Kinetic effects of methyl-β-cyclodextrin and 2-hydroxypropyl-β-cyclodextrin and their mixtures on the reaction [Fe(CN)5(4-phepy)]3- + [Co(NH3)4(H2O)2]3+

Article abstract of DOI:10.1016/j.cplett.2009.02.033

Oxidation (electron transfer) of [Fe(CN)₅(4-phepy)]^3- with [Co(NH₃)₄(H₂O)₂]³⁺ has been studied in solutions containing methyl-β-cyclodextrin (MβCD) and 2-hydroxypropyl-β-cyclodex

Full text of DOI:10.1016/j.cplett.2009.02.033

Chemical Physics Letters 471 (2009) 234–238
Contents lists available at ScienceDirect
Chemical Physics Letters
journal homepage: www.elsevier.com/locate/cplett
Kinetic effects of methyl-b-cyclodextrin and 2-hydroxypropyl-b-cyclodextrin
and their mixtures on the reaction [Fe(CN)5(4-phepy)]3ꢀ + [Co(NH3)4(H2O)2]3+
*
Maria Marchena, Francisco Sanchez
Department of Physical Chemistry, University of Seville, C/Profesor Garcia Gonzalez s/n, 41012 Sevilla, Spain
a r t i c l e i n f o
a b s t r a c t
Oxidation (electron transfer) of [Fe(CN)5(4-phepy)]3ꢀ with [Co(NH3)4(H2O)2]3+ has been studied in solu-
tions containing methyl-b-cyclodextrin (MbCD) and 2-hydroxypropyl-b-cyclodextrin (HbCD) as well as
mixtures of these CDs. The results can be explained by using the Brönsted equation after an adequate for-
mulation of the activity coefficients appearing in this equation. The non-additivity of the effects of the
CDs in these mixtures is also shown here.
Article history:
Received 20 November 2008
In final form 11 February 2009
Available online 15 February 2009
Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction
We are currently interested in the changes of reactivity caused
by the so-called restricted geometry conditions [13], including the
Cyclodextrins (CDs) are macrocyclic oligosaccharides formed by
(1?4)-linked glucopyranose monomers. The number of these
kinetic changes observed when one, or both reactants, are included
in the CD cavity. As we, and others, have shown, these changes can
be described through a two state model [14], provided that the
partition process between these two states is rapid in relation to
the reactive process. This model describes the reactivity changes
through the following expression:
a
monomers determines their size and properties [1,2]. The three-
dimensional shape of the cyclodextrins is similar to that of a trun-
cated cone, where all hydroxyl groups point outside of the mole-
cule, conferring high polar characteristics to the molecular
exterior, while the interior is somewhat hydrophobic. Interactions
of CDs with their guest molecules depend mainly on the inclusion
of the guest in the hydrophobic cavity of the host [3]. However,
outer-sphere complexes between CDs and guest, through hydrogen
bonding, are also formed [4,5]. This implies the possibility of form-
ing a variety of complexes which gives a great number of applica-
tions to CDs. Thus, the possibility of using CDs as drug carriers [6],
or to mimic enzyme behaviour [7,8] has been described. In this re-
gard, the inclusion of a molecule in a CD cavity can change the
mechanism and/or the kinetics of the reactions in which the mol-
ecule participates [9,10]. These changes can be modulated by using
CDs of different sizes. This difference produces changes on the de-
gree of inclusion and on the guest orientation, as has been known
kw þ kCDK½CDꢁ
k ¼
ð1Þ
1 þ K½CDꢁ
where kw is the rate constant for the free (non-encapsulated) reac-
tant, kCD is the rate constant for the reactant interacting with the
CD, K is the equilibrium constant corresponding to the inclusion
process and [CD] the concentration of CD.
Alternatively, one can describe any reactivity change in con-
densed phases through the Brönsted equation [15]:
cAcB
cz
k ¼ ko
ð2Þ
after an adequate formulation of the activity coefficients of the reac-
tants (A and B) and the transition state (à) [16]. Thus, in the case of
CDs, it can be shown that (see Appendix 1):
for some time. Thus, for example, the inclusion of 2-naphthol in a-,
b- and
c-CD presents binding constants of 126, 625 and
53 molꢀ1 dm3, respectively [11]. Therefore, implying that the
inclusion free energy of the 2-naphthol in these CDs, with respect
1
ci
¼
ði ¼ A; B; zÞ
ð3Þ
to the free energy in water, increased in the order
c > a > b. As a
1 þ Ki½CDꢁ
consequence, the relative changes of reactivity are expected to de-
pend on CD nature. Moreover, the different orientations of the
guest as a consequence of different CDs’ sizes, can lead to a confor-
mational control of the reactivity, because of the exposition of dif-
ferent reactive moieties of the guest to a second reactant [12], this
producing different reaction products.
