- Email: [email protected]
Thermodynamics of phase transitions at electrosorbed layers
Elecmchimica Acm. Vol. 36, No. 314, pp. 441457. Printed in Great Britain.
001~-4686/91 $3.00 + 0.00 Pergamon PnsS Qk.
1991
THERMODYNAMICS OF PHASE TRANSITIONS ELECTROSORBED LAYERS
AT
P. NIKITAS Laboratory of Physical Chemistry, Department of Chemistry, University of Thessaloniki, 54006 Thessaloniki, Greece (Received 29 January 1990; in revised form 23 April 1990)
Ahstraet-T&a’s theory for critical phases is applied to describe phase transitions of electrosorbed layers formed by dilute solutions of strongly adsorbed substances. Within the framework of this theory we determine relationships which define critical points and establish rigorous criteria for an order-disorder phase transition to occur. An approximate treatment baaed on the bulk phase approach is also presented. The implications of the electrode charge or the potential drop across the adsorbed layer being used as independent electrical variables are indicated. Finally, it is shown that, during an order-disorder phase transition of an electrosorbed layer, we have a separation of the interface into two new phases each one saturated with either adsorbate or solvent molecules. Key words: thermodynamics,
phase transition.
1. INTRODUCTION Experimental observations have shown that a number of interesting phase transitions can take place during adsorption at charged interfaces, which may be classified into three main categories[l]. (a) Phase transitions analogous to those appearing in multi-component liquid systems. (b) Orientational transitions. (c) Surface crystallization. The typical feature of the first type of phase transitions is the appearance of characteristic abrupt changes in the capacitance 11spotential curves at the desorption potentials associated with phenomena of hysteresis[ l-31. In addition, capacitance peaks are usually not observed at these potentials. The case exhibits symptoms of two-dimensional condensation and for this reason Rangarajan[l] considers as an additional type of phase transition, that which is observed in one component systems. However, the thermodynamic treatment presented below shows that in the case of electrosorption the condensationdilution of the interface is always accompanied with phase separation phenomena, like those appearing during first order phase transitions in multicomponent liquid systems. Adopting Tisza-Callen’s terminology[4], we shall call them order-disorder phase transitions throughout the paper. Other observable criteria for an order-disorder phase transition to occur are discussed in [2]. Orientational transitions are characterized by one or two “pits” in the capacitance-potential curves [1,5,6]. The pits correspond to certain orientations of the adsorbate. Finally, the surface crystallization is a different kind of phase transformations and is observed in the case of adsorption of cations[7,8]. In particular, this phenomenon has been observed during the adsorption of Tl+ and Pb* + in aqueous halides and it is detected
from the abrupt changes in the surface excess of Tl+ and Pb2+, when it is plotted against the concentration of anions in solution. The theoretical work for the description of phase transitions cannot be considered as adequate. The two-dimensional condensation was firstly treated on the basis of the Frumkin isotherm model[9]. It is well known that the model predicts a phase transition when the attraction constant a is greater than 2. This model has been also adopted by Sathyanarayana et al. [2] to interpret experimental capacitance data when a two-dimensional condensation takes place. The effect of temperature on phase transitions has been analysed by Gurevich et a1.[10] on the basis of a Flory-Huggins type isotherm. The case of orientational transitions has been also treated within the framework of this model[ll]. Recently, a more advanced model, that developed by Sangaranarayanan-Rangarajan[ 121, was used to describe the dependence of the capacitance pits upon temperature and bulk adsorbate concentration[l3]. In the present paper we focus our attention only on order-disorder phase transitions. Experimentally the phenomenon has been widely studied[l, 31. The most thorough work has been done by de Levie et aZ.[3, 14-181 and Buess-Hennan et a1.[19-251. Their experimental data show that the interfacial tension is a continuous function of potential, whereas discontinuities appear in the surface excess-potential, capacitance-potential and charge density-potential curves. For the last case de Levie appears to doubt whether his experimental data give the precise shape of the charge-potential curve in the region of phase transition[3]. Perhaps de Levie has observed that the existence of discontinuities in the charge densitypotential curves leads possibly to a continuous variation of the surface excess when it is plotted against charge density! This matter is discussed in the present paper, in which we try to examine whether 447
P. NIKITAS
448
the experimental behaviour observed in systems which undergo an order-disorder phase transition is predicted thermodynamically. For this reason we apply Tisza’s theory[4,26,27] to study this phenomenon in electrosorbed layers, in an attempt to establish rigorous criteria for these phase transitions to take place.
Here, Q is the total charge on the Hg electrode and F is Faraday’s constant. If the reference electrode is reversible to the anions X-, for example Ag]AgX(X-, then we again obtain equation (2) but with terms L_ dQ and PMxdn”+ instead of E+dQ and /~~xdnO_, respectively, where E_ = E_ - (&
- p;g)/F
and 2. RIGOROUS
2.1. Basic equations for the internal energy We consider the interfacial region (a) which is formed between a Hg electrode and an electrolyte solution composed of the electrolyte MX, the organic compound A and the solvent S. We further consider that the interface is extended up to the homogeneous regions of the phases which form it (Fig. 1). The total differential of the internal energy U” of the interface is given by dU” = TdS” - PdV” + ydd + pidni
,
Adn: + FHg+dn”,,+ = @g - FrpHg)dn; + ol$+ + WHg) dnB, = kmn:
+p$dn&+
(1)
p_dnL +P+dn”,
and
=(pb_ - Ft#~~)dn: +(/h
+F+b)dC
=Pbdnb+pb+dn: - 4bdQ,
(6)
equation (1) becomes dU”= TdS”+ &lnI
PdV”+ydd
+ A+“dQ +P”,dn:
+ Prgdn: + &;+dn&g+
+&dn:
+&dn”,,
(7)
where 4 denotes the inner potential of the phase indicated by the superscript and A@ = +Hg - C#J~ is the potential drop across the interface. The superscripts (Hg) and (b) denote the Hg electrode and the bulk electrolyte solution, respectively. Note that the chemical potentials ~‘5, p(“care not uniform throughout the system. 2.2. Tisza’s criteria for a critical point Equation (2) shows that U” = U”(S”, V”, d, C, nI, Q, nLg+, n$ ),
+Pc,,dn!.,
(2)
E+=($-jY&)/F.
(3)
where c+=E,-p$/F
(5)
+ 9”‘dQ
TdS”-PdV”+ydd+&dni+p;dn; +e+dQ
+ p::+dGg+
and
where, S” is the entropy of the interface, V” its volume, T is the temperature, P the pressure inside the interface, y the interfacial tension, d the area of the interface, ni, rzgare the number of moles of A and S, respectively, at the interface, ~1, & their chemical potentials, nr, n&+ are the moles of electrons and Hg+ on the Hg electrode, A, Fur+ their electrochemical potentials, n”_ , n: are the number of moles of the ions X- and M+ at the interface and ji_, p+ their electrochemical potentials. On the chemical or electrochemical potentials no superscript indicating the phase is necessary, because they are uniform throughout a system under equilibrium conditions. However, we use a superscript in the case of pi, Pg just to distinguish them from the corresponding chemical potentials calculated on the basis of the bulk phase approximation[28]. In real systems the potential of the ideally polarized Hg electrode is controlled with respect to a reversible charge transfer electrode. If this reference electrode is reversible to the cations M+, then eauation (1) can be written in the form[29-311 ’ a . ’ dU”=
(4)
For reasons of generality equation (2) is used with the symbols &* and n; instead of 6+ and n”_ throughout the paper. In addition, the applied potential is denoted by E, instead of either E, or E_. An alternative expression for dU” may be obtained as follows: since
+ &dng
+ P&C + Png+dnOH,+ +fi_dnT. +P+dn”,
E_ = (iitg - p,)/F.
TREATMENT
(8)
that is, the internal energy depends only upon extensive parameters. In addition, it is a first order homogeneous function of its arguments. Therefore, if we introduce the specific parameters e=
U’/.G~,
CI= S”/szl,
rA = nil.&,
2, = Vu/&,
Ts = rig/d,,
Q = Q/d,
rHg+ = nC+/d,
f* =n”,/&,
(9)
a = &, v, O, PA, Ps, Pug+, r,)
(10)
we obtain[4,26,32]
and da = Tdg - Pdv + C+da + p;dPA + &drs Fig.
+ !‘tigdPng+ + P,xdP,
.
(11)
Thermodynamics of phase transitions at electrosorhed layers Equations (10) and (11) can be written in the more general form 16=&(x,,x2,x~,..
.9x,)
da = P,dx, + P2dx2 f.
WI
+. + P,dx,,
) .
(14)
\"rri/xj
Now, we define the thermodynamic potential $ from r-l (15) 1 PiXi. *(pit p2, . . . . P~_,,X,)=Ui=l
The total differential of $ is given by r-1 d$ = - c xidP,+ P&q.
(16)
i-l
Then, according to Tisza’s theory[4,26,27], point is defined from the relationships
ax; PI ,....
