J. Phys. Chem. 1985,89, 880-883
880
of the Fourier series expansion of the distribution. Moreover, a slight decrease of the scattered intensity is observed at low q which means that the interference term has been misestimated or that the methyl groups are distributed in a hollow sphere. From the energies given in Table I and the geometrical confinement in a sphere, we can calculate the order parameter of the 3 cos2 8, chain, relative to that of the first carbon, as Si = 1 ), where Bi is the angle between the Ci-l-Ci+l vector and the axis of the fully extended chain (Figure 8). The order parameter deduced from a deuterium N M R quadrupole splitting measurement for the same conformation in a lamellar mesophase is Sco = - 1/2S.60The obtained order parameter profile differs from that of a lamellar mesophase as obtained from 2H N M R . This (60) Edholm, 0. Chem. Phys. 1982, 65, 259.
behavior has been predicted by G r ~ e n ,but ~ . here ~ again the model proposed by Dill and Flory27,28fails in predicting the order parameters profile. Conclusions The present study has shown that the hydrocarbon chains of a micelle are confined in the hydrocarbon core. Their exist out of the micelle is allowed only with the polar group ahead. The conformation of the first segments of the chains is very trans as a consequence of the opposite forces applied on the two parts of the molecules. The last segments have a conformation similar to that of a liquid hydrocarbon. The distribution of the carbons in the micellar core is more and more random as the number of the carbon increases. Registry No. Mn, 7439-96-5; C8H,,0P0,H-.Na+, 30410-34-5.
Chemical Equilibrium In Complex Organic Systems Robert A. Alberty Chemistry Department, Massachusetts institute of Technology, Cambridge, Massachusetts 021 39 (Received: August 20, 1984; in Final Form: October 22, 1984)
The fact that isomer groups may be considered to be single components in equilibrium calculations leads to a great simplification and makes it possible to extrapolate thermodynamic properties of isomer groups to higher carbon numbers. At fixed hydrogen partial pressure this concept may be extended to include isomer groups from several homologous series at the same carbon number. At fixed ethylene partial pressure all of the isomer groups in a single homologous series may be combined into a single extended isomer group. The use of Gibbs energies of formation calculated for these larger groups of substances which are in equilibrium greatly simplifies calculations of equilibrium compositions for ideal gases and provides insight into the nature of the equilibrium in complex systems.
Introduction The calculation of equilibrium compositions of organic systems a t relativeIy high temperatures in the presence of catalysts is complicated a t higher carbon numbers by the large number of isomers and the lack of data. The numbers of isomers of successive members of a homologous series increase about threefold with each additional carbon atom, and exceed 100 for (28 alkenes, clo alkanes, Clo cyclopentanes, C l l cyclohexanes, and C12alkylChemical thermodynamic data for isomers in these homologous series are complete through c6 alkenes, cloalkanes, C7 cyclopentanes, c8 cyclohexanes, and c9alkylben~enes.~ The problem of calculating equilibrium compositions of complex systems has been solved in the sense that there are a number of computer codes that will do this for quite large system^.^ However, even when the data are available, there is the problem of dealing with so many species and trying to gain chemical insight into the equilibrium behavior of these systems. One method for simplifying equilibrium calculations on complicated systems of organic compounds has been known for some time. Smith6 pointed out that such calculations can be carried out in two steps: (1) the isomers in a group are treated as a single component and the equilibrium mole fractions of the various isomer groups are calculated; (2) the equilibrium mole fraction of an isomer group is distributed between the various individual species in the group. Isomers in equilibrium can be treated as a single component in (1) J. Lederberg, G. L. Sutherland, G. G. Buchanan, E. A.Feigenbaum, A. V. Robertson, A. M. Duffield, and C. Djerassi, J. Am. Chem. Sac., 91, 2973 (1969). (2) R. C. Read in "Chemical Applications of Group Theory", A. T. Balaban, Ed., Academic Press, New York, 1976, Chapter 4. (3) J. C. Nourse, J . Am. Chem. SOC.,110, 1210 (1979). (4) D. R. Stull, E. F. Westrum, and G. C. Sinke, "The Chemical Thermodynamics of Organic Compounds", Wiley, New York, 1969. (5) W. R. Smith and R. W. Missen, "Chemical Reaction Equilibrium Analysis: Theory and Algorithms", Wiley-Interscience, New York, 1982. (6) B. D. Smith, AIChE J., 5 , 26 (1959).
