Silica-Promoted Diels-Alder Reactions in Carbon Dioxide from Gaseous to Supercritical Conditions

Article abstract of DOI:10.1021/jp984227g

Amorphous fumed silica (SiO₂) was shown to increase yields and selectivities of several Diels-Alder reactions in gaseous and supercritical CO₂. Pressure effects on the Diels-Alder reaction were explored using methyl vinyl ketone and penta-1,3-diene at 80°C. The selectivity of the reaction was not affected by pressure/ density. As pressure was increased, the yield decreased. At the reaction temperature, adsorption isotherms at various pressures were obtained for the reactants and the Diels-Alder adduct. As expected when pressure is increased, the ratio of the amount of reactants adsorbed to the amount of reactants in the fluid phase decreases, thus causing the yield to decrease. The Langmuir adsorption model fit the adsorption data. The Langmuir equilibrium partitioning constants all decreased with increasing pressure. The effect of temperature on adsorption was experimentally determined and traditional heats of adsorption were calculated. However, since supercritical CO₂ is a highly compressible fluid, it is logical to examine the effect of temperature at constant density. In this case, entropies of adsorption were obtained. The thermodynamic properties that influence the real enthalpy and entropy of adsorption were derived. Methods of doping the silica and improving yields and selectivities were also explored.

Full text of DOI:10.1021/jp984227g

2878
J. Phys. Chem. B 1999, 103, 2878-2887
Silica-Promoted Diels-Alder Reactions in Carbon Dioxide from Gaseous to Supercritical
Conditions
Randy D. Weinstein,†,| Adam R. Renslo, Rick L. Danheiser, and Jefferson W. Tester*
‡
‡
,†,§
Departments of Chemical Engineering and Chemistry and Energy Laboratory, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139
ReceiVed: October 27, 1998
Amorphous fumed silica (SiO
in gaseous and supercritical CO
2
) was shown to increase yields and selectivities of several Diels-Alder reactions
. Pressure effects on the Diels-Alder reaction were explored using methyl
2
vinyl ketone and penta-1,3-diene at 80 °C. The selectivity of the reaction was not affected by pressure/
density. As pressure was increased, the yield decreased. At the reaction temperature, adsorption isotherms at
various pressures were obtained for the reactants and the Diels-Alder adduct. As expected when pressure is
increased, the ratio of the amount of reactants adsorbed to the amount of reactants in the fluid phase decreases,
thus causing the yield to decrease. The Langmuir adsorption model fit the adsorption data. The Langmuir
equilibrium partitioning constants all decreased with increasing pressure. The effect of temperature on adsorption
was experimentally determined and traditional heats of adsorption were calculated. However, since supercritical
2
CO is a highly compressible fluid, it is logical to examine the effect of temperature at constant density. In
this case, entropies of adsorption were obtained. The thermodynamic properties that influence the real enthalpy
and entropy of adsorption were derived. Methods of doping the silica and improving yields and selectivities
were also explored.
Introduction
and reactants is impractically low in carbon dioxide. The
solvating ability of high-density carbon dioxide is often
The development of “environmentally friendly” chemical
processes has been a subject of increasing interest in recent
compared to that of nonpolar organic solvents such as hexane,
2
c
1
carbon tetrachloride, and fluorocarbon. Recent studies aimed
at improving the solubility of materials in CO2 include
Johnston’s investigation of water in carbon dioxide microemul-
years. One major goal of studies in this area has been the
identification of environmentally benign reaction solvents, with
considerable attention focusing on the potential utility of
supercritical carbon dioxide (scCO2) as a medium for performing
6
sions and DeSimone et al.’s work on self-assembled micelles
7
2
of block copolymer amphiphiles. The use of phase transfer
important synthetic reactions. Although a number of standard
catalysis has also been explored to improve the solubility of
synthetic transformations have been demonstrated to proceed
uneventfully in scCO2, to date there have been relatively few
reports of reactions that exhibit dramatic rate enhancement or
selectivity changes when carried out in supercritical carbon
dioxide. Hrnjez et al. observed a modest pressure effect on the
stereochemical course of the photochemical dimerization of
isophorone in scCO2,3 and Weedon and co-workers have
observed an interesting increase in the ratio of rearrangement
to cage-escape products in the photo-Fries rearrangement of
8
9
some materials in CO2. Curran and co-workers have recently
used fluorinated alkylstannanes to mediate radical cyclization,
coupling, and reduction reactions in scCO2, and Burk et al.10
have reported the rhodium-catalyzed asymmetric hydrogenation
of enamides in supercritical carbon dioxide.
In addition to the solubility issues discussed above, a second
limitation in performing synthetic reactions in scCO2 is that
many standard organic transformations suffer from impractically
slow reaction rates in carbon dioxide as compared to conven-
4
naphthyl acetate in near critical CO2. Noyori and co-workers
11
have shown that the homogeneous catalytic hydrogenation of
CO2 to formic acid and its derivatives proceeds significantly
faster in scCO2 (in the presence of a cosolvent) than in traditional
tional organic solvents. While resorting to various Lewis acid
promoters might provide a remedy to this problem, the use of
such additives would greatly diminish the environmental benefits
5
liquid solvents. Although carbon dioxide has not yet been
associated with the use of scCO . For this reason, we have
2
shown to provide dramatic advantages over conventional
solvents with respect to improved reaction rates and selectivites,
the potential environmental and economic advantages of scCO2
continue to fuel interest in this area.
investigated the use of environmentally benign, reusable solids
such as silica and alumina as promoters for organic reactions
in scCO . Among the many attractive features of these materials
2
are their low cost, minimal environmental impact, easy separa-
A significant obstacle to the application of scCO2 as a reaction
solvent is the fact that the solubility of many common reagents
tion from CO2, and potential for reuse. In this paper we report
on the use of amorphous silica (SiO2) as an environmentally
benign promoter for the Diels-Alder reaction in scCO2.
*
Corresponding author. Phone (617) 253-3401. Fax: (617) 253-8013.
The use of silica to promote reactions including ozonolyses,
alkylations, acylations, cycloadditions, halogenations, oxidations,
E-mail: testerel@mit.edu.
†
Department of Chemical Engineering.
Current address: Department of Chemical Engineering, Villanova
|
12
and reductions in traditional organic solvents is well-known.
University, Villanova, PA 19085.
‡
Although these reactions can be carried out on solid silica
surfaces without the need for a solvent, extraction with an
Department of Chemistry.
Energy Laboratory.