=o
w 4ax” > r
h-l
a’*
(4
ax; PI . . . . . h-l
a critical
> 0.
axi axj d4qjyp4ji
= 0,
-
pHg+ rHg+
--L*Q -jl;r, -
hdXr*
(22)
is coherent with equation (15). Its total differential is given by - r* dp,x -Tsd&+&dT* and it is reduced to d$ = &dr,
(17)
(18)
axi
I( > I
apipk+, +*.
(19)
Relationships (17) and (19) can be used to determine the critical point of a phase transition. Moreover, since x, may be any of the specific variables x, , x2, . . ., x,, writing down all the possible functions for $ we can derive all the possible relationships valid at a critical point. However, in the case of surfaces and interfaces the above relationships are not always applicable. The properties of these systems are usually studied as a function of the composition of the bulk solution. Under equilibrium conditions at constant T and P, the chemical potential of any uncharged species i is uniform throughout the system and therefore (i=A,S,MX). potentials
(23) (24)
when T, P, Q, p,, and & are constant. However, in this case, according to equations (20) and (21), the chemical potential ~1 is also constant and consequently r, is constant too. Thus, when the intensive parameters T, P, L*, p,, and & are kept constant, equation (22) does not define any function of the variable r, . In what concerns the elements of the compliance matrix, they may be physically meaningless for the same reason. For example, the element ar,
becomes infinite. That is
In addition, the chemical Gibbs-Duhem equation
Td+Pv
Pl,...,P,-I
At the same time all the elements of the compliance matrix C = [+J, where
p;=pF,
JI =a-
drj = - cldT + vdP - adc, - rur+ dpHg
Pi=%(
av (3
It is easy to show that equation (15) may not define a thermodynamic potential rj. For example, the function
(13)
where xi = X,/X,+, and Xi, X,, I are extensive properties of the system. To each specific variable x, corresponds the intensive quantity Pi,
449
(20)
( % > ~.P,~+,P$PMx implies a variation of r, with respect to pi at constant T, P, c+, pMx and pg, which, as it has been noted above, is impossible due to equations (20) and (21). These problems can be easily overcome, when we study dilute solutions of strongly adsorbed substances. In this case &( = ~8) can be approximately considered as constant, whereas r,, varies with the bulk concentration of A. Since this approximation is used throughout this paper, the present treatment is applied only to charged interfaces formed by dilute solutions of strongly adsorbed substances. 2.3. Rigorous relationships valid at a critical point We define the thermodynamic potential tj from *
=ll-Td++Pu-p;rA-jq-s - hgrHg+ - hd-*
which under constant d$=
(25)
T, P and pMx yields
-rAdp;-Tsd&++dda
= -(Ta--rsN,/Ns)d&++da.
(26)
Equation (25) is coherent with the definition of $ from equation (15) and therefore it can be used in (17). Since
p! fulfil the
=o where Ni is the mole fraction of i in the bulk solution. A close inspection of the problem shows that equations (20) and (21) impose certain restrictions to the direct application of Tisza’s theory to phase transitions at surfaces or interfaces. The restrictions concern the definition of the thermodynamic potential I(/ from equation (15) and the elements of the compliance matrix.
(27)
T,P,.+,n: we
obtain
> 0,
T,P.a*,a~
(28)
P. NIKLITAS
450
where a, and uk are the mean ionic activity of the salt MX and the activity of the adsorbate A in the bulk solution, respectively. In addition, from (19) we obtain a number of interesting relations which are also valid at the critical point. Thus, the effect of temperature is described by
homogeneous function of the first degree in np. Hence by Euler’s theorem[33]
where CQis the partial mole area of i( = A, S) at the interface. Then equation (2) is written with the form dU” = T dS” - P dV” + (& + a,+y) dn$ +&+e,y)dG++dQ (37)
+ /JngdG&+ + L&xb”, *
Now we can define the specific internal energy and the function $ from
For the effect of pressure we have
du=Tds-Pdu+(&+aAy)dc + Q dq +
PHg dclfg+
+
(38)
@MX dc,
$ = a - Ts + Pu - 6 * 4 - ,&Q+Q+ - p~xC*, (39) respectively, where where we have assumed that the effect of pressure on the potentials E* and E, is negligible. The effect of the applied field at a critical point leads to the expressions
(31)
The last relation is equivalent to C *co, where C is the differential capacitance of the interface. In addition, from (28) and (31) we obtain
u = W/n&
s = S”/n&
v = Vu/n&
cHg+= n&+ In:,
c = no&,
q = Q/n&
c* = n$ In;.
w
The specific variables introduced by equation (40) are not common in experimental studies. However, they are necessary to determine relationships [(43) below] which will be found useful in clarifying the nature of surface transitions. In addition, they help to define the critical point in the plot of the surface pressure II us PAI which is a common plot, especially in studies of uncharged interfaces. Equation (39), under constant T, P, pMx, results in d$=
(ii&) =(~)T,p,.,,.:+~. (32)
-q dE, +(p:+CIAY)dC
= -q dE, + PX dc,
(41)
where
Finally, the bulk adsorbate concentration the critical point through the relationships
affects
p: = p: +
(42)
aAy.
In this case, the critical point for phase separation is defined from the relationships a*pz
(33) Note that if we use the function $ defined from equation (22) then we obtain for the critical point the relations
>
0.
(43)
T,P,a*,E+
For symmetry reasons we also have
= 0, 7,P.a, ,E*
T,Pa*
.E*
> 0, T.P.a*,Ek
where c’ = ng/ni = l/c. Since which under equilibrium
conditions
rA=
become
ni
c
EAn: + xsn;
=-=A as + caA
(45)
then arA -= ac
(35) Other rigorous relationships valid at a critical point can arise as follows: The area of the interface is a
as
>o
(a,+caA)'
and (46)
Thermodynamics of phase transitions at electrosorhed layers provided that the partial areas uA, as are independent of the surface composition. Irrespective of this constraint, the physical content of the variables r,, c, c’ entails that r, should increase with increasing c values and decreasing c’ values. Therefore, relationships (43) and (44) can be rewritten as
m
(3 ari
T,P,o*:,E*
=”
(47)
>o
approximately &fined from the thickness of the adsorbate molecules. Now we can distinguish two parts in the interface. The compact part includes an homogeneous region of the Hg electrode, and it is extended up to the plane 00’ and the “diffuse” part which is extended from the plane 00’ up to the homogeneous electrolyte solution. The area of the interface is again a linear homogeneous function of the number of moles, n;, of i( = A, S) in the compact part. Therefore, we can write[34] a=SAn~+Ssn~=(SA/lA)n;+(Ss/ls)n~
m (4 ari
<
(48)
0.
T,P&E+
We denote by n the ratio of the partial areas aA, as. That is, n = tlA/as. This ratio expresses the number of solvent molecules which are displaced from the surface solution by one adsorbate molecule and it is usually called size ratio parameter. In this case we have
aab,
(>
=-
arA
T,P,o*,E&
=
’ (50)
the criteria for a critical point to exist are given again by (35) provided that n is constant. This means that relationships (47), (48) and (34), (35) define the same critical point. In this case, making use of (34), (42) and (47), we obtain
= 0,
=
0, (51)
where n = y” - y is the surface pressure of the interface and y” the interfacial tension of the base solution, without the adsorbate A.
3. APPROXIMATE
TREATMENT
3.1. Internal energy of an a&orbed layer In the treatment presented above, the interface had a considerable thickness and it included homogeneous regions of the phases which form it. However, in studies of adsorption of organic compounds on electrode surfaces the attention is usually focused on the “compact” part of the interfacial region, which is formed from the monolayer of the adsorbed organic molecules. For this reason, let us consider the plane 00’, the position of which inside the interface is
(52)
where ny is the number of moles of i( = A, S) when it forms a monolayer on the electrode, Sj is the effective area covered by i on the electrode and li the average number of layers of i which form the compact part. When the effective area S, and the number of layers Ii do not alter during the adsorption process, equation (52) yields d~=((SA/IA)dn~+(ss/rs)dn~.
(53)
The infinitesimal electrical work Ac%“dQ in equation (7) may be split into two terms A&‘dQ = 84” dQ + Ab* dQ,
(54)
where A@, A4* are the potential drop across the compact and the diffuse part, respectively. The quantity Ad” dQ expresses the work done on the interface when under constant A4# the charge on the Hg electrode is increased by dQ. Evidently, a constant value of Ac$O does not entail constant values of A@, Ad*. However, we can approximately consider that the quantity A#J”dQ is the infinitesimal electrical work done on the layer s which is bounded by the Hg electrode and the plane 00’, that is the work required to increase the charge on the Hg electrode by dQ under constant A@. Note that the layer s may be identified to the adsorbed layer. If we assume that there is no specific adsorption of ions, then the adsorbed layer is free of ions. Therefore, by Gauss’s theorem of electrostatics[35] over the surface indicated by the dotted line in Fig. 2 we obtain c,A+“.cJ/I = Q - Q’,
> 0,
451
(55)
where Q’ is the induced charge on the surface of the adsorbed layer which is adjacent to the electrode, co the permittivity of vacuum and I the thickness of the adsorbed layer. The total polarization P of the adsorbed layer s is equal to[35] P =
VEQ'/d=lQ',
Fig. 2. Model of the adsorbed layer s.
