1930
-
-
J . Phys. Chem. 1987, 91, 1930-1935
S1 T, intersystem crossing and TI So quenching by collision. The average ratios of the observed product quantum yields at 334 and 313 nm, (f$x)334/(&)313, for the only matched experiments at the lowest pressures of added air (runs 1 , 2 and 12, 13 of Table I) are 0.27, 0.07, 0.07, 0.16, 0.56, and 0.38, where x = acrolein loss, C2H4,COz, CO, CH20, and CH30H, respectively. The same quantum yield ratios for experiments near 400 Torr, obtained by interpolation of the 313-nm data, give 0.76,0.67,0.63,0.68,0.63, 0.67, and 0.75 with x = acrolein loss, C2H4,C 0 2 , CO, C H 2 0 , HCHCHO, and C H 3 0 H , respectively. The different ratios for the two pressure regimes may reflect the lifetime differences between SI and Tl states; the decomposition of the excited singlet, the dominant source of products at the higher air pressure, is much more rapid than the decomposition of the triplet, and hence it is much less likely to be quenched than the excited triplet; see Figure 7. Although the 334-nm data are insufficient to perform any detailed analysis of the reactive channel distribution, a comparison with the 313-nm data suggests that at 334 nm decomposition of the excited acrolein triplet processes I, 111, IV, and V is more strongly suppressed than that by process 11. This is unexpected in view of our considerations of the thermochemistry of the various processes.
The Apparent First-Order Photolytic Rate Constant for the Decay of Acrolein in the Lower Troposphere. We may estimate the approximate apparent first-order rate constant (J) for the photodecomposition of acrolein in the lower atmosphere from the present quantum yield data, the absorption cross sections (Figure l), and the estimated actinic flux calculated by Demerjian et al. (best estimate surface albedo).53 The wavelength dependence of the quantum yield cannot be well defined from our experiments at only two wavelengths within the absorption band. However, qualitatively useful J estimates can be obtained by assuming that the two data points for & at 313 (0.0066) and 334 nm (0.0044) define a linear relationship with A, extending over the wavelength region of significance to the lower troposphere (380 > h > 290 nm). Following this procedure we estimate J (s-') X lo6 1.7, 1.6, 1.4,0.84, 0.22, and 0.05 for solar zenith angles ( Z ) of 0, 20, (53) Demerjian, K. L.; Schere, K. L.; Peterson, J. T. Adv. Enuiron. Sci. Technol. 1980, 10, 369.
40,60, 78, and 86O, respectively. Therefore, at a solar zenith angle of 40° the lifetime of acrolein is approximately 10 days. This is comparable to that which we estimate for acetone under similar conditions (14 days) and significantly longer than that for acetaldehyde (3 days) and formaldehyde (0.2 days). The quantum yields of acrolein loss observed in this work at 3 13 nm exhibit a marked dependence on the concentration of air ([MI, molecule ~ m - which ~ ) is described reasonably well (for 8 X lo1' < [MI < 2.6 X l O l 9 by the following relation: I / ( & - 0.00400) = 0.086 -k 1.613 x 10-"[M] For 3 13 nm & increases from 0.0065 at 1 atm (sea level) to about 0.014 at 0.26 atm (10 km). Accordingly, if we consider the 3 13-nm data to be representative of acrolein excitation throughout the 290-380-nm region, the lifetime of acrolein is expected to decrease to less than 5 days near the top of the troposphere (Z = 40O). Therefore, if photodissociation were the only loss mechanism for acrolein in the troposphere, we would infer a reasonably long residence time. However, the measured rate constant for HO-radical reaction with acrolein54is reasonably large, k = 1.9 X lo-" cm3 molecule-'s-]. Assuming an average [HO] of lo6 molecule cm3, representative of the lower troposphere near 40° N, we estimate that the lifetime of the acrolein molecule with respect to HO-radical attack is about 0.6 days. Therefore it is clear that the major sink for acrolein in the lower troposphere, and probably at all elevations within the troposphere, is its reaction with H O radicals.
