J. Phys. Chem. 1993,97, 9488-9492
9488
Effect of Pressure on the Equilibrium e2,ZDimethylbutane
+ C02 F= C02-in 2-Methylbutane and
Shiro Ninomiya, Kengo Itoh, and Masaru Nishikawa Department of Pure and Applied Sciences, University of Tokyo, Tokyo, Japan
Richard Holroyd' Department of Chemistry, Brookhaven National Laboratory. Upton, New York 1 1 973 Received: May 7 , 1993; In Final Form: June 28, 1993'
+
The reaction e- COz * COz- was studied in two alkane solvents for temperatures from 22 to 100 OC and for pressures up to 3000 bar. The equilibrium constants and thermodynamic quantities for this reaction are reported from which the energies and entropies of solution of the electron in 2-methylbutane and 2,2-dimethylbutane are obtained. The free energy of reaction decreases as the solvent density increases. The significance of the volume changes for this reaction, which are between -190 and -300 cm3/mol, is discussed. The entropy of reaction decreases largely due to the entropy of polarization of the solvent by COz-. Mobilities of excess electrons in 2-methylbutane are also reported for temperatures from 23 to 100 OC and for applied pressures from 1 to 2500 bar. The decrease in mobility with increasing pressure is attributed to increased probability of electron trapping at high pressure.
Introduction Measurements of equilibrium constantsfor electron attachment reactions can be used to derive thermodynamic quantities such as the free energy of solution of the electron, AGs(e). How this energy varies with the density of the liquid can also be determined from measurements as a function of applied pressure. Recently, the effect of pressure on the energy of the electron in the quasifree state (V,) for some nonpolar liquids was reported.' The effect of pressure on the attachment and detachment rates for reaction 1 was reported earlier for three other solvents.2 ka
e-
+ CO, + C0,-
(1)
kd
This reaction is very sensitive to pressure because of the large negative volume change, due in part to the electrostriction of the solvent by the product C02-ion. Thevolume change also includes the partial molar volume, V,, of the electron,which may be positive in some nonpolar alkanes3 but is not known for most alkanes. Quantities like AG,(e) and may change with pressure but also are a function of the particular nonpolar solvent used. This study of reaction 1 in 2-methylbutane (2MB) and 2,2-dimethylbutane (DMB) is part of an effort to measure these fundamental electron properties. A previous study4 of electron attachment to COZin 2MB at pressures less than 1 bar and temperatures below 10 OC showed that the attachment rate is almost diffusion controlled. A key factor in studies of this type is to know the energetics of reaction 1 in the gas phase. A recent related work led to a value for AHr for reaction 1 in the gas phase of +0.61 eVa5 This is in good agreement with both experimental6 and theoretical estimates' of the electron affinity. Here, we report the results of our study of the equilibrium reaction of electrons with CO2 in DMB and 2MB as a function of pressure. The results are compared with those obtained previously in TMS.2
ve
Experimental Section The DMB used was Wiley Reagent Grade (>99%); the 2MB was Fischer 99.9%. These were degassed in vacuo followed by passage through a silica gel-molecular sieves column and Abstract published in Advance ACS Abstracts, August 15, 1993.
0022-365419312097-9488$04.00/0
treatment with NaK. The electron lifetime in the liquid was about 0.5 ms. Impurity rates of electron attachment were determined at all temperatures and pressures prior to addition of COa (Matheson 99.99%) which was used without further purification. The concentration of C02 was estimated by assuming the total amount of C02 transferred from a calibrated volume, in which the pressure was measured with an MKS Baratron pressure transducer, is dissolved in the liquid. A metal cell equipped with bellows and concentric electrodes2 was used. Electrons were generated by irradiating the cell with X-ray pulses converted from 2-MeV electron pulses from a Van de Graaff accelerator by means of a lead target. The pressure container and the pressurizing apparatus have been described.2 The temperature was controlled within 1 OC by means of a thermostat. The technique for the collection of the current decay signal and the method of analysis to determine the attachment (k,) and detachment (kd) rates have been described.2 The analysis was modified to account for reactions occurring during the pulse in the case of 2MB because a longer (1 ps) pulse of X-rays was used. During this time, some buildup of C02- anions occurs. This was negligible in the DMB case where a shorter (0.2 ps) pulse was used. The density and compressibility of 2MB and DMB at various pressures were calculated by fitting a Tait equation (2) to the data of Bridgman.a
In eq 2, P is the pressure and V the volume, and A and B are constants. The subscript r refers to a reference state. Results
DMB. In DMB, electrons react reversibly with CO2 and the attachment rate constant k, is temperature-independent, as in TMS.2 However, k, increases with pressure (see Table I) as is the case in all other nonpolar liquids studied so far. The detachment rate constant kd increases with temperature; the activation energy E, for detachment is 49 f 1 kJ/mol. Figure 1 shows the plot of In K1(K1is the equilibriumconstant of reaction 1) as a function of pressure for DMB. The reaction was studied 0 1993 American Chemical Society
Effect of Pressure on e1oo
+ COZ
1
o
0 .