(56)
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452
3.2. Approximate relationships valid at a critical point
where V” is its volume. In addition, we have P=P,+P,,
(57)
where the term P, is due to translation effects and the second one P,, to the orientation of the permanent dipoles. The term P, may be written as P. = (@,a,
+ n:a,)
A&/l,
(58)
P c = nXF* + nifs,
(59)
where Giis the average value of the permanent dipole vector normal to the electrode surface. If the molecules of an adsorbed substance can not be reoriented, as is the usual case for the majority of organic adsorbates, then fA is constant. On the contrary, if a reorientation takes place, as we accept for the case of solvent, then the value off depends upon the strength of the applied field. At low fields the quantity P is proportional to the applied field A#“/1 and we may write[35] j = AA@/&
B = (1 + O(A@)) Ab”/l,
(61)
where O(Ad*) denotes terms of order A4” or higher. Use of equations (5)-(61) yields Q = {niTA + n&%?,+ n;O(A$“) + n;O(A@)} A@/12, (62) Consequently
dQ = {@A+ ‘W~4”)‘1) W + G’s+ W4”)31) WHMW2.
Ts”+ Pv”,
$ = us-
(68)
where us = Us/n$,
s’= F/n;,
us = Vs/ni,
c = ni/nH. (69)
Then d$ = -s”dT+v”dP+&dc,
(70)
and the relationships which define a critical point for phase separation may be easily written with the form
and
(g)T,p =0, (z),, =0, (g)T,p <0,
(72)
where now PA = nil&. These relationships lead to the following criteria for a critical point to exist: (a) under equilibrium conditions between bulk and adsorbed molecules, we have p;=p~=p~b+RTlna~,
(73)
where pFb is the chemical potential of i( = A, S) in its standard state in the bulk solution. If this equation is introduced into equation (65), we obtain ly = pFb + RT In a! + y(S,/l,) + ~i(A~“/l)2 + 0[(A@)3], i = A, S
(63)
Therefore, if we introduce equations (53) and (63) into equation (7) we obtain the following expression for dU” dU”=
(67)
(60)
where I is a constant characteristic of the adsorbed substance. When the field increases v also increases and at high field strengths it tends to become independent of the applied field. Finally, at moderate fields we may write in general[35]
A&
du”=Tds”-Pdv=+&dc
of i. For the
where ai is the average polarizability term P, we have
where Xi is a constant.
If we accept that a phase transition is determined almost exclusively from the properties of the adsorbed layer, then we can use the above expression of dU” to derive the relationships which define a critical point. In this case we can work as previously and define the specific internal energy us and the function $ from
which in combination
(74)
with (72) yields = 0,
TdS’-PdV”+r~dn~+r~dn~ (75)
+TdSd-PdVd+A4ddQ+/&dnd_ +&dnd+ + T dS”g-
(b) In this model the size ratio parameter n is defined from n = S,/Ss . Then, equation (74) leads to
+&dn:+&dn: P dV”g+
p,“‘dnr*
+ PI;+ dn::+,
(W
where
c;\ - n& = (xA - n~s)(A~s/1)2 + RT ln{ai/(ag)n} + O[(A@)3]+ (~2~ - nptb)
l: = &’ + Y(S,/M + X,(AM)*
+
WW”)31,
i = A, S.
Now we can quantity dU”,
(65) approximately
assume
dU”=TdS”-PdV”+&dn~+~~dn’,
that
the
= 0, T, P, [email protected];
(66)
expresses the differential of the internal energy of the adsorbed layer s.
(76)
and if it is differentiated with respect to PA under constant T, P, Ads, ai, we obtain
= ”
a3ab >o H ar
f
T, P,A&!g
provided that n is constant.
(77)
Thermodynamics of phase transitions at electrosorbed layers (c) Differentiation of equation (76) with respect to P., at constant T, P, ai (and consequently at constant a,b) leads to
(78) provided that A& # 0. Finally, if we do not express the area d as a function of ni, ng and write the specific internal energy as da” = Td2 - Pdu” + A@da + p:dP,
+ &dP,,
(79)
vs = v=/&d
(80)
the measured potential the potential of zero charged and the potential drop across the diffuse layer calculated from the Gouy-Chapman theory. In this case, the constraint a, is constant should be added in the relationships of the approximate treatment. In general, almost all the relationships obtained by means of the approximate treatment may derive from the corresponding relationships of the rigorous treatment, if we replace E, by A&. The only exceptions are relationships (78) which appear not to be obtained, at least directly, by means of the rigorous treatment. The treatment leads to expressions (31) from which we obtain
(aE,>
arA r,p,o*,ai=
where CL= = V/d,
6== s=/&/,
then, following precisely the same procedure to that adopted for the determination of relationships (28), we obtain
453
O.
(83)
However, if we take into account that the function P, = P,(E,) is strictly increasing (or decreasing) up to the point of maximum adsorption, then the above equation necessarily entails that this point is a point of inflection and therefore at the critical point we have
= 0, (81) In this case, at the critical point relationships (29x33) are also valid provided that E, is replaced by A@. Note that the mean ionic activity a, does not appear in the relationships obtained by means of the approximate treatment. However, its variations alter the values of 4b and therefore affect A4$, since A$,“=+nr-A@-#b.
(82)
In practice, Ad” is calculated when r#~ b and consequently when a, are constant, by subtracting from
(84)
4. DISCUSSION Table 1 summarizes the most important relationships for the determination of a critical point. They have been obtained by means of the rigorous treatment. These relationships seem to define only one critical point, when T, P and a, are constant. However, if we take into account that the relation E, = E, (r,) is a double-valued function of P,, we
Table 1. Relationships valid at a critical point 1.
2. 3. a97 H ar:
EA 36/3-4-F
= 0, T,P,a*,E*,&a,
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454
conclude that there are two critical points on both sides of maximum adsorption. A critical point is closely related to the abrupt changes of certain variables observed during a phase transition. It is well known that a critical point is the limit where the connodals, ie the straight lines joining the two co-existing phases, tend, when one of the variables tends to its critical value. For example, the critical point in the plot of rA against ui is the limit of the abrupt changes of rA in the same plot, as ai tends to its critical value. Thus, a criterion for a phase transition to occur is
variables, 6, and u. It is seen that, according to (28) or (33), e changes abruptly during a phase transition, whereas 6, remains constant. More generally, E, always changes continuously, while u shows discontinuities during a phase transition. However, the most interesting point is that if we consider the electrode charge density e as an independent electrical variable, then it is very likely that a phase transition cannot be observed! The facts which support this prediction are the following: (a) Plots of rA trs r~do not show critical points. This is equivalent to the fact that neither
= 0,
(85) since at the critical point the above relation is also valid. However, we should point out the different meaning which this criterion acquires when it is applied at the critical point and at the transition region far from it. Relationship (85) shows that the tangent line to the curve of rA us ui at the critical point is parallel to the Y axis, whereas at the transition region it shows that rA varies abruptly at a certain value of ai. Therefore, the criteria for a phase transition can be obtained directly from the treatment of the critical points presented above. The most important observable criteria are listed in Table 2. It is seen that the thermodynamic treatment presented in this paper predicts abrupt changes of the adsorbate surface concentration rA when it is plotted against ai and E, in the transition region. Abrupt changes are also expected in the plots of cr and C us E, . These predictions have been already verified experimentally. For example, they are in agreement with the experimental data obtained by de Levie et a1.[3, 161 and Buess-Hennan et a1.[19-251. Moreover, the theory predicts that the differential capacitance tends to infinity at the transition region. This feature has not been observed yet. However, the lack of sharp and very pronounced peaks on the capacitance curves in the transition region is attributed to experimental limitations[3]. These peaks could be observable only if the capacitance was measured at a sufficiently low frequency[3]. Finally, studies at uncharged interfaces [36-391 show abrupt changes of rA with respect to the surface pressure ll in agreement with our results. The results of the present treatment raise some questions about the equivalence of the two electrical
(
2. 3.
#
(86)
0,
) T.PP&
nor
=0, (87) are valid at a critical point. If relationships (86) were valid, then
($)=(ff)(&J+3@)($!j x($)+(q(&g=o (88) which is inconsistent with (84). The quantity (CJ’E,/aI’i) is different from zero at a critical point only if au/ar, # 0. In addition, if (87) were valid, then at the critical point we would have
which is also inconsistent with (84). (b) The theory does not allow to clarify rigourously whether a critical point appears in the plot of rA us
Table 2. Observable criteria for an order-disorder 1.