Acknowledgment. This work was supported in part by an Interagency Agreement (DW 499303 19-01- 1) between the Environmental Protection Agency and the National Science Foundation. We are grateful to Dr. Sasha Madronich for his calculation of the J values employed in this work. Registry No. C2H4,74-85-1; CO, 630-08-0; CO,, 124-38-9; C H 2 0 , 50-00-0; (HCO),, 107-22-2; CHJOH, 67-56-1; CH,=CH', 2669-89-8; N2, 7727-37-9; 02,7782-44-7; H2, 1333-74-0; CHI, 74-82-8; C2H2, 7486-2; CH3C02H,64-19-7; acrolein, 107-02-8. (54) Atkinson, 1983, 15, 75.
R.; Aschmann, S. M.; Pitts, Jr., J. N. Inr. J . Chem. Kine?.
Density of the Electronic States of Graphlte: Derlvation from Olfferential Capacitance Measurements H. Gerischer,* R. McIntyre,* D. Scherson,+ and W. Storck Fritz-Haber-Institut der Max-Planck-Gesellschaft. 0-1000 Berlin 33, West Germany (Received: March 25, 1986)
Differential capacitance data for stress-annealed pyrolytic graphite in acetonitrile containing tetrapropylammonium tetrafluoroborate have been measured over an energy range of 3 eV. The data are interpreted within a model which has successfully been applied to semiconductor electrolyte contacts. In this model the overall capacitance is considered as a space charge capacitance inside the solid in series with a Helmholtz double-layer capacitance at the interface. In contrast to normal metals, the relative low density of electronic states of the semimetal graphite results in the measured capacitance being dominated by the space charge contribution of the graphite. From a correlation between the local position of the Fermi level and the local density of excess charge, the density of states around the Fermi surface can be calculated from such data. The result is compared with theoretical calculations of the electronic properties of graphite.
Introduction It has been shown in a preceding that the unusually small differential capacitance of stress-annealed pyrolytic graphite compared to metallic conductors2 can be related to the relatively
small density of electronic states at the Fermi surface of this material. A correlation between the local positi-on of the Fermi level and the local density of excess charge, based On electrostatics the density was used in Order to and energy band
Permanent address: Department of Chemistry, Case Western Reserve University, Cleveland, OH 44106.
(1) Gerischer, H. J . Phys. Chem. 1985, 89, 4249. P.; Yeager, E. J. Electrochem. SOC.1971, 118, 711. J . Elecrroanal. Chem. 1972, 36, 251. 1974, 54, 93.
0022-3654/87/2091-1930$01.50/0
(2) Randin, J.
0 1987 American Chemical Society
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1931
Graphite Electronic State Density
Relationship between Capacity, Excess Space Charge, and Density of States. In the preceding paper' the following relationship has been derived in order to correlate the space charge capacitance Csc with the excess charge on the electrode:19
css
i 4P
CH
csc Figure 1. Representation of the interfacial capacitance by the Helmholtz double-layercapacity, C,, in series with two capacitors in parallel, representing the contribution of the space charge, Csc,and the contribution of surface states, Css.