-
The Journal of Physical Chemistry, Vol. 97, No. 37, 1993 9489
8 V l
n
C02-
.
v v
O
1o8
0
-
V I
A A -
n
v v
A
V I
A
V I
W
11,
2.5 n
V
s
-
N
1
i 2.0
E
0
1.5
A
A
i m
A
0
1 .o
0
A
I
A A
0
4 200
I
I
400
600
1 800
PRESSURE ( b a r s )
+
Figure 1. Equilibrium constants for e- C02 + C0,-in DMB: (0) 24 OC, (0)32 "c,(v)40 OC,('11 44 "c,( 0 )60 O C , (A) 80 OC, (m) 60 OC mobility data, (A) 80 OC mobility data.
TABLE I: Rates of Electron Attachment to COi in DMB k, (lo1*M-Is-l) at temp ("C) pressure (bar) 24 44 80 1 100 200 400 600
800 lo00 1250 1500 2000
2.02 3.08 4.33 6.30 7.76 9.76 10.8 11.6 12.4 13.6
1.75 2.17 3.20 5.90 8.42 10.3 11.6 13.6 15.1 16.5
-
6 -
--t
Here, ps is the electron mobility in the pure alkane solvent and K1 the equilibrium constant of reaction 1. This latter mobility method is most useful at the lower end of the pressure range, but combinationof the two methods gives results over a wider pressure range, as shown in Figure 1. 2MB. Measurement of the mobility of electrons in pure 2MB under various conditions was necessary for the analysis of the rate data, but it is also of interest in itself. The results are shown in Figure 2. The mobility decreases with increasing pressure at all temperatures. The mobility is 0.73 cmZ/V-s at 1 bar and 25 OC; the observed behavior is similar to that reported for n-pentane and 3-methylpentanegfor which the mobility is also low. In that earlier study, the decrease was attributed to a shift in the equilibriumof quasi-freeand trapped electronsto favor the trapped state at high pressure. The rate of electron attachment (ka) to C02 in 2MB as a function of temperature and pressure is shown in Figure 3. At 25 OC, k, is between 2 and 3 X 1012 M-* s-1 at pressures up to 2500 bar. At 70 OC and above,the attachment rate is accelerated at low pressures and then tends to level off above 1 kbar. The attachment rate is approaching the diffusion limit at high pressures; values of k, at high pressure correspond to a reaction
A
Ai
-U
Q1
0
A
I
A
A
e
~
V V
2.70 5.22 7.90 9.67 13.0 15.0 17.6
at 0.14 and 0.19 pm C02; the results given are from the average rate constants. Values of the equilibrium constant of reaction 1 were obtained in two ways. One involved measuring both the attachment and detachment rate and taking the ratio (4 = k,/ k d ) . The other utilized eq 3 and apparent mobility data ( p ) , where the fraction 1/(1 + K1[CO2]) is the probability that the electron is not on C02.
A
0
0
0
0
c 1
~ 0
"
"
~ 500
"
"
~ " 1000
"
~ 1500
PRESSURE( b a r s ) Figure 3. Electron attachment rate (k,) to C02 in 2MB: (0)25 OC, (0) 40 "C, (V) 47 O C , (V) 55 OC, (m) 70 O C , (0) 85 O C , (A) 100 O C .
radiusrof0.3-0.4nmin theequation k, = 4 ~ r p k ~ T f Areaction e. radius of 0.55 nm was reported for 3-methylpentane, in which the reaction is diffusion controlled.2 Figure 4 shows the equilibrium constants (KI) for 2MB. The open points are from k,/ka measurements; filled points are from mobility data.
