Journal of
Electroanalytical
Chemistry
Journal of Electroanalytical Chemistry 576 (2005) 57–63
www.elsevier.com/locate/jelechem
A reaction route graph analysis of the electrochemical
hydrogen oxidation and evolution reactions
I. Fishtik
a
a,*
, C.A. Callaghan a, J.D. Fehribach b, R. Datta
a,*
Department of Chemical Engineering, Fuel Cell Center, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609, USA
b
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA
Received 30 June 2004; received in revised form 24 August 2004; accepted 13 October 2004
Abstract
The reaction route (RR) graph approach recently developed by us for complex, non-linear kinetic mechanisms is applied to the
hydrogen oxidation and evolution reactions. A corresponding RR graph is constructed and translated into an equivalent electrical
circuit network by associating each elementary step with a characteristic resistance for the steady-state case and considering the
overall reaction as a power source. It is further shown that the steady-state kinetics of the reaction can be investigated employing
the conventional methods of the electrical network theory. Using a set of rate constants for the hydrogen evolution reaction (her) in
alkaline solutions from the literature, the dominant RRs are identified and simplified mechanisms and kinetics derived.
Ó 2004 Elsevier B.V. All rights reserved.
Keywords: Hydrogen oxidation reaction; Hydrogen evolution reaction; Graph theory; Electrical networks
1. Introduction
Graph theoretical methods are being increasingly utilized to study the topology of chemical reaction networks [1]. In electrochemical reaction systems, an
appropriate topological description of the mechanism
is especially important since it is directly related to the
corresponding electrical circuit that is used to analyze
and interpret impedance spectroscopy data [2]. It is,
therefore, of interest to have a methodology for translating a reaction mechanism into an equivalent circuit.
Graph theoretical methods are of interest in this regard.
Normally, a chemical or electrochemical reaction
mechanism is depicted graphically by placing individual
species at the nodes of a graph while the connectivity of
the nodes reflects the stoichiometry of the mechanism
*
Corresponding authors. Tel.: +50 883 15445; fax: +50 883 15853.
E-mail addresses: ifishtik@wpi.edu (I. Fishtik), rdatta@wpi.edu
(R. Datta).
0022-0728/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jelechem.2004.10.007
[1]. Although such an approach is useful in studying
many topological aspects of certain reaction mechanisms, it presents difficulties for the more general case
and leaves open several significant kinetic issues such
as reduction and simplification.
We have recently proposed [3–6] a new, more general,
graph theoretical approach applicable to complex, nonlinear kinetic mechanisms. The main idea is to represent
a kinetic mechanism by a reaction route (RR) graph
such that the branches are directly associated with the
elementary steps while the nodes represent simply their
connectivity and also satisfy the steady-state conditions
for the reaction intermediates and terminal species. As a
result, we obtain a graph that is particularly useful and
has a remarkable property, namely, any walk between
two terminal nodes in this graph is a full RR (FR),
i.e., an RR that produces an overall reaction (OR).
For the steady-state case, we further employed the concept of the resistance (impedance for the more general
case) of a chemical reaction defined as the ratio between
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I. Fishtik et al. / Journal of Electroanalytical Chemistry 576 (2005) 57–63
its dimensionless affinity and rate and, thus, showed that
there is a complete analogy between RR graphs and
electrical networks. Altogether, this is a powerful new
approach for comprehension, analysis, reduction and
simplification of complex kinetic mechanisms, especially
in relation to impedance spectroscopy. In this contribution, we apply the RR graph approach to the steadystate analysis of the electrochemical hydrogen oxidation
(hor) and evolution (her) reactions.
!