of states around the Fermi energy. This was done by using an approximation which neglected the thermal distribution of electrons. We have now performed capacitance measurements over a wider range of applied potential and derived from these data, using a more precise method, the density of states in graphite around the Fermi surface. The Model and the Method. The model used in this paper has been derived for semiconductor/electrolyte contacts3s4 and has successfully been applied to many semiconductors in the depletion region of the space charge. Assuming an ideally blocking contact and disregarding relaxation processes in the charge distribution, which can be shown to be negligible when using low enough frequencies for the capacity measurements, one can represent the capacity by three capacitors, shown in Figure 1. The two capacitors in parallel represent the space charge ( C , ) and the charge in surface states (Css), if the occupancy of surface states is controlled by the position of the Fermi level in the solid.s These capacitors are in series with the Helmholtz capacitor (CH)which represents the fact that there is a thin charge free dielectric layer between the surface and the electrolyte due to the finite radius of the solvation shell around the ions in solution. For semiconductors with a doping level I1019cmW3,the space charge capacitance is in a depletion range much smaller than the Helmholtz layer capacitance and therefore controls the series capacitance. In the accummulation range, the space charge capacitance can reach or even exceed C,. Even in this case, the model has satisfactorily been applied with a reasonable assumption for the Helmholtz layer capacitance and its dependence on the voltage drop.6 We have a similar situation for graphite, where both capacitances in series are of the same order of magnitude. Quantitative data for CH are therefore necessary in order to calculate the sum of Csc + C, from capacity measurements. We shall use in this paper as in the previous one6 the capacity of a mercury electrode in the same electrolyte as a reference for the Helmholtz layer capacity, which should be a good approximation for the negative branch of the excess charge on the solid. The steep increase of CHat a mercury electrode with positive excess charge renders this assumption more questionable for the positive branch. l8 After having calculated within this model the magnitude of the capacitance due to the charge distribution in the solid, we need a relation between the excess charge, the voltage drop in the solid, and the density of states in the electron bands around the Fermi surface in order to relate these measurements to the electronic properties of graphite. We shall derive such a relation for the space charge capacity in the following section, neglecting a possible contribution of surface states. The possible effect of surface states will be considered later in the discussion of the results. (3) Cf. Gerischer, H. In Advances in Electrochemistry and Electrochemical Engineering, Vol. 1 , Delahay, P., Ed.; Interscience: New York, 1961; p 136. (4) Cf. Pleskov, Yu.V. In Progress in Surface and Membrane Science, Vol. 7; Academic: New York, 1973; p 57. (5) Reference 3, p 159. (6) Gerischer, H.; McIntyre, R. J . Chem. Phys. 1985, 83, 1363.
@ is the local electric potential related to the bulk where @ = 0,
4, is the electric potential at the surface, t is the dielectric constant, to is the permittivity of free space, and p ( @ ) is the local charge density at the potential 4. The electric potential shall be defined here as a potential energy in units of electronvolts. The overall excess charge in the space charge layer is related to the integral of the charge from the bulk to the surface by'
(esc)
It is also related to the differential capacitance by Qsc(@,) = - ~ m 6 ~ s c ( @ d@, s )= KSC(@,)
(3)
where Ksc is the integral space charge capacitance. Combining eq 1 and 3 gives 1
A@,)= -Csc(@J Ksc(@s) et0
(4)
Qsc and p are positive if 4, has a negative sign. For measured values of CSc(&),over a range including @s = 0, KSc(@,)can be obtained by integration and therefore p(@,) can be determined from eq 4 if the static dielectric constant is known. A correlation with the density of states can be derived in the following way. In the bulk where @ = 0 there is no excess charge. All occupied electronic states are electrically compensated by the positive charge on the atomic nuclei. The number of the occupied electronic states is given by the integral S__D(E)f(E - EF)d E
~(EF)
E
(5)
where D ( E ) is the density of electronic quantum states in units cm-3 (eV)-l and f(E - E F )is the Fermi distribution function.
The Fermi level in the bulk may be at the energy EFo.When the solid contains excess charge, the potential @(x)varies from the bulk to the surface, x being the distance from the surface. The local position of the Fermi level relative to the vacuum level varies accordingly20 EF(X) = EF(x=O) = E,'
-k @(XI
+ 4,
EFS
(7) Combination of eq 5 with eq 6 and 7 gives the concentration of occupied electronic states as a function of @ E
The excess charge density at any point in the space charge layer is then related to the'difference of the integrals over the density of states between the values for @(x) and @ = 0. Since the charge density at the surface p(@,) is the experimentally available parameter (cf. eq 4) we shall concentrate on this value:
~ ( 4 , )= e [ I ( @ = o )- 1(4,)1
(9)
It can be shown (cf. Appendix and eq A6) that the integrals I(@) can be approximated by
Gerischer et al.