Discussion Thermodynamic Results. The thermodynamic quantities for reaction 1 are obtained from a plot of In KIversus 1/ T according to eq 4
In K1 = -AHJRT
+ ASJR
(4) Such a plot for the 2MB data at various pressures is shown in Figure 5 for illustration; the results for DMB are similar. The resulting thermodynamicvaluesare given inTable 11. Theaverage values of AHro are -99 kJ/mol for 2MB and -102 kJ/mol for DMB. Similar values were obtained in other solvents.* The free energy of reaction at 25 OC decreases as the pressure increases; this change is compensated by a change in the entropy of reaction, which leaves AH," pressure-independent over the range of pressures studied.
The Journal of Physical Chemistry, Vol. 97, No. 37, 1993
9490
1 on
n 7
I0
-
H W
v
Y
lo7
lo6t *
r
**:r r
0
500
1000
PRESSURE(bars) Figure 4. Equilibrium constant for e- + C02 ==C02- in 2MB: (0)47 OC,(A)55 O C , (0)70 O C , ( 0 )85 O C , (v)100 O C . Open points from k,/kd; filled points from mobility data.
n 7
1 o7
Y
1oe
2.600
2.800
1/ T * l O O O
3.000
3.200
(K-')
Figure 5. Plot of log K1 versus 1 / T for 2MB data: ( 0 ) 100 bar, (0) 200 bar, (V) 300 bar, (v)400 bar, (a) 500 bar, (0)600 bar.
The thermodynamic quantities for solution of the electron, AX,(e), can be calculated from the values of AXr since
+
(5) AX, = AX,(gas) AX,(CO;) - AXs(e) where AX,is AXs(C02-) -AXs( COz). lo This requiresinformation about the gas-phase reaction: we take TAS,(gas) = +0.05 eV as beforell and since AHf (equal to -EA) is +0.61, AGr(gas) = +0.56 eV. To calculate the free energy of solution of the electron, the polarization energy P for C02- is needed and is estimated with the Born equation: P = -(e2/2R,)( 1 - 1/e) using R, = 0.23 r ~ m .The ~ dielectric constant (e) was calculated from density data assuming the Clausius-Mosotti equation. Resulting values of AG,(e) in 2MB and DMB are given in Table 11. Also included in Table I1are thermodynamicdata for reaction 1 in TMS solvent. The data here are derived from rate data in molal units whereas the values in ref 2 were in molar units. It can be seen from examination of Table I1 that the decrease in AGr that occurs in both liquids as the presssure increases is accounted for by two effects: a decrease in P and an increase in AGs(e). In 2MB and DMB, the free energy of reaction decreases by about 0.1 eV for an increase in pressure of 400 bar; this corresponds to a 50-fold increase in K1. The decrease in AGf is due to a decrease in P of -0.06 eV and an increase in AG,(e)
Ninomiya et al. of -0.04 eV. As is shown in Figure 6, AG,(e) increases with density in both hydrocarbons. In TMS by comparison there is only a small increase in AG,(e) with pressure, and the change in AGf is due mainly to a change in P. The effect of solvent on this equilibrium is also shown by the data in Table 11. There is a dramatic shift between TMS and 2MB. At 500 bar, ACr is -0.40 eV in TMS and -0.63 eV in 2MB; Kq is consequently much larger in 2MB. The energies of polarization for C0,-differ by only 0.02 eV in the two solvents. Thus, the main cause of this solvent effect is the difference in the energy level of the electron, which is higher in 2MB (-0.30 eV) than in TMS (-0.55 eV) at 500 bar (see Table 11). It as noted empirically that for each solvent the free energy of reaction, AG, = -RTln Kl, at all temperatures and pressures, when plotted versus density, fell on a common line.5 Such a plot for three solvents is shown in Figure 7; the data for each solvent are represented by a separate line. The data for all temperatures fall on the same line. That is, for a given density AG, is approximatelythesame regardless of temperature. The rationale for this dependence is embodied in eq 5 with X = C. Values of AGfo for 298 K were calculated from this equation; for P we used the density equivalent of the Born equation, and AG,(e) was calculated from the least-squares line through the data in Figure 6. Thus