2
rT ¼ k T h2H k T ð1 hH Þ ;
ð8Þ
!
b FE
ð1 bV ÞFE
rV ¼ k V ð1 hH Þ exp V
k V hH exp
;
RT
RT
ð9Þ
!
bH FE
ð1 bH ÞFE
k H ð1 hH Þ exp
:
rH ¼ k H hH exp RT
RT
ð10Þ
2. The mechanism and elementary reactions
The hor and her reactions are by far the most thoroughly investigated electrochemical reaction systems
[7,8] and have assumed a new importance by virtue of
renewed interest in fuel cells. The mechanisms of these
processes are fairly well established and, as is widely accepted, are described by three elementary reactions referred to as the Volmer (sV), Heyrovsky (sH) and Tafel
(sT) steps. Thus, in acidic solutions the mechanism of
the hor is
sT: H2 þ 2M¡2HM
ð1Þ
sV: H2 O þ HM¡M þ H3 Oþ þ e
ð2Þ
sH: H2 O þ H2 þ M¡HM þ H3 Oþ þ e
ð3Þ
where sq is a shorthand notation for an elementary step
q and M is an active site on the electrode surface. The
OR for the hor in acidic solutions is
OR: H2 þ 2H2 O $ 2H3 Oþ þ 2e
ð4Þ
The elementary steps in the her are the reverse of
those describing the hor. For instance, in alkaline solutions, the mechanism of the her is
sT: 2HM¡2M þ H2
ð5Þ
sV: M þ H2 O þ e ¡HM þ OH
ð6Þ
sH: HM þ H2 O þ e ¡M þ H2 þ OH
Here, hH is the hydrogen surface coverage,
!
!
k V ; k V ; k H ; k H ; k T and k T are the chemical pseudo-rate
constants (in mol cm2 s1) as defined in [9], E is the
potential referred to the standard hydrogen electrode
scale (SHE) and the other notation is conventional.
The composition dependence of fluid species is included
in the pseudo-rate constants. Thus,
!
ð7Þ
Hence, for the her in alkaline solutions the OR is
OR: 2H2 O þ 2e ¡H2 þ 2OH
Thus, while H3O+ is the proton source for the her
in acidic media, H2O is the proton source in alkaline
media. In neutral solutions, both acidic and basic hydrogen
reactions can occur.
Since the mechanisms of both the her and the hor are
similar, in what follows, for simplicity, we will consider
only the her in the alkaline solutions, i.e., elementary
steps (5)–(7). The net rates of these steps, i.e., the rates
!
of forward minus reverse steps, rq ¼ r q r q , on nickel,
in alkaline solutions are described by the following
equations [9]:
!0
!
0
k V ¼ k V c H2 O ;
k V ¼ k V cOH ;
!
!0
k H ¼ k H c H2 O ;
0
k H ¼ k H cOH ðP H2 =P 0 Þ
!0
0
where k q and k q are the partial standard reaction rate
constants (in cm s1).
The elementary steps are characterized by their
(dimensionless) affinities defined as [10],
!
1
rq
Aq ¼ ln :
Aq ¼
RT
rq
ð11Þ
Thus,
!
AT ¼
1
k T h2H
AT ¼ ln
;
2
RT
k T ð1 hH Þ
ð12Þ
!
V FE
k V ð1 hH Þ exp bRT
1
AV ¼ ln
AV ¼
h
i ;
RT
k h exp ð1bV ÞFE
ð13Þ
!
k H hH exp bHRTFE
1
AH ¼ ln
AH ¼
h
i:
RT
k H ð1 hH Þ exp ð1bRTH ÞFE
ð14Þ
V H
RT
We also define the resistances of the individual elementary steps as [3,11],
!
Rq Aq lnð r q = r q Þ
¼ !
;
rq
rq rq
ð15Þ
i.e.,
!
ln
k T h2H
AT
k T ð1hH Þ2
RT ¼!
;
rT
2
k T hH k T ð1 hH Þ2
ð16Þ
I. Fishtik et al. / Journal of Electroanalytical Chemistry 576 (2005) 57–63
RV AV
rV
!
ln
ð1b
k V hH exp
while FRVH and FRVT are familiar, FRHT is less so.
