J. Phys. Chem. 1987, 91, 4639-4643
4639
Excess Electron Mobility in Hydrocarbon Liquids at High Pressure Raul C. Muiioz,+ Richard A. Holroyd,* Chemistry Department, Brookhaven National Laboratory, Upton. New York 1 1 973
Kengo Itoh, Kazumichi Nakagawa, Masaru Nishikawa, Department of Pure and Applied Science, University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo, 153 Japan
and Kenji Fueki Department of Synthetic Chemistry, Nagoya University, Furo-cho, Chikusa- ku, Nagoya, 464 Japan (Received: February 25, 1987)
Excess electron mobilities (p)are reported as a function of pressure (1-2500 bar) and temperature (2C-120 "C) for n-pentane, 3-methylpentane, and cyclopentane for which p at 20 OC and low pressure is 51.0 cm2/(V.s), and for 2,2-dimethylbutane (2,2-DMB) for which p 10 cm2/(V.s) at 1 atm and room temperature. For the first three liquids the mobility decreases with pressure at all temperatures studied. The results are interpreted in terms of an equilibrium between quasi-free and trapped electrons, and application of pressure shifts the equilibrium in favor of trapped electrons thus lowering the mobility. The volume changes, associated with trapping, which are of the order of -20 cm3/mol at standard conditions are attributed to electrostriction of the solvent by the trapped (or localized) electron. For 2,2-DMB, the mobility increases with increasing pressure at low temperature and decreases with pressure at high temperature. This behavior is not understood but suggests that traps are unimportant in 2,2-dimethylbutane.
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Introduction Studies of the mobility (p) of excess electrons in liquids subjected to high pressure (1-2500 bar) provide an opportunity to obtain new insight into the states and transport of electrons. If, for example, these electrons exist in more than one state then high pressure may induce a shift in the population of these states with a consequent effect on the mobility. Further, the dependence of the mobility on density can be determined at constant temperature. Most of the existing data on temperature effects also involve simultaneous variation of density. Data a t high pressure permit deriving activatbn energies at constant density. Most importantly these measurements allow a test of theories of electron transport. From our earlier measurements's2 a trend emerged in the pressure-temperature dependence of the electron mobility; so far, the liquids studied can be classified into three types in which (1) the mobility either decreases substantially with pressure at any temperature in the range 20-120 OC; or (2) the mobility decreases slightly with pressure at any temperature but the degree of decrease is somewhat greater at higher temperature; (3) the mobility increases with pressure near room temperature, but decreases at higher temperatures. To type 1 belongs the liquids in which the mobility at atmospheric pressure and room temperature is low (p < 0.1 cm2/(V.s)), and the results are consistent with the existence of traps.2 Tetramethylsilane (p = 100 cm2/(V-s)) belongs to type 2 and the results are in rough accord with the BasakCohen3 deformation potential theory. The intermediate-mobility liquids, where p N 10 cm2/(V.s), constitute type 3. Type 3 behavior is the most enigmatic and poses a challenge to the currently available theoretical models on electron transport. We have extended our study on the pressuretemperature dependence of mobility in type 1 and type 3 liquids in the hope of approaching a fuller understanding of the nature of electron transport in nonpolar liquids, especially in low- and intermediate-mobility liquids. Experimental Section n-Pentane, cyclopentane, 3-methylpentane (3-MP), and 2,2dimethylbutane (2,2-DMB) are all of Wiley 99.9% purity and were purified as described elsewhere: involving degassing, storing over silica gel and molecular sieves, and finally stirring over NaK. The +Presentaddress: Group EP/UAl, CERN, CH-1211 Geneva 23, Switzerland.
0022-3654/87/2091-4639$01.50/0
TABLE I: Activation Energies and Preexponential Factors at Constant Density
density, dcm' 3-MP
n-pentane 2,2-DMB
cyclopentane
0.670 0.704 0.731 0.641 0.672 0.704 0.667 0.698 0.740 0.757 0.819 0.840
Ea, eV 0.141 0.138 0.142 0.116 0.123 0.133 0.028 0.013 0.0 0.08 1 0.073 0.070
FO,
cm2/(v.s) 54 41 42 11 11 13 33 20 13
25 16 13
construction of the pressure cell with a concentric electrode geometry and the pressurizing apparatus have been described.' Temperature was maintained by heating tapes and was regulated within f0.5 OC by a thermostat (Cole Palmer Model 2156). The temperature inside the cell was measured by a Pt thermometer.2 Electrons were generated by irradiating the cell with an X-ray pulse produced by bombarding a tungsten target with a 100-ns electron pulse from a 2-MeV van de Graaff accelerator. The electron mobility was determined by drift time measurements. Details are described elsewhere.' Density and compressibility of liquids at a given pressure and temperature were estimated by fitting existing data5-' to the Tait equation. This gave density values in the liquids which agreed within a few percent when compared with available experimental data.2 Dielectric constants were derived from densities by using the Clausius-Mosotti equation.