-ab ar:
= 0,
phase transition to occur
Thermodynamics of phase transitions at electrosorbed layers
ai at various c values. If we assume that a critical point does appear in this plot, that is if
aa:
(ar,> r,PP*
,o =
O
is valid, then the same equation is expected to be valid in the whole region of phase transitions. In this case of plot of rA against rr at various ai values should show abrupt changes of I’, at certain values of u. However, this means that (86) are valid, which has been shown to be incorrect. In addition, if we graphically eliminate E, from the plots rr us E, and rA us E, , then the resulting plot of r,, against CJpresents no abrupt changes of r,. There is only an ambiguity about the precise shape of the r, us u curve in the region where c and r, changes abruptly in the original plots (cr us E, and rA us E, ), since in this region u and rA do not depend on E,. Therefore, we have to conclude that a critical point appears neither in r, us CJnor in rA us ai (at various Q values). In addition, no abrupt changes of rA in the above plots are expected to appear when a phase transition takes place far from the critical point. It is still an open question what would be the precise shape of the above curves in the region of phase transitions. We anticipate that these curves are continuous in this region, because we speculate that the function 5 = y + aE, can be calculated from y against E, data at every electrode charge density u and therefore r, can be calculated at every charge and bulk adsorbate concentration[40,41]. The available experimental data do not clarify this point. Thus, according to Buess-Herman et a1.[19], the plots of the surface concentration of iso-quinoline against u and ai (at constant u) show characteristic vertical steps and the same feature is observed in the plots of u us E, , rA us E, and rA us ai at constant E, .However, for reasons explained above, the plots of u us E,, rA us E, and r, us u cannot show simultaneously vertical steps. In a subsequent paper, Buess-Herman et a1.[21] reported their results on the adsorption of 3-methyl-iso-quinoline on a Hg electrode. Although there are sharp, vertical discontinuities in the plots of r, us E, and u us E, , the plots of rA us u and rA us ai at constant u are rather continuous. In our opinion, the differentiation used for the calculation of Ta usually amplifies the experimental error and may lead to vague results. Thus, there is need for a further study on this topic based on experimental data of high accuracy. Such data already exist. They are those obtained by de Levie et al.[ 161.Therefore, a proper re-examination of these data may clarify, without ambiguity, whether a phase transition can be observed, when we use the surface charge density as an independent electrical variable. In addition, we should point out the difference between the relationships obtained from the two treatments presented above. The relationships derived by means of the rigorous treatment contain the applied potential E, as the independent electrical variable. Substitution of this potential by the potential drop Ad” across the adsorbed layer leads to approximate relationships. Thus, it cannot be assured that the plot of rA us A&” shows abrupt changes of r, in the
transition
455
region, since the relevant relationship
(78)
is only approximately valid. This means that at least in the transition region the use of A@ instead of E, , as the independent electrical variable, is inappropriate. The same holds for the surface charge density, as we proved above. In the past there was a long debate whether the electrode potential or the charge density is the primary electrical variable, ie the variable which is appropriate to express the properties of the electrical double layer[9,42-48]. Later on, Trasatti[49] introduced the potential drop Ac$” as an independent electrical variable and argued that this variable as well as the electrode charge density are the most appropriate variables to describe adsorption phenomena. On the contrary, the present results show that the use of these variables instead of E, tends to mask the phenomenon of phase transition, since the plots of r, us u or Ab” and rA us ai at constant u or Ac#J”are unlikely to show characteristic abrupt changes. Therefore, our results tend to favour the view of the Russian school[9], which is that the most suitable electrical variable is the applied potential. Finally, conclusions concerning the physical content of an order-disorder phase transition of a charged interface can be drawn by comparing the basic relationships (43) derived above with the corresponding ones valid for an orderdisorder phase transition of a bulk binary liquid mixture. This comparison is facilitated when the meaning of the quantities ~2, pd defined from equation (42) is clarified. Making use of Euler’s theorem, equation (37) yields U”= TS”-PV”+~Xn~+~dn~+s,Q +
~H&Q+
This equation can be differentiated compared with equation (37) gives
+
k”;.
cg2)
and the result
n:dpX + ntdpz = 0,
(93)
provided that T, P, a, and E, are constant. That is, we obtain a Gibbs-Duhem type equation, which is valid in bulk phases. From this point of view, we can consider that at a constant potential E, the quantities ~1, pb express the chemical potentials of A and S at the surface solution, when it is treated as a bulk one (bulk phase approach). In a liquid bulk mixture of two components, A and S during an order-disorder phase transition the homogeneous solution separates into two liquid phases each one saturated with one of the two constituents. In addition, the relationships which determine the critical point are given by[4,26]
(gg,*=0, (Z),, =0, ($),, >0.
(94)
It is seen that these relationships are almost identical with (43). Moreover, in (94) the specific concentration c can be replaced by the mole fraction of A[4,26] and the same can be done for (34), where the surface mole fraction of A can replace the surface concentration
P. NIKITAS
456
rA. Then, the transformed relationships become similar. Therefore, we have to conclude that, during an order-disorder phase transition at an electrode surface, a separation of the interface into two new phases, each one saturated with either adsorbate or solvent molecules, occurs. In this case, we can speculate that the new phases either form patches on the electrode or one of them is distributed within the other forming a micro-heterogeneous system, which might be a kind of surface suspension or emulsion extended to several molecular diameters from the electrode. Interfacial micellization cannot be ruled out as well. We believe that there are experimental evidences for these assumptions. During a phase transition the surface concentration of the adsorbate increases abruptly to values which are close to saturation. Thus, the formation of patches on the electrode is expected to lead to a more or less strong inhibition of electrode reactions-a phenomenon which is frequently observed. On the other hand, if a surface suspension or emulsion is formed or an interfacial micellization takes place, then it is possible the increase of the bulk adsorbate concentration to decrease the inhibition of electrode reactions, due to a decrease of the compactness of the adsorbed monolayer on the electrode surface. This very rare phenomenon has been already observed[SO, 5 11. It is also interesting to add that the phase separation described above appears to be accompanied by a kind of condensation of the interface. At the critical point we have [see relationships (31) and (3311 +co
and (95)
These relations, which are also expected to be valid far from the critical point in the region where phase transition occur, show that the volume of the interface per unit area changes abruptly. In addition, this abrupt change of v values takes place at the same potential where rA and u also change abruptly. Although an interface has no real boundaries and therefore no precise volume, the above results may indicate that the phase separation is accompanied with a kind of condensation of the interface. Note that this interesting feature of the order-disorder phase transitions at charged interfaces is not observed in bulk uncharged phases[4,26].
REFERENCES 1. S. K. Rangarajan, in Specialist Periodical Reports, Eiectrochemisty (Edited by H. R. Thirsk), Vol. 7. The Chemical Society, London (1980). 2. S. Sathyanarayana and K. Baikerikar, J. electroanal. Chem. 21, 449 (1969). 3. R. de Levie, Chem. Rev. 88, 559 (1988). 4. A. Miinster, Classical Thermodynamics. WileyInterscience, London (1970). 5. V. Brabec, M. Kim, S. Christian and G. Dryhurst, J. electroanal. Chem. 100, 111 (1979). 6. Y. M. Temerk, P. Valenta and H. Numberg, J. electroanal. Chem. 100, 77 (1979). 7. H. B. Herman, R. L. McNeely, P. Surana, C. M. Elliott and R. W. Murray, Anal. Chem. 46, 1258 (1974).