1932 The Journal of Physical Chemistry, Vol. 91, No. 7 , 1987 I
The first term corresponds to the number of occupied states at T = 0 K; the second term gives the correction due to the Fermi distribution function at finite T. Since ( k T ) 2 = 6.25 X at T = 298 K, one can neglect the second term if the gradient dD(E)/dE is not extremely large. Neglecting the second term, one obtains for the charge density at the surface from eq 9 and 10
N
k > 6.0 -u7O: LL
1
w
0
2
F 5.0
I
I
I
\
0
b
$
; i
4
0
s
_1
a
This result invites us to derive a more direct relation between the measurable charge density at the surface and the density of states by differentiation with respect to 4,:
On the other hand, one obtains from eq 4
5 L.0 w w
[L
0-65HZ
LL
LL
~-200Hz
3.0-
A
- 2000Hz
I
I
I
I
+IO
0
-1.0
-2.0
or
eD(EFo + 4s) (13) For a constant CSc, one obtains the previously given result:' 1 D(EFo+ 4,) = -Csc2(+s), woe
if
dCsc(4,) ~
d4S
=0
(13a)
Experimental Section The highly orientated pyrolytic graphite specimen used in this study was obtained from Union Carbide [Cleveland]. Before each experiment the crystal was cleaved by pressing a piece of adhesive tape to one of the edges and then pulling gently so as to expose a new basal plane. A scanning electron micrograph of a typical surface was shown to be free from surface imperfections for a magnification of 1 wm. The sample was then mounted into a Kel-F holder by using a silicon rubber gasket, which blocked most of the crystal front face, exposing only a high-quality circular area of -0.06 cm2. The holder was then inserted into an all-glass electrochemical cell, with separate counter and reference compartments and an auxiliary compartment for the preparation of the electrolyte. A known amount of tetrapropylammonium tetrafluoroborate, purified as described earlier: was introduced into the auxiliary compartment and a small crystal of AgN03 was introduced into the reference compartment. The cell was then Torr, and baked out overnight at sealed, evacuated to 4 X 110 OC. The cell was connected via an all-glass vacuum line, using Teflon taps (Rotaflo), to a glass ampule containing the acetonitrile (ACN) purified as described earlier.' The ACN was first deoxygenated by performing a sequence of four freeze-thaw cycles, each time allowing the solid to equilibrate with argon (Linde, 99.9999). The argon was introduced into the system via a stainless steel line which contained an oxysorb catalyst cartridge (Messer Griesheim) close to the inlet. Subsequently, the solvent was transported through the vapor phase and condensed directly into the auxiliary compartment by cooling the latter to liquid nitrogen temperatures. The frozen solvent/salt mixture was allowed to warm to room temperature and then magnetically stirred until the solution became homogeneous. Transfer of the electrolyte to the main cell was affected simply by tilting the cell to allow the electrolyte to flow into the working, counter, and reference (7) Brown, 0. R.; McIntyre, R. Electrochim. Acta 1984, 24, 995.
(8) Fawcett, W. R.; Loutfy, R. 0. Can. J . Chem. 1973, 5 1 , 230.
The Journal of Physical Chemistry, Vol. 91, No. 7, 1987 1933
Graphite Electronic State Density
i -
I
'
1
'
1
1 I
2 5 U1
. U
I
-
20G W
u
L -1.0 0 +to cathodic
anodic
E - E; = 0, I
I
I
1
I
I
-1.0
-0.5
0
+a5
+IO
+IS
anodic
cathodic
ELECTRODE POTENTIAL ($1 vs PZC / V
Figure 3. Differential capacitance (CH) and integral charge (QH) curves for the system Hg/acetronitrile, (0.1 M) tetraethylammonium perchlorate. Differential capacitance (Cexpt)and integral charge for the system stress-annealed pyrolytic graphite/acetonitrile, (0.2 M) tetrapropylammonium tetrafluoroborate. The broken curves represent the hypothetical values for CH and QH on the anodic branch symmetrical to the cathodic branch.