§
1
0.1021/jp984227g CCC: $18.00 © 1999 American Chemical Society
Published on Web 03/26/1999

Silica-Promoted Diels-Alder Reactions in CO2
J. Phys. Chem. B, Vol. 103, No. 15, 1999 2879
organic solvent is still required in order to recover the products
from the solid silica phase at the conclusion of the reaction.
The use of “doped” silica and alumina to promote reactions in
organic solvents has also been reported. For example, Kropp
dioxide was passed through the system while the temperature
was raised to the desired reaction temperature. The reactor was
then pressurized, causing the ampules to burst. At the conclusion
of the experiment, the contents of the reaction vessel were
released slowly through a long narrow tube whose end was
immersed in ca. 70 mL of ethyl acetate. Additional ethyl acetate
(30 mL) was used to rinse the inside of the reactor and the
sampling tubing, and the silica remaining in the reactor was
transferred to the ethyl acetate solution. Approximately 100 mL
of ethyl acetate was used for the collection process; dichlo-
romethane could also be used as a collection solvent with no
effect on the results. Control experiments verified that the
quantity of solvent used was sufficient to desorb all materials
from the solid silica surface. The combined ethyl acetate solution
was passed through a 0.2 µm filter, dried over MgSO4, filtered,
and concentrated by rotary evaporation (at ca. 20 mmHg).
1
3
and co-workers have used silica in combination with protic
acids to promote a variety of reactions in solvents such as
dichloromethane. A solid-phase promoter composed of silica-
supported ZnCl2 and alumina-supported K2CO3 has recently
been employed for Friedel-Crafts allylation of aromatic rings
1
4
in dichloroethane.
Despite this considerable body of literature, the use of silica
or alumina to promote reactions in scCO2 has not been
previously reported. Very recently, Poliakoff and co-workers
reported Friedel-Crafts15 and hydrogenation reactions in
scCO2 using sulfonylated polysiloxane and polysiloxane-sup-
ported palladium promoters, respectively. These reactions
involve the use of gaseous reagents (propene and hydrogen),
thereby taking advantage of the high solubility of these gases
in scCO2. Compared to the polysiloxane polymers used in this
work, silica is significantly less expensive and more readily
available.
16
1
p-Dimethoxybenzene was used as an internal standard for H
NMR analysis of the products in CDCl3. In the case of
experiments carried out at pressures of 0-33 bar, reactants were
placed directly into the reactor since at these pressures the glass
ampules would not burst.
The aim of the investigation reported herein was to explore
temperature and pressure/density effects on the silica-promoted
Diels-Alder reaction in CO2. To characterize the reaction
system, the partitioning of species between solid and fluid phases
was first determined as a function of operating conditions and
then models for the adsorption process were developed. The
application of protic and Lewis acid dopants to further enhance
reaction rates was also explored.
Representative Phase Distribution Experiments. Various
ratios of solid silica and a reactant or cycloadduct were prepared
as described above. The reactor vessel was sealed, heated, and
pressurized. After 6-8 h, a six-way sample valve was used to
remove a 0.5 mL of sample from the fluid phase of the reaction
medium. For each experiment, two samples were taken, and
the second was deemed to be more representative of the fluid
environment. The sample was depressurized by releasing the
contents of the 0.5 mL sample loop through a long narrow tube
whose end was immersed in diethyl ether. For volatile compo-
nents (methyl vinyl ketone and penta-1,3-diene), the ether used
for collection was cooled by an acetone-dry ice bath to aid in
recovery. Tests without silica present were performed to ensure
that greater than 95% recovery of all chemicals occurred. The
sample loop was washed with ether and low-pressure CO2. To
prevent silica particles from clogging the inlet to the sample
valve, stirring was terminated just prior to taking a sample, thus
allowing the particles to settle away from the inlet to the sample
valve. Analysis was performed by GC FID. Equilibrium was
verified by taking another sample 24 h later. With the known
amount of solvent used for collection, the known volumes of
the sample loop and the reactor, and the GC results, the amount
of the material in the CO2 phase could be obtained. The amount
of species on the silica was calculated by a material balance
because this phase could not be sampled.
Experimental Section
Materials. Isoprene, methyl vinyl ketone, methyl acrylate,
penta-1,3-diene, and acrylonitrile were purified by distillation
and stored at -10 °C until use. Cyclopentadiene was prepared
by thermal cracking of the dimer and was stored at -60 °C
until use. Amporphous fumed silica (Alfa Aesar) with a surface
2
area of 400 m /g was dried at 200 °C to constant weight and
sealed in a vial under a CO2 atmosphere. Grade 5 CO2 was
purchased from BOC Gases. Doped silica was prepared by
charging a round-bottomed flask with 25 g of dried silica and
7
0 mL of dichloromethane and stirring at room temperature
while 40 mmol of the dopant (either phosphoric acid, sulfuric
acid, AlCl3, or TiCl4) was added dropwise over 2 min. The
resulting suspension was stirred for 30 min and then concen-
trated by rotary evaporation and then at 0.1 mmHg for 2 h to
provide material with ca. 1.35 mmol dopant per gram of silica.
1
Instrumentation. H NMR spectra were recorded on Varian
Results and Discussion
XL-300 and Unity 300 spectrometers using a delay time of 10
s. Gas chromatographic analyses were performed on a Hewlett-
Packard 6890 gas chromatograph with an auto injector, FID
detector, and BD WAX (J&W Scientific) column. All GC data
reported are the result of averaging at least three measurements.
Reactor. All CO2 experiments were performed in 25-mL
Silica-Promoted Diels-Alder Reactions. Table 1 sum-
marizes the results of our initial investigation of silica-promoted
Diels-Alder reactions in carbon dioxide. All of these reactions
were performed at 1500 psi (103 bar) in scCO . In the case of
2
reactions employing cyclopentadiene as diene, cycloadditions
11b
stainless steel view cell reactors described previously. Agita-
tion was provided by a magnetic stirbar. A six-way sample valve
allowed for sampling of the fluid phase of the vessel, and control
experiments were carried out to verify the validity of the
sampling procedure.
were carried out for 4 h at 50 °C. Reactions using penta-1,3-
diene and isoprene were performed at 80 °C for 24 h. In all
cases, 3.75 mmol of diene and 2.5 mmol of dienophile were
used as feed, and 0.5 g of silica was used in experiments
involving this material as a reaction promoter.
