a
q u a n t i t a t i v e m e a s u r e of the r e l a t i v e difference i n r e duction potentials, the p a r a m e t e r , hEAB, is defined as
59
AEAB =
Table I. Variation of AEp values as a function of ~1' (~I -~ 0.5)
~Ep (mV)
~'
EE mechanism
Ref. (4)
1.0
0.5
0.2
0.1
45
61
107
156
42
53
79
106
AE
--
log %s
~
[I0]
al
This equation is derived from the fact that an irreversible reduction is shifted to more negative potentials by
a rate of 59/~i log #s. This parameter corrects AE for
the fact that irreversible reductions do not occur at
their thermodynamically defined potential but at some
more negative potential depending upon al and ksA.
Similar to reversible electron transfer, only one wave
is seen if AEAB is positive and two waves are seen if
AEAB is negative enough. The exact values of these
regions are shown later. By combining Eq. [9] and
[I0], AEAB can be defined in terms of #s',which is experimentally measured
59
A E A B --~ A E / 2
-- - log #s'
[ii]
aS
B y using digital simulation (8), it was found that the
theory for irreversible electron transfers derived by
Nicholson and Shain (ii) can be used if AEAB is
greater than 75/~i mY. In this case, the following diagnostic criteria for the cathodic peak potential, Epc, the
cathodic peak current, /pc, and the peak Epp/2 values
can be calculated
Quasireversible electron transfer.--For q u a s ir e ver si b l e
electron transfers, the shape of the w a v e depends upon
r and hE, and is not v e r y d e p e n d e n t upon ~1. If ~E is
l a r g e r than 180 mV, the v a l u e of ks' can be calculated
f r o m the t h e o r y for q u a s i r e v e r s i b l e w a v e s given by
Nicholson (4). But e v e n for AE > 180 m~/, noticeable
deviations occur w h e n the waves are s i m u l a t e d by the
digital simulation t e c h n iq u e (8, 9). The results are
shown in Table I for ~1' less than 1. T h e deviations are
due to the g r e a t e r effect of ~1 on the peak potentials for
the same v al u e of ~.
To calculate ks.z, 5E and as must be known. The
v a l u e of ~1 cannot be accurately d e t e r m i n e d unless
scan rates large e n o u g h to m a k e the w a v e i r r e v e r s i b l e
are used. At these large scan rates, the theory p r e sented in the n e x t subsection on i r r e v e r s i b l e electron
transfers can be used to calculate ~.1. The v a l u e of hE
can be calculated using the r e v e r s i b l e t h e o r y if slow
enough scan rates can be achieved or by using the i r r e v e r s i b l e t h e o r y that is described for fast scan rates.
F o r AE values less than 180 mV, the w o r k i n g cu r v e
A p r i l 1978
/~c :- 0.992 FACA* -V/alDAal
[12]
Epp/2 = Epc -- Ep/2 -: 47.7/as m V
[13]
dEpc/d log v ----30/al m V
[14]
as = F v / R T
[15]
and
and w h e r e Ep/2 is the h a l f - p e a k potential.
If AEAB is less t h a n 75/~1 mV and only one w a v e is
seen, the shape of the w a v e is changed from that p r e dicted for an i r r e v e r s i b l e electron t r a n s f e r and depends
upon the scan rate. This is shown in Fig. 1 for t hr e e
different values of hEAB. As the scan rate is increased
(nEAB l a r g e r ) , the value of AEA~ will increase so as to
m a k e the w a v e approach the i r r e v e r s i b l e t h e o r y as
g i v en in Eq. [12]-[14]. The Epp/2 v a l u e is most strongly
d e p e n d e n t upon AEAB for AEAB values less t h a n 75/~i
mV. The v ar i at i o n of Epp/2 values w i t h hEAB is shown
in Fig. 2 for various values of al, w h e r e the Epp/2
values have been normalized in the following m a n n e r
E*pp/2 :
alEpp/2
[16]
For large n e g a t i v e values of AEAB tWO w av es are seen
and each w a v e can be analyzed i n d i v i d u a l l y because A
will reduce prior to the potential for the B wave. For
i n t e r m e d i a t e values of AEAB, the A r ed u ct i o n w a v e
will overlap the B w a v e and hence a b r o ad en w a v e
will be seen. Th er e is a m a x i m u m v al u e for which hE
can h a v e and still see the b e h a v i o r as described in
Fig. 2. F o r the w a v e to begin to broaden, AEAB must be
less than 75/~1 m V and %1' must be less t h a n 0.1. Thus,
r e a r r a n g i n g Eq. [11], we obtain
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Voi I25, No. 4
CYCLIC V O L T A M M O G R A M S
t
l
I
I
I
549
waves can be seen on the r e t u r n scan. The e x p e r i m e n tal canditions for the n u m b e r of peaks that are seen
are given i n the n e x t section.