Results The results showing the effect of pressure (from 1 to 2500 bar) on electron mobility are shown in Figures 1-4. For n-pentane ~~~~
~
(1) Mufioz, R. C.; Holroyd, R. A.; Nishikawa, 89, 2969.
M. J . Phys. Chem. 1985,
(2) Muiioz, R. C.; Holroyd, R. A J . Chem. Phys. 1986, 84, 5810. (3) Basak, S.; Cohen, M. H. Phys. Rev. 1979, B20, 3404. (4) Muiioz, R. C.; Ascarelli, G. J . Phys. Chem 1984, 88, 3712. (5) Bridgman, P. B. The Physics of High Pressure; Bell: London, 1949; 128. (6) Brazier, D. W.; Freeman, G. R. Can. J Chem. 1969, 47, 893. (7) Takagai, T. Kagakukogaku Ronbunshu 1978, 4, 1
0 1987 American Chemical Society
4640
The Journal of Physical Chemistry, Vol. 91, No. 17, 1987
Muiioz et al.
TABLE 11: Mobilities and Densities in n-Pentane P, bar
T , "C 23
LC'
30
w
250 0.1 11 0.650 0.135 0.643 0.199 0.629 0.259 0.621 0.368 0.605 0.428 0.597
Pa P
50
Lc
P
60
w
80
P Lc P
90
w P
Op,
cm*/(v.s).
p.
500 0.096 0.671 0.1 17 0.664 0.172 0.653 0.217 0.646 0.307 0.632 0.345 0.626
750 0.084 0.688 0.104 0.680 0.153 0.671 0.186 0.666 0.260 0.653 0.295 0.648
1000 0.077 0.702 0.095 0.694 0.140 0.686 0.168 0.681 0.234 0.670 0.263 0.665
1250 0.072 0.715 0.087 0.706 0.126 0.700 0.155 0.695 0.217 0.683 0.241 0.679
1000 0.165 0.737 0.201 0.731 0.241 0.725 0.28 1 0.720 0.339 0.715 0.428 0.704 0.477 0.698
1250 0.152 0.748 0.187 0.743 0.225 0.737 0.269 0.733 0.317 0.728 0.407 0.7 18 0.439 0.712
1500 0.067 0.726 0.083 0.7 17 0.118 0.71 1 0.144 0.707 0.192 0.696 0.223 0.692
1750 0.064 0.736 0.077 0.727 0.111 0.722 0.135 0.718 0.183 0.707 0.204 0.703
2000 0.062 0.745 0.074 0.736 0.107 0.731 0.127 0.727 0.173 0.7 17 0.195 0.714
2250 0.059 0.753 0.070 0.744 0.100 0.740 0.122 0.736 0.161 0.726 0.189 0.723
2500 0.056 0.761 0.066 0.752 0.098 0.748 0.1 16 0.744 0.158 0.735 0.182 0.732
1500 0.151 0.759 0.183 0.753 0.214 0.748 0.259 0.744 0.303 0.139 0.384 0.729 0.414 0.724
1750 0.147 0.768 0.180 0.763 0.204 0.758 0.243 0.753 0.287 0.749 0.365 0.740 0.396 0.734
2000 0.142 0.776 0.170 0.772 0.197 0.767 0.233 0.762 0.276 0.758 0.353 0.749 0.382 0.744
2250 0.139 0.784 0.164 0.780 0.192 0.775 0.229 0.77 1 0.265 0.766 0.333 0.758 0.366 0.753
2500 0.135 0.792 0.165 0.787 0.187 0.782 0.223 0.778 0.255 0.774 0.327 0.766 0.347 0.761
g/cm3.