0022-3654/85/2089-0880$01.50/0
an ideal gas system because the distribution of isomers depends only on the temperature. This method leads to a very large reduction in the number of components that have to be included in a computer calculation on a complex organic system. The equations for making these calculations, which are discussed in the next section, have been used by Dantzig and D e H a ~ e nDuff ,~ and Bauer,8 and Smith and M i ~ s e n . ~The , ~ applicability of this method may be extended in several ways which further simplify equilibrium calculations on complex systems and provide insight into the nature of these equilibria. The concepts developed in this paper are illustrated by examples connected with the conversion of methanol to gasoline using zeolite catalysts.I0 The catalyst opens up kinetic pathways to many thousands of species of alkanes, cyclopentanes, cyclohexanes, alkylbenzenes, and alkenes, but not to graphite and molecular hydrogen to which most of these substances can be converted in principle. Calculations of equilibrium compositions to be expected must include species that are found experimentally and species of higher carbon number that may be present only at quite low mole fractions. The methods described here are applicable in general, but are illustrated with specific examples to clarify the concepts. In all cases the gases are assumed to be ideal. Equilibrium Compositions within Isomer Groups The formation reaction for isomer i in an isomer group with molecular formula C,H, is represented by
nC(graphite)
+ (v/2)H2(g) = C,H,(g,i)
K, = P i / P ~ , " / 2 (1)
The equilibrium mole fraction r, of this isomer within the isomer group is given by (7) G. B. Dantzig and J. D. DeHaven, J . Chem. Phys., 36, 2620 (1962). (8) R. E. Duff and S. H. Bauer, J . Chem. Phys., 36, 1754 (1962). (9) W. R. Smith and R. W. Missen, Can. J . Chem. Eng., 52,280 (1974). (10) D. C. Chang and A. J. Silvestri, J. Curd., 47, 249 (1977).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 5, 1985 881
Chemical Equilibrium in Complex Organic Systems
r,=Pi = Ki
'
N,
exp(-AfGio/RT)
-
is
Nn
A&* = A@'
+ R T In (Pc-H~/PH,'I~) = A@' - (v/2)RT In PHI ( 6 )
where NIis the number of isomers in the group and A@' (I), the standard Gibb energy of formation of the isomer group, is given by N,
A p o ( I ) = -RT In
[X ~X~(-A@~'/RT)] i= 1
(3)
This equation may be rearranged to a form that is more familiar by use of eq 2. The standard Gibbs energy of formation for an isomer group is equal to the weighted average standard Gibbs energy of formation for the isomers in the group plus the Gibbs energy of mixing of the isomer group, assuming ideal gases. N1
+
NI
AGo(I) = ~ r j A f G ~RTCr, In ri i=l
i= 1
(4)
Since the Gibbs energy of mixing is necessarily negative, AfG'(1) is more negative than the weighted average standard Gibbs energy of formation given by the first term. Since eq 3 gives the standard Gibbs energy of formation of the isomer group as a function of temperature, it may be used to derive the corresponding equations for the enthalpy of formation, entropy, and heat capacity at constant pressure for the isomer group in the ideal gas state." These equations have been used to produce tablesI2 of A@'(I), AfHo(I), S'(I), and C,'(I) for the alkanes up to C10H22from Scott's tablesI3 and for the alkylbenzenes up to Cl2HI8.l4The data for the thermodynamic properties of the alkylbenzenes are complete4 only through C9HI2,and so the Benson" method has been used to estimate the thermodynamic properties for all isomers of CI0Hl4,C,,Hl6, and CI2Hls. For both of these homologous series, the increments in the various thermodynamic properties per carbon atom approach constant values as the carbon number increases, and so these limiting increments may be used for extrapolation to higher carbon numbers. In this way isomer group properties make another contribution to equilibrium calculations on complex organic mixtures because they provide a way to extrapolate beyond the highest carbon numbers for which thermodynamic data are available. Equations 2 and 3 apply to all species that have the same molecular formula, even though they may have different types of structures. For example, the alkenes, cyclopentanes, and cyclohexanes all have the formula CnH2,, and so the mole fractions of all of these species with a given number of carbon atoms may be calculated by using a single Gibbs energy of formation for the isomer group including species from the three homologous series. The applicability of these equations may be extended to include fixed-ratio subgroups with the same average composition as the isomers in the group.I6 If, because of the selectivity of the catalyst, certain isomers are excluded from the equilibrium, they may be excluded from the isomer group, and eq 2 and 3 may be applied to the isomers which are in equilibrium. Equilibrium Compositions at Fixed Hydrogen Pressures Krambeck has pointed out that, when the partial pressure of hydrogen is fixed, an adjusted Gibbs energy of formation may be used in equilibrium calculation^.