With the i n f o r m a t i o n from the reverse scan, it is
possible to d e t e r m i n e all the p a r a m e t e r s for the twoelectron transfer. There are four u n k n o w n s which need
to be evaluated from the e x p e r i m e n t a l data." They are:
E~, E~, ks,1 a n d ~1. There are two other parameters
that are related to these and are calculated d u r i n g the
analysis. They are
I
AEAB
2OO
"T
EL2 = (E, -5 E~)/2
50
-I00
[20]
and AE. It is the aim of the rest of this section to develop the equations necessary to calculate these parameters from the data.
If two anodic peaks are seen, E2 can be found directly from the most negative peak potential, Epa(2),
because that w a v e will occur at its thermodynamically
defined potential
E2
E,
MV VS E01
~iox*2
Epa(2)
:
--
0.0285
[21]
-~2.00 The value of al can be d e t e r m i n e d from several m e t h '
ods depending on the value of AEAB. If Eq. [12]-[14]
are obeyed, al can be d e t e r m i n e d from the shift i n
Fig. 1. The simulated cathodic scan for three different values
Ep~ or by the Epp/2 value. If hEAB is less t h a n 75/al mV
of ~EAB (first-electron transfer irreversible).
(in other words, the Shape depends u p o n scan r a t e ) ,
I
I
I
I
I
the value of al can be d e t e r m i n e d from the more positive anodic peak potential, Epa(1). Thus
9O0
l
9 00
-2. O0
- 4 , O0
I
1
- 6 , O0
I
- 8 , O0
1
-~0.00
70
dEpa(1)
30
d log v
1 -- al
mV
[22]
The cathodic peak potential can be related to the
E1,2 value by the following equation (11) if hEAd >
75/al m V
Epc
= E1,2 -- -~IF
and Epa(1) iS
Epa = E1.2 -~
(1
0.339 -- log41'
--~,~F
~J
30
[
-~0
I
I
0
100
[
I
200
B y t a k i n g the difference i n the peak potentials, w e obtain
Fig. 2. The variation of Epp/2* as a function of AEAB for vario-s
values of al in the case where the first electron transfer is irreversible.
0.059
~Ep _ a 1 ( l -- a l )
(0.339 - - l o g ~ { )
0.059 log . ~ 1--aI__ -5 0.059
.... l o g ~ /
32
aE < - - m V
[17]
-5 1 -- ~I
~z
a~.
[25]
al
Thus, since al is k n o w n , @I' can be calculated f r o m the
Otherwise, Eq. [12]-[14] will always hold w h e n the
wave is irreversible. I n the limit of v e r y slow scan
rates, the Epp/2 value is limited by the value derived
by Myers and Shain (2). The peak c u r r e n t function
goes through a m a x i m u m at the same point that the
Epp/2 values go through a m i n i m u m b u t the change in
this p a r a m e t e r is not very large with the m a x i m u m
change on the order of 20%, m a k i n g this p a r a m e t e r
less useful.
I n the cyclic experiment, on the anodic scan, the
reverse of reactions [1] and [2] occurs
E2
C ~- B + e
[18]
AEp values. If @1' is now substituted i n Eq. [23], El,2
can be found. The value of E2 is k n o w n from Eq. [21]
so E1 can be calculated by using Eq. [20].
If only one anodic peak is seen, ~1 can be found from
the anodic or cathodic wave. If the anodic peak potential is used to calculate ~1, then as will be described
in the next section, Eq. [26] must be used
dEpa
----30/(2 -- ~1)mV
[26]
d logv
The value of 41' can be found, as before, from the AEp
v a l u e b u t Eq. [28] m u s t be used because Epa is given
by
El
B ~-- A -5 e
[19]
For the r e t u r n peak, reaction [19] is rate limiting,
which is the second-electron transfer. A full discussion
of this case is given in the n e x t section and only a few
p e r t i n e n t details are given here. Either one or two
Epa : E1,2 -5 ~
0.339 + log 41' -5 log
[27]
Using Eq. [27], the variation of •
by
with 41' is given
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J. Electrochem. Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
550
0.118
~Ep _
,~i
(2
-
(0.339 + log ~l')
,,~i)
0,059 log
-t- 2 - al
V l--al +
0.059 log
al
2~
V ~al
[28]
Irreversibility of the second electron transfer makes
the second electron transfer occur at more negative
potentials. In order to correct for this, we can define a
parameter, hEBc, which takes into account the effect
of the slow electron transfer
59
The value of Ei,2 can be found from Eq. [23]. The final
problem is to find a method to determine E1 and E2
from EI,2. To do this, either fast or slow scan rates
must be used to separate either the anodic or cathodic
peaks into two waves. It is theoretically possible to
cause the cathodic peak to split apart if hE is less
than --60 mY. If this is not possible, then scan rates
fast enough to split fhe anodic peak apart must be
used. If the cathodic peak begins to split apart, Fig. 2
can be used to calculate the value of AEAB from the
Epp/2 values. From Eq. [11], it is possible to determine
hE because
and ~ i ' a r e k n o w n
hEBc : hE~2 + - -
bE
= 2
AEAB
59
+
This p a r a m e t e r normalizes the combined effects of bE
and ~2' on the relative positions of the two waves.