TABLE 111: Mobilities and Densities in 3-Methylpentane P, bar
T, "C 18
fica pb
30
w
40
P Lc
P
50
w
60
P Lc P
80
w
90
w
P P O p ,
250 0.208 0.690 0.257 0.68 1 0.300 0.674 0.366 0.667 0.437 0.659 0.602 0.644 0.68 1 0.636
500 0.192 0.709 0.234 0.701 0.281 0.695 0.333 0.688 0.395 0.682 0.552 0.670 0.597 0.662
750 0.178 0.724 0.215 0.7 17 0.260 0.7 1 1 0.302 0.706 0.364 0.700 0.460 0.689 0.522 0.682
cm2/(v-s). P , g/cm3.
05
OO
50
,
100
I50
,
200
I
1
50
!OO 150 P/MPa
P/MPa
Figure 1. Electron mobility isotherms for n-pentane as a function of pressure at the temperatures ("C) indicated.
the mobility decreases by a factor of 2 as the pressure increases at 20 'C. At the highest temperature a decrease of about 4 is observed. The results for 3-MP (Figure 2 ) and n-hexane2 are similar to the n-pentane results. For cyclopentane the mobility decreases only slightly at 20 O C but a factor of 2 decrease is observed at 120 OC. For 2,2-DMB, on the other hand, the mobility actually increases with increasing pressure at low temperatures, is roughly independent of pressure at intermediate temperatures, and decreases at high temperatures. For the low-mobility liquids, the temperature dependence of g is given by fi = go exp(-E,/kT). Arrhenius plots of log g vs. 1 / T were made for different densities for data between 20 and 90 O C . For n-pentane the isochoric (constant density) activation energies increase from 0.105 0.003 eV at 0.626 g/cm3 to 0.133 f 0.003 eV at 0.704 g/cm3 (seeFigure 5 ) . Similar plots for 3-MP and cyclopentane are linear in this temperature interval but the activation energies (see Table I) are constant at 0.14 & 0.005 eV and 0.075 f 0.005 eV, respectively. These activation energies
,
I
,
250
,
200
! 250
Figure 2. Electron mobility isotherms in 3-MP as a function of pressure at the temperatures ("C) indicated.
*
I
0
50
1
!
1
150
IO0
200
250
P/ M Pa
Figure 3. Electron mobility isotherms in cyclopentane as a function of pressure at the temperatures ("C) indicated.
The Journal of Physical Chemistry, Vol. 91, No. 17. 1987 4641
Electron Mobility in Hydrocarbon Liquids TABLE I V Volume Changes for Trapping
-AT',,.,
n-pentane
n-hexane' 18 o c 14.9 10.1 6.9 6.0 4.6
P, bar 250 500 1000 1500 2000
cm3/mol
90 OC
23 'C 15.1 12.2 7.9 5.4 4.5
28.6 19.4 11.6 7.8 5.6
3-MP
90 O C 34.6 23.1 11.5 8.6 6.9
18 o c 9.2 7.8 5.3 4.0 2.4
cyclopentane (30 OC)
90 o c 22.3 15.4 10.1 6.6 5.3
3.8 3.4 3.2 3.1 2.9
'Derived from data in ref 2. I
I
I
I
I
0.4
I
I
I
I
I
(
0.3
.1
17
0.2
I
>
N
E
15
i
13 0.1
0
, 50
I
IO0
I
I50
I
I
200
250
P/ M Pa
Figure 4. Electron mobility isotherms in 2,2-DMB as a function of pressure at the temperatures ("C) indicated.
are considerably less than the values measured along the coexistence line. For example, values of 0.2 eV have been reported for n-pentane and 3-MP.8 Also by contrast isobaric activation energies decrease with increasing pressure in all liquids. See Tables I1 and 111 for measured mobilities and calculated densities of n-pentane and 3-MP. For 2,2-DMB the isochoric activation energies are quite small, of the order of 0.03 eV or less, and the uncertainty in these values is larger. The value of E, at high pressure is -0.