8. C. M. Elliott and R. W. Murray, Anal. Chem. 48,259 (1976). 9. B. B. Damaskin, 0. A. Petrii and V. V. Batrakov, A&sorption of Organic Corn@ on Electrodes. Plenum Press, New York (1971). 10. Yu. Ya. Gurevich and Yu. I. Kharkats, J. electroanal. Chem. 86, 245 (1978). 11. Yu. I. Kharkats, J. electroanal. Chem. 115, 75 (1980). and S. K. Rangarajan, 12. M. V. Sangaranarayanan J. electroanal. Chem. 176, 119 (1980). 13. R. Sridharan. R. de Levie and S. K. Rangarajan, - _ Chem. Phys. Len. 142, 43 (1987). 14. R. Sridharan and R. de Levie. J. electroanal. Chem. 2Q1, 133 (1986). 15. R. Srinivasan and R. de Levie, J. electroanal. Chem. 201, 145 (1986). 16. M. Saffarian, R. Sridharan and R. de Levie, J. electroanal. Chem. 218, 273 (1987). 17. R. Sridharan and R. de Levie. J. electroanal. Chem. 218. 287 (1987). 18. R. Sridharan and R. de Levie, J. electroanal. Chem. 230, 241 (1987). 19. C. Buess-Herman, N. Vanlaethem-Meuree, G. Quarin and L. Gierst, J. electroanal. Chem. 123, 21 (1981). 20. G. Ouarin. C. Buess-Herman and L. Gierst, J. electroanal: Chek 123, 35 (1981). 21. C. Buess-Herman, G. Quarin, L. Gierst and J. Lipkowski, J. electroanal. Chem. 148, 79 (1983). 22. C. Buess-Herman, G. Quarin and L. Gierst, J. electroanal. Chem. 148, 97 (1983). 23. C. Buess-Herman, J. electroanal. Chem. 186,41 (1985). 24. C. Buess-Herman, C. Franck and L. Gierst, Electrochim. Acta 31, 965 (1986). 25. C. Buess-Herman, L. Gierst and M. Gonze, J. electroanal. Chem. 226, 267 (1987). 26. L. Tisza, Generalized Thermodynamics. M.I.T. Press (1977). 27. L. Tiara, in Phase Transformation in Solids (Edited by R. Smoluchowski, J. E. Mayer and W. A. Weyl). Wiley, New York (1961). 28 P. Nikitas, Electrochim. Acta 34, 697 (1989). 29. J. E. Randles, B. Behr and Z. Borkowska. J. electroanal. Chem. 65, 775 (1975). 30. D. M. Mohilner, in Electroanalytical Chemistry (Edited by A. J. Bard) Vol. 1. Marcel Dekker, New York (1966). 31. R. Parsons, in Comprehensive Treatise of Electrochemistry (Edited by I.O’M. Bockris, B. E. Conway and E. Yeaaer) Vol. 1. Plenum Press, New York (1980). 32. J. G. I&kwood and I. Oppenheim, Chemical .Therhodynamics. McGraw-Hill, New York (1961). 33. R. Defay, I. Prigogine, A. Bellemans and D. H. Everett, Surface Tension and Adsorption. Longmans (1966). 34. P. Nikitas, Electrochim. Aka 34, 293(1989). 35. C. F. J. Bdttcher, Theory of Electric Polarization. Elsevier, Amsterdam (1959). 36. N. Matubayasi, K. Motomura, M. Aratono and R. Matuura. Bull. them. Sot. Jtm 51. 2800 (1978). 37. N. Matubayasi, M. Dohzonq M.. Aratono, K. Motomura and R. Matuura, Bull. them. Sot. Jpn 52, 1597 (1979). 38. M. Aratono, M. Yamanaka, K. Motomura and R. Matuura, Colloid Polymer Sci. 260, 632 (1982). 39. M. Aratono, S. Uryu, Y. Hayami, K. Motomura and R. Matuura, J. Colloid Interface Sci. 98, 33 (1984). 40. R. Parsons, Trans. Faraday Sot. 51, 1518 (1955). 41. J. Lawrence, R. Parsons and R. Payne, J. electroanal. Chem. 16, 193 (1968). 42. J. O’M. Bockris. M. A. V. Devanathan and K. Miiller. Proc. R. Sot. (And) A274, 55 (1963). 43. R. Parsons, J. electroanal. Chem. 7, 136 (1964). 44. B. B. Damaskin, J. eleccroanal. Chem. 7, 155 (1964). 45. R. Parsons, J. electroanal. Chem. 8, 93 (1964).
Thermodynamics
of phase transitions at ele-ctrosorbed layers
46. E. Dutkiewitcz, J. D. Garnish and R. Parsons, J. electroanal. them. 16, 505 (1968). 47. A. Frumkin. B. B. Damaskin and A. A. Survila. J. eIectroanai. Chem. 16, 493 (1968). 48. J. O’M. Bockris, E. Gileadi and K. Miiller, Electrochim. Acta 12, 1301 (1967).
457
49. s. Trasatti, J. electroanal. Chem. 53, 335 (1974). 50. A. Anastonouios. A. Christodoulou and I. Poulios. J. electrokal. C&m. 262, 235 (1989). 51. N. Batina and B. Cosovib, J. electroanal. Chem. 227,129 (1987).
001~-4686/91 $3.00 + 0.00 Pergamon PnsS Qk.
1991
THERMODYNAMICS OF PHASE TRANSITIONS ELECTROSORBED LAYERS
AT
P. NIKITAS Laboratory of Physical Chemistry, Department of Chemistry, University of Thessaloniki, 54006 Thessaloniki, Greece (Received 29 January 1990; in revised form 23 April 1990)
Ahstraet-T&a’s theory for critical phases is applied to describe phase transitions of electrosorbed layers formed by dilute solutions of strongly adsorbed substances. Within the framework of this theory we determine relationships which define critical points and establish rigorous criteria for an order-disorder phase transition to occur. An approximate treatment baaed on the bulk phase approach is also presented. The implications of the electrode charge or the potential drop across the adsorbed layer being used as independent electrical variables are indicated. Finally, it is shown that, during an order-disorder phase transition of an electrosorbed layer, we have a separation of the interface into two new phases each one saturated with either adsorbate or solvent molecules. Key words: thermodynamics,
phase transition.
1. INTRODUCTION Experimental observations have shown that a number of interesting phase transitions can take place during adsorption at charged interfaces, which may be classified into three main categories[l]. (a) Phase transitions analogous to those appearing in multi-component liquid systems. (b) Orientational transitions. (c) Surface crystallization. The typical feature of the first type of phase transitions is the appearance of characteristic abrupt changes in the capacitance 11spotential curves at the desorption potentials associated with phenomena of hysteresis[ l-31. In addition, capacitance peaks are usually not observed at these potentials. The case exhibits symptoms of two-dimensional condensation and for this reason Rangarajan[l] considers as an additional type of phase transition, that which is observed in one component systems. However, the thermodynamic treatment presented below shows that in the case of electrosorption the condensationdilution of the interface is always accompanied with phase separation phenomena, like those appearing during first order phase transitions in multicomponent liquid systems. Adopting Tisza-Callen’s terminology[4], we shall call them order-disorder phase transitions throughout the paper. Other observable criteria for an order-disorder phase transition to occur are discussed in [2]. Orientational transitions are characterized by one or two “pits” in the capacitance-potential curves [1,5,6]. The pits correspond to certain orientations of the adsorbate. Finally, the surface crystallization is a different kind of phase transformations and is observed in the case of adsorption of cations[7,8]. In particular, this phenomenon has been observed during the adsorption of Tl+ and Pb* + in aqueous halides and it is detected
from the abrupt changes in the surface excess of Tl+ and Pb2+, when it is plotted against the concentration of anions in solution. The theoretical work for the description of phase transitions cannot be considered as adequate. The two-dimensional condensation was firstly treated on the basis of the Frumkin isotherm model[9]. It is well known that the model predicts a phase transition when the attraction constant a is greater than 2. This model has been also adopted by Sathyanarayana et al. [2] to interpret experimental capacitance data when a two-dimensional condensation takes place. The effect of temperature on phase transitions has been analysed by Gurevich et a1.[10] on the basis of a Flory-Huggins type isotherm. The case of orientational transitions has been also treated within the framework of this model[ll]. Recently, a more advanced model, that developed by Sangaranarayanan-Rangarajan[ 121, was used to describe the dependence of the capacitance pits upon temperature and bulk adsorbate concentration[l3]. In the present paper we focus our attention only on order-disorder phase transitions. Experimentally the phenomenon has been widely studied[l, 31. The most thorough work has been done by de Levie et aZ.[3, 14-181 and Buess-Hennan et a1.[19-251. Their experimental data show that the interfacial tension is a continuous function of potential, whereas discontinuities appear in the surface excess-potential, capacitance-potential and charge density-potential curves. For the last case de Levie appears to doubt whether his experimental data give the precise shape of the charge-potential curve in the region of phase transition[3]. Perhaps de Levie has observed that the existence of discontinuities in the charge densitypotential curves leads possibly to a continuous variation of the surface excess when it is plotted against charge density! This matter is discussed in the present paper, in which we try to examine whether 447
P. NIKITAS
448
the experimental behaviour observed in systems which undergo an order-disorder phase transition is predicted thermodynamically. For this reason we apply Tisza’s theory[4,26,27] to study this phenomenon in electrosorbed layers, in an attempt to establish rigorous criteria for these phase transitions to take place.
Here, Q is the total charge on the Hg electrode and F is Faraday’s constant. If the reference electrode is reversible to the anions X-, for example Ag]AgX(X-, then we again obtain equation (2) but with terms L_ dQ and PMxdn”+ instead of E+dQ and /~~xdnO_, respectively, where E_ = E_ - (&
- p;g)/F
and 2. RIGOROUS
2.1. Basic equations for the internal energy We consider the interfacial region (a) which is formed between a Hg electrode and an electrolyte solution composed of the electrolyte MX, the organic compound A and the solvent S. We further consider that the interface is extended up to the homogeneous regions of the phases which form it (Fig. 1). The total differential of the internal energy U” of the interface is given by dU” = TdS” - PdV” + ydd + pidni
,
Adn: + FHg+dn”,,+ = @g - FrpHg)dn; + ol$+ + WHg) dnB, = kmn:
+p$dn&+
(1)
p_dnL +P+dn”,
and
=(pb_ - Ft#~~)dn: +(/h
+F+b)dC
=Pbdnb+pb+dn: - 4bdQ,
(6)
equation (1) becomes dU”= TdS”+ &lnI
PdV”+ydd
+ A+“dQ +P”,dn:
+ Prgdn: + &;+dn&g+
+&dn:
+&dn”,,
(7)
where 4 denotes the inner potential of the phase indicated by the superscript and A@ = +Hg - C#J~ is the potential drop across the interface. The superscripts (Hg) and (b) denote the Hg electrode and the bulk electrolyte solution, respectively. Note that the chemical potentials ~‘5, p(“care not uniform throughout the system. 2.2. Tisza’s criteria for a critical point Equation (2) shows that U” = U”(S”, V”, d, C, nI, Q, nLg+, n$ ),
+Pc,,dn!.,
(2)
E+=($-jY&)/F.