(eexpt
TABLE I
-1.2 -1.1 -1 .o -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 +o. 1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +1.3 +1.4
+!.5
6.85 5.95 5.16 4.60 4.25 3.95 3.70 3.50 3.30 3.15 2.95 2.80 2.80 2.80 2.85 2.85 2.95 3.15 3.30 3.50 3.65 3.90 4.15 4.40 4.80 5.30 5.75 6.30
4.35 3.90 3.45 3.05 2.65 2.25 1.85 1S O 1.15 0.80 0.50 0.20 0.00 -0.15 -0.40 -0.70 -1 .oo -1.35 -1.70 -2.10 -2.50 -2.90 -3.35 -3.75 -4.25 -4.80 -5.35 -6.05
-0.25 -0.23 -0.21 -0.20 -0.17 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.01 0.03
0.05 0.07 0.10 0.12 0.15 0.18 0.21 0.25 0.29 0.33 0.40 0.45 0.53
(0.35) (0.30) (0.26) (0.23) (0.19) (0.16) (0.13) (0.10) (0.08) (0.06) (0.04) (0.02)
-0.95 -0.87 -0.79 -0.70 -0.63 -0.56 -0.48 -0.40 -0.32 -0.24 -0.17 -0.08 0.00
0.09 0.17 0.25 0.33 0.40 0.48 0.55 0.62 0.69 0.75 0.8 1 0.87 0.90 0.95 0.97
(0.85) (0.80) (0.73) (0.66) (0.62) (0.53) (0.46) (0.40) (0.32) (0.24) (0.17) (0.08)
.
phite/acetonitrile interface (Can)and the corresponding integrated charge values (Qexpt)as a function of & From this figure we can now determine the contribution of dH to &, for the condition Q = QH = Qs, and obtain 4sas the difference. The values determined in this way are shown in Table I. It appears questionable whether the Helmholtz capacitance data for mercury can be used for graphite in the anodic range of the excess charge on the solid since these data may contain a specific interaction between the mercury
1e V
vs. pzc
Figure 4. Calculated differential capacitance of the space charge layer in graphite plotted as a function of the potential energy difference of electrons between the surface and the bulk, @s, Broken line: data, calculated with the alternative assumption for the anodic branch (cf. Figure 3).
atoms and the electrolyte. We have therefore employed, for comparison, another rather extreme alternative for C, in this range. We have used the image of the cathodic branch, shown in Figure 2 as broken lines, for the differential and the integral Helmholtz capacities. Table I contains the data, obtained with this assumption for the anodic branch, in brackets. We can now conveniently read of the values of CeXpt(+J and CH(+H) needed for the calculation of CSc(4,)according to eq 14. The calculated data for Csc(q$) obtained in this way are tabulated in Table I1 together with the integral capacitance values Ksc(ds). Csc is also plotted in Figure 4 as a function of (-dS). Differentiating CSc(4Jwith respect to 4, gives the dCsc(+,)/d4, values which are also shown in Table 11. Given the value of the dielectric constant e = 3.28: we can now determine the density of electronic states at the surface of graphite as a function of 4, from eq 13. The calculated values are shown in Table I1 in units of (states/cm3) eV and (states/atom) eV.
