5605
J. Phys. Chem. 1991,95,5605-5610
Solvation of Excess Electrons In Apolar Solvents S.Petrucci* and E.E.Kuuhardt Weber Research Institute, Polytechnic University, Farmingdale, New York I I735 (Received: June 11, 1990)
Statistical thermodynamic theories for the solvation of excess electrons in apolar solvents, in particular,straightchain hydrocarbons are developed. Relations between activation energies for drift mobility and solvent properties are found by using a quasichemical approach. The Bjenum-like method calculates the "chemical" formation constant of an exocss electronsolvent species formed under the influence of the induced polarization energy (due to the presence of the ex- electron). A simplified model based on the induced potential of a tetrahedral polarized solvent trap surrounding the electron is also discussed. This second method also yields values of the activation energies for drift mobility of excess electrons in straight-chain hydrocarbons.
TABLE I: Experimental Data for Electrical Mobillties c (cm' V-' d)at Temperatures T (K) and Comepondlog Enegics of Aetiratioa E , (kcal/mol) for Excess Electrow in Lipuid Hydrocarboas cmpd c, cm2 V-I s-l T,K E., kcal/mol
The problem of understanding the large differences in drift mobilities of cxccss electrons in solvents of similar structure such as linear-chain hydrocarbons is still largely unresolved on a quantitative basis. Data reported in the literature' of drift mobility of excess electrons in liquid hydrocarbons show evidence of the following: In methane, electrons appear to occupy delocalized or desolvated states and thus are free to move. In ethane, on the other hand, electrons also appear to occupy localized or solvated states, and even more so, in hydrocarbons of longer chain length such as propane, n-pentane, and n-hexane, with the drift mobility decreasing in the above order. Table I reports literature data2 for mobility and activation energies illustrating the above point. Yet, no quantitative explanation is given, in the literature, of this phenomenon, with the exception of a correlation between mobility p and the energy of the quasi-free electron state Vo in a given solvent.3 We have found evidence of the nature of solvation of excess electrons in these hydrocarbon media, stemming from the correlation between the activation energy of conduction E,, and the molar polarization P. Linear correlation between these values gives the relation (Figure 1) E,, = 0.275 0.127P.with the determination coefficient, 9 = 0.983. In other words, the activation energy for conduction of solvated electrons is linearly dependent on the molar polarization of the solvent. On the basis of the above empirical observation, we have tried to express the differences in the mobilities of excess electrons in linear hydrocarbons by calculating the solvation energy of an electron as a function of the solvent polarization. We have used a "quasi-chemical" statistical approach resembling the Bjerrum theory of ionic association' to develop a theory of solvation of excess electrons in apolar liquids. The potential energy is taken as due to an electron-induced dipole interaction. Thermodynamic parameters such as the free energy change AGO, the enthalpy change AHo, and the entropy change ASo associated with the quasi-chemical solvation process are calculated. One relevant feature is that the calculated ASo shows a correlation with the molar polarization of the hydrocarbon solvents. Further, the values of AGOcan be matched to the experimental barrier of energy for excess electron mobility with reasonable values ((3.45-3.65) X 10-8 cm) of the parameter d , the minimum approach distance between the electron and the induced dipoles.
methand methaneb ethaneC ethaneC propane n- butanea n-pentanea n-pentane" n- hexanea n- hexaned
n- hexaned
* 50 1.7 x 10-3
40 0.4 0.4 0.075 0.16 0.09 0.07 0.09
120 111 111 294 208 296 298 295.5 307 295.2 295.2
-0 -0 1.85 2.6 3.6 4.06 4.3 5.5
'Kestner, N. R.; Jortner, J. J . Chem. Phys. 1973,59,26. *Schmidt, W.; Bakale, G. Chem. Phys. L ~ I I1972, . 17, 617. eDoeldissen, W.; Bakale, G.; Schmidt, W. F. Chem. Phys. Lett. 1978, 56, 347. dMinday, R.;Schmidt, L. D.; Davis, H.T. Phys. Rev. Lett. 1971, 26, 360.