Representative Procedure for Diels-Alder Reactions in
Carbon Dioxide. Samples of 3.75 mmol of the diene and 2.5
mmol of the dienophile were sealed in separate oven-dried glass
ampules. The ampules, a magnetic stirbar, and 0.50 g of silica
were placed in the reactor, which was sealed, placed on a
stirplate, and wrapped with heating tape. Low-pressure carbon
As shown in Table 1, the use of silica in scCO2 led to
increased yields in a number of Diels-Alder reactions. In all
cases, an increase in selectivity (favoring the ortho or para
regioisomer over the meta isomer and/or the endo stereoisomer
over the exo isomer) was observed. The exact mechanism of
these silica promotion effects is not well understood.17 Silica

2880 J. Phys. Chem. B, Vol. 103, No. 15, 1999
Weinstein et al.
TABLE 1: Yields and Selectivities of Diels-Alder Reactions
in Supercritical Carbon Dioxide with and without Silica
Present
yield
entry
conditionsa
promoterb (%)c
selectivityd
W ) COMe
e
e
1
CO2, 50 °C, 4 h no
CO2, 50 °C, 4 h yes
29 82:18
82 92:8
e
2
W ) CN
3
4
CO2, 50 °C, 4 h no
CO2, 50 °C, 4 h yes
5
57:43
e
14 59:41
W ) CO2Me
5
6
e
e
CO2, 50 °C, 4 h no
CO2, 50 °C, 4 h yes
5
72:28
21 85:15
Figure 1. Total yield of the Diels-Alder reaction of methyl vinyl
ketone and penta-1,3-diene after 24 h at 80 °C in sub- and supercritical
carbon dioxide as a function of pressure with 0.5 g of silica present.
The point at 0 bar pressure corresponds to reaction conditions with no
f
7
8
CO2, 80 °C, 24 h no
CO2, 80 °C, 24 h yes
5
85(72:28):15(66:33)
f
35 92(87:13):8(76:24)
2
CO present.
own affinity for the reactants, and consequently, the phase
distribution of the reactants and products is expected to be a
major factor influencing these reactions. In this connection it
should be noted that scCO2 is often used in chromatographic
separations because of its ability to alter the partitioning of
species from a solid to the supercritical phase. It is also important
to note that silica can adsorb CO2 molecules onto its surface,
further suggesting that the solvent likely plays a critical role in
the adsorption process.
W ) COMe
g
9
CO2, 80 °C, 24 h no
CO2, 80 °C, 24 h yes
5
67:33
g
10
35 73:27
W ) CO2Me
g
g
1
1
1
2
CO2, 80 °C, 24 h no
CO2, 80 °C, 24 h yes
2
3
68:32
73:27
23
a
. b 0.5 g of SiO
All reactions were conducted at 103 bar in CO
2
2
2
c
1
d
(
400 m /g). Yield estimated by H NMR. Ratios of isomers deter-
mined by H NMR analysis except for entries 3 and 4 (GC analysis).
Endo:exo. Ortho(endo:exo):meta(endo:exo). Para:meta.
1
Several studies concerning adsorption from a scCO2 phase
onto a solid support have appeared in the literature. Ethyl acetate
adsorption equilibria on silica in the presence of scCO2 have
e
f
g
promotion of the Diels-Alder reaction probably involves
hydrogen bonding to (or protonation of) the dienophile by acidic
SiOH groups on the silica surface, thereby lowering the energy
of the LUMOdienophile and facilitating the cycloaddition. How-
ever, it is unlikely that the observed rate and selectivity
enhancements are due solely to the presence of Lewis acid sites
on silica surfaces, since these sites are rarely found on the
surface of untreated silica.18 Vesselovskii et al. have proposed
that the adsorption of reactants on the surface of silica particles
may facilitate the formation and stabilization of prereaction
complexes by bringing the reacting moieties within each reactant
into closer proximity. A related theory has been advanced by
Menger,20 suggesting that the rigid anchoring of functional
groups can fix reactants in proximity to each other less than
the critical bonding distance, thus leading to significant rate
enhancements. An alternative theory introduced by Parlar and
2
4
been explored, and studies have also been reported on the
phase partitioning of sweet orange and lemon essential oils
2
5
between silica and scCO2. Acetone and benzene adsorption
2
6
on silica in scCO2 has been examined recently, and activated
2
7
28
carbon has been studied as an adsorbent for phenol, toluene,
2
9
and benzene in the presence of scCO2. In all of these
experiments, temperature and pressure were found to have a
significant influence on the phase partitioning, consistent with
our suggestion that the scCO2 phase plays a significant role in
the reactions presented in Table 1. In addition, several infrared
spectroscopic studies have revealed how some typical solvents
and carbon dioxide are affected when they are adsorbed on the
19
2
6,30
surface of silica in the presence of scCO2.
In summary, the rates of our Diels-Alder reactions in the
scCO2/SiO2 system are expected to be lower in scCO2 than on
the surface of the SiO2 promoter. Any factor that influences
the partitioning of reactants between the solid and solution phase
could therefore influence the rate of the reaction. The solubility
of organic compounds in scCO2 is a strong function of pressure/
density, with the solvating power of scCO2 being greater at
higher density. Reactions in scCO2 under heterogeneous condi-
tions (i.e., with silica promotion) might therefore be expected
to exhibit a pressure effect on reaction rates and yields.
Pressure Effects. The effects of CO2 pressure on the silica-
promoted Diels-Alder reaction was examined using the reaction
of methyl vinyl ketone and penta-1,3-diene at 80 °C for 24 h.
In each experiment, 0.5 g of silica was employed with 3.75
mmol of pentadiene and 2.5 mmol of the ketone. Figure 1 plots
the yield of cycloadduct as a function of pressure. As a reference
point, note that the critical pressure of CO2 is 74 bar. This
2
1
Baumann suggests that only a small fraction of reactants are
actually strongly adsorbed on the silica surface and that the
enhanced rates and selectivities are caused by symmetry-
controlled secondary orbital interactions between strongly
adsorbed species and species that are free to move about on
the surface. Experimental evidence has been reported for lateral
interactions between adsorbed alkanes and alkenes on silica,
which greatly influence the absorption process.22
In addition to the adsorbate-adsorbent interactions outlined
above, there are additional factors that must be considered when
evaluating the results of the silica-promoted Diels-Alder
reactions reported in Table 1. Previous theories to account for
the promotion of reactions on the surface of silica were
developed for processes taking place in the absence of solVent.
In our system, there is a fluid phase (CO2) present that has its

Silica-Promoted Diels-Alder Reactions in CO2
J. Phys. Chem. B, Vol. 103, No. 15, 1999 2881
Figure 2. Percent of initial feed amount of methyl vinyl ketone (mvk),
penta-1,3-diene, and their Diels-Alder adduct present in the carbon
dioxide phase at 80 °C when 2.5 mmol of one of the species is added
to 0.5 g of silica in the 25 mL reactor at various carbon dioxide loadings.