Discussion The theoretical models usually invoked to interpret electron transport in low- and intermediate-mobility liquids are the hopping model and the two-state modeL8v9 In the two-state model the electron is considered to be in thermal equilibrium between a set of traps and the conduction band, and the electron moves in the conduction band when thermally activated from traps. In the hopping model, electrons are thought to reside in traps and transport is effected by a series of phonon-assisted jumps between neighboring traps. In both models the temperature dependence of the electron mobility can be described by an Arrhenius-type equation which is usually confirmed by experimental observations. Thus in the absence of Hall mobility measurements in these liquids, a clear distinction beween the two models could not be made. The hopping model is reported to describe the electron transport in 3-MP at very low temperature.'O In the hopping madel the pressure effect modifies p through changes in the hopping frequency, the jump distances, and the height of barriers. In the two-state model pressure affects the traps and the magnitude of the mobility in the conduction band to some extent, but more importantly it would shift the equilibrium distribution of electrons between the two states. Previously it was concluded that either model was capable of describing the normal (type 1) behavior of mobilities vs. pressure.* (8) Schmidt, W. Can. J. Chem. 1977,55, 2197. (9) Davis, H. T.; Brown, R. G . Ado. Chem. Phys. 1975, 31, 329. (10) Kevan, L. Int. J . Radiat. Phys. Chem. 1974, 6, 291.
0.05
/
I
'
2.8
I
3.2
3.0
I
3.4
K-I
Figure 5. Arrhenius plots for the mobility in n-pentane at the densities (in g/cm3) indicated.
The decrease in p was attributed mainly to an increase in activation energy for the case of hexane. Here we observe that the isochoric activation energy also increases with pressure for n-pentane and is nearly constant for 3-MP. As is shown in the following the results are in accord with the two-state model. A . Volume Changes. The two-state model presupposes the presence of the following equilibrium in order to describe electron transport in liquids eqf + trap e e,
(1)
where K = [e,]/( [eqf][Tr]) and the subscripts qf and t designate quasi-free and trapped states, respectively; Tr denotes preexisting traps. The observed mobility is the mobility of quasi-free electrons, pf, times the fraction of electrons in the quasi-free state; thus P = Pr/(l
+ K[Trl)
(2)
For cases when K[Tr] >> 1, and assuming the ratio [Tr]/p, is slowly varying with pressure d In K / d P = d In ( I / p ) / d P
(3)
Since from thermodynamics
AV = -RT d In K / d P
(4)
AV,,, = R T d In p / d P
(5)
then Assuming the above is true, we can obtain values of AV,,, associated with reaction 1 from mobility data for liquids when electron trapping occurs. Some of the values of AV,,, derived this way for n-pentane, 3-MP, n-hexane? and cyclopentane at two temperatures are given in Table IV. Values for intermediate temperatures are shown in Figures 6-8. The volume changes are negative and increase
4642
The Journal of Physical Chemistry, Vol. 91, No. 17, 1987 I
I
"
/
h
I
I
A
l
I
:
1
I
k
Muiioz et al.
I
i
i , , , , , , I 2 3 4 5 6
'0
3
Figure 6. Plot of AVe?, (from d In r / d P ) vs. ( x / 3 ) [ ( c - l ) ( c + 2)/c2] for n-hexane. Calculation of abscissas as explained in the text. Mobility data are from ref 2. (0)18 'C; (0)30 'C: (A)50 'C; (V)7 0 'C; (0) 90 ' C . 20 r
0
I
1
I
I
I
I
I
I
I
I
1
I
I
2
3
4
5
6
1
7
Figure 7. Plot of AVcsxp(from d In p/dP) vs. ( x / 3 ) [ ( c - l ) ( c + 2)/c2] for 3-MP. Calculation of abscissas is explained in the text. ( 0 )18 'C; (V) 30 ' c ; (A)40 OC; (0)50 ' c ; (H) 60 ' c ; (0) 80 (A) 90 "c.
'c;
in magnitude with increasing temperature and decrease in magnitude with increasing pressure. We associate these changes with
Figure 8. Plot of AV,,, (from d In p/dP) vs. ( ~ / 3 ) [ ( -e ] ) ( e + 2)/c2] for n-pentane. Calculation of abscissas is explained in the text. (V)23 ' c ; (A)30 ( 0 )50 'c; (0) 60 ' c ; ( 0 )80 'c; (0) 90 O C .