It is interesting to note that Eq. (2) is more general than Eq. (1)
because the latter can only be applied to unimolecular processes,
or to biomolecular reactions if only one of the reactants forms a
complex with the host. Eq. (2), however, can be applied to any
reaction. Moreover, Eq. (1) can be applied only to a biphasic sys-
tems (in fact, pseudo-biphasic) whereas the interpretation based
on the Brönsted model is flexible enough to be generalized
whatever the number of pseudo-phases are present in the system.
* Corresponding author. Fax: +34 954 55 71 74.
E-mail address: gcjrv@us.es (F. Sanchez).
0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2009.02.033

M. Marchena, F. Sanchez / Chemical Physics Letters 471 (2009) 234–238
235
In this regard, a mixture of two CDs in water behaves as a pseudo-
triphasic system, in such a way that our results can be employed to
demonstrate this fact.
To check the previous ideas we have studied the reactivity
changes observed in the reaction:
changes at 405 nm. This wavelength corresponds to the absor-
bance maximum of the iron complex in water (it was also checked
that changing the observation wavelength does not change the va-
lue of the rate constant).
Kinetic runs were performed in a stopped-flow apparatus from
[Fe(CN)5(4-phepy)]2- + [Co(NH3)4(H2O)2]2+
[Fe(CN)5(4-phepy)]3- + [Co(NH3)4(H2O)2]3+
ð4Þ
Co2+ + 4NH3 + 2H2O
when two CDs, methyl-b-cyclodextrin (MbCD) and 2-hydroxypro-
pyl-b-cyclodextrin (HbCD), are present in the reaction medium. As
will be seen, kinetic effects of these two CDs are non-additive.
Moreover, although MbCD and HbCD produce, separately, an in-
crease in the rate of reaction 4, the opposite effect can be observed
in the presence of both of them, depending of their concentrations.
Applied Photophysics. The concentrations of the iron and cobalt
complexes were 10ꢀ4 and 6 ꢂ 10ꢀ3 mol dmꢀ3, respectively. That
is, we worked under pseudo-first order conditions. The estimated
uncertainty in the values of the rate constants was about 5%. The
concentration ranges of both CDs studied appear on the corre-
sponding figures. The temperature was maintained at
298.2 0.1 K.
2. Experimental
3. Results and discussion
2.1. Materials
The maximum wavelength of the iron complex spectra change
in CD solutions as shown in Fig. 1, indicating the expected interac-
tion of the complex with the receptors. Changes in kmax are well
described by an equation similar to Eq. (1):
[Co(NH3)4(H2O)2]3+ was prepared according to published proce-
dures [17,18], as a perchlorate salt. [Fe(CN)5(4-phepy)]Na3 was
prepared in situ from a solution of the commercially available
(from Aldrich) [Fe(CN)5(NH3)]Na3, after adding the stochiometric
amount of the 4-phenylpyridine ligand (from Fluka). The rapid
hydrolysis of the ammonia complex produces the corresponding
kw þ kCDK½CDꢁ
k ¼
ð5Þ
1 þ K½CDꢁ
aquocomplex [19] [Fe(CN)5(H2O)]3ꢀ
. Complete formation of
This equation describes well the changes in the spectrum as a con-
sequence of the interaction with a receptor, provided that the con-
ditions described by Nelsen [20] hold. The fact that Eq. (5) can be
applied permits us to get the values of the equilibrium constants
corresponding to the inclusion of the iron complex in the CDs. These
constants were 262 molꢀ1 dm3 and 149 molꢀ1 dm3 for MbCD and
HbCD, respectively [21].
No changes in the spectra of the cobalt complex were observed.