These relations imply a similar set of equations between
the affinity AOR of the OR and the affinities AV ; AH
and AT of the elementary steps, i.e.,
bV FE
RT
V ÞFE
RT
k V ð1hH Þ exp ¼!
h
i;
V FE
k V ð1 hH Þ exp bRT
k V hH exp ð1bRTV ÞFE
AV AH þ AT ¼ 0;
ð17Þ
RH AH
rH
!
ln
b FE
H
RT
ð1b
ÞFE
k H hH exp k H ð1hH Þ exp
H
RT
¼!
h
i:
k H hH exp bHRTFE k H ð1 hH Þ exp ð1bRTH ÞFE
ð18Þ
It may be noticed that the resistances are necessarily
positive quantities, and are the same for both the her
and the hor, i.e., they are independent of the direction
of the net reaction. The direction depends on the sign
of the affinity, determined largely by the potential E
along with species concentrations. Since the affinities
are defined as dimensionless quantities, the units of the
resistances are rate1.
3. Reaction routes and KirchhoffÕs laws
At steady-state, the rate rOR of the OR, expressed as
the rate of H2 production for the her, is
rOR ¼ rH þ rT ;
ð19Þ
while the steady-state rate of production and consumption of the surface intermediate HM is equal to zero,
i.e.,
rV rH 2rT ¼ 0:
59
ð20Þ
Solving Eq. (20) for rH and rT and substituting the results into Eq. (19) we arrive at two alternate expressions
for the rate of the OR:
rOR ¼ rV rT ;
ð21Þ
2rOR ¼ rV þ rH :
ð22Þ
Eqs. (19)–(22) are the equivalent of KirchhoffÕs current law (KCL) [3].
As shown by Milner [12] and Happel and Sellers [13],
the her involves one cycle, or empty route (ER), and
three FRs. These are:
ER: sV sH þ sT ¼ 0;
ð23Þ
FRVH: sV þ sH ¼ OR;
ð24Þ
FRVT: 2sV þ sT ¼ OR;
ð25Þ
FRHT: 2sH sT ¼ OR;
ð26Þ
ð27Þ
and
AOR ¼ AV þ AH ¼ 2AV þ AT ¼ 2AH AT ;
ð28Þ
which are the equivalent of KirchhoffÕs voltage law
(KVL) [3]. An appropriate RR graph for the system
must be consistent with KCL and KVL relations given
above. Further, the affinity Aq and reaction rate rq for
a given reaction step sq must remain the same, even if
the step occurs more than once in an RR graph, as in
the case for non-minimal RR graphs.
4. The RR graph
By definition [3,5], an RR graph involves two types of
nodes. One of these, referred to as terminal nodes (TNs),
satisfies the steady-state condition for the OR. In other
words, at TNs, the algebraic sum of the rates of the elementary steps leaving or entering the node should equal
the rate of the OR. In our case, these conditions are
alternatively expressed by Eqs. (19), (21) and (22). Thus,
the possible connectivity of the reactions at the TNs is
TN1: OR sH sT
ð29Þ
TN2: OR sV þ sT
ð30Þ
TN3: 2OR sV sH
ð31Þ
The other type of nodes, referred to as intermediate
nodes (INs), must satisfy the steady-state conditions
for the intermediates. That is, the algebraic sum of the
rates of the elementary steps leaving or entering an IN
should be equal to zero. In our case, this condition is described by Eq. (20). Since for the case of the her, there is
only one linearly independent intermediate (HM), we
have only one independent IN, and the connectivity of
the reactions at this IN is
IN : sV þ sH þ 2sT
ð32Þ
It should be noticed that, although the nodes are defined in terms of the rates of the reactions entering and
leaving the nodes at quasi-steady state, i.e, KCL, they
are represented in terms of connectivity of reactions sq
and OR, i.e., their number and direction.