Quasi-Chemical Theory of Solvation of Excess Electrons in Apolar Liquids The induced polarization energy, due to the presence of excess electrons in an apolar liquid, and a radial probability function are used to calculate an equilibrium constant of formation of an electron-solvent species. By use of the solvent polarizability and the static permittivity (and only one adjustable parameter, Le., the electronsolvent distance of closest approach d), the value of the free energy is matched to the energy of activation of the excess electron mobility in straight-chain liquid hydrocarbons. Expressions for the changes AHo and ASo for the polarization of the solvent due to the presence of excess electrons have been derived, and the values of these thermodynamic parameters have been evaluated. We use a quasi-chemical approach based on finding the "chemical" formation constant of an e x e s electronsolvent species formed under the influence of the induced polarization energy (due to the presence of the excess electron). We therefore search for the constant K related to the formal reaction scheme: e+SseS where e, S,and eS represent a molecule of electron, solvent, and solvated electron species eS, respectively. In the following, we shall consider the solvent in large excess, hence the concentration [SI = Cs is taken to be largely unperturbed by the presence of the excess electrons. On the contrary, if Coois the bulk concentration of the excess electrons (mol/dm3) then Ceo= [e] + [eS] Then the equilibrium constant expressing the formation of the species eS will be
+
(1) Kertner, N. R.; Jortner, J. J. Chem. Phys. 1973, 59, 26. (2) Reported in ref 1 Sse Table I for specific references. (3) Springett, B. E.; Jortner, J.; Cohen, M.H.J . Chem. Phys. 1968,18, 2720. Jortner, J. Ber. Bunsen-Ges Phys. Chem. 1971,75,696. Holroyd, R. A.; Allen, M.J . Chem. Phys. 1971, 51, 5014. (4) Hamed, H.; Owen, 8.Physicul Chemistry of Electrolyte Solurions, 2nd ed.; Van Natrand Rdnhold: New York,1967; FUOSS, R. M.;Accaecina, F. Electrolyric Conductuncc; Intencience: New York, 1959. Robinson, H.; Stokes, R. Electdyte Solutiotu. 2nd ed.; Butterwath: Stoneham, MA, 1967. I
Petrucci, s.In Ionic Inteructions; Petrucci, S.,Ed.; Academic Press: New York, 1971; Vol. 1. Farbcr, H.;Petrucci, S . In The Chemicul Physics of Soluution; Dogonadze, R. et al., Edr.; Elwier: Amsterdam, 1986; Part B.
OO22-3654/9~/2095-5605S02.50/0
300
450
neglecting the activity coefficient ratio YoS/Yer taken to be unity. (0
1991 American Chemical Society
5606 The Journal of Physical Chemistry, Vol. 95, No. 14, 1991
Petrucci and Kunhardt one would remain with a constant concentration of induced dipoles n in virtual absence of the inducing electron. The difficulty is circumvented below, by defining as induced dipoles only molecules contained in a spherical shell of radius q. q is the distance where the induced potential energy U becomes equal to 0.5kT. Integrating between 0 and 2r,one finds the number of dipoles in a volume annulus of width r dB and thickness dr
of excess electrons E,+ vs. molar polarization P of straight chain hydrocarbons.
dN(r,B) = 2 ~ n ( & / ~ " ~ 3 $dr d ( - m
e)
By integrating 6 between 0 and r,one fmds the number of induced dipoles in a spherical shell of radius between r and r + dr dN(r) = 27rn(&lzrk39
dr (COS
el:)
dN(r) = 47rn(&/e'rkT)rZ dr It is now convenient to change variables. Let Y = (e%t/c2r"kT)
(IV)
then r = ( & ~ / c * k T ) * / ~ r ~d /r ~= (8a/c2kT)1/4(-f/,r5/4)dY dN( Y) = - 4 ~ n e ~ ( e 2 a / e ~ k T )Y1/2/4)Y-514 ~/'( dY
~/Y'/~) dN(Y) = - u n ( e % ~ / e ~ k T ) ~ / ~ ( edY ob
I
I
I
I
10
20
30
40
p(cm3/mo0-
Figure 1. Activation energy of electrical mobility of excess electrons E,, vs molar polarization P of straight-chain hydrocarbons.
Before integrating, it will be convenient to find the minimum with r of the distribution function (IV). The minimum in the number of dipoles AN(r) for a given finite and fixed range Ar is found from eq IV as follows:
which gives 1 = (2e2a/e2kT)rm4,or 112 = e2a/e2rm4kT.The minimum in AN(r) occurs for a ratio (potential energy/kT) = 0.5; that is, the distance rdn
corresponds to an induced potential energy equal to 0.5kT. We shall take (after similar ion-ion and dipole-dipole theories') this energy to be sufficiently small for r > q to make the entity eS unstable. Therefore, for r > q, the entity eS does not need to be taken into consideration, and both the electron and solvent are defined as free. Thus, the number of induced dipoles is N(Y) = - ~ o ' s n n ( P a / c 2 k T ) 3 / 4 ( e ydY / ~ / 4 ) (VI) where
Figure 2. Polar coordinate representation of an excess electron at the center of the coordinate system and of an induced dipole in the volume element d Y = 9 dr sin 0 dB d$. We shall assume that the induced moment in the molecules of the liquid is m = ea/€$, where e is the electron charge, a the polarizability of the liquid, e the static permittivity, and r the distance between the electron and solvent molecules being polarized. More specifically, r is the distance between the electron taken at the origin of a polar coordinate system (Figure 2) and the volume element dvcontaining the induced dipole rn,which, being induced, is always oriented toward the electron. The induced potential energy U will then be:
U = -em/€? = -e2a/e2r"
b = e2u/c2&kT and d = minimum distance of approach between electron and solvent molecules. Similarly Y(q) = 0.5. The integral in eq VI can be solved by expanding e y in a series
ey= 1
and integrating each term to yield
(11)
The number of induced dipoles dN in a volume element dV = rZ dr sin 0 dB dt$, depends on the concentration of the induced dipoles
ne-"lkT, as follows, where n is the bulk concentration (density) with L