^^^^^ In effect the standard state pressure for hydrogen is changed from 1 bar to the fixed value, while the standard state pressure for the organic substance is kept at 1 bar. The change in Gibbs energy A@* for the reaction nC(graphite) + (v/2)Hdg,P~,) = CnHp(g,l bar) (5) (11) R. A. Alberty, Ind. Eng. Chem. Fundam., 22, 318 (1983). R. A. Alberty, and C. A. Gehrig, J. Phys. Chem. Ref.Data, in press. D. W. Scott BuZl.,U.S., Bur. Mines, No.666 (1974). R. A. Alberty, J. Phys. Chem. Ref.Data, in press. S.W. Benson, 'Thermochemical Kinetics", Wiley, New York, 1976. R. A. Alberty, Ind. Eng. Chem. Fundam., 23, 129 (1984). A. M. Kugelman, Hydrocarbon Processing, 1 , 95 (1976). M. P. Ramage, K. R. Graziani, and F.J. Krambeck, J . Chem. Eng. Sci., 35, 41 (1980). (12) (13) (14) (15) (16) (17) (18)
When the hydrogen pressure is fixed, the general equilibrium calculation does not have to take conservation of hydrogen into account, and A@* values for reactants and products are used rather than A@' values. This point is also discussed in Smith and Missems The equations for AfH*, S*,and C,* are readily derived. This change in standard state may be applied to the Gibbs energy of formation, enthalpy of formation, entropy, and heat capacity of an isomer group as well. When the hydrogen pressure is fixed, alkanes, alkenes, and alkylbenzenes with the same number of carbon atoms essentially become isomers; we can call them pseudoisomers. The standard Gibbs energies of formation of alkanes, alkenes, and alkylbenzenes with n carbon atoms at a fixed partial pressure of hydrogen are given by ArG*(I,CnH2n+J = A@'(I,CnH2n+2) - (n + 1 ) R T In P H ~(7) A@*(I,CnH2,) = A@'(I,CnH2,) - n R T In P H ~ (8) AfG*(I,CnH2,6)
= AfG*(ACnH2,-6) - (n - 3)RT
P H ~ (9)
These Gibbs energies of formation at a fixed partial pressure of hydrogen can be used to calculate the standard Gibbs energy of formation of the C, alkane plus alkene plus alkylbenzene isomer group by using AfG*(I,C,) = -RT In (~X~[-A~G*(I,C,H~,+~)/RT] + exp[-A.fG*(1,CnH2n) / RT] + exp[-A@* (19CnH2n-6) / RT]1 (10) The other standard thermodynamic properties of this extended isomer group may, of course, also be calculated.Il The successive extended isomer groups C,, C,+], ... are in effect successive polymers and so the equilibrium composition of a system containing extended isomer groups with different carbon numbers will depend on the total partial pressure P h c of the hydrocarbons that are in e q ~ i l i b r i u m .The ~ reactions forming the successive polymers may be represented by nCl(g) = cn(g)
Kn
= ynPhc/blPhc)n
(1 1)
It is more convenient to use an equilibrium constant K,, written in terms of mole fractions only. It may be expressed in terms of the Gibb energies of the extended isomer groups at constant hydrogen pressure and the total partial pressure P h c of the hydrocarbons by K,' = y,/yln = KZh2-I = lexp[-(A@n* - na@1*)/RT]tbp[(n - 1) Phclt = exp(-[(A@,* + R T In Phc) - n(AfC1* + R T In P h C ) ] / R g (12) Since the sum of the mole fractions of the hydrocarbons is unity y,
+ y2 + y, + ... = 1 = y1 + K;yl2 + K3'y13+ ... = eK,,'yl"
(13)
i= 1
where K,' = 1. According to Descartes' rule of signs this polynomial in y1 has a unique, positive, real root. Thus for a hydrocarbon system at a fixed partial pressure of hydrogen, the calculation of the equilibrium composition comes down to the solution of a polynomial equation. Once the equilibrium mole fractions for the various degrees of polymerization have been calculated, the mole fractions of the alkane, alkene, and alkylbenzene isomer groups with these carbon numbers may be calculated from the analogue of eq 2. The equilibrium mole fraction Y C ~ H ~of~ the + ~ CnH2n+2isomer group, for example, may be calculated from YC.H2+2
- YnrC.H2+2= Yn
tA@*(I,Cn) - A@*(I,CnH*n+dl /RTt (14)
882 The Journal of Physical Chemistry, Vol. 89, No. 5, 1985
(n/2)C2H,(g) = CflH,n(g)