Figure 3. Concentration on the solid surface of methyl vinyl ketone
(mvk), penta-1,3-diene, and their Diels-Alder adduct at 80 °C when
2.5 mmol of one of the species is added to 0.5 g of silica in the 25 mL
reactor at various carbon dioxide loadings.
Diels-Alder reaction was also performed without silica at 1500
psi (103 bar), in which case the isolated yield was only 5%
with an ortho(endo:exo):meta(endo:exo) ratio of 85(72:28):15-
Above 100 bar, the amount of each reactant transferred to the
fluid phase appears to level off, and the affinity of these species
for the silica surface thus remains fairly constant at pressures
above 100 bar.
(66:33). In the case of the silica-promoted reaction, the yield
of cycloadduct increases markedly as the pressure is decreased
below 100 bar but appears to remain constant at pressures above
The observed phase behavior helps to explain the yield data
shown in Figure 1. As soon as CO2 is added to the system (at
33 bar), the yield begins to drop off. We attribute this to the
fact that penta-1,3-diene is beginning to be removed from the
solid surface although the methyl vinyl ketone remains adsorbed.
When the reaction is conducted at higher pressures, the yield
continues to drop due to the increased desorption of the
pentadiene. At pressures above 74 bar, the yield falls even more
because the dienophile (MVK) also begins to be desorbed at
that point. Finally, when the reaction is carried out at pressures
above 100 bar, the amount of the reactants on the surface
remains fairly constant with the result that the yield remains
constant as pressure is increased.
1
00 bar. The stereo- and regioselectivity of the reaction was
found to be independent of the system pressure within experi-
mental error and was determined to be 92(87:13):8(76:24),
ortho(endo:exo):meta(endo:exo).
As expected, pressure/density has a dramatic effect on the
yield of the reaction. This pressure effect is consistent with an
increased concentration of reactants on the silica surface at low
CO2 pressures (<100 bar) where the solubility of organic
compounds in CO2 is low. As the pressure/density of the system
is increased, the solubility of organic molecules in the CO2 phase
increases3 and more of the reactants are desorbed from the
surface of the silica particles, thus slowing the reaction.
Phase Partitioning Studies. To gain a better understanding
of the solvating power of carbon dioxide for each of the reactants
and products, we next investigated the phase partitioning of these
species under the reaction conditions. Initially, phase distribution
experiments were performed using 0.50 g of silica together with
either one of the reactants (MVK or pentadiene) or with the
product (2.5 mmol; as a mixture of isomers). Procedures for
these experiments are described in the Representative Phase
Distribution section of the Experimental Section. These reactions
were performed at 80 °C at various pressures in order to mimic
the reaction conditions employed for the experiments shown in
Figure 1. Phase partititoning results for methyl vinyl ketone,
penta-1,3-diene, and their Diels-Alder adduct are shown in
Figures 2 and 3.
1
These phase partitioning experiments reveal that at higher
pressures there is a significant portion of the reactants in the
fluid phase and not on the surface of the silica particles. The
effect of pressure on the rate and selectivity of uncatalyzed,
homogeneous Diels-Alder reactions in scCO2 has been previ-
1
1,32
ously examined.
An increase in pressure or fluid density
tends to increase the reaction rate and selectivity, but to a lesser
degree than that of silica. The fact that the selectivities do not
change as a function of pressure suggests that the majority of
the reaction is most likely still taking place on the surface of
the particles, since if reaction was occurring to a significant
extent in the bulk fluid phase, the selectivies would be expected
to be lower than we observe.
Adsorption isotherms were next obtained for the reactants
and the Diels-Alder adduct in order to better understand the
phase behavior of these compounds. Adsorption isotherms were
obtained at constant temperature (the reaction temperature of
80 °C) and at constant pressure (several different pressures for
each system) by varying the amount of silica and adsorbate fed
into the reactor. Procedures for these experiments are described
in the Representative Phase Distribution section of the Experi-
mental Section. Constant pressure adsorption isotherms for
penta-1,3-diene, methyl vinyl ketone, and their Diels-Alder
adduct are shown in Figures 4-6, respectively.
As revealed in these experiments, all of the Diels-Alder
adduct remains adsorbed on the surface of the silica until the
pressure of the system reaches 100 bar. As the pressure is
increased, more adduct is transferred from the solid surface into
the scCO2 phase. No maximum in the solubility of this
compound in the CO2 phase appears to be reached. Penta-1,3-
diene begins to desorb from the surface as soon as some CO2
is added (at the first point of 33 bar); lower pressures could not
be examined because reproducible sampling of the fluid phase
was not possible. Methyl vinyl ketone begins to desorb at a
slightly higher pressure (around the critical pressure of CO2).
The equilibrium adsorption isotherms clearly indicate that
adsorption loading decreases with increases in pressure. This

2882 J. Phys. Chem. B, Vol. 103, No. 15, 1999
Weinstein et al.
TABLE 2: Calculated Langmuir K
L
and q
∞
Values for
Methyl Vinyl Ketone (mvk), Penta-1,3-diene, and Their
Diels-Alder Adduct in Carbon Dioxide at 80 °C
mvk
pentadiene
adduct
pressure
K
L
q
∞
K
L
q
∞
K
L
q
∞
(bar) (L/mol) (mmol/g) (L/mol) (mmol/g) (L/mol) (mmol/g)
3
7
03
10
3
6
813
4.9
782
754
248
4.7
2.7
0.7
1
3
404
179
1.5
0.5
1892
1432
4.5
2.1
changes. Other investigators have also observed changes in the
2
4,25a,26,28,29,33
solvent power of CO2 in adsorption processes.
There are two interactions that dominate the amount of adsorbed
reactant or adduct: (a) the interactions between the adsorbed
species, adsorbed CO2, and silica and (b) the interactions
between a dissolved species and CO2 in the fluid phase. As
pressure is increased, the solubility of the species in the fluid
phase is increased and more CO2 is adsorbed on silica. Both of
these factors in combination cause the reactant and adduct
loadings to decrease as pressure increases. From the adsorption
isotherms in scCO2, it appears that silica has a higher affinity
for the adduct over methyl vinyl ketone and penta-1,3-diene.
These data may also be interpreted as indicating that CO2 has
a lower affinity for the adduct than the reactants in the presence
of silica. Either interpretation of the adsorption processes is
reasonable, and the actual process is most likely a combination
of the effects.
Figure 4. Constant pressure adsorption isotherms for penta-1,3-diene
on silica at 80 °C in carbon dioxide.