'c;
the difference in partial molar volume of the quasi-free and trapped electrons. E . Role of Electrostriction. Volume changes observed in ionic reactions are usually associated with electro~triction.~~-~~ Because this effect is important in ionic reactions we suggest that an electron localized by a cavity will polarize the solvent as an ion does, causing electrostriction. An essential difference to be kept in mind is that the wave function of the trapped electron does extend beyond the boundary of the cavity, whereas for an ion the charge is considered to be totally within the cavity, at least classically. Drude and NernstI4 first derived from classical theory the volume change or contraction of a medium around a charged sphere. As given in a recent reviewI3 this volume change is
where R is the radius of the sphere carrying a charge ze and E is the dielectric constant. Other authors1'-l2arrive at the same equation in other ways. To compare with experiment we are interested in the volume change for reaction 1. In our case a "preexisting" trap or cavity is changed into a "trapped" electron. This part of the reaction we simulate as a sphere gaining a charge ze. If the partial molar volume of the quasi-free electron is taken to be zero then the volume change for trapping is just AV,. Thus;it is appropriate to compare our observed volume changes with that predicted by (1 1) Haman, S. D. Physico-Chemical Effects of Pressure; Butterworths: London, 1957; p 55. ( 1 2 ) Bradley, R. S. High Pressure Physics and Chemistry; Academic: New York, 1963; Vol. 2, p 147. (13) LeNoble, W. J.; Kelm, H. Angew. Chem. 1980, 19, 841. (14) Drude, P.; Nernst, W. 2. Phys. Chem. 1984, 15, 79.
Electron Mobility in Hydrocarbon Liquids
eq 6. This equation can be converted into a more convenient form by using the Clausius-Mosotti equationI5 and the isothermal compressibility, x.
The Journal of Physical Chemistry, Vol. 91, No. 17, 1987 4643 energy. The above discussion of volume changes on trapping is easily understood in terms of the two-state model as described by p
Equation 7 provides a way of comparing with the experimental AVvalues and also gives some insight into trapping. First some general observations are in order. This equation predicts AVwill be negative as is observed. Further AV should be proportional to the quantity (x/3)[(e - 1)(c 2)/c2]. Figures 6-8 are such plots for our data for hexane,2 pentane, and 3-MP, respectively, where data from 18 to 90 “C are included. The values of the abscissa are calculated from the compressibilities and dielectric constants at various temperatures and pressures (see Experimental Section). A linear relationship is indicated in support of this model. It should also be noted the term in square brackets in eq 7 changes very slowly with pressure, remaining close to 1.O;thus, the volume changes are a function mainly of the compressibility of the liquid. This is one reason why electrostriction effects are more important in nonpolar than in polar solvents; hexane is much more compressible than water, for example. Of course the Coulombic field of the ion also penetrates further in nonpolar liquids and can affect more solvent molecules causing larger effects. From Figures 6-8 the volume change involved in trapping can be estimated for standard conditions, that is, at 1 atm and 25 O C . The values of the abscissa are 5.2, 7.1, and 6.0 X bar-’ for n-hexane, n-pentane, and 3-MP, respectively under these conditions. Linear extrapolation leads to values of A P of -22, -28, and -19 cm3/mol for n-hexane, n-pentane, and 3-MP, respectively. We attribute these reaction volumes to solvent contraction around the trapped electron. The quasi-free electron is in an extended state and its partial molar volume is taken to be zero. Thus it follows that the trapped electron in hydrocarbons has a negative contribution to the volume. For liquid ammonia as solvent the solvated electron has a positive partial molar volume.16 Some additional insight into the nature of electron trapping in hydrocarbons is provided by this data. Although traps are considered to preexist as cavities in the liquid, the results show that “occupied” traps differ from unoccupied traps. The solvent around the trapped electron is constricted, corresponding to the negative reaction volume, whereas the solvent around the unoccupied trap is normal. Finally, does this model of electrostriction account for the observed volume changes? The slopes of the lines in Figures 6-8 are 4.8, 4.3, and 3.5 X lo5 cm3-bar/mol for n-hexane, n-pentane, and 3-MP, respectively. According to eq 7 the slopes should be given by z2e2/2R. Semicontinuum model calculations of Fueki et al.” indicate a value for the radius of the cavity in n-hexane to be 4.1 A and that only a fraction of the charge is within the cavity. Thus z , instead of being 1 as it would be for an ion, is a fraction which is estimated to be 0.7.’’ With R = 4.1 and z = 0.7 the slope is predicted to be 8.3 X lo5 cm3.bar/mol for n-hexane, which is greater than the observed value. More detailed corrections could be introduced to account for the fact that part of the charge of the trapped electron remains outside the cavity. This would decrease somewhat the effect of electrostriction below the crude estimate based on eq 7, when z is taken to be a fraction less than 1, thus bringing the calculated value closer to the observed one. However, our aim here is to show that the electrostriction model accounts at least semiquantitatively for the negative volume changes derived from mobility data. C. Actiuation Energies. In an earlier paper2 we attributed the decrease in mobility with pressure to an increase in the activation
+
(15) The Clausius-Mosotti equation is not strictly valid at high pressure, (see ref 6) but for hydrocarbons it is accurate to 1-2% in the pressure range investigated. (16) Boddeker, K. W.; Lang, G.; Schindewolf, U. Angew. Chem., In?.Ed. Engl. 1969, 8, 138. (17) Kimura, T.; Fueki, K.; Kevan, L. J . Chem. Phys. 1978, 68, 3945.