TABLE I: Equilibrium Mole Fractions of Alkane, Alkene, and Alkylbenzene Isomer Groups at 700 K alkanes alkenes alkylbenzenes total a. PH2= 1 bar, Phc = 1 bar c6 0.001 0.000 0.256 0.257
c7 C8
total
0.000 0.000 0.001
0.000 0.000 0.000
0.457 0.286 0.999
c7 cs total
0.688 0.104 0.018 0.810
0.006 0.002 0.001 0.009
0.019 0.071 0.09 1
0.181
0.457 0.286 1.000 0.714 0.177 0.109 1.ooo
c. PH2= 10 bar, Phc= 10 bar c6
c7 cs total
0.324 0.093 0.074 0.491
0.003 0.002 0.004 0.009
0.009 0.064 0.427 0.500
0.336 0.159 0.505 1.ooo
where y, is the sum of the mole fractions of the alkanes, alkenes, and alkylbenzenes calculated from eq 12 and 13. The use of eq 7-13 is illustrated by the calculation summarized in Table I. This table gives calculated equilibrium mole fractions of the Cg, C7, and csisomer groups of alkanes, alkylbenzenes, and alkenes at 700 K. The first part of the table is for a hydrogen partial pressure of 1 bar and a total hydrocarbon partial pressure of 1 bar. The second part of the table is for a hydrogen partial pressure of 10 bar and a total hydrocarbon partial pressure of 1 bar. In the third part of the table both of these partial pressures are 10 bar. It is evident that raising the partial pressure of hydrogen favors the species containing more hydrogen and raising the total partial pressure of the hydrocarbons increases the degree of polymerization. The next step of the calculation would be to use equations 2 and 14 to obtain the equilibrium mole fractions of the individual species of alkanes, alkylbenzenes, and arenes, but that step has not been taken here. In making these calculations the standard Gibbs energies of formation of the C7HI4and CsH16 alkene isomer grups were estimated by linear extrapolation. Equilibrium Compositions within Homologous Series For homologous series other than the alkenes, the calculation of the equilibrium composition within the series currently requires the use of a general equilibrium program. For ideal gases the equilibrium composition within a homologous series, other than those of the alkene type, is independent of the total partial pressure of the hydrocarbons in that series. The reason for this is that equilibrium is reached through disproportionation and isomerization reactions in which there is no change in the number of molecules; for the alkylbenzenes
2CnH2n-6 = Cn-lH2n-8 + Cn+lH2n-4
(17)
The fact that fixing the hydrogen partial pressure simplifies the calculation of equilibrium compositions across homologous series suggests that fixing the ethylene partial pressure will simplify the calculation of equilibrium compositions within a homologous series. The following derivation shows that fixing the ethylene partial pressure leads to the definition of a Gibbs energy of formation AfG*(HSG) of a homologous series group, just like fixing the hydrogen partial pressure in the preceeding section led to the definition of A@*(I,C,). This new type of Gibbs energy of formation makes it easy to calculate the equilibrium mole fractions of isomer groups within a homologous series. At a fixed partial pressure of ethylene the isomer groups within a homologous series become pseudoisomers. The alkylbenzene homologous series is used as an example, but the following derivation may be adapted to other homologous series. Ethylene is used in the formation reactions for the carbon and hydrogen atoms beyond those in benzene, but of course any other alkene could be used.
b. P H =~ 10 bar, p h c = 1 bar c6
Alberty
(15)
However, the equilibrium composition for a homologous series does depend on the H / C ratio. Although the H / C ratio is a suitable independent variable for the discussion of equilibrium in systems containing homologous series, it is more convenient to use a variable which is common to several subsystems, just as the partial pressure of molecular hydrogen was used in the preceeding section. A suitable variable for this purpose is the partial pressure of ethylene, since the next higher homologue can be formed by reaction with ethylene, or with any other alkene. For the alkylbenzenes Therefore, the distribution of equilibrium mole fractions in a homolgous series can be taken as a function of temperature and the partial pressure of ethylene and is independent of pressure for ideal gases. This statement applies to all homologous series except the alkenes, or other homologous series containing carbon and hydrogen in the proportion 1 to 2. For homologous series of the alkene types the equilibrium distribution depends on the pressure since there is a change in the number of moles of gas in the polymerization reaction.