where PMis the molar polarization, equal to 25.29 for 2MB and 30.08 for DMB, and Mis the molecular weight of the alkane. The values calculated at 298 K from eq 6 are shown by the dashed lines in Figure 7 and are in reasonable agreement with the rest of the data. To determine the entropy of solution of the electron it is necessary to calculate the entropy associated with polarization of the solvent by C02-. If the Born equation is assumed then AS,(CO,-) = (e2/2R8$)(de/dT). Valuesofthedielectricconstant were calculated from densities at each pressure and at incremental temperatures around 25 OC to evaluate de/dT. Most of the entropy of reaction in solution is accounted for by the entropy of solvation of C02-. The entropies of solution of the electron at each pressure obtained with eq 5 are shown in the final column of Table 11. The values are positive; no trend is apparent in 2MB and DMB. The average value for 2MB is +24 J/mol.K, which is less than the average for DMB of +73 J/mol.K. The value for TMS is around 60 J/mol.K at low pressure, comparable to that for DMB. Volume Changes for Attachment to COz. The rate of electron attachment to C02 increases with pressure in DMB at all temperatures and in 2MB at high temperature (see Table I and Figure 3). The observed increases indicate the attachment reaction has a negative activation volume; that is, there is a decrease in the volume going from reactants to the transition state. The values of these activation volumes AVa*are shown in Table I11 (column 3) for various temperatures and pressures. The values of AV,* decreases in magnitude as the pressure increases, but at low pressures AV,* is near -96 cm3/mol for DMB and near -62 cm3/mol for 2MB. The smaller change in 2MB is consistent with the conclusion reached earlier2 (in the case of 3-methylpentane) and below that (since the solvent is constricted around the trapped electron in 2MB) the volume change from reactants to transition state should be less in 2MB than in DMB. From the change of the equilibrium constant with pressure, the overall volumechanges for reaction 1 under various conditions can be deduced; that is
AVf = -RT d In K,/dP
(7)
Since K1 changes rapidly with pressure, the volume changes (see
-0.2
-0.3
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-
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-0.30
-
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\ -0.40
-
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%
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,
of which the first, the electrostriction of the solvent by C02-, dominates. The second term is the actual volume of the C02-ion which, assuming the radius of 0.23 nm, is 3 1 cm3/mol. The partial molar volume of C02 in alkanes is about 60 cm3/ mol.I2 Thus, the difference V,(C02-) - V(C02) is about -30 cm3/mol. Our earlier studygas well as this one indicates avolume change near 1 bar of -20 cm3/mol for the transition of electrons from the quasi-free to the trapped state. If we take this as V(e) for 2MB, then the last three terms of eq 8 contribute only -10 cm'/mol to the volume change of attachment to COz. The electrostriction volume of an ion in a solvent is often estimated by eq 9,13914 which is readily derived Vel = -(e2/2R,e2)de/dP (9) from the Born equation and the thermodynamic relationship AV = dAG/dP. We calculated (l/eZ)de/dP as (XT/3E2)(€+ 2)(e 1),where XT is the isothermal compressibility.3 The contribution of this volume term is given for our experimental conditions in
47 55 70 85 100 24 32 40 44 60 a0 a
125 175 325 450
625 75 100 200 175 400 575
"
~
'
I
-
-59 -66 -5 1 -5 1 4 6 -96 -110 -8 1
-
-84 -a3
% p s
i
0
:
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-
%&yB
% qy I
I
I
I
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0, I
I
I
I
I
A. Solvent: 2MB -298 -251 -255 -246 -228
6.86 6.575 5.51 4.90 4.20
-207 -198 -166 -148 -127
B. Solvent: DMB -23 1 -27 1 -248 -254 -219 -193
5.08 5.26 4.82 5.20 4.22 3.82
-153 -158 -145 -156 -127 -115
Values are X lo5 bar'.