Discussion The density of states for graphite has been calculated theoretically by several techniques. These theories have been reviewed by Tatar and Rabii.Io The structure of the electronic energy states in a narrow energy range around the Fermi surface has been calculated within a tight binding model by McClureI' and by Slonczewski and Weiss'* and over a wider energy range by Johnson and Dre~selhaus.'~Tatar and Rabii'O applied a variational KKR method for a calculation of the total band structure and found, in comparison, that the approach of McClure, Slonczewski, and Weiss agreed well with their ab initio calculations in the region of the Fermi energy. The result of all these calculations is that the Fermi surface covers only a small volume in k-space around the H-K axis. A picture of the Fermi surface can be found in ref 10 and also two curves are given therein for the density of states around EFo,calculated from the two theoretical approaches referred to a b o ~ e . ' ~ ~ ' ~ The results of our measurements are plotted in Figure 5 for comparison with the theoretical calculations. One sees that according to our data the minimum is not as deep and as sharp as (9) Ergun, S.; Yasinsky, J. B.; Townsend, J. R. Carbon 1967, 5, 403. (10) Tatar, R. C.; Rabii, S . Phys. Reu. B 1982, 25, 4126. (11) McClure, J. W. Phys. Reu. 1957, 108, 612. (12) Slonzcewski, C.; Weiss, P. R. Phys. Reu. 1958, 109, 272. (13) Johnson, L. G.; Dresselhaus, G . Phys. Reu. B 1973, 7, 2275.
1934
The Journal of Physical Chemistry, Vol. 91, No. 7 , 1987
Gerischer et al.
TABLE I1
-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 +o. 1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0
8.18 6.64 (10.16) 5.86 (6.95) 5.32 (5.88) 4.89 (5.08) 4.48 (4.55) 4.04 (3.97) 3.83 (3.80) 3.56 (3.56) 3.42 3.50 3.58 3.67 3.99 4.34 4.77 5.48 6.73 9.10 13.52
4.40 3.67 (3.91) 3.07 (3.19) 2.50 (2.52) 2.01 (2.02) 1.52 (1.53) 1.12 (1.11) 0.71 (0.71) 0.35 (0.35) 0 0.35 0.7 1 1.06 1.44 1.85 2.31 2.80 3.41 4.14 5.23
6.69 4.41 (10.32) 3.43 (4.83) 2.83 (3.46) 2.39 (2.58) 2.01 (2.07) 1.63 (1.57) 1.47 (1.44) 1.27 (1.26) 1.17 1.23 1.28 1.35 1.59 1.88 2.28 3.00 4.53 8.28 18.3
18.30 11.60 6.60 (21.4) 4.85 (9.35) 4.20 (6.65) 4.25 (5.55) 3.25 (3.75) 2.40 (2.05) 1.05 (2.00) 0.00 0.80 1.45 2.05 3.35 3.90 5.70 9.80 18.10 33.95
31.6 18.63 11.23 (25.06) 8.69 (1 1.96) 6.95 (8.44) 5.11 (6.24) 4.28 (4.26) 3.53 (3.40) 2.88 (2.86) 2.52 2.70 2.88 3.37 4.46 5.59 7.73 12.3 23.0 48.0
27.86 16.43 10.34 (22.10) 7.66 (10.55) 6.13 (7.44) 5.03 (5.50) 3.28 (3.76) 3.11 (3.00) 2.54 (2.52) 2.22 2.38 2.54 2.97 3.93 4.93 6.82 10.8 20.3 42.3
the calculation leads one to believe, but the order of magnitude is the same. The question arises whether the presence of surface states could be the reason for this difference since we have neglected a possible contribution from C, (cf. Figure 1) in our calculation. This appears, however, impossible since surface states can only increase the capacitance contribution of the solid which would result in larger values of the apparent density of states. Our values are below the theoretical values with the exception of the minimum and values in the immediate neighborhood. However, our result for the density of states at the Fermi level, EFo,agrees well with data from conductivity measurements to which electrons and holes contribute to approximately the same extent. According to KittelI4 and Dillon et al.I5 the concentration of electrons and holes together at 300 K is of the order of (5-6) X l o t 8~ m - ~ . Spaint6gives a value between 10 X lot8and 12 X 10l8 cm-3 for stress-annealed graphite. With a density of states from Table I1 for E = EFoof 2.5 X l e o cm-3 (eV)-l, we can calculate the number of mobile charge carriers by assuming a constant density of states D(EFo)over an energy range of EFof 4kT and multiplying this value twice with the integral I;(1 exp[E/kq)-' dE, once for the electrons and once for the holes. This integral has the value of 0.693kT" and the result for the concentration of mobile charge This . value is in excellent carriers at T = 298 K is 8.6 X loi8~ m - ~ agreement with the other experimental data and thus excludes the possibility that our capacitance data could contain a