Based on Eqs. (23)–(32), the RR graph may be obtained following the algorithm for RR graph construction described previously [3,5]. Namely, in the
particular case of reaction mechanisms in which the
number of elementary reactions and ORs in every FR,
ER, IN and TN does not exceed ±2, we noticed that
the RR graphs are symmetrical and involve each
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I. Fishtik et al. / Journal of Electroanalytical Chemistry 576 (2005) 57–63
elementary reaction and OR twice [5]. As a result, the
realization of the RR graphs may be performed based
on the following considerations. The starting point is a
set of linearly independent RRs comprising a single minimal smallest FR, the balance being the smallest ERs.
The FR is drawn next, along with another variation in
which the order of the reactions is reversed. Further,
the ERs are placed starting with the smallest ones such
that none of the reactions is involved more than once
in each of the sub-graphs. The sub-graphs are next fused
only via the reactions that are not present in the subgraphs. If all of the smallest ERs have been placed
and all nodes, except two, are balanced, the graph is
complete. It remains only to balance the remaining
two nodes by attaching the OR according to an appropriate TN from the complete list of TNs. Finally, the
resulting RR graph should be checked to ensure that
all enumerated direct FR and ERs are depicted as walks.
The number of linearly independent RRs for the her
is equal to 2. Thus, according to the algorithm, these are
selected as FRVH, and the ER. The RR graph construction is depicted graphically in Fig. 1. We begin by
depicting twice the FRVH, the second with the steps in
reverse order. Next, we add symmetrically ER to each
of the graphs. At this stage it is seen that, because in
each individual sub-graph the ER cannot be introduced
into the graph due to the direction of the existing
branches, it is necessary to connect the graphs by fusing
two nodes. Finally, we identify the balanced INs and
potential TNs. The TNs are then balanced via the addition of the OR yielding the RR graph. It may be seen
that the resulting graph is an RR graph in that it satisfies
all of the conditions imposed on the RR graph. That is,
the graph involves the ER, Eq. (23), all of the three FRs,
Eqs. (24)–(26), the INs satisfy Eq. (32), while the TNs
satisfy Eq. (31). Notice that every FR in the graph is involved twice. This is a consequence of the fact that the
mechanism is non-minimal [5], i.e., the elementary steps
are involved more than once in an FR. Nonetheless, the
affinity and the rate of a step (e.g., sH or sV) remain unchanged regardless of their placement, as, of course, is
required in any non-minimal graph. This imparts a certain symmetry to the RR graph, i.e., the branches may
be replaced by resistors for steady-state kinetic analysis
or by a combination of resistors and capacitors for the
general case, depending upon the objective.
5. Reaction kinetics
Next, we employ the electrical circuit analogy to analyze the steady-state kinetics of the her. More specifically, our goal is to determine the dominant FRs and,
eventually, if possible, to reduce and simplify the mechanism and kinetics. In doing this, the elementary steps
are viewed as resistances while the OR is viewed as a
power source [3]. The resulting electrical network is presented in Fig. 2. Thus, we are now in a position to employ the conventional electrical network analysis [14],
e.g., KirchhoffÕs laws, to evaluate the currents (rates)
and resistances through various FRs. It should be noticed that the representation of the reaction steps simply
as resistances means that the resulting equivalent circuit
is suitable for steady-state dc analysis rather than for an
ac analysis. The electrical network equivalent of the RR
graph may also be used to interpret and analyze integral
impedance spectroscopy measurements, provided that
+
O
R
-
RH
RV
RT
RT
RV
RH
+
Fig. 1. The realization RR graph for her and hor.
O
R
-
Fig. 2. The electrical circuit analog of the RR graph.
I. Fishtik et al. / Journal of Electroanalytical Chemistry 576 (2005) 57–63
the reactions are represented as an appropriate combination of resistors and capacitors.
First, let us show that the electrical network adequately describes the mechanism. Consider the overall
resistance ROR of the network. Employing the standard
procedures, e.g., a D–Y conversion [14] (Fig. 3), the overall network resistance is readily shown to be equal to:
ROR ¼ RA þ
ðRV þ RB ÞðRH þ RC Þ
;
RV þ RH þ RB þ RC
Numerical evaluations of the polarization curve
according to Eqs. (38) and (39) are presented in Fig. 4.