It was desirable to describe the adsorption isotherms math-
3
4
ematically and there are many models to do this. However,
2
4,28,33b
previous investigators
have found that the Langmuir
model fits adsorption data from solutions and CO2 fairly well
and this model was selected. The Langmuir expression is shown
in eq 1.
K C
q
L
)
(1)
q∞ 1 + K C
L
Figure 5. Constant pressure adsorption isotherms for methyl vinyl
ketone on silica at 80 °C in carbon dioxide.
where q is the amount of adsorbed material, q∞ is the maximum
amount the surface can hold, KL is the Langmuir equilibrium
partitioning constant, and C is the concentration in the bulk CO2
phase. KL is equivalent to the initial slope of the adsorption
isotherm in terms of mass of solute adsorbed per unit of surface
sites available as a function of concentration in the bulk fluid
phase. At constant system pressure and temperature, a plot of
C/q versus C should give a straight line with slope m ) 1/q∞
and intercept b )1/(q∞KL). KL is equal to m/b. The calculated
values for KL and q∞ are given in Table 2.
In the Langmuir model, KL and q∞ can be used to calculate
σ , the effective area occupied by a molecule on the surface,
o
when the specific area of the solid is known. It is also possible
to calculate a heat of adsorption with all the species in their
+
L,ads
standard states, ∆ Hh , for a system that follows the Lang-
muir model by using eq 2, the van’t Hoff equation.
+
∂
ln KL
(1/T)
∆Hh
L,ads
)
-
(2)
(
)
∂
P
R
Figure 6. Constant pressure adsorption isotherms for the Diels-Alder
adduct of penta-1,3-diene and methyl vinyl ketone on silica at 80 °C
in carbon dioxide.
where R is the universal gas constant, T is temperature, and the
derivative is taken at constant pressure, P.
is contrary to the general behavior observed in adsorption from
gases near ambient pressure, where the carrier gas is believed
to be inert and unaffected by pressure and temperature changes.
However, in our systems there is no inert carrier, and the solvent
power of CO2 is greatly affected by pressure and temperature
We explored the effect of temperature on the adsorption
isotherms at 103 bar for penta-1,3-diene and methyl vinyl
ketone. The results are shown in Figures 7 and 8, respectively.
Figure 9 shows the effect of temperature on the adsorption

Silica-Promoted Diels-Alder Reactions in CO2
J. Phys. Chem. B, Vol. 103, No. 15, 1999 2883
L
TABLE 3: Calculated Langmuir K Values for Methyl
Vinyl Ketone (mvk), Penta-1,3-diene, and Their Diels-Alder
Adduct in Carbon Dioxide at Fixed Pressures
adduct
mvk
(L/mol)
pentadiene
(L/mol)
K
L
K
L
K
L
(L/mol)
L
K (L/mol)
temp (°C)
at 103 bar
at 103 bar
at 103 bar
at 310 bar
8
6
4
0
0
0
754
1249
3120
389
526
825
1892
2886
9999
1432
2366
5686
+
TABLE 4: Calculated Values of ∆Hh
(kJ/mol) for Methyl
L,ads
Vinyl Ketone (mvk), Penta-1,3-diene, and Their Diels-Alder
Adduct in Carbon Dioxide at Fixed Pressures
mvk
at 103 bar
pentadiene
at 103 bar
adduct
at 103 bar
adduct
at 310 bar
-33
-17
-39
-32
+
TABLE 5: Estimated Values of ∆Sh
(kJ/mol K) for
L,ads
Figure 7. Effect of temperature on the adsorption isotherms for penta-
Methyl Vinyl Ketone (mvk), Penta-1,3-diene, and Their
Diels-Alder Adduct in Carbon Dioxide at a Fixed Density of
1
,3-diene on silica at 103 bar in carbon dioxide.
0
.6 g/cm3
mvk at 0.6 g/cm3
pentadiene at 0.6 g/cm3
0.03
adduct at 0.6 g/cm3
0.05
0
.07
quantify the partitioning of components. These results are shown
+
L,ads
in Table 3. ∆Hh
at constant pressure is calculated by using
eq 2 and is tabulated in Table 4. Both the enthalpy of adsorption
and the equilibrium partitioning constants provide similar
information about the adsorption process.
However, in scCO2, density has actually been found to be a
more representative property than pressure for explaining
3
1
changes in solubility and phase partitioning. Therefore, it is
appropriate to examine the equilibrium constant at conditions
of constant density, F. Although it is tempting to equate (∂ ln
+
KL/∂T)F with ∆Hh
, it is strictly only true for ideal fluids
L,ads
where the enthalpy of mixing is zero and there are no Poynting
correction effects between the real state at T and P and the
+
standard states at T and P . Variations in the equilibrium
Figure 8. Effect of temperature on the adsorption isotherms for methyl
vinyl ketone on silica at 103 bar in carbon dioxide.
partitioning coefficient with temperature are more closely related
to the entropy of the process. If we still assume the Langmuir
model describes the adsorption process, then
∂
(T ln KL)
+
R
) ∆Sh
(3)
(
)
L,ads
∂(T)
F
Although adsorption data were not obtained at constant
+
densities, we can still estimate ∆ Sh . This was done by
interpolating the experimental KL’s at 103 bar and 40 °C (pure
L,ads
3
CO2 F ) 0.6 g/cm ) from Table 3 and estimating the KL’s at
3
2
01 bar and 80 °C (pure CO2 F ) 0.6 g/cm ) from the data in
Table 2. When the two KL’s at constant density and different
temperature are applied to eq 3, we get a rough estimate of the
+
value of ∆ Sh . These estimates are shown in Table 5.
L,ads
Although the adsorption data can be mathematically repro-
duced using the Langmuir model (eq 1), the exact physical
interpretation of the model parameters is probably not accurate
o
for our system. This can be quickly verified by calculating σ
from our adsorption isotherms. σ is on the order of 10 nm
Figure 9. Effect of temperature on the adsorption isotherms for the
Diels-Alder adduct of penta-1,3-diene and methyl vinyl ketone on silica
at 103 and 310 bar in carbon dioxide.
o
2
2
for our species, which is unrealistically large for a monolayer
of the reactants or adduct. Most likely, we do not have complete
monolayer coverage because the surface of the silica is not
homogeneous. The Langmuir model assumes a two-dimensional
surface adsorption phase in equilibrium with the bulk fluid
solution. The bulk solution may also be assumed to be dilute
isotherms for the adduct at 103 and 310 bar. In all cases the
equilibrium adsorptive capacity of silica decreases as temper-
ature is increased. Using the data presented in Figures 7-9,
the equilibrium adsorption isotherms can be estimated to

2884 J. Phys. Chem. B, Vol. 103, No. 15, 1999
Weinstein et al.