considanodic cot hodic erable contribution from surface states. The alternative asENERGY ( E -E; 1 / eV sumption for CH on the anodic branch of the capacity gives a steeper increase of D(E)for E C EFoand makes the curve D(E) Figure 5. Curves representing the density of electronic states for more symmetrical around EFo. This appears reasonable and shows stress-annealed pyrolytic graphite determined from our experiment in a range of uncertainty of the data in this region. A similar comparison with calculated curves for graphite obtained by using the full zone *-band Johnson-Dres~elhaus~~ (JD) model or by using the uncertainty, however, will not exist at the minimum and for the Slonzcewski-Weiss-McClure11.'2 (SWMC) model for the energy bands cathodic branch. near the H-K axis. The broken line represents the experimental result In conclusion, the unusual double-layer capacity of graphite for the states below EFowith the alternative assumption for the capacity with the basal plane in contact with an electrolyte can quantiof the Helmholtz double layer on the anodic branch. tatively be interpreted as a quantum effect due to the relatively
+
(14) Kittel: C . Introduction to Solid State Physics, 5th ed; Wiley: New York, 1976; p 238. (15) Dillon, R. 0.;Spain, I. L.; McClure, J. W. J . Phys. Chem. Solids 1977, 38, 635. (16) Spain, I. L. In Chemistry and Physics of Carbon Walker, Jr., P. L., Thrower, P. A., Ed., Vol. 8; Marcel Dekker: New York, 1973; D 111. (17) Blakemore Semiconductor Statistics; Pergamon: Oxford,' 1962; Appendix B. (18) NB: For the positive branch of an n-type semiconductor, C,, is so small that this uncertainty does not matter. (19) In the preceding paper, the minus sign in front of the integral was omitted. It appears here, also in eq 2 and 3, in order to make the notation consistent with the reference state chosen for 4. (20) EFo is per definition negative in a solid since it is referred to the vacuum level.
small number of electronic states at the Fermi energy. It is possible to derive the density of states for an energy range around the Fermi energy from capacity measurements at a solid if the density of states is below (5/t) X 1O2I cm-3 (eV)-'. This is the condition for the space charge capacity being smaller than the Helmholtz layer capacity which will be fulfilled only for some semimetals. Insofar, graphite is a singular case. Bismuth may be another candidate for the application of this method unless e is unusually large therein. The accuracy of the interpretation with respect to the absolute values of D(E)depends on the value for the static dielectric constant t and to some extent on the assumption for C,. The relative dependence of the density of states on the energy,
J . Phys. Chem. 1987, 91, 1935-1941 however, should be quite accurate.
k T dz 1 exp(z) k T dz IOD-(E - EF) 1 + exp(-z) dz kT Sm 0 D+(E-EF) + exp
AI = l m D + ( E- EF) 0
Appendix
In order to take into account the Fermi distribution function a t finite T for the number of occupied states in eq 8 and 9, it is useful to divide the integral eq 5 into three parts:
1935
+
-
+ dz
l:D(E)
dE
+ S EFm D ( E )f(E - EF) d E -
-m F D ( E )d E SE -m F D ( E ) f ( E F - E ) d E = S E
+ AZ ( A l )
The first integral gives the number of occupied states for the temperature equal to 0 K. The second integral represents the occupied states above E F and the third the vacant states below EF. SinceflE - EF) andf(EF - E ) decay quickly with increasing values of IE - EFI, one can approximate the second and third integral by developing D(E) in a power series around D(EF). D*(E
- EF)
i=
D(EF)
z
d
D ( E ) E = E+~
z2 d2 -D(E)E=E~ 2 dz2 (A21
-
with z = E - EF/(kT) and the minus sign valid for the third integral. Introducing the new parameter z, the difference between the second and third integral AZ is
x m D - ( E- EF) 1
+ exp z
('43)
One sees that by inserting the series of eq A2 into eq A3 all terms cancel except those of the second term of the series:
The last integral has the value 0.8225'' and replacing dz by dE, one obtains as the final approximation
The integral over all occupied states in an electronic energy band, Z(EF) of eq 5 , can therefore be approximated by
Registry No. Graphite, 7782-42-5.