This fact points out that this complex remains in the aqueous pseu-
do-phase. This is corroborated by the fact that the rate constants
for the oxidation of the iron complex by the cobalt complex in
the presence of CDs are described by Eq. (1) (see Fig. 2). This equa-
[Fe(CN)5(4-phepy)]3ꢀ complex was checked spectrophotometri-
cally. Cyclodextrins (see Scheme 1) were commercial products
from Fluka and were used as received. The average degree of sub-
stitution per glucose unit is 1.6–2 units of methyl (CH3) in MbCD
and in HbCD it is 0.6–0.8 units of 2-hydroxypropyl (C3H7O). Puri-
fied water used in the preparation of solutions was obtained from
a Millipore Milli-Q water system. Its conductivity was less than
10ꢀ6 S mꢀ1, and was deoxygenated before use.
2.2. Absorbance measurements
Absorbance measurements of solutions containing the iron
complex and CDs were carried out in
spectrophotometer.
a Hitachi U-2800
440
435
430
425
2.3. Kinetic measurements
The kinetics of the reaction between [Fe(CN)5(4-phepy)]3ꢀ and
[Co(NH3)4(H2O)2]3+ was followed by measuring the absorbance
M
β
CD
CD
420
415
410
405
400
Hβ
0.00
0.01
0.02
0.03
0.04
[CD]Total /mol·dm-3
Fig. 1. Plot of kmax/nm of the [Fe(CN)5(4-phepy)]3ꢀ spectrum against the
concentration of MbCD (d) and HbCD (s).The dots are experimental data and the
continuous lines (—) are the best fits obtained by using Eq. (5).
Scheme 1. Representations of methyl-b-cyclodextrin (MbCD) and 2-hydroxypro-
pyl-b-cyclodextrin (HbCD) structures.

236
M. Marchena, F. Sanchez / Chemical Physics Letters 471 (2009) 234–238
40
35
30
25
20
15
behaviour that one would expect on intuitive grounds: the two cat-
alysts, apparently, produce additive effects. However, this additive
character does not always occur: the behaviour of the CD mixture
depends on the position of the arrows, that is, of the concentration
of MbCD when one starts to add the HbCD. Thus, for a MbCD con-
centration of 3.82 ꢂ 10ꢀ3 mol dmꢀ3 no effect of HbCD is observed
(Fig. 3b). Moreover, when one adds HbCD on a dissolution of
0.025 mol dmꢀ3 fixed concentration of MbCD, a decrease of the
reaction rate is observed.
HβCD
MβCD
The results in Fig. 3 can be fitted to the following equation:
kw þ kCD1K1½CD1ꢁ þ kCD2K2½CD2ꢁ
k ¼
ð6Þ
1 þ K1½CD1ꢁ þ K2½CD2ꢁ
which can be derived from the Brönsted equation, as shown in
Appendix 2.
0.00
0.01
0.02
0.03
0.04
The values of the fitting parameters are listed in Table 1. We can
see that these values are in good agreement with those obtained
with Eq. (1) for each CD. It is interesting to note that the limiting
values of k in the plots of Fig. 3 are the same. This extrapolated va-
lue corresponds to the limiting value when only HbCD is present in
the reaction medium. Thus, for any concentration of MbCD (CD1),
adding HbCD (CD2) at the limit of K2[CD2] ꢃ K1[CD1] one would ob-
serve the same kCD2 limiting value. Even the apparently additive ef-
fect observed in Fig. 3a, is not truly additive. This behaviour is
clearly seen by considering how Eq. (6) simplifies when
K2[CD2] ꢃ K1[CD1].
[CD]/mol·dm-3
Fig. 2. Plot of the rate constant k/sꢀ1 for the reaction [Fe(CN)5(4-phe-
py)]3ꢀ + [Co(NH3)4(H2O)2]3+ for different concentrations of MbCD (d) and HbCD
(s). The dots are experimental data and the continuous lines (—) are the best fits
obtained by using Eq. (1).
tion is obeyed only if just one of the reactants of the bimolecular
processes interacts with the receptor (see Section 1). So, the ob-
served kinetic and the spectroscopic behaviour correspond to the
case in which only one of the reactants forms a complex with
the receptor. The values of kw, kCD and K obtained from the kinetic
data fit to Eq. (1) are given in Table 1. The table also contains the
values of K obtained from the spectral changes of the iron complex.
As can be seen, K values obtained by the two procedures outlined
previously are in good agreement.