(3)
where c+=E,-p$/F
(5)
+ 9”‘dQ
TdS”-PdV”+ydd+&dni+p;dn; +e+dQ
+ p::+dGg+
and
where, S” is the entropy of the interface, V” its volume, T is the temperature, P the pressure inside the interface, y the interfacial tension, d the area of the interface, ni, rzgare the number of moles of A and S, respectively, at the interface, ~1, & their chemical potentials, nr, n&+ are the moles of electrons and Hg+ on the Hg electrode, A, Fur+ their electrochemical potentials, n”_ , n: are the number of moles of the ions X- and M+ at the interface and ji_, p+ their electrochemical potentials. On the chemical or electrochemical potentials no superscript indicating the phase is necessary, because they are uniform throughout a system under equilibrium conditions. However, we use a superscript in the case of pi, Pg just to distinguish them from the corresponding chemical potentials calculated on the basis of the bulk phase approximation[28]. In real systems the potential of the ideally polarized Hg electrode is controlled with respect to a reversible charge transfer electrode. If this reference electrode is reversible to the cations M+, then eauation (1) can be written in the form[29-311 ’ a . ’ dU”=
(4)
For reasons of generality equation (2) is used with the symbols &* and n; instead of 6+ and n”_ throughout the paper. In addition, the applied potential is denoted by E, instead of either E, or E_. An alternative expression for dU” may be obtained as follows: since
+ &dng
+ P&C + Png+dnOH,+ +fi_dnT. +P+dn”,
E_ = (iitg - p,)/F.
TREATMENT
(8)
that is, the internal energy depends only upon extensive parameters. In addition, it is a first order homogeneous function of its arguments. Therefore, if we introduce the specific parameters e=
U’/.G~,
CI= S”/szl,
rA = nil.&,
2, = Vu/&,
Ts = rig/d,,
Q = Q/d,
rHg+ = nC+/d,
f* =n”,/&,
(9)
a = &, v, O, PA, Ps, Pug+, r,)
(10)
we obtain[4,26,32]
and da = Tdg - Pdv + C+da + p;dPA + &drs Fig.
+ !‘tigdPng+ + P,xdP,
.
(11)
Thermodynamics of phase transitions at electrosorhed layers Equations (10) and (11) can be written in the more general form 16=&(x,,x2,x~,..
.9x,)
da = P,dx, + P2dx2 f.
WI
+. + P,dx,,
) .
(14)
\"rri/xj
Now, we define the thermodynamic potential $ from r-l (15) 1 PiXi. *(pit p2, . . . . P~_,,X,)=Ui=l
The total differential of $ is given by r-1 d$ = - c xidP,+ P&q.
(16)
i-l
Then, according to Tisza’s theory[4,26,27], point is defined from the relationships
ax; PI ,....
=o
w 4ax” > r
h-l
a’*
(4
ax; PI . . . . . h-l
a critical
> 0.
axi axj d4qjyp4ji
= 0,
-
pHg+ rHg+
--L*Q -jl;r, -
hdXr*
(22)
is coherent with equation (15). Its total differential is given by - r* dp,x -Tsd&+&dT* and it is reduced to d$ = &dr,
(17)
(18)
axi
I( > I
apipk+, +*.
(19)
Relationships (17) and (19) can be used to determine the critical point of a phase transition. Moreover, since x, may be any of the specific variables x, , x2, . . ., x,, writing down all the possible functions for $ we can derive all the possible relationships valid at a critical point. However, in the case of surfaces and interfaces the above relationships are not always applicable. The properties of these systems are usually studied as a function of the composition of the bulk solution. Under equilibrium conditions at constant T and P, the chemical potential of any uncharged species i is uniform throughout the system and therefore (i=A,S,MX). potentials
(23) (24)
when T, P, Q, p,, and & are constant. However, in this case, according to equations (20) and (21), the chemical potential ~1 is also constant and consequently r, is constant too. Thus, when the intensive parameters T, P, L*, p,, and & are kept constant, equation (22) does not define any function of the variable r, . In what concerns the elements of the compliance matrix, they may be physically meaningless for the same reason. For example, the element ar,
becomes infinite. That is
In addition, the chemical Gibbs-Duhem equation
Td+Pv
Pl,...,P,-I
At the same time all the elements of the compliance matrix C = [+J, where
p;=pF,
JI =a-
drj = - cldT + vdP - adc, - rur+ dpHg
Pi=%(
av (3
It is easy to show that equation (15) may not define a thermodynamic potential rj. For example, the function
(13)
where xi = X,/X,+, and Xi, X,, I are extensive properties of the system. To each specific variable x, corresponds the intensive quantity Pi,
449
(20)
( % > ~.P,~+,P$PMx implies a variation of r, with respect to pi at constant T, P, c+, pMx and pg, which, as it has been noted above, is impossible due to equations (20) and (21). These problems can be easily overcome, when we study dilute solutions of strongly adsorbed substances. In this case &( = ~8) can be approximately considered as constant, whereas r,, varies with the bulk concentration of A. Since this approximation is used throughout this paper, the present treatment is applied only to charged interfaces formed by dilute solutions of strongly adsorbed substances. 2.3. Rigorous relationships valid at a critical point We define the thermodynamic potential tj from *
=ll-Td++Pu-p;rA-jq-s - hgrHg+ - hd-*
which under constant d$=
(25)
T, P and pMx yields
-rAdp;-Tsd&++dda
= -(Ta--rsN,/Ns)d&++da.
(26)
Equation (25) is coherent with the definition of $ from equation (15) and therefore it can be used in (17). Since
p! fulfil the
=o where Ni is the mole fraction of i in the bulk solution. A close inspection of the problem shows that equations (20) and (21) impose certain restrictions to the direct application of Tisza’s theory to phase transitions at surfaces or interfaces. The restrictions concern the definition of the thermodynamic potential I(/ from equation (15) and the elements of the compliance matrix.
(27)
T,P,.+,n: we
obtain
> 0,
T,P.a*,a~
(28)
P. NIKLITAS
450
where a, and uk are the mean ionic activity of the salt MX and the activity of the adsorbate A in the bulk solution, respectively. In addition, from (19) we obtain a number of interesting relations which are also valid at the critical point. Thus, the effect of temperature is described by
homogeneous function of the first degree in np. Hence by Euler’s theorem[33]
where CQis the partial mole area of i( = A, S) at the interface. Then equation (2) is written with the form dU” = T dS” - P dV” + (& + a,+y) dn$ +&+e,y)dG++dQ (37)
+ /JngdG&+ + L&xb”, *
Now we can define the specific internal energy and the function $ from
For the effect of pressure we have
du=Tds-Pdu+(&+aAy)dc + Q dq +
PHg dclfg+
+
(38)
@MX dc,
$ = a - Ts + Pu - 6 * 4 - ,&Q+Q+ - p~xC*, (39) respectively, where where we have assumed that the effect of pressure on the potentials E* and E, is negligible. The effect of the applied field at a critical point leads to the expressions
(31)
The last relation is equivalent to C *co, where C is the differential capacitance of the interface. In addition, from (28) and (31) we obtain
u = W/n&
s = S”/n&
v = Vu/n&
cHg+= n&+ In:,
c = no&,
q = Q/n&
c* = n$ In;.
w
The specific variables introduced by equation (40) are not common in experimental studies. However, they are necessary to determine relationships [(43) below] which will be found useful in clarifying the nature of surface transitions. In addition, they help to define the critical point in the plot of the surface pressure II us PAI which is a common plot, especially in studies of uncharged interfaces. Equation (39), under constant T, P, pMx, results in d$=
(ii&) =(~)T,p,.,,.:+~. (32)
-q dE, +(p:+CIAY)dC
= -q dE, + PX dc,
(41)
where
Finally, the bulk adsorbate concentration the critical point through the relationships
affects
p: = p: +
(42)
aAy.
In this case, the critical point for phase separation is defined from the relationships a*pz
(33) Note that if we use the function $ defined from equation (22) then we obtain for the critical point the relations
>
0.
(43)
T,P,a*,E+
For symmetry reasons we also have
= 0, 7,P.a, ,E*
T,Pa*
.E*
> 0, T.P.a*,Ek
where c’ = ng/ni = l/c. Since which under equilibrium
conditions
rA=
become
ni
c
EAn: + xsn;
=-=A as + caA
(45)
then arA -= ac
(35) Other rigorous relationships valid at a critical point can arise as follows: The area of the interface is a
as
>o
(a,+caA)'
and (46)
Thermodynamics of phase transitions at electrosorhed layers provided that the partial areas uA, as are independent of the surface composition. Irrespective of this constraint, the physical content of the variables r,, c, c’ entails that r, should increase with increasing c values and decreasing c’ values. Therefore, relationships (43) and (44) can be rewritten as
m
(3 ari
T,P,o*:,E*
=”
(47)
>o
approximately &fined from the thickness of the adsorbate molecules. Now we can distinguish two parts in the interface. The compact part includes an homogeneous region of the Hg electrode, and it is extended up to the plane 00’ and the “diffuse” part which is extended from the plane 00’ up to the homogeneous electrolyte solution. The area of the interface is again a linear homogeneous function of the number of moles, n;, of i( = A, S) in the compact part. Therefore, we can write[34] a=SAn~+Ssn~=(SA/lA)n;+(Ss/ls)n~
m (4 ari
<
(48)
0.