Here, and in the discussion below, the numerical calculations were performed using a set of chemical rate constants (mol cm2 s1) and partial standard chemical rate
constants (cm s1) for individual steps on Ni in 1.0
mol dm3 NaOH at 20.0 °C [8]. Under these conditions
the constants are [9]:
ð33Þ
!
k V ¼ 7:8 1017 ;
where
k H ¼ 3:2 1019 ;
ð34Þ
RB ¼
RH ðRT =2Þ
;
RV þ RH þ ðRT =2Þ
ð35Þ
RC ¼
RV ðRT =2Þ
:
RV þ RH þ ðRT =2Þ
ð36Þ
AOR
AV þ AH
:
¼ 2ROR 2 RA þ ðRV þRB ÞðRH þRC Þ
RV þRH þRB þRC
k T ¼ 1:3 109
F ðAV þ AH Þ
B ÞðRH þRC Þ
RA þ ðRRVV þR
þRH þRB þRC
!
kV kT
j ¼ F ðrV þ rH Þ:
The proof that Eq. (38) is equivalent to Eq. (39) is elementary but cumbersome and, for brevity, is not presented here. In particular, the equivalence of Eqs. (38)
and (39) may be easily shown by substituting Eqs.
(16)–(18) into Eqs. (33)–(36) and further into Eq. (38),
additionally employing Eqs. (20) and (27).
RT
-
+
O
R
O
R
-
+
O
R
-
RC
RV
RV
RA
RT/2
RH
+
ð40Þ
:
kH
-
RH
RT
RV
kH
According to the electrical circuit theory, the dominant FRs under a given set of conditions are those that
have the lowest resistances. To determine which of the
three possible FRs are dominant, we have performed
numerical calculations of the resistances of the elementary steps, as a function of potential E, employing the
experimental conditions and rate constants given above.
The results are presented in Fig. 5.
As can be seen, the Volmer step has the smallest
resistance over the entire range of potential. On the
other hand, the Tafel step has the highest resistance in
the potential range E < 1.20 V while the resistance of
the Heyrovsky step is the highest in the potential range
E > 1.20 V. It may be further noted that RT is also a
function of potential, even though Eq. (16) may lead
one to believe otherwise. The reason is the variation of
hH with potential, as obtained by solving Eq. (20).
ð39Þ
RV
¼
6. Dominant FRs
ð38Þ
RH
!
ð37Þ
:
O
R
!
kV kT
Now, in order to prove that the electrical network
quantitatively describes the kinetics of the her it is necessary to show that Eq. (38) is identical to the conventional rate, i.e., a quasi-state current,
+
k T ¼ 7:8 1012 :
and
As can be seen from Fig. 4, there is an exact match between the current density evaluated from the electrical
network and the current density evaluated via the conventional way.
KVL also provides thermodynamic consistency of the
rate constants involved in an ER [9],
Notice that the rate has been divided by 2 since the
network involves two ORs. The corresponding current
density is
j ¼ 2FrOR ¼
k H ¼ 3:7 106 ;
!
According to OhmÕs law, the rate rOR of the OR may
be expressed via the overall affinity AOR and overall
resistance ROR of the network via
rOR ¼
k V ¼ 1:5 101 ;
!
RV RH
;
RA ¼
RV þ RH þ ðRT =2Þ
61
RV
RH
+
O
R
-
RB
RH
+
O
R
Fig. 3. The evaluation of the overall resistance of the electrical network.
-
62
I. Fishtik et al. / Journal of Electroanalytical Chemistry 576 (2005) 57–63
Fig. 4. Polarization curves evaluated via different equations (see the
text for details).