+
and therefore approximated by an ideal solution. Also, adsorbed
solvent (CO2) and solute molecules are assumed to occupy equal
areas on a homogeneous surface that is characterized by a single
parameter, the fractional coverage, Θ, which varies from 0 to
taken generally to represent the standard state condition of T ,
P+, and xi (mole fraction) in each phase π. If the standard
states are taken as pure components at T and P of the system,
then x ) 1 and the “overbars” are not needed. In addition, it
was assumed that â is a constant unaffected by temperature.
+
+
i
1
. The surface is also believed to be homogeneous. Many of
these assumptions are probably not valid in a system where a
highly compressible nonideal supercritical fluid is present. Using
eqs 2 and 3 for exploring temperature effects of adsorption
provides useful scaling information. However, these equations
are only approximate representations of the adsorption process,
which includes the effects of nonideal solution behavior. In the
limit of an ideal solution, eqs 2 and 3 would result, but in the
real system significant nonideal interactions need to be included
to place physical significance on the adsorption process. This
will be done by deriving a more rigorous form of the van’t Hoff
equation for our system.
By examining eq 10, we can see which terms will affect the
adsorption equilibrium as a function of temperature at constant
pressure. The traditional heat of adsorption is actually made up
of four terms, the partial molar enthalpies of the two species in
(s)
the two phases following the reaction in eq 4, ∆Hads ) Hh
-
1
(b)
(s)
2
(b)
2
Hh
+ â(Hh
- Hh
). If we assumed the behavior required for
1
the Langmuir model, â would be set to 1. Also, we would have
to assume ideal solution behavior so that the activities can be
replaced by the mole fractions. The “overbars” may then be
dropped because the partial molar enthalpy is just equal to the
pure component enthalpy in the ideal solution limit. These
assumptions are required to obtain a heat of adsorption from
the Langmuir model. Consequently, when supercritical fluids
are involved in the process, an unrealistic heat of adsorption
would be extracted from the data. This can be shown by relating
the concentration-based constant (or continued product of
concentrations) to the true activity-based thermodynamic equi-
librium constant. The Langmiur model uses a concentration-
based partitioning coefficient. The activity coefficient relates
activity to concentration directly from its definition:
At equilibrium, the rate of adsorption equals the rate of
desorption. We can consider the process of adsorption and
desorption as an equilibrium chemical reaction shown in eq 4
for each species separately.
â(adsorbed solvent) + solute in solution a
adsorbed solute + â(solvent in solution) (4)
where â equals the number of solvent molecules displaced from
the surface for each solute molecule that is adsorbed. The
equilibrium constant can be written in terms of the species
activities, ai, as
(
π) (π)
Ci γi
(π)
(π) (π)
(
1
s) (b) â
ai ≡ xi γi
)
(11)
a (a )
(π)
2
F
K )
(5)
a
(
s) â (b)
(
a ) a
2 1
(π)
where γi refers to component i in phase π relative to the
+(π)
(π)
where the subscripts denote the species (1 ) solute, 2 ) CO2)
and the superscripts denote the phase (s ) adsorbed solid phase,
b ) bulk fluid phase). The activity of a species is related to the
selected standard state Ci , Ci is the concentration of
species i in phase π (moles of i per liter of fluid for the bulk
fluid phase and moles of i per gram of solid for the adsorbed
chemical potential of the species, µi, and the chemical potential
(π)
phase), and F is the molar density of phase π. For the bulk
+
i
of the species in its standard state, µ , by the following:
fluid phase, the density is the true molar density of the phase
in moles per liter. For the adsorbed phase, it is more convenient
to use the maximum amount of i that the surface can hold,
+
RT ln a ) µ - µ
(6)
(7)
(8)
i
i
i
(
s)
F
, as the “density of the adsorbed phase” for use in an
max
At equilibrium at a uniform temperature and pressure,
equation of the form of eq 11 for defining the mole fraction in
(s)
(
b)
(s)
the solid phase, x . For the adsorbed phase, the mole fraction
i
µi ) µi
is more of a percentage of maximum coverage rather than a
true mole fraction interpretation. The concentration-based
constant is defined as
and
(
π)
νiµi ) 0
∑
(
1
s) (b) â
1
,2
C (C )
2
K ≡
(12)
C
(
s) â (b)
where νi is the stoichiometric coefficient for the reaction shown
in eq 4 (reactants are negative and products are positive) and π
represents the phase of the species. Thus, the equilibrium
constant can now be written as
(C
) C
2 1
and can be related to the activity-based equilibrium constant
given in eq 5 using eq 11.
(
1
s)+
(b)+
(s)+
(b)+
RT ln K ) µ + âµ2 - âµ2 - µ1
(9)
(s) â-1
a
F
max
-1
γ
K ) K K
(13)
C
a
(
)
(b)
Taking the derivative of both sides of eq 9 with respect to T at
constant P, we get
F
Here Kγ is defined as the continued product of activity
coefficients and is mathematically similar to eq 12 with species
concentrations replaced by species activity coefficients. If the
solution is ideal in both phases then KC ) Ka.
∂
ln Ka
(s)+
(b)+
(s)+
(b)+
+
ads
-R
) Hh 1 + â Hh 2 - â Hh 2 - Hh 1 ) ∆ Hh
(
(1/T))
∂
P
(
10)
Now we can take the natural log of both sides of eq 13 and
then take the partial derivative with respect to temperature at
where the “overbar” denotes a partial molar property that is

Silica-Promoted Diels-Alder Reactions in CO2
J. Phys. Chem. B, Vol. 103, No. 15, 1999 2885
constant pressure. Again, â was assumed to be a constant
independent of temperature.