T,P&E+
We denote by n the ratio of the partial areas aA, as. That is, n = tlA/as. This ratio expresses the number of solvent molecules which are displaced from the surface solution by one adsorbate molecule and it is usually called size ratio parameter. In this case we have
aab,
(>
=-
arA
T,P,o*,E&
=
’ (50)
the criteria for a critical point to exist are given again by (35) provided that n is constant. This means that relationships (47), (48) and (34), (35) define the same critical point. In this case, making use of (34), (42) and (47), we obtain
= 0,
=
0, (51)
where n = y” - y is the surface pressure of the interface and y” the interfacial tension of the base solution, without the adsorbate A.
3. APPROXIMATE
TREATMENT
3.1. Internal energy of an a&orbed layer In the treatment presented above, the interface had a considerable thickness and it included homogeneous regions of the phases which form it. However, in studies of adsorption of organic compounds on electrode surfaces the attention is usually focused on the “compact” part of the interfacial region, which is formed from the monolayer of the adsorbed organic molecules. For this reason, let us consider the plane 00’, the position of which inside the interface is
(52)
where ny is the number of moles of i( = A, S) when it forms a monolayer on the electrode, Sj is the effective area covered by i on the electrode and li the average number of layers of i which form the compact part. When the effective area S, and the number of layers Ii do not alter during the adsorption process, equation (52) yields d~=((SA/IA)dn~+(ss/rs)dn~.
(53)
The infinitesimal electrical work Ac%“dQ in equation (7) may be split into two terms A&‘dQ = 84” dQ + Ab* dQ,
(54)
where A@, A4* are the potential drop across the compact and the diffuse part, respectively. The quantity Ad” dQ expresses the work done on the interface when under constant A4# the charge on the Hg electrode is increased by dQ. Evidently, a constant value of Ac$O does not entail constant values of A@, Ad*. However, we can approximately consider that the quantity A#J”dQ is the infinitesimal electrical work done on the layer s which is bounded by the Hg electrode and the plane 00’, that is the work required to increase the charge on the Hg electrode by dQ under constant A@. Note that the layer s may be identified to the adsorbed layer. If we assume that there is no specific adsorption of ions, then the adsorbed layer is free of ions. Therefore, by Gauss’s theorem of electrostatics[35] over the surface indicated by the dotted line in Fig. 2 we obtain c,A+“.cJ/I = Q - Q’,
> 0,
451
(55)
where Q’ is the induced charge on the surface of the adsorbed layer which is adjacent to the electrode, co the permittivity of vacuum and I the thickness of the adsorbed layer. The total polarization P of the adsorbed layer s is equal to[35] P =
VEQ'/d=lQ',
Fig. 2. Model of the adsorbed layer s.
(56)
P. NIKITAS
452
3.2. Approximate relationships valid at a critical point
where V” is its volume. In addition, we have P=P,+P,,
(57)
where the term P, is due to translation effects and the second one P,, to the orientation of the permanent dipoles. The term P, may be written as P. = (@,a,
+ n:a,)
A&/l,
(58)
P c = nXF* + nifs,
(59)
where Giis the average value of the permanent dipole vector normal to the electrode surface. If the molecules of an adsorbed substance can not be reoriented, as is the usual case for the majority of organic adsorbates, then fA is constant. On the contrary, if a reorientation takes place, as we accept for the case of solvent, then the value off depends upon the strength of the applied field. At low fields the quantity P is proportional to the applied field A#“/1 and we may write[35] j = AA@/&
B = (1 + O(A@)) Ab”/l,
(61)
where O(Ad*) denotes terms of order A4” or higher. Use of equations (5)-(61) yields Q = {niTA + n&%?,+ n;O(A$“) + n;O(A@)} A@/12, (62) Consequently
dQ = {@A+ ‘W~4”)‘1) W + G’s+ W4”)31) WHMW2.
Ts”+ Pv”,
$ = us-
(68)
where us = Us/n$,
s’= F/n;,
us = Vs/ni,
c = ni/nH. (69)
Then d$ = -s”dT+v”dP+&dc,
(70)
and the relationships which define a critical point for phase separation may be easily written with the form
and
(g)T,p =0, (z),, =0, (g)T,p <0,
(72)
where now PA = nil&. These relationships lead to the following criteria for a critical point to exist: (a) under equilibrium conditions between bulk and adsorbed molecules, we have p;=p~=p~b+RTlna~,
(73)
where pFb is the chemical potential of i( = A, S) in its standard state in the bulk solution. If this equation is introduced into equation (65), we obtain ly = pFb + RT In a! + y(S,/l,) + ~i(A~“/l)2 + 0[(A@)3], i = A, S
(63)
Therefore, if we introduce equations (53) and (63) into equation (7) we obtain the following expression for dU” dU”=
(67)
(60)
where I is a constant characteristic of the adsorbed substance. When the field increases v also increases and at high field strengths it tends to become independent of the applied field. Finally, at moderate fields we may write in general[35]
A&
du”=Tds”-Pdv=+&dc
of i. For the
where ai is the average polarizability term P, we have
where Xi is a constant.
If we accept that a phase transition is determined almost exclusively from the properties of the adsorbed layer, then we can use the above expression of dU” to derive the relationships which define a critical point. In this case we can work as previously and define the specific internal energy us and the function $ from
which in combination
(74)
with (72) yields = 0,
TdS’-PdV”+r~dn~+r~dn~ (75)
+TdSd-PdVd+A4ddQ+/&dnd_ +&dnd+ + T dS”g-
(b) In this model the size ratio parameter n is defined from n = S,/Ss . Then, equation (74) leads to
+&dn:+&dn: P dV”g+
p,“‘dnr*
+ PI;+ dn::+,
(W
where
c;\ - n& = (xA - n~s)(A~s/1)2 + RT ln{ai/(ag)n} + O[(A@)3]+ (~2~ - nptb)
l: = &’ + Y(S,/M + X,(AM)*
+
WW”)31,
i = A, S.
Now we can quantity dU”,
(65) approximately
assume
dU”=TdS”-PdV”+&dn~+~~dn’,
that
the
= 0, T, P, [email protected];
(66)
expresses the differential of the internal energy of the adsorbed layer s.
(76)
and if it is differentiated with respect to PA under constant T, P, Ads, ai, we obtain
= ”
a3ab >o H ar
f
T, P,A&!g
provided that n is constant.
(77)
Thermodynamics of phase transitions at electrosorbed layers (c) Differentiation of equation (76) with respect to P., at constant T, P, ai (and consequently at constant a,b) leads to
(78) provided that A& # 0. Finally, if we do not express the area d as a function of ni, ng and write the specific internal energy as da” = Td2 - Pdu” + A@da + p:dP,
+ &dP,,
(79)
vs = v=/&d
(80)
the measured potential the potential of zero charged and the potential drop across the diffuse layer calculated from the Gouy-Chapman theory. In this case, the constraint a, is constant should be added in the relationships of the approximate treatment. In general, almost all the relationships obtained by means of the approximate treatment may derive from the corresponding relationships of the rigorous treatment, if we replace E, by A&. The only exceptions are relationships (78) which appear not to be obtained, at least directly, by means of the rigorous treatment. The treatment leads to expressions (31) from which we obtain
(aE,>
arA r,p,o*,ai=
where CL= = V/d,
6== s=/&/,
then, following precisely the same procedure to that adopted for the determination of relationships (28), we obtain
453
O.
(83)
However, if we take into account that the function P, = P,(E,) is strictly increasing (or decreasing) up to the point of maximum adsorption, then the above equation necessarily entails that this point is a point of inflection and therefore at the critical point we have
= 0, (81) In this case, at the critical point relationships (29x33) are also valid provided that E, is replaced by A@. Note that the mean ionic activity a, does not appear in the relationships obtained by means of the approximate treatment. However, its variations alter the values of 4b and therefore affect A4$, since A$,“=+nr-A@-#b.
(82)
In practice, Ad” is calculated when r#~ b and consequently when a, are constant, by subtracting from
(84)
4. DISCUSSION Table 1 summarizes the most important relationships for the determination of a critical point. They have been obtained by means of the rigorous treatment. These relationships seem to define only one critical point, when T, P and a, are constant. However, if we take into account that the relation E, = E, (r,) is a double-valued function of P,, we
Table 1. Relationships valid at a critical point 1.