RVH ¼ RV þ RH ;
ð42Þ
RVT ¼ 4RV þ RT ;
ð43Þ
RHT ¼ 4RH þ RT :
ð44Þ
Numerical simulations of these quantities, presented
in Fig. 6, clearly show that the FRVH dominates the
kinetics at low potentials while FRVT is dominant at
higher potentials. Further, RVH RH and RVT RT,
i.e., these steps dominate the FR total resistances.
Based on the numerical evaluations, the electrical network may be reduced and simplified as shown in Fig. 7.
More specifically, for potentials E > 1.20 V the kinetics are dominated by the RVT, i.e.,
sV :
M þ H2 O þ e ¡HM þ OH
2
sT :
2HM¡2M þ H2
1
—————————————————
OR : 2H2 O þ 2e ¡H2 þ 2OH
where the numbers on the right are stoichiometric numbers. The current density corresponding to this FR may
be accurately described by
j ¼ 2FrOR ¼
Fig. 5. Resistances of the elementary steps vs. potential.
According to the principle of extremum resistance [6],
thus, it may be expected that at E < 1.20 V the Tafel
step may be neglected since its resistance is much higher
as compared to the resistances of the Volmer and Heyrovsky steps. Furthermore, under these conditions, the
resistance of the Volmer step is lower than the resistance
of the Heyrovsky step, which means that the rate of the
reaction is controlled by the Heyrovsky step.
A different situation is observed for E > 1.20 V. As
can be seen from Fig. 5, in this region the Heyrovsky
step has the highest resistance and, consequently, the effect of this step can be neglected. As a result, the kinetics
of the reaction may be accurately described by the Tafel
and Volmer steps, the Tafel step being rate controlling
since its resistance is higher. These conclusions are in
complete agreement with those of Krstajic et al. [9].
The same conclusions result for the her in alkaline
solutions on Ni if the total resistances of the individual
FRs are considered [6]. In defining these we take into account the fact that the elementary steps in an individual
RR are connected in series and each elementary step sq
in the sequence occurs rq times, where rq is the stoichiometric number of a step in an FR. Thus, the resistance
RFR of an FR is [6],
X
RFR ¼
r2q Rq :
ð41Þ
2F ðAV þ AT Þ
:
2RV þ RT =2
ð45Þ
On the contrary, at higher potentials, e.g., E <
1.20 V, the kinetics are dominated by RVH, i.e.,
sV :
M þ H2 O þ e ¡HM þ OH
1
sH : HM þ H2 O þ e ¡M þ H2 þ OH
1
———————————————————
OR :
2H2 O þ 2e ¡H2 þ 2OH
Respectively, under these conditions, the overall current density, Eq. (38), is reduced to
j ¼ 2FrOR ¼
2F ðAV þ AH Þ
:
RV þ RH
The performance of these reduced rate equations is
compared in Fig. 4. Thus for E < 1.20 V, Eq. (46) is
accurate, while for E > 1.0 V, Eq. (45) is accurate. In
the range of 1.20 V < E < 1.0 V the complete rate
equation, Eq. (38), is required.
q
The resistances of the three FRs that describe the her
are, thus:
ð46Þ
Fig. 6. Resistances of the FRs vs. potential.
I. Fishtik et al. / Journal of Electroanalytical Chemistry 576 (2005) 57–63
+
O
R
63
-
RH
RV
RT
RT
RV
RH
+
O
R
-
RT
RV
+
-1.5
RV
RH
O
R
-
RT
+
O
R
RV
-
-1.1
-0.9
E/V
Fig. 7. Reduction and simplification of the electrical network as a function of potential.
7. Conclusions
From the analysis presented above, it follows that the
RR graph approach may be successfully applied to
electrochemical reaction systems. Thus, the three elementary steps describing the mechanism of the hor
and her are represented as an RR graph. The latter
can be further translated into an equivalent electrical
network. Thereupon, one may employ the well-developed and powerful methods used in conventional electrical networks to study the kinetics of the hor and her
mechanisms, e.g., to determine the dominant FRs.
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