the last column of Table I11 for an assumed radius of C0,-of 0.23 nm. The difference AV,I - AV,,,,~ is around 80-100 cm3/ mol. In evaluating the significance of this volume difference, one should keep in mind the following reservations about the use of eq 9. It is clear from the results that there is a large negative
~
9492 The Journal of Physical Chemistry, Vol. 97, No. 37, 1993 TABLE IV: Electron Trapping Volumes in 2MB AYlo (cm3/mol)
prcssure (bar)
25 O C
100 O C
250
-8.1 -5.1 -4.4 -2.9 -1.9
-27.6 -18.0 -10.4 -6.2 -4.2
500 1000 1500
2000
volume change associated with compression of the solvent around the ion. If this is so, then (1) the assumption of a dielectric continuum in the use of the Born equation, from which eq 9 can be derived, is not validI2 and (2) the value of (l/e2)dc/dP used in eq 9 should be a value for a more compressed solvent rather than for the unperturbed solvent as we have assumed. Hopefully, future theoretical studies will permit more rigorous estimates of electrostriction volumes, and future experiments will be devised to measure such quantities. Then the partial molar volume of the trapped electron may be estimated from reaction volumes. VolumeChangeforTrapping. Thedecreaseinelectron mobility in 2MB is assumed to result from a shift in the equilibrium: e,,
+ tr +e,,
A negative volume change of about -20 cm3/mol was reporetd for this transition in other alkane^.^ If Klo is the equilibrium constant, the mobility can be written a s p = pqf/(1 KIO). Since Klo >> 1, we can define the volume change as
+
If the second term in eq 11, the one involving pqf,is ignored, then the volume associated with trapping can be calculated from the change in observed mobility with pressure. This procedure leads to the values for AV10 shown in Table IV. These volume changes are attributed to electrostriction of the solvent by the trapped electron (see eq 9). The same reservations about the use of this equation, as expressed above for the C02- ion, must be kept in mind. The experimental volume changes were plotted versus the quantity (I/$)dc/dP. A straight line regression through the points gave a small intercept, +3.7 cm3/mol and a slope -z*e*/ 2R, = (-3.7 f 0.5) X IO5 cm3.bar/mol. Here, R, refers to the cavity radius. Within the uncertainty, this slope is the same as that reported for 3-methylpentane.9 From eq 9 and these parameters, the volume change for reaction 10 can be calculated at 1 bar and 25 OC, where (l/e2)de/dP is 7.38 X 10-5 bar', to be -25 cm3/mol.
Ninomiya et al. The second term in eq 1 1 was also ignored in our earlier work9 because of lack of information on the magnitude of pqf and on how it changes with pressure. Recently, however, the change in the conduction band energy, VO,with pressure was measured as a function of pressure for several liquids. From the derivative dV,/dP, the quasi-free mobility was calculated using the deformation potential model.' This model predicted an increase in pqf with pressurefor 2,2-dimethylbutaneand 2,2,4-trimethylpentane, which agreed qualitatively with the experimental data. This introduces some uncertainty into the value of AVIOdeduced here.
Summary The free energy change upon electron attachment becomes more exoergic as the pressure increases. This is attributed to two factors: anincreaseinstabilityofCO2-andadecreaseinstability of the electron as the pressure increases. The attachment process involves a very large decrease in reaction volume which is largely due to electrostriciton of the solvent by CO,-. However, the classical continuum theory accounts for only two-thirds of the observed volume change. Acknowledgment. This research was carried out at the Brookhaven National Laboratory. R.H. is supported by Contract No. DE-AC02-76CH00016 with the U.S.Department of Energy and its Division of Chemical Sciences, Office of Basic Energy and S.N. are supportedby the Japan Society Sciences. M.N., K.I., for the Promotion of Sciencesunder International Joint Research Projects. References and Notes (1) Holroyd, R. A.; Nishikawa, M.; Nakagawa, K.; Kato, N. Phys. Rev. B 1992,45, 3215. (2) Nishikawa, M.; Itoh, K.; Holroyd, R. A. J. Phys. Chem. 1988,92, 5262. (3) Itoh, K.; Nishikawa, M.; Holroyd, R. A. J. Phys. Chem. 1993, 97, 503. (4) Itoh, K.; Tada, S.;Susami, S.;Nishikawa, M. IEEE Trans. Elec. Ins?. 1991, 26,518. (5) Nisbikawa, M. Nuclear Ins?. Methods A 1993,A327, 3. (6) Compton, R. N.; Reinhardt, P. W.; Cooper, C. D. J. Chem. Phys. 1975,63,3821. ( 7 ) Yu,D.; Rauk, A.; Armstrong, D. A. J . Phys. Chem. 1992,96,6031. (8) Bridgman,P.W. The PhysicsofHigh Pressure; Bell & Sons: London, 1949;p 1928. (9) Mufioz, R. C.; Holroyd, R. A.; Itoh, K.; Nakagawa, K.; Nishikawa, M.; Fueki, K. J. Phys. Chem. 1987,91,4639. (IO) Conway, B. E. Annu. Rev. Phys. Chem. 1966,17,498and refs cited therein. (11) Holroyd, R. A.;Gangwer, T. E.; Allen, A. 0. Chem. Phys. Lett. 1975,31, 520. (12) Smith, E. B.; Walkley, J. J. Phys. Chem. 1962,66,597. (13) Whalley, E.J . Chem. Phys. 1963, 33,1400. (14) Hamann, S. D. Physico-Chemical Eflects of Pressure; Butterworths: London, 1957;Vol. 55.