In conclusion, as shown in Fig. 3, the Brönsted equation can be
used to account for not only the CDs’ effects, but also to rationalize
the non-additivity of the effects of CD mixtures. Of course, this idea
can be extended to mixtures of other receptors, the Brönsted equa-
tion being the basis for interpretation.
It is clear from the results in Fig. 2 and Table 1 that MbCD and
HbCD act as catalysts; that is, inclusion of the iron complex pro-
duces an increase in the reaction rate. Moreover, it follows from
Eqs. (2) and (3) that this rate increase is a consequence of the fact
that the inclusion of the transition state is more favourable than
the inclusion of the initial state. Thus, CDs act as true catalysts be-
cause they interact more favourably with the transition state than
with the initial state. In fact, as shown in Appendix 2 Eq. (A.22)
Acknowledgements
This work was financed by D.G.I.C.Y.T. (CTQ2005-01392/BQU)
and the Conserjería de Educación y Ciencia de la Junta de
Andalucía.
Appendix A
kwKz ¼ kCD
K
Obtaining the expression of the activity coefficient
where K is the equilibrium constant corresponding to the iron com-
plex and Kà is the same constant for the transition state. As kCD > kw
this implies that Kà > K.
As to the data corresponding to CD mixtures, the results are gi-
ven in Fig. 3. These data were obtained as follows: the first part of
the curves (up to the point marked by an arrow) corresponds to a
constant concentration of HbCD (in fact, zero concentration) and
variable concentrations of MbCD. The portion of the curve after
the arrow corresponds to a fixed concentration of MbCD, whereas
the concentration of HbCD is increased. Fig. 3a represents the
As mentioned in the Introduction section of the Letter, reactiv-
ity in systems containing CDs can be explained by taking the Brön-
sted equation (Eq. (2)) as a starting point. It suffices to substitute
into this equation the activity coefficients of reactants and the
transition state. In this first Appendix we derive the expression
of these activity coefficients.
Consider a solution containing a species A which can associate
with two receptors,
a1 and a2, forming two supramolecular aggre-
gates. It is understood that the concentrations of a1 and a2 are in
Table 1
Values of the parameters obtained from the fitting of the spectroscopic and kinetic data to Eqs. (5), (1), and (6).
kmax vs [CD1]
kmax vs [CD2]
k vs [CD1]
k vs [CD2]
k vs [CD1], [CD2]
2 ꢂ 10ꢀ3
[CD1] (mol dmꢀ3) marked by arrows in Fig. 3a
3.82 ꢂ 10ꢀ3
320
165
38
30
16
0.025
K1 (molꢀ1 dm3)
K2 (molꢀ1 dm3)
262
294
41
385
141
36
30
16
282
151
41
32
16
149
162
kCD1 (sꢀ1
kCD2 (sꢀ1
)
)
29
16
kw (sꢀ1
)
16
CD1 = methyl-b-cyclodextrin (MbCD).
CD2 = 2-hydroxypropyl-b-cyclodextrin (HbCD).
a
See text.

M. Marchena, F. Sanchez / Chemical Physics Letters 471 (2009) 234–238
237
ni being the total numbers of moles of i in the solution and li its
chemical potential. The symbol S refers to the solvent.
Taking into account the possibility of the association of A with
both a1 and a2, the solution free energy can be written as:
30
28
26
24
22
20
18
16
14
a
G ¼ nAð1 ꢀ h1 ꢀ h2ÞlꢄA þ nAh1lA 1 þ ðna1 ꢀ nAh1Þlꢄa
a
1
þ nAh2lA 2 þ ðna2 ꢀ nAh2Þlꢄa þ nSlS
ðA:2Þ
a
2
lꢄi being the chemical potential of free i (i = A, a1 and
chemical potential of the pair A
A with j. Decomposing the chemical potential of the pair as:
a
2), lA the
a
a
and hj the association degree of
a
lA
¼
lA= j þ la
ð
aj
¼
a1 or a2
Þ
ðA:3Þ
aj
a
j=A
where lA/ is the chemical potential of A bound to aj and la the
a
j
j/A
chemical potential of aj bound to A, Eq. (A.2) becomes:
G ¼ nAð1 ꢀ h1 ꢀ h2ÞlꢄA þ nAh1lA= 1 þ nAh1la1=A þ ðna1 ꢀ nAh1Þlaꢄ
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
a
1
[CD]Total/mol·dm-3
þ nAh2lA= 2 þ nAh2la2=A þ ðna2 ꢀ nAh2Þlaꢄ þ nSlS
ðA:4Þ
a
2
32
30
28
26
24
22
20
18
16
14
The chemical potential of the solution must be the same
whether we consider, or not, existing associations. Thus, from
Eqs. (A.1) and (A.4) it follows:
b
nAlA þ na1la þ na2la ¼ nAð1 ꢀ h1 ꢀ h2ÞlꢄA þ nAh1lA=
1
2
a1
þ nAh1la1=A þ na1laꢄ ꢀ nAh1laꢄ
1
1
þ nAh2lA= 2 þ nAh2la2=A þ na2laꢄ ꢀ nAh2lꢄa
a
2
2
ðA:5Þ
But lꢄA
¼
lA= (and lꢄa
¼
laj=A), because free and bound A (and aj
)
aj
are in equilibrium, so that:
nAlA þ na1la þ na2la ¼ nAlAꢄ þ na1lꢄa þ na2laꢄ
ðA:6Þ
1
2
1
2
But la
ꢅ
lꢄaj; that is, the chemical potential of aj in solution is
j
approximately the same as the chemical potential of free
is true because aj is present in great excess over A. Consequently,
lꢄA, that is:
aj. That
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
[CD]Total/mol·dm-3
lA
¼
loA þ RT lnfcA½Aꢁg ¼ lAoꢄ þ RT lnfcꢄAð1 ꢀ h1 ꢀ h2Þ½Aꢁg
ðA:7Þ
40
35
30
25
20
15
c
It is clear that loA
cA½Aꢁ ¼ cꢄAð1 ꢀ h1 ꢀ h2Þ½Aꢁ
¼
loAꢄ and thus:
ðA:8Þ
According to this equation, the experimentally measured activ-
ity coefficient of A, cA would be the product of the activity coeffi-
cient of free A, cꢄA; and the overall dissociation degree. If the
reference state of A is considered to be the free A (and thus
cꢄA ¼ 1), Eq. (A.8) results in:
cA ¼ 1 ꢀ h
ðA:9Þ
h being the overall association degree, defined as:
h ¼ h1 þ h2
ðA:10Þ
And the individual association degrees are given by:
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
[CD]Total/mol·dm-3
½A=a1
½AꢁT
ꢁ
ꢁ
K1½a1ꢁ½Aꢁw
½AꢁT
h1 ¼
h2 ¼
¼
¼
Fig. 3. Plot of the rate constant k/sꢀ1 for the reaction [Fe(CN)5(4-phe-
py)]3ꢀ + [Co(NH3)4(H2O)2]3+. The data before arrows were obtained for varying
MbCD concentration. The arrow point corresponds to: (a) [MbCD] = 2 ꢂ
ðA:11Þ
½A=a2
½AꢁT
K2½a2ꢁ½Aꢁw
½AꢁT
10ꢀ3 mol dmꢀ3, (b) [MbCD] = 3.82 ꢂ 10ꢀ3 mol dmꢀ3, (c) [MbCD] = 0.025 mol dmꢀ3
.
Points followings the arrows were obtained by increasing HbCD concentration. Dots
Introducing Eq. (A.11) into Eq. (A.10) leads to Eq. (A.12):
are experimental data and continuous lines (—) are the best fits to Eq. (6).
K1½a1ꢁ½Aꢁw K2½a2ꢁ½Aꢁw ½Aꢁw
h ¼
þ
¼
ðK1½a1ꢁ þ K2½a2ꢁÞ
ðA:12Þ
ðA:13Þ
½AꢁT
½AꢁT
½AꢁT
So, the overall dissociation degree in given by
great excess with respect to A. The solution free energy, without
considering association, is given by:
½Aꢁw
1 ꢀ h ¼ 1 ꢀ
ðK1½a1ꢁ þ K2½a2ꢁÞ
½AꢁT
G ¼ nAlA þ na1la þ na2la þ nSlS
ðA:1Þ
1
2

238
M. Marchena, F. Sanchez / Chemical Physics Letters 471 (2009) 234–238
0
½Aꢁ
and taking into account that
¼ 1 ꢀ h, Eq. (A.14) is obtained
ðA:14Þ
lo ðiÞ is the standard (formal) chemical potential of species i in
½AꢁwT
water, and lo0 (i/CD) is the standard (formal) chemical potential of
the inclusion complex formed by species i,
1
1 ꢀ h ¼
1 þ K1½a1ꢁ þ K2½a2
ꢁ
ꢀ
ꢁ
0
0
0
0
kBT
lo ðz=CDÞ ꢀ lo ðFeÞw
ꢀ
lo ðCoÞw
ꢀ
ꢀ
lo ðCDÞw
kwKz ¼
exp ꢀ
Now, introducing Eq. (A.14) into Eq. (A.9), the desired expression of
the activity coefficients is obtained:
2
hðCoÞ
RT
ðA:20Þ
1
ꢀ
ꢁ
cA
¼
ðA:15Þ
o0
o0
o0
o0
kBT
hðCoÞ
l
ðz=CDÞ ꢀ
l
ðFeÞw
ꢀ
l
ðCoÞw
l
ðCDÞw
1 þ K1½a1ꢁ þ K2½a2
ꢁ
kCDK ¼
exp ꢀ
2
RT
This equation can easily be generalized for n receptors:
ðA:21Þ
1
cA
¼
ðA:16Þ
n
Consequently,
P
1 þ Ka½
aꢁ
kwKz ¼ kCD
K
ðA:22Þ
a
Appendix B
From Brönsted equation to Eq. (6)
References
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It is important to remember that when the activity coefficients
are formulated as in Eq. (A.15) (or Eq. (A.16)), the reference state
for i is the free i. This means that if for a given species there is
no interaction with the receptors, this species is always in its free
state, ci = 1. This is precisely the case of the cobalt complex for the
case studied here. Thus, Brönsted equation reduces in the present
case to:
cFe
cz
k ¼ kw
ðA:17Þ
where cFe and cà represent, respectively, the activity coefficients of
the iron complex and transition state. Notice that, in Eq. (A.17) the
rate constant corresponding to the reference state (ko in Eq. (2)) has
been written as kw, in agreement with the fact that the reference
state of a given species is its free state.
From the previous arguments, for the case in which the two CDs
are present in the reaction media, one can write:
[18] T.W. Newton, J.D. White, J. Phys. Chem. 75 (1971) 2117.
[19] R. Juretic, D. Paulovic, S. Asperger, J. Chem. Soc., Dalton Trans. (1979) 2029.
[20] S.F. Nelsen, R.F. Ismagilov, J. Phys. Chem. A 103 (1999) 5373.
[21] Commercial modified CDs are (or can be) mixtures. However, this does not
have any influence on our conclusions. Simply, the binding constants would be
average binding constants, K, defined as follows:
1 þ Kz ½CD1ꢁ þ Kz ½CD2ꢁ
1
2
k ¼ k
ðA:18Þ
w 1 þ K1½CD1ꢁ þ K2½CD2ꢁ
Obviously, this equation becomes identical to Eq. (6) if
X
kwKz ¼ kCD1K1 and kwKz ¼ kCD2K2. These identities can easily be
K ¼
a
iKi
1
2
i
proved. Thus,
where ai is the fraction of the i th component of the mixture and Ki its corre-
sponding binding constant. Notice that K is only a constant provided that ai
are constant, as happened in our case. In fact, we have employed different sam-
ples of CDs and we do not observed changes in K.
ꢀ
ꢁ
0
0
0
kBT
lo ðzÞw
ꢀ
lo ðFeÞw
ꢀ
lo ðCoÞw
kw
¼
exp ꢀ
hCo
kBT
RT
ꢀ
exp ꢀ
ꢀ
ꢁ
0
0
0
lo ðz=CDÞ ꢀ lo ðFe=CDÞ ꢀ lo ðCoÞw
kCD
¼
ðA:19Þ
hCo
RT
ꢁ
0
0
0
1
lo ðz=CDÞ ꢀ lo ðzÞw
ꢀ
lo ðCDÞw
Kz ¼
K ¼
exp ꢀ
Co
1
RT
ꢀ
ꢁ
0
0
0
lo ðFe=CDÞ ꢀ lo ðFeÞw
ꢀ
lo ðCDÞw
exp ꢀ
Co
RT