Volumetric Properties of Aqueous Solutions of Bovine Serum Albumin, Human Serum Albumin, and Human Hemoglobin Mohammad Iqbal and Ronald E. Verrall* Department of Chemistry, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S 7 N 0 WO (Received: April 18, 1986; In Final Form: November 20, 1986)
Proteinsolvent and protein-protein interactions in aqueous solutions are interpreted from calculated partial specific volumes (&O), adiabatic compressibilities (&O). and expansibilities ( & O ) obtained from high-precision density and sound velocity measurements at 20,25, 30, 35, and 40 OC. The results provide some information about the structural and dynamic features of these proteins and show a greater dependence of volumetric data on hydrophobicity than on any other protein characteristic. Hemoglobin has a larger content of apolar amino acids in a quaternary structure and this is reflected by larger changes in dVo,&.so, and deowith temperature, relative to the serum albumins which have fewer apolar amino acids and a tertiary structure. Subtle differencesbetween the globular structure of BSA and HSA are revealed from compressibility and expansibility data which not only support some previous observations but also provide new information about the surface polarity and distribution of hydrophobicity within these molecules. A slight but systematic concentration dependence of &, $J~,~, and is observed in this temperature range which is interpreted in terms of protein dynamics. It is argued that, with increasing temperature, microunfolding of the protein molecule increases the probability of exposing some of the buried apolar residues to the solvent, thereby altering the surface hydration. The mechanism of solutesolute interactions thus changes as the hydration shells of interacting protein molecules begin to overlap.
Introduction One convenient way to study intermolecular interactions is by thermodynamic methods. A number of studies of protein solutions have been carried out in the past using thermodynamic techniques. Notable among these are those in which volume changes associated with processes like ionization,l ligand and relaxation of various molecular equilibriaIO were observed. As well, partial ~ - ~expansibility" ~ data specific volume,' 1-22 c o m p r e s ~ i b i l i t y , ~and have been reported and found useful in attempts to elucidate the structural features of the protein interior.24 In order to advance the current state of knowledge about intramolecular interactions
* To whom correspondence should be addressed. 0022-3654/87/2091-1935$01.50/0
in solutions, thermodynamic studies of a number of drug and protein systems are being carried out in our laboratory. Improved (1) Rasper, J.; Kauzmann, W. J . A m . Chem. SOC.1962, 84, 1771. (2) Breuer, M. M. J . Phys. Chem. 1964, 68, 2074. (3) Breuer, M. M. J . Phys. Chem. 1964, 68, 2081. (4) Krausz, L. M. J . A m . Chem. SOC.1970, 92, 3168. (5) Kauzmann, W.; Bodanszky, A.; Rasper, A. J . A m . Chem. SOC.1962, 84, 1777. (6) Katz, S.; Shan, M. E.; Shillag, S.; Miller, J. E. J . Biol. Chem. 1972, 247, 5228. ( 7 ) Katz, S.; Crissman, J. K.; Roberson, L. C. J . Biol. Chem. 1974, 249, 7892. (8) Katz, S.; Shinaberry, R. G.; Heck, E. L.; Squire, W. Biochemistry 1980, 19, 3805.
0 1987 American Chemical Society