2. 3. a97 H ar:
EA 36/3-4-F
= 0, T,P,a*,E*,&a,
P. NIKITM
454
conclude that there are two critical points on both sides of maximum adsorption. A critical point is closely related to the abrupt changes of certain variables observed during a phase transition. It is well known that a critical point is the limit where the connodals, ie the straight lines joining the two co-existing phases, tend, when one of the variables tends to its critical value. For example, the critical point in the plot of rA against ui is the limit of the abrupt changes of rA in the same plot, as ai tends to its critical value. Thus, a criterion for a phase transition to occur is
variables, 6, and u. It is seen that, according to (28) or (33), e changes abruptly during a phase transition, whereas 6, remains constant. More generally, E, always changes continuously, while u shows discontinuities during a phase transition. However, the most interesting point is that if we consider the electrode charge density e as an independent electrical variable, then it is very likely that a phase transition cannot be observed! The facts which support this prediction are the following: (a) Plots of rA trs r~do not show critical points. This is equivalent to the fact that neither
= 0,
(85) since at the critical point the above relation is also valid. However, we should point out the different meaning which this criterion acquires when it is applied at the critical point and at the transition region far from it. Relationship (85) shows that the tangent line to the curve of rA us ui at the critical point is parallel to the Y axis, whereas at the transition region it shows that rA varies abruptly at a certain value of ai. Therefore, the criteria for a phase transition can be obtained directly from the treatment of the critical points presented above. The most important observable criteria are listed in Table 2. It is seen that the thermodynamic treatment presented in this paper predicts abrupt changes of the adsorbate surface concentration rA when it is plotted against ai and E, in the transition region. Abrupt changes are also expected in the plots of cr and C us E, . These predictions have been already verified experimentally. For example, they are in agreement with the experimental data obtained by de Levie et a1.[3, 161 and Buess-Hennan et a1.[19-251. Moreover, the theory predicts that the differential capacitance tends to infinity at the transition region. This feature has not been observed yet. However, the lack of sharp and very pronounced peaks on the capacitance curves in the transition region is attributed to experimental limitations[3]. These peaks could be observable only if the capacitance was measured at a sufficiently low frequency[3]. Finally, studies at uncharged interfaces [36-391 show abrupt changes of rA with respect to the surface pressure ll in agreement with our results. The results of the present treatment raise some questions about the equivalence of the two electrical
(
2. 3.
#
(86)
0,
) T.PP&
nor
=0, (87) are valid at a critical point. If relationships (86) were valid, then
($)=(ff)(&J+3@)($!j x($)+(q(&g=o (88) which is inconsistent with (84). The quantity (CJ’E,/aI’i) is different from zero at a critical point only if au/ar, # 0. In addition, if (87) were valid, then at the critical point we would have
which is also inconsistent with (84). (b) The theory does not allow to clarify rigourously whether a critical point appears in the plot of rA us
Table 2. Observable criteria for an order-disorder 1.
-ab ar:
= 0,
phase transition to occur
Thermodynamics of phase transitions at electrosorbed layers
ai at various c values. If we assume that a critical point does appear in this plot, that is if
aa:
(ar,> r,PP*
,o =
O
is valid, then the same equation is expected to be valid in the whole region of phase transitions. In this case of plot of rA against rr at various ai values should show abrupt changes of I’, at certain values of u. However, this means that (86) are valid, which has been shown to be incorrect. In addition, if we graphically eliminate E, from the plots rr us E, and rA us E, , then the resulting plot of r,, against CJpresents no abrupt changes of r,. There is only an ambiguity about the precise shape of the r, us u curve in the region where c and r, changes abruptly in the original plots (cr us E, and rA us E, ), since in this region u and rA do not depend on E,. Therefore, we have to conclude that a critical point appears neither in r, us CJnor in rA us ai (at various Q values). In addition, no abrupt changes of rA in the above plots are expected to appear when a phase transition takes place far from the critical point. It is still an open question what would be the precise shape of the above curves in the region of phase transitions. We anticipate that these curves are continuous in this region, because we speculate that the function 5 = y + aE, can be calculated from y against E, data at every electrode charge density u and therefore r, can be calculated at every charge and bulk adsorbate concentration[40,41]. The available experimental data do not clarify this point. Thus, according to Buess-Herman et a1.[19], the plots of the surface concentration of iso-quinoline against u and ai (at constant u) show characteristic vertical steps and the same feature is observed in the plots of u us E, , rA us E, and rA us ai at constant E, .However, for reasons explained above, the plots of u us E,, rA us E, and r, us u cannot show simultaneously vertical steps. In a subsequent paper, Buess-Herman et a1.[21] reported their results on the adsorption of 3-methyl-iso-quinoline on a Hg electrode. Although there are sharp, vertical discontinuities in the plots of r, us E, and u us E, , the plots of rA us u and rA us ai at constant u are rather continuous. In our opinion, the differentiation used for the calculation of Ta usually amplifies the experimental error and may lead to vague results. Thus, there is need for a further study on this topic based on experimental data of high accuracy. Such data already exist. They are those obtained by de Levie et al.[ 161.Therefore, a proper re-examination of these data may clarify, without ambiguity, whether a phase transition can be observed, when we use the surface charge density as an independent electrical variable. In addition, we should point out the difference between the relationships obtained from the two treatments presented above. The relationships derived by means of the rigorous treatment contain the applied potential E, as the independent electrical variable. Substitution of this potential by the potential drop Ad” across the adsorbed layer leads to approximate relationships. Thus, it cannot be assured that the plot of rA us A&” shows abrupt changes of r, in the
transition
455
region, since the relevant relationship
(78)
is only approximately valid. This means that at least in the transition region the use of A@ instead of E, , as the independent electrical variable, is inappropriate. The same holds for the surface charge density, as we proved above. In the past there was a long debate whether the electrode potential or the charge density is the primary electrical variable, ie the variable which is appropriate to express the properties of the electrical double layer[9,42-48]. Later on, Trasatti[49] introduced the potential drop Ac$” as an independent electrical variable and argued that this variable as well as the electrode charge density are the most appropriate variables to describe adsorption phenomena. On the contrary, the present results show that the use of these variables instead of E, tends to mask the phenomenon of phase transition, since the plots of r, us u or Ab” and rA us ai at constant u or Ac#J”are unlikely to show characteristic abrupt changes. Therefore, our results tend to favour the view of the Russian school[9], which is that the most suitable electrical variable is the applied potential. Finally, conclusions concerning the physical content of an order-disorder phase transition of a charged interface can be drawn by comparing the basic relationships (43) derived above with the corresponding ones valid for an orderdisorder phase transition of a bulk binary liquid mixture. This comparison is facilitated when the meaning of the quantities ~2, pd defined from equation (42) is clarified. Making use of Euler’s theorem, equation (37) yields U”= TS”-PV”+~Xn~+~dn~+s,Q +
~H&Q+
This equation can be differentiated compared with equation (37) gives
+
k”;.
cg2)
and the result
n:dpX + ntdpz = 0,
(93)
provided that T, P, a, and E, are constant. That is, we obtain a Gibbs-Duhem type equation, which is valid in bulk phases. From this point of view, we can consider that at a constant potential E, the quantities ~1, pb express the chemical potentials of A and S at the surface solution, when it is treated as a bulk one (bulk phase approach). In a liquid bulk mixture of two components, A and S during an order-disorder phase transition the homogeneous solution separates into two liquid phases each one saturated with one of the two constituents. In addition, the relationships which determine the critical point are given by[4,26]
(gg,*=0, (Z),, =0, ($),, >0.
(94)
It is seen that these relationships are almost identical with (43). Moreover, in (94) the specific concentration c can be replaced by the mole fraction of A[4,26] and the same can be done for (34), where the surface mole fraction of A can replace the surface concentration
P. NIKITAS
456
rA. Then, the transformed relationships become similar. Therefore, we have to conclude that, during an order-disorder phase transition at an electrode surface, a separation of the interface into two new phases, each one saturated with either adsorbate or solvent molecules, occurs. In this case, we can speculate that the new phases either form patches on the electrode or one of them is distributed within the other forming a micro-heterogeneous system, which might be a kind of surface suspension or emulsion extended to several molecular diameters from the electrode. Interfacial micellization cannot be ruled out as well. We believe that there are experimental evidences for these assumptions. During a phase transition the surface concentration of the adsorbate increases abruptly to values which are close to saturation. Thus, the formation of patches on the electrode is expected to lead to a more or less strong inhibition of electrode reactions-a phenomenon which is frequently observed. On the other hand, if a surface suspension or emulsion is formed or an interfacial micellization takes place, then it is possible the increase of the bulk adsorbate concentration to decrease the inhibition of electrode reactions, due to a decrease of the compactness of the adsorbed monolayer on the electrode surface. This very rare phenomenon has been already observed[SO, 5 11. It is also interesting to add that the phase separation described above appears to be accompanied by a kind of condensation of the interface. At the critical point we have [see relationships (31) and (3311 +co
and (95)
These relations, which are also expected to be valid far from the critical point in the region where phase transition occur, show that the volume of the interface per unit area changes abruptly. In addition, this abrupt change of v values takes place at the same potential where rA and u also change abruptly. Although an interface has no real boundaries and therefore no precise volume, the above results may indicate that the phase separation is accompanied with a kind of condensation of the interface. Note that this interesting feature of the order-disorder phase transitions at charged interfaces is not observed in bulk uncharged phases[4,26].
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