Kn =
PC~H”/PH~~PC~H,‘”-~’’~
The equilibrium mole fraction of a particular isomer group within the alkylbenzene homologous series group is given by ‘Wz-
KnPC2H4(n-6’’2 C Pc,HC KnPC”4‘n-6’’2 n=6 n-6 exP(-[A@o(1,CnH2n_6) - ( ( n - 6)/2)(AfGoC2H4 +
-
-
‘CJh
m
R T In PC2H4)]/RT]/exp[-AfG*(HSG)/RT] = exp([AfG*(HSG) - A@*(I,CflH,ndl/RTJ (19) where the Gibbs energy of formation A@*(I,C,H2,,J for the isomer group CnHZd for standard state with a fixed value for the partial pressure of ethylene is given by A@*(1,CnH2~6) = AfG0(1,CnH2n-6) - ( ( n - 6)/2)(AfGOC2H4 + R T In PC2H4) (20) and the Gibbs energy of formation A@*(HSG) for the alkylbenzene homologous series group is given by 0)
AfG*(HSG) = -RT In
~ X ~ [ - A @ * ( I , C ~ H ~ ~ ~ )(21) /RT] n=6
In order to calculate the equilibrium mole fractions within a homologous series a multiple of AfC°C2H4 R T In PCzH4 is subtracted from the standard Gibbs energy of formation of each isomer group. The Gibbs energy of formation AfG*(HSG) of the homologous series group is then calculated with eq 21, and the equilibrium mole fractions of the various isomer groups are calculated with eq 19. These last two equations are again simply analogues of eq 2 and 3. In the last step eq 2 is used to calculate the mole fractions of the individual molecular species. As an illustration the equilibrium mole fractions of alkylbenzene isomer groups have been calculated for six partial pressures of ethylene at 700 K by this method. The calculated values given in Table I1 are in agreement with those c a l ~ u l a t e d with ’ ~ EQUCALC, a general equilibrium program written by Krambeck20 which uses the Netwon-Raphson method. The standard Gibbs energies of
+
(19) R. A. Alberty Ind. Eng. Chem. Fundam. in press.
(20) F. J. Krambeck, 71st Annual Meeting of AIChE, Miami Beach, FL, Nov 16, 1978.
J . Phys. Chem. 1985,89, 883-886 TABLE Ik Equilibrium Mole Fractions of Alkylbenzene Isomer Groups for Six Partial Pressures of Ethylene at 700 K Plbar 0.003 0.006 0.012 0.024 0.048 0.096 0.328 0.426 0.194 0.043 0.008 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.222 0.409 0.263 0.081 0.022 0.003 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0,134 0.347 0.316 0.139 0.053 0.009 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.069 0.254 0.327 0.203 0.109 0.026 0.009 0.002 0.001 0.000 0.000 0.000 0.000 0.000 0.000
0.029 0.152 0.277 0.243 0.185 0.061 0.031 0.012 0.005 0.002 0.001 0.000 0.000 0.000 0.000
0.009 0.069 0.177 0.219 0.235 0.111 0.079 0.043 0.025 0.014 0.008 0.005 0.003 0.002 0.001
formation of higher isomer groups were estimated by linear extrapolation. Since eq 21 gives the Gibbs energy of formation of A,C*(HSG) the use of for an homologous series group at a particular PCIHI, thermodynamic derivatives makes it possible to calculate the enthalpy of formation AfH*(HSG), the entropy S*(HSG), and the heat capacity at constant pressure C,* (HSG) under these same conditions. These quantities are each a function of the partial pressure of ethylene, but tables may be prepared by using known values for the lower isomer groups and using linear extrapolations to higher isomer groups. Equilibrium between Homologous Series Homologous series of hydrocarbons which are in equilibrium with the same partial pressure of ethylene are in equilibrium with each other, and so a series of possible equilibrium compositions for a system containing alkanes, alkenes, and alkylbenzenes can be calculated from a series of CzH4pressures. this method has the advantage that various homologous series are handled separately and the method of calculation is straightforward. The relative numbers of moles from various homologous series must be set so that element balances are satisfied. The conversion of methanol to gasoline using zeolite catalysts provides an illustration of the applicability of these equations. When the dehydration of methanol and any ethers formed from it is complete, the system consists essentially of alkenes, alkanes, and alkylbenzenes, with the later two homologous series having a mole ratio of 3:l in order to preserve element balances. The