= poe-&lRT = pLr(Ne/ N,)e-EalR T
(8)
where Ne is the effective density of states at the bottom of the conduction band and Nt is the effective density of trap sites. Pressure shifts the ratio of concentrations [eqf]/[e,], which is equal to (Ne/Nt)e-EalkT. Thus the small increase observed in isochoric Ea’s, for n-pentane and n-hexane, with increasing pressure is equivalent to a decrease in the ratio [eqf]/[ef]and to an increase in K. For 3-MP the isochoric E, is, within experimental error, independent of pressure. This suggests that in this case there is a change in the preexponential (Ne/Nt) or that the change in E, is simply too small to be detectable.’* D. Cyclopentane and 2,2-DMB. Whether or not the cyclopentane results are consistent with the two-state model is not clear. The effect of high pressure is to decrease the mobility but the percent decrease is quite small. Cyclopentane is similar in this respect to tetramethylsilane2 but on the other hand quite dissimilar since the mobilities are 100-fold less than in tetramethylsilane. However, if the data are treated like those for the low-mobility liquids then values of AV,,, can be derived from the data at 30 “C. (The lack of density data precludes analysis at other temperatures). The results are shown in Table IV. The volume changes are considerably smaller and are consistent with the lower trap depth, 0.075 eV for cyclopentane. In the case of 2,2-DMB, as well as 2,2,4-trimeth~lpentane,~ the electron mobility increases with pressure near room temperature and at pressures > 100 MPa the mobility is nearly constant, independent of temperature. An explanation of this behavior is not readily apparent. A significant point is that at high pressures and at high densities the observed activation energies are close to zero. A relevant finding is that recent measurements at low pressure of the Hall mobility of electrons in 2,2-DMB indicate pHallN pdriftbetween 20 and 90 OC.I9 These facts show that traps cannot be very important under these conditions. On the other hand the magnitude of the mobility is less than that usually associated with quasi-free band motion; Le., 100 cmz/(V.s). If one uses the Lorentz equation, a mobility of 11 cm2/(V.s) would correspond to a mean free path of only 8 A. Thus the mean free path is less than the wavelength of a thermal electron and the concept of charge transport by itinerant electrons undergoing occasional scattering is inappropriate.
-
Conclusions The effect of high pressure has given additional insight into the mechanism of electron transport in nonpolar hydrocarbon liquids. The explanation of the results we suggest for low-mobility liquids (type 1) requires an equilibrium of the electron between a quasi-free state and a trapped state. The latter is favored at high pressure because of electrostriction of the solvent by the trapped electron. Type 3 liquids, e.g., 2,2-DMB, appear not to fit this model. Acknowledgment. We are grateful to H. Schwarz and S. Ehrenson for helpful discussions. This research was carried out at Brookhaven National Laboratory under contract DE-AC0276CH00016 with the U S . Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences. We also thank the Japan Society for the Promotion of Sciences, under the US-Japan Cooperative Science Program, for their support. Registry No. 3-MP, 96-14-0; 2,2-DMB, 75-83-2; n-pentane, 109-66-0; cyclopentane, 287-92-3. (18) Since smaller volume changes are observed for 3-MP, the energy difference P A V would be correspondingly less for 3-MP and this would be reflected in E,. (19) Mufioz, R.; Holroyd, R. A,; Itoh, K.; Nishikawa, M., to be published.