TABLE 6: Isolated Yields and Selectivities of the
Diels-Alder Reaction of Methyl Vinyl Ketone and
Penta-1,3-diene with Several Doped and Untreated Silica
Promoters
∂
ln KC
∂ ln Ka
∂T
∂ ln Kγ
∂T
)
(
)
-
(
)
+
yield
(
)
entry
conditions
promoter (%)a
selectivityb
∂T
P
P
P
(
s)
(
b)
∂
ln F
max
∂ ln F
∂T
(
â - 1)
(
-
)
(14)
∂
T
P
Using eq 10 and noting that the maximum coverage on the solid
is constant,
1
2
3
4
5
6
7
8
toluene, 100 °C, 5 d
no solvent, 25 °C, 0.5 h SiO2-H3PO4 83 99(93:7):1(N.D.)
no solvent, 25 °C
no solvent, 25 °C
no solvent, 25 °C
CO2, 33 bar, 80 °C, 24 h SiO2
CO2, 103 bar, 80 °C, 24 h SiO2
none
77 82(69:31):18(60:40)
+
(
b)
SiO2-H2SO4
SiO2-AlCl3
SiO2-TiCl4
c
c
c
∂
ln KC
∆Hh
∂ ln Kγ
ads
2
∂ ln F
∂T
)
-
- (â - 1)
(15)
(
)
(
)
(
)
∂T
P
∂T
P
P
RT
78 92(87:13):8(76:24)
35 92(87:13):8(76:24)
In a nonideal system, like ones containing supercritical fluids,
CO2, 103 bar, 40 °C, 4 h SiO2-H3PO4 57 98(84:16):2(N.D.)
it is really the activity-based equilibrium constant that must be
used to obtain the true heat of adsorption. Using concentration-
based constants, like the ones used in most adsorption models
including the Langmuir model, does not produce the heat of
adsorption when the derivative of ln KC is taken with respect
to temperature. Instead, a term that is influenced by the heat of
adsorption, the activity coefficients, as well as the density of
the bulk phase is obtained. When the bulk phase consists of a
supercritical fluid, variations in both activity coefficients and
density with changes in temperature are often large, especially
near the critical point of the fluid, and consequently, neither
can be neglected. A similar approach may be used to show how
KC changes with temperature at constant density.
a
Isolated yields. b Selectivity, ortho(endo:exo):meta(endo:exo), was
1
c
determined by H NMR analysis. Diene polymerization predominated.
AlCl3, and TiCl4. The reaction of methyl vinyl ketone and penta-
,3-diene was selected as a test case to evaluate the reactivity
1
of these new acid-doped promoters. The reaction was first
investigated under solid-phase reaction conditions (no solvent)
and then, if successful, the reactions were attempted in scCO2.
The results of the initial studies are shown in Table 6.
Phosphoric acid doped silica was found to be an exceptionally
effective promoter, producing an 83% isolated yield of cycload-
duct after only 30 min at room temperature in the absence of
solvent. Furthermore, the regio- and stereoselectivity of the
reaction was excellent. When other doped promoters were used
in the reaction, violent exotherms developed, and the starting
materials decomposed almost immediately. Presumably, cationic
diene polymerization is a favorable process under these highly
acidic reaction conditions. Apparently, the phosphoric acid
promoted cycloaddition is sufficiently accelerated as to be
competitive with diene polymerization. In scCO2, the use of
SiO2-H3PO4 provides acceptable yields (57%) of Diels-Alder
cycloadduct after 4 h at 40 °C. Longer reaction times and higher
temperatures did not improve the yield of this reaction. By
comparison, the use of untreated silica requires longer reaction
times and higher temperatures.
+
∂
(T ln KC)
∆Sh
∂(T ln Kγ)
ads
(b)
)
-
- (â - 1) ln F
(
)
(
)
∂T
F
R
∂T
F
(16)
Equations 15 and 16 provide a physical basis for interpreting
adsorption behavior. However, since it is often difficult to
measure KC with high precision, an alternative approach for
examining phase partitioning in supercritical fluids should be
used. It is possible to express variations in the bulk concentration
of the solute with respect to temperature, without introducing
the concept of an equilibrium constant. This approach is
presented in detail in the Appendix, as it deviates from the
traditional method of examining adsorption behavior but seems
more appropriate when supercritical solvents are present. With
an accurate volumetric PVTN equation of state and the adsorp-
tion data obtained in Figures 7-9, it is possible to estimate true
heats of adsorption for the process without actually having to
calculate all the activity coefficients and without explicitly using
the â term. The final result expresses the derivative of solute
concentration as a function of temperature as
Concluding Remarks
In gaseous and supercritical CO2, the yield and selectivity of
several Diels-Alder reactions are enhanced by fumed silica with
a high specific area (≈400 m /g). Acid doping of silica
2
promoters has been shown to greatly enhance their activating
effect. The initial studies with these promoters lay the ground-
work for their application to a wide range of other acid and
Lewis acid promoted reactions in scCO2. Although the selectiv-
ity of these silica-enhanced reactions are not affected by changes
in pressure or fluid density, product yields are altered. In general,
as pressure is increased, the yield decreases. This was discovered
to be caused by the change in concentration of the reactants on
the surface of the silica particles. At the reaction temperature,
adsorption isotherms were obtained at selected pressures. As
expected, when pressure is increased, the ratio of the amount
of reactants adsorbed on the silica surface to the amount of
reactants in the fluid phase decreases, thus causing the yield to
decrease. The Langmuir adsorption model fits the data well,
primarily as a result of the decrease in the Langmuir equilibrium
partitioning constant that occurs with increasing pressure.
The effect of temperature on the measured Langmuir equi-
librium constants was correlated using a traditional heat of
(
1
b)
(s)
1
(s)
1
(b)
1
(s)
2
(s)
2
(b)
2
∂
x
-x ( Hh - Hh ) - x ( Hh - Hh )
(
)
)
(17)
(
2
s) (b)
(b)
∂
T
P
x
x
1
∂ ˆf 1
2
(s)
1
RT x -
(
(b) )( (b)
)
x
∂x1
2
T,P
where the terms in eq 17 are defined in the Appendix.
Doped Silica. We have found the activation provided by
untreated silica promoters to be only modest. By comparison,
Lewis acid promoted Diels-Alder reactions typically proceed
at or below room temperature. The identification of highly
activated solid-phase promoters with reactivity comparable to
Lewis acids is desirable and we have initiated an investigation
of the reactivity of protic and Lewis acid doped silica promoters.
To date, we have prepared silica doped with H2SO4, H3PO4,

2886 J. Phys. Chem. B, Vol. 103, No. 15, 1999
Weinstein et al.
adsorption approach. However, because of the highly nonideal
fluid state in this reacting system, the true heat of adsorption
for the process in CO2, ∆ Hh ads, is really a combination of the
of components, π is the number of phases, and r is the total
number of reactions and other constraints. In this case n ) 2,
π ) 2, and r ) 1 (constant P), so I ) 1. Since we are dealing
with a monovariant equilibrium system, variations in chemical
potential only need to be examined with respect to one mole
fraction parameter. Variations with the other mole fraction term
are constrained by the multiphase reaction equilibrium and by
the fact that the sum of the mole fractions of any phase must
equal unity. We can now apply the Gibbs-Helmholtz equation
for the temperature derivative of the chemical potential divided
by temperature in eq A3 to produce
+
partial molar enthalpies of the solvent and reactant or product
in the CO2 phase and on the solid surface and can only be
obtained when an activity-based equilibrium constant is used.