883
equilibrium composition may be calculated as described above without the use of a general equilibrium program. The method described here may also be used to calculate compositions at earlier stages of the reaction on the assumption that the alkylation of alkenes present is at equilibrium and the alkylation of alkylbenzenes present is at equilibrium. The alkanealkylbenzene ratio is set at 3:l and the polynomial for the alkene polymerization is solved for PCtHIat the total partial pressure of the alkenes. The equilibrium mole fractions of the various alkane and alkylbenzenes isomer groups can then be calculated with eq 19. Discussion Two methods of simplifying equilibrium calculations have been discussed in rather specific terms to clarify the way they operate, but these methods can be generalized. They are extensions of the concept of isomer groups. A Gibbs energy of formation for isomers or pseudoisomers in equilibrium with each other is calculated. As with isomer groups other thermodynamic properties may be calculated for this equilibrium group. These methods lead to considerable simplifications in equilibrium calculations on complex organic systems and provide an insight into the nature of the equilibrium. At constant hydrogen pressure a Gibbs energy of formation of isomer groups in different homologous series having the same carbon number can be calculated. This reduces the system discussed to a polymerization reaction for which the equilibrium mole fractions can be calculated by solving a polynomial. At constant ethylene pressure the equilibrium mole fractions of isomer groups in an homologous series is readily calculated. Then the equilibria between different homologous series can be related through the ethylene partial pressure. Selective catalysts may not provide catalytic pathways to all isomers. In that case equilibrium calculations should use standard Gibbs energies of formation of isomer groups that exclude isomers that are not formed. The standard Gibbs energy of a subgroup is always more positive than that of the whole isomer group. If a catalyst produces only a subgroup, the standard thermodynamic properties calculated for that subgroup can be used to calculate the effects of temperature, pressure, and composition for that subgroup.
Acknowledgment. This research was supported by a grant from the Office of Basic Energy Sciences of the Department of Energy. The author is indebted to Fredrick J. Krambeck for introducing him to eq 6 and the importance of simplifying equilibrium calculations for complex systems.
Pressure-Induced Phase Transitions and Structural Changes in Aqueous Dipalmitoyiand Distear0y IphosphatIdyIcholined P. T. T. Wong* and H. H. Mantsch Division of Chemistry, National Research Council of Canada, Ottawa, Ontario, Canada K I A OR6 (Received: March 9, 1984; In Final Form: October 2, 1984) From the pressure dependence of the Raman spectra of dipalmitoyl- and distearoylphosphatidylcholine bilayers at 30 OC we found two gel-gel phase transitions, at 1.7 and 4.8kbar in DPPC, and at 1.1 and 3.1 kbar in DSPC. The Raman spectral changes observed at these critical pressures suggest a change in the acyl chain packing from a distorted hexagonal to a monoclinic lattice at 1.7 (DPPC) or 1.1 kbar (DSPC) and then to an orthorhombic lattice at 4.8 (DPPC) or 3.1 kbar (DSPC). The critical pressures of both phase transitions are shown to decrease with increasing chain length. Introduction Aqueous d i s p e ~ i of o ~dimyristoy~~h~phatidylcholine (DMX) exhibit three pressure-induced structural phase transitions with critical pressures at 0.15, 1, and 2.6 kbar.' The transitions at NRCC No. 23088.
0.15 and 1 kbar correspond to the well-known temperature-induced transitions at 24 and 14 OC, respectively, whereas the transition at 2.6 kbar involves a structural change from a distorted hexagonal ( 1 ) Wong, P. T. T.; Murphy, W. F.; Mantsch, H. H. J . Chem. Phys. 1982, 76, 5230.
0022-3654/85/2089-0883$01.50/00 1985 American Chemical Society