F r o m digital simulation, it was found that, if hEBc is
greater than 50 mV, the shape of the wave is given by
the theory derived by Nicholson and S h a i n (11). S u b stituting n = 2 and an = 1 + a2, the following diagnostic parameters can be calculated
)
--log
~l'
[29]
Unless there is some change i n the shape of either the
anodic or cathodic wave it is impossible to determine
E~ and E2.
Second-electron transSer limiting.--No disproportionation.--If the disproportionation reaction (reaction
[3]) does not occur, the only way of producing C is by
the direct reduction of B at the electrode surface. As
with the previous case, an a p p a r e n t electron t r a n s f e r
rate constant, ks', is observed e x p e r i m e n t a l l y rather
t h a n the actual ks value of the second step. The value
of ks' is given by
k s, "- ks,2 exp
(-
F
(1 - - a 2 ) -RT
-
or
2
[30]
F hE]
~2' : ~2exp [ -- (1 - - a 2 ) -RT
-
2
[31]
As before, reversible behavior is observed if @2' is
greater than 7. If ~2' is less t h a n 0.1, a n irreversible
wave is seen. For reversible waves, the shape of the
wave (2) depends only on hE and is i n d e p e n d e n t of
hE if hE is greater t h a n 180 mV.
Quasireversible electron transSer.--For quasireversible waves, the shape of the wave depends upon @2'
and hE and is not very d e p e n d e n t upon a2. If hE is
larger t h a n 180 mV and @2' greater t h a n 0.5, the value
of ks' can be calculated from the theory for quasireversible waves given by Nicholson (4). The hEp values
as calculated b y digital s i m u l a t i o n for different a2
values are given i n Table II for several ~ ' values less
t h a n 1.
If hE is less t h a n 180 mV, the hEp values are larger
t h a n predicted by Ref. (4). To a close approximation,
the working curves for ~E < 180 mV can be obtained
by adding the excess broadness, Eexc, as defined i n
Eq. [32] to the data i n Table II
Eexc :
hEpr
--
29 mV
[33]
log ~2'
1 -{-a~
ip = 0.992 F A %/(1 -t- a2)DalCA*
[34]
Epp/2 : 47.7/(1 + a2)mV
[35]
dEp/d log v -- 30/(1 -{- a2)mV
[36]
~i
(
April 1978
If hEBc is less t h a n 50 mV the wave will b r o a d e n
and the peak c u r r e n t function will decrease. This
effect can be seen i n Fig. 3 for three different values
of hEBc. The variation i n Xp and Epp/2 values can be
calculated as a function of a2 by the following n o r malizations
xp* : xpl(1 + ~2) li~
[37]
Epp/2* = Epp/2 (1 + a2)
[38]
The effect of hEBc on Xp* and Epp/2* is shown i n Fig.
4 and 5. The variations in these parameters are qualitatively similar to the ECE and DISP mechanisms. It
is only at fast scan rates that the Xp and E p p / 2 values
differ significantly from the ECE and DISP mechanism.
But c h r o n o a m p e r o m e t r y will easily distinguish between these mechanisms because a t w o - e l e c t r o n diffusion-controlled wave can be obtained if a negative
enough potential is used. I n contrast, an ECE or DISP
mechanism will show the slow kinetic step regardless
of applied potential.
I n a cyclic experiment, since the B to C reduction is
limiting in this case, the C to B oxidation will be
limiting on the reverse scan. Thus, the theory for
the first-step limiting must be used to evaluate the
oxidation peak.
As before, there are four parameters which must be
determined: El, E2, ks,e, and a2. Thus, four i n d e p e n -
I
[
I
I
100
1,2
0
0.8
[32]
where hEpr is the hEp value for a given hE (reversible
case).
0.4
Irreversible case.--With a single scan, as defined earlier, the wave is irreversible if ~ ' is less t h a n 0.1.
Table II. Variation of hEp values as a function of ~2'
~ll'
(~i :
0.35
AEp (mV)
0.5
0.63
Ref. (4)
i
20o
i
i
0
I
-200
F-E1, mV
1.0
O.fi
0.2
0.1
44
58
91
128
44
.~6
88
135
43
53
85
147
42
53
79
106
Fig. 3. The simulated cathodic scan for three different values
of AEBc (second-electron transfer irreversible).
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Vol. 125, No. 4
CYCLIC V O L T A M N I O G R A M S
~E--2
f
551
59
~EBc
/
log r
1-~
[42]
~
If ~E is such that the wave does not begin to split
apart, then a lower limit on ~E can be established because AEBC must be greater t h a n 50 mV
118
~E > 100
I
I
I
I
I
i
i
a. Ee~/r~
Fig. 4. The variation of Xp* als a function of AEBc for the case
where the second electron transfer is irreversible.
i
J
j
r
- log r
l+a~
[43]
Thus, from Elq. [35] or [36], [39], and [41] and [42],
we have four i n d e p e n d e n t equations which allow one
to calculate ks,2, ~ , E1 and E2. The solution is simple
because there is only one n e w u n k n o w n in each equation. It is i m p o r t a n t to obtain a good d e t e r m i n a t i o n of
a2 since this is critical i n all the equations. If the wave
is b e g i n n i n g to split apart, it is p r o b a b l y more accurate
to use the anodic peak potential to d e t e r m i n e a2 because that will not be affected. But fortunately, it t u r n s
out that unless the wave has broadened considerably,
the calculation of c,z from Eq. [36] is also not seriously
affected even though Eq. [35] cannot be used.
Disproportionation.--For ~g > 0, the disproportionation reaction is favorable and can compete with the
electron transfer if it is fast enough. I n order to discuss
this m e c h a n i s m quantitatively, the rate constant, kd, is
normalized as follows
~cl : kaCA */a
70
I
- 40
I
i
0
I
i
40
I
i
80
I
/', EBC, mV
Fig. 5. The variation of Epp/2* as a function of AEBc for the
case where the second electron transfer is irreversible.
dent equations must be derived. Only one anodic peak
can be seen u n d e r practically a n y conditions w h e r e
only one cathodic peak is seen as a consequence of
Eq. [17]. This limits the existence of two peaks to •
values which are close to zero or negative. The value
of a2 can be calculated from Eq. [35] or [36]. The second equation that can be used to calculate these parameters is the cathodic peak potential (11), which is
0.059
Epc : El,2
i ~- a2
[44]
where ~d is the disproportionation kinetic parameter.
It is i m p o r t a n t to keep i n m i n d that ~ depends u p o n
CA* while ~2 does not. Thus, it is possible to change ~d
without changing ~ at the same time. In Fig. 6, the
effect of the disproportionation reaction can be seen
quite clearly. In this figure, the wave becomes d e p e n d ent upon the disproportionation reaction and not on
the second electron transfer. The same type of behavior is demonstrated i n Fig. 7, where the electron
transfer predominates if it is large enough in spite of
a fairly significant rate for the disproportionation reaction. Qualitatively, the DISP mechanism is most i m p o r t a n t for negative values o~ ~iEsc. If ~EBc is positive,
the difference in the shape and height of the wave due
to these two mechanisms is not large.
The effect of the disproportionatio,n reaction on the
EE mechanism was d e t e r m i n e d by the use of digital
simulation (8). The results are shown i n Fig. 8 and 9
for the Xp and Epc values for a2 = 0.5 as a f u n c t i o n of
AEBc. It is assumed i n this case that the wave is i r J
J
l
I
I
I
[ 0,339 _ log +, _i_log.~/_~_
l
~ J
[39]
I n addition, Epa is given by (11)
00 0[
Epa : El;2 nu ~
0.339 -- log ~z' -? log
2~
[40]
Combining Eq. [39] and [40], the AEp values can be
derived as a function of ~ '
fl=
0.118
0.059
~
~lEp -- - - -1 a2 (0.339 -- log ~2') ~- 1 + a~ log " V
0.059
/
~- 1 -- a2 log ~ /
1-
2~
a2
2"-~"
a2
[41]
Once r is calculated, E1;2 can be calculated from Eq.
[39]. The final d e t e r m i n a t i o n of E~ or E2 requires that
~E~c be less t h a n 50 mV so that the wave wilI begin
to split apart. If AE~c can be d e t e r m i n e d from the Epp/2
or ~p values, t h e n
E-El. MV
}O.O0
200. O0
~00, O0
900
7100.00
-200.00~
Fig. 6. The simulated cathodic scan for -~EBc =
(a) ~d = O; (b) ~d ---- 10; and (c) ~d = 30.
~300.00
--20 mV and
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J. Electrochem. Soc.: E L E C T R O C H E M I C A L S C I E N C E A N D T E C H N O L O G Y
552
L
1
I
I
I
I
I
I
I
April 1978
I
I
c
50
@
DE
~.-E i, MV
0,00
200.00
I
IO0.O0
,OO
I
J
~O0, O0
:
-200. OO
-~00. O0
~
I
Fig. ?. The simulated cathodic scan for ld ~ 10, and AEBc
equal to (a) --100 mV; (b) --50 mV; (c) O; (d) -I-75 inV.
I
I
I
I
I
....
-50
T - -
d-f
!
I.o
b
-I
!
os
I
i
I
l
l
_
3
Fig. 10. Values of AEBe and ld where various mechanisms are
applicable. E = EE Mechanism, D : - DISP Mechanism, DE =
Mixture of EE, and DISP Mechanism.
0.1t
-So
o
5o
~EBC, mV
Fig. 8. The variation of Xp as a function of AEBc for various
values of td (second-electron transfer irreversible). (a) Xd = 1000;
(b) t d ~--- 300; (c) ~-d = 100; (d) ;~d :
30; (e) ~-d = 10; (f)
;Ld ---- 0.1.
reversible (42' < 0.1). For quasireversible a n d r e v e r s ible waves, the effect of the DISP mechanism is small.
Using the criteria that the DISP mechanism predominates if xp is w i t h i n 5% of the value for the wave w h e n
42' - : 0, a range of AE~c values can be calculated and
the results are shown in Fig. 10. At the other extreme,
the wave is defined by the EE mechanism if the Epc
value is w i t h i n 2 mV of the value obtained when Ld ---0. These values are also shown in Fig. 10. Between
these two limits, the wave is controlled by both the
EE and DISP mechanism and Fig. 8 and 9 can be used.
These limits can be expressed as follows
DISP m e c h a n i s m predominates if
I
I
I
I
AEBc < 25 log Ld - - 70 mV
[45]
EE mechanism predominates if
AEBc > 25 log kd -- 5 m V
b
'5
I
I
-25
0
I
I
25
SO
9 ~EBC, mV
Fig. 9. The variation of Epc -- E1 as a function of AEBc for
various values of ;~d- Values of Xd given in Fig. 2.
[46]
While these results were obtained for a2 = 0.5, the
same conclusions were obtained for ae = 0.35 and 0.65.
The Epc curve shown i n Fig. 9 is almost i n d e p e n d e n t of
~2. The Xp curve (Fig. 8) does depend upon a2 because
the limiting value of xp w h e n AEBc is large does depend upon a2. I n general, this dependence is small and
varies by only 10%, at most, for ~2 b e t w e e n 0.35 and
0.65. But the limits given by Eq. [45] do not depend
upon ~2. This can be seen in Fig. 11 for a2 -~ 0.35, 0.5,
and 0.65 and kd = 10.
Diagnostic C r i t e r i a and D a t a Analysis
The key to the analysis of cyclic voltammetric data
for this case is a systematic approach. The analysis can
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Vol. I25, No. 4
1.4
f
l
I
CYCLIC V O L T A M M O G R A M S
I
I
I
I
I
I
c
!
~6
I
I
I
I
-50
I
0
I
50
I
I
100
AEBc, eM
Fig. 11. Variation of Xp as a function of AEBc for ~.d = 10 and
a~ equal to (a) 0.35; (b) 0.50; end (c) 0.65. Dashed fine indicates
the value where Xp has increased by 5%.
be segregated into the following steps:
1. d e t e r m i n a t i o n that EE m e c h a n i s m is occurring
2. d e t e r m i n e if disproportionation is occurring
3. d e t e r m i n a t i o n of the reversibility of electron
transfer
4. identification of the slow step
5. calculation of electron transfer p a r a m e t e r s
S t e p / . - - T h e d e t e r m i n a t i o n that an EE m e c h a n i s m is
occurring is most easily done by d o u b l e - p o t e n t i a l step
chronoamperometry. By comparison of the diffusion
c u r r e n t with compounds of k n o w n electrochemical b e havior, the n value can be calculated. Second, from the
ia/ic ratio, the stability of the product can be v e r i f i e d
and one can verify that no irreversible chemical reactions are occurring. Third, the ic ~fr/CA* value
should be constant, indicating that no new electroactive products are formed. The value of the final potential m u s t be negative enough so that the slow electron transfer step does not affect the ic %/~'/CA* value.
Step 2.--Since the value of ~.d depends u p o n CA*, the
shape and position of the wave should be dependent
upon CA*. Conversely, 4' is i n d e p e n d e n t of CA*, so one
can d e t e r m i n e if disproportionation is i m p o r t a n t u n d e r
the conditions used.
Step 3.--If the shape or position of the wave is i n dependent of scan rate, t h e n the electron transfers are
reversible, if the values of hEp are b e t w e e n 30 and 150
mV, at least one of the electron transfers is slow and
the wave is quasireversible. If the AEp values are
greater t h a n 150 mV, the electron transfer is irreversible.
Step 4.--The i d e n t i t y of the slow step can be surmised from several ways depending upon the rate of
the electron transfers. If the wave is quasireversible,
it is difficult to d e t e r m i n e the slow step because of the
similarities in w o r k i n g curves. Only if scan rates large
enough so that 4' is ( 0.1, (i.e., the wave is i r r e v e r s ible) are used can one d e t e r m i n e which step is slow.
If the electron transfer is irreversible, the i d e n t i t y of
the slow step can be easily d e t e r m i n e d from the ratio
of the peak currents, ip,a/ip.c, where ip,a is measured
from a baseline calculated from the t -~/2 decay of the
cathodic current. If the first step is slow, t h e n from Eq.
[12] and [34]
ip.a/~p,c - - %/(2 - - ~1)/el
[47]
Since al ~ 1
ip,a/ip,c > 1
[48]
for all values of ~1. Conversely, if the second electron
t r a n s f e r is slow
ip,a/ip, c = %/(1 -- ~2)/(1 + ~2)
[49]
Once again, ~2 is less t h a n one. Thus
ip.j~p.~
< z
[50]
553
S u b s t i t u t i o n of typical values of ~ into Eq. [47] or
[49] will show that there will be significant deviations
from u n i t y for the peak c u r r e n t ratio. I n order to use
this criterion with confidence, 4' should be less t h a n
0.1 (hEp ~ 150 mV) and reduced product must be
stable.
Step 5.--The c a l c u l a t i o n of the parameters related to
the electron transfer can now be d e t e r m i n e d once the
previous steps have been completed. If the wave is
reversible, only the EI,2 value can be d e t e r m i n e d for
sure. If hE is less t h a n 180 mV, E1 and E2 can also be
found. If the wave is quasireversible, then 4' and E1,2
can be calculated. Once again, if hE is less t h a n 180 mV,
E1 and E2 can also be found. If scan rates large enough
to make the wave irreversible are used, ~ can be det e r m i n e d (and hence ~ if E1 and E2 are k n o w n ) . If the
wave is irreversible, ~ can be calculated from the shape
or shift in the wave and 4' can be calculated from the
AEp values. Finally, El,2 can be found. If the values of
AEA~ or hEBC are appropriate, then El, E2, and ~ can be
found. In all this analysis, it is assumed that the other
electron transfer remains reversible. This, i n practice,
limits the m a x i m u m scan rates which can be used to
analyze the data b y this method.
Results and Conclusions
The reduction of benzil (Bn) i n 0.10F b a r i u m perchlorate with DMF as solvent is a two-electron quasireversible electron transfer. The cyclic v o l t a m m e t r i c
data for this solution is shown i n Table III. Using the
criteria discussed earlier, single and double step
c h r o n o a m p e r o m e t r y verified that the reduction was a
two-electron transfer for times at least as short as 1
msec and that the reduction product was stable. The
second step i n the analysis was to v a r y the concentration of benzil to determine if disproportionation was
occurring. It was found that for CBn equal to 0.75 and
5.0 mM no change in the shape of the wave was observed (6). Thus, one could analyze the data i n Table
III without including the disproportionation reaction.
The third s t e p is to determine the reversibility of the
electron transfer. Since hEp is greater t h a n 150 mV for
all scan rateS: studied, the electron transfer is irreversible. Finally, because the iv,a//~.~ ratios are m u c h less
t h a n 1.0, the second electron t r a n s f e r m u s t be the slow
step.
For the data below 1 V/sec, the e x p e r i m e n t a l data
fits quite well with the theory for a n EE mechanism.
F r o m the ~E~ values, it was possible to find ~2, which is
equal to 0.66. Equation [39], [41] and [42] can then be
expressed in the following m a n n e r
log ~2'
=
--4.78 (hEp --~ 0.021)
[47]
E~,2 = Epc ~- 0.0355 (0.35.5 -- log ~ ' )
[48]
hE -- 2(AEBc -- 0.0355 log ~2')
[49]
The results obtained by these equations are shown i n
Table IV. Using these values as estimates, it was possible to generate theoretical v o l t a m m o g r a m s and compare them with the e x p e r i m e n t a l data. This is shown
i n Fig. 12 for benzil at 100 mV/sec. The best values
given in Table IV are obtained from the best fits of
Table III. Cyclic voltammetry of 0.75 mM benzil in DMF
containing 0.10F Ba(CI04)2
V (V/sec)
Epc*
AEp (mV)
0.05
0.1
0.2
0.5
1.0
5.0
lO.O
20.0
- 1.419
- 1.428
1.438
-- 1.448
-- 1 . 4 5 7
-- 1 . 4 7 7
-- 1.500
- - 1.528
156
190
221
260
285
365
* V vs.
- -
Epp/2
(mY)
39
44
48
54
58
81
63
68
SItE (silver reference electrode).
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April 1978
J. Electrochem. Sac.: E L E C T R O C H E M I C A L S C I E N C E AND T E C H N O L O G Y
554
o
Table IV. Determination of second-electron transfer rate
in 0.1F Ba(CI04)2
V (V/see)
0.05
0.1
0.2
0.5
Best value
AEBe
AE
(raV)
,,b'
Ez*
0
- 12
- 18
- 26
0.143
0.0983
0.0699
0.0455
-1.407
- 1.404
- 1.412
- 1.410
--1.408
-1.415
Ref. (6)
K
(mV)
]
I
I
I
I
I
k,,s
(M -~)
(era/see)
60
48
46
43
55
1200
1350
1000
1060
1150
i00
880
5.5
5.0
4,9
5.0
5.0
5 X
x
x
•
•
x
N
d
10 ~
10 -~
10 -~
10-'
i0-'
10.4
* V vs. S H E .
the e x p e r i m e n t a l data with the simulated v o l t a m m o grams.
There occur significant deviations at scan rates above
1V/sec due to the slow rate of the first electron t r a n s fer. The values of ks,1 and at were d e t e r m i n e d by digital simulation, using the values already calculated for
the second electron transfer. It was found that ~1 was
equal to 0.7, and k~.l was equal to 6.5 X 10 -~ cm/sec.
The e x p e r i m e n t a l data along with the theoretical
curves are shown in Fig. 13 for the Epp/2 values. A n exp e r i m e n t a l v o l t a m m o g r a m with the simulated wave is
shown in Fig. 14 for benzil at 1.0 V/sec.
The rate of the first electron transfer appears to be
u n u s u a l l y slow w h e n compared to the rate of reduction
with t e t r a b u t y l a m m o n i u m perchlorate (TBAP) as a
supporting electrolyte. But once again we are dealing
with a n apparent electron t r a n s f e r rate constant, ks, l*,
I
I
I
I
I
o
~5
E, V VS SRE
[.00
1. [0
[
-I.20
I
-I,30
[
1.40
I
1.50
I
-[.60
[
-1.70
I
Fig. 14. Cyclic voltammogram for the reduction of 0.75 mM benzil in DMF with 0.10F Ba(CI04)2. Scan rate = 1 V/sec; solid
line, theoretical; x, experimental points.
instead of the true ks.1. This is caused by the shift of
the benzil
Bn + e ~ Bn'-
[50]
K
+
B n - + B ~ 2+ ~ B n B a "
[51]
reduction from its equilibrium position being due to
the ion-pairing reaction. Since the wave occurs positive of the potential where benzil alone is reduced, the
heterogeneous forward rate constant, khf, at the peak
potential is less than ks,[ due to the exponential relationship between kaf and ks,,
I
khf = ks,1 e x p
--o~1~-~ (E -- E o)
[52]
If one solves the b o u n d a r y value p r o b l e m for the above
reaction, it t u r n s out that k~,t* is related to ks,1 b y the
following equation
ks.1 = ks,l* (I ~- KCBa)~I
-2
1
-I.0
I
I
-1.2
I
I
I
I
-1.6
-14
E, V vs SRE
Fig. 12. Cyclic vohammogram for the reduction of 0.75 mM
benzil in DM,F with 0.10F Ba(CI04)2. Scan rate = 100 mV/sec;
solid line, theoretical; x, experimental points.
I
[
I
I
I
I
I
/I
/
/
0
I
I
-I
I
I
I
I
1
o
LOG v, V/s
Fig. 13. The variation of Epp/2 {is a function of scan rate. Solid
line, ks,1 = 0.10 cm/sec, a 1 = 0.7, ks,2 : 5,0 X 10 - 4 cm/sec,
a~, = 0.66; dashed line, first-electron transfer reversible; closed
circles, CBn = 0.75 mM; open circles, CBn = 5.0 mM.
[53]
where CBa is the concentration of the b a r i u m ion. For
KCBa = 115 and =1 ---- 0.7, ksn is 0.18 cm/sec. This value
of ks.1 is w i t h i n the range that has been observed for
several quinones recently (12). This value is still
smaller t h a n the m i n i m u m value estimated i n the presence of T B A P only. This difference cannot be due to
ion pairing b e t w e e n the b a r i u m and perchlorate ions.
While this will reduce [Ba2+], it will at the same time
increase the value of K because the K [ B a 2+] value
was observed experimentally. I n addition, estimations
of the Ba-C104 ion pair constant from B j e r r u m ' s theory
(13) and from data for Mg-C10r i n acetonitrile a n d
acetone (14) indicate that ion p a i r i n g of the b a r i u m
perchlorate electrolyte is insignificant in DMF. This
lower rate constant could be due, though, to double
layer changes caused by the presence of b a r i u m ions.
Using the information in Table IV, it is possible to
estimate the m a x i m u m value for the disproportionation rate constant, kd, using Eq. [45]. T h e m a x i m u m
value of kd depends u p o n scan rate, with the disproportionation reaction being more i m p o r t a n t at faster
scan rates because AEBc is decreasing. For CBn = 0.75
mM, it was found that kd must be less t h a n Y00 M -1
sec-1. This m a x i m u m v a l u e may be too small to see. A
more practical value is p r o b a b l y 3 • 103 M -1 sec -1.
Since no disproportionation reaction is seen for the
5 mM solution, kd must be less t h a n 750 M -1 sec -1. It is
interesting to note that if ~2' were zero, a value of
kd = 750 M -1 sec -1 could be easily seen a n d calculated. Thus, for CBn = 5 m M and v = 50 mV/sec;
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Vol. 125, No. 4
CYCLIC VOLTAMMOGRAMS
would be 2.7. But this reaction, if it occurs, is o b scured b y the second e l e c t r o n transfer.
The d e t e r m i n a t i o n of ks,2 for the r e d u c t i o n of benzil
in the presence of s t r o n t i u m ion can be accomplished
in the s a m e way. The a~ value was again found to be
0.65. Thus, Eq. [47]-[49] could be used. The results are
shown in Table V. The c a l c u l a t e d values for the benzil
r e d u c t i o n w i t h t h e s t r o n t i u m ion p r e s e n t agree well
w i t h the values d e t e r m i n e d b y t r i a l - a n d - e r r o r s i m u l a tion. As was seen before (6), the s e c o n d - e l e c t r o n t r a n s fer is s o m e w h a t slower w i t h t h e s t r o n t i u m ion p a i r
t h a n with the b a r i u m ion pair. This difference, though,
could be due to double l a y e r effects.
The simplicity of this m e t h o d of analysis is quite
evident. Previously, it was necessary to p e r f o r m a
considerable n u m b e r of t r i a l and e r r o r simulations b e fore the best fit could be o b t a i n e d in o r d e r to d e t e r m i n e the electron t r a n s f e r p a r a m e t e r s for t h e EE
mechanism. But b y use of the equations given in the
paper, it was possible to d i r e c t l y calculate the p a r a m eters. A second a d v a n t a g e is t h a t it is possible to
i d e n t i f y significant deviations from the EE mechanism,
as was shown in this case for scan rates in excess of
1 V/sec w i t h b a r i u m ions present. If an EE m e c h a n i s m
is occurring w i t h only one of t h e ks values being slow,
t h e shape of the w a v e should follow t h e c r i t e r i a given
previously. In p a r t i c u l a r , a good diagnostic c r i t e r i a is
the shift in the anodic and cathodic p e a k potentials
w i t h scan rates. Consistent ,~ values should be obtained
f r o m these calculations. A factor complicating the
analysis is the situation w h e r e both electron transfers
m a y be slow. The diagnostic criteria will obey q u a l i t a t i v e l y the analysis given in this paper, b u t will fail
seriously u p o n q u a n t i t a t i v e analysis. W o r k is in p r o g -
Table V. Cyclic voltammetry of benzil in DMF containing
O.IOF Sr(C104)2
Epv/~
AEp
V (V/sec)
(mV)
(mV)
El*
AE
(mV)
0.05
0.1
0.5
Best value
R e f . (6)
47
49
58
285
322
3~0
-1.372
-1.370
-1.383
- 1.375
-1.397
72
76
80
80
135
* V vs. SITE.
ks,~
(em/sec)
1.4
1.4
1.7
1.6
2.0
x
x
•
x
x
10 -4
10 -~
10 -4
10 -~
10-~
K
(M -I)
4800
5140
3100
4300
1800
555
ress in this l a b o r a t o r y to p r e s e n t s i m i l a r methods for
the analysis of this case.
Acknowledgment
This r e s e a r c h was s u p p o r t e d b y the P e t r o l e u m R e search Fund, a d m i n i s t e r e d b y the A m e r i c a n Chemical
Society. I w o u l d also t h a n k Dennis Evans for his help.
In addition, the C o m p u t e r Services Division of M a r quette is a c k n o w l e d g e d for p r o v i d i n g c o m p u t e r time
for this work.
M a n u s c r i p t s u b m i t t e d A p r i l 11, 1977; r e v i s e d m a n u script r e c e i v e d Oct. 31, 1977.
A n y discussion of this p a p e r will a p p e a r in a Discussion Section to be p u b l i s h e d in the D e c e m b e r 1978
JOURNAL. A l l discussions for the D e c e m b e r 1978 Discussion Section should be s u b m i t t e d b y Aug. 1, 1978.
Publication costs 05 this article were assisted by
Petroleum Research ]und.
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