The Langmuir equilibrium constant, although providing useful
scaling information, does not physically describe the true process
of adsorption in these systems. Since supercritical CO2 is a
compressible dense fluid, the effect of temperature on the
equilibrium partitioning constants is more clearly described at
a fixed fluid density rather than at a specified constant pressure.
+
(
1
b)
(b)
At fixed density, one could obtain ∆ Sh , from a combination
ads
µ1
Hh
∂µ1
1
T
(b)
of the partial molar entropies of the solvent and the reactant or
product in both phases. Again, the activity-based equilibrium
constant must be used. Our work suggests that for adsorbing
d
) -
dT +
dx1
(A4)
2
(b)
(
T
)
(
)
T
∂x1
T,P
reacting systems in supercritical fluids it might be best to utilize
Equation A2 can be combined with eq A4 to give the variation
of the fluid phase mole fraction with respect to temperature.
+
the standard state reaction free energy, ∆ Gh , which is equal
ads
+
+
+
to ∆ Hh - T ∆ Sh . Such standard state Gibbs free energies of
adsorption capture both pressure and density effects in the
reacting system. There is no reason to believe that ∆Gh
Hh , and ∆ Sh are actually constant over a wide range of
temperatures and pressures (or fluid densities).
ads
ads
(s)
(s)
2
(b)
2
(s)
1
(b)
1
(s)
2
(b)
2
H
H
x
x
(b)
x
x
x
x
+
ads
,
+ Hh 1
(b)
(b)
|
|
|
|
+
+
∂x1
∆
ads ads
) -
P
(A5)
(
)
(
1
(
1
s)
(s)
2
(b)
1
∂T
x
x
x
∂µ
T
b)
(b)
2
(b)
1
|
|
(
)
x
∂x
Appendix
T,P
Here we present an alternative approach for examining phase
distributions of species between a solid surface and a bulk
supercritical fluid phase. The true heats of adsorption can be
obtained easily with this approach. In a two-component (1 )
solute, 2 ) CO2) system with two phases (s ) adsorbed solid,
b ) bulk fluid phase), the Gibbs-Duhem equation can be
written independently for each phase.
It is also possible to introduce the fugacity of species 1 in the
fluid phase mixture, ˆf , at this point.
(b)
1
(
1
b)
(b)
µ
) RT ln ˆf 1 + λ (T)
(A6)
1
where λ1 is a function that depends on temperature only. Now
eq A6 can be used with the following equations to simplify eq
A5:
(
s)
(s)
H
V
(s)
(s)
i
dT -
dT -
dP + xi d(µ /T) ) 0
(A1)
∑
2
T
1,2
T
(
s)
(s) (s)
H )
∑xi
Hh i
(b)
(b)
H
V
(A7)
(b)
(b)
i
dP + xi d(µ /T) ) 0
(b)
(b) (b)
∑
H
)
∑x
Hh i
2
i
T
1,2
T
which yields,
With the phase in equilibrium, the chemical potential of each
(s)
(b)
species must be equal in each phase. Thus, µi ) µ . For a
i
(
1
b)
(s)
1
(s)
1
(b)
1
(s)
2
(s)
2
(b)
2
constant pressure case (dP ) 0), Cramer’s rule can be used to
∂x
-x ( Hh - Hh ) - x ( Hh - Hh )
simplify eq A1.35 Using a determinant format, eq A1 becomes
)
P
(A8)
(
)
(
2
s) (b)
(b)
∂T
x
x
1
∂ ˆf 1
2
(s)
1
RT x -
(
s)
(s)
2
(b)
2
(s)
1
(b)
1
(s)
2
(b)
2
(
(b) )( (b)
)
H
x
x
x
x
x
x
x
∂x1
1
2
T,P
dT +
d(µ /T) ) 0
(A2)
1
2
(b)
|
|
|
|
T H
Equation A8 contains the true heats of adsorption for the isobaric
monovariant system. The change in partial molar enthalpy for
each species from the solid phase to the fluid phase is effectively
the differential heat of adsorption for that species. With an ideal
solution or ideal gas mixture, the composition of each phase as
well as the change in the bulk phase composition with respect
to temperature at constant pressure is all that is required to
calculate the heats of adsorption. In these ideal cases the fugacity
can be replaced by the mole fraction of the species times
pressure (for ideal gas) or the pure component fugacity (for ideal
solution). However, in a nonideal solution, the mixture fugacity
must be used, which is given by the continued product of the
mole fraction of the species, the pressure of the system, and
Now, the total differential of the chemical potential over
temperature needs to be expanded. For the constant pressure
case it is explicitly expanded as a function of T and xi.
Arbitrarily, choosing the bulk fluid phase (b)
(
1
b)
(b)
1
µ1
µ
∂(µ /T)
d
) d
)
dT +
(
T
)
(
T
) ( ∂T
)
P,x(
b)
2
(
1
b)
2
∂µ
1
(b)
dxi (A3)
∑
(
(b)
)
i)1
T
∂
xi
T,Px j( b)[i,k]
(b)
the fugacity coefficient, φˆ1 . In such nonideal systems, the
fugacity coefficient does not equal unity but can be obtained
with an appropriate PVTN equation of state for the mixture:35
The variance of this system (I ) needs to be evaluated. From
the Gibbs phase rule, I ) n + 2 - π - r where n is the number

Silica-Promoted Diels-Alder Reactions in CO2
J. Phys. Chem. B, Vol. 103, No. 15, 1999 2887
V
PV
(15) Hitzler, M. G.; Smail. F. R.; Ross, S. K.; Poliakoff, M. J. Chem.
Soc., Chem. Commun. 1998, 259.
(
1
b)
∂P
RT
V
RT ln φˆ ) -
-
dV - RT ln
∫
(
(∂N
)
)
(
)
NRT
(16) Hitzler, M. G.; Poliakoff, M. J. Chem. Soc., Chem. Commun. 1997,
∞
1 T,N ,V
2
1
667.
17) Veselovskii, V. V.; Gybin, A. S. Lozanova, A. V.; Moiseenko, A.
M.; Smit, W. A.; Caple, R. Tetrahedron Lett. 1988, 29, 175.
18) (a) Little, L. H. Infrared Spectra of Adsorbed Species; Academic
(
(
A9)
(
where N1 and N2 are mole numbers.
Press: New York, 1966. (b) Iler, R. K. The Chemistry of Silica; John Wiley
and Sons: New York, 1979.
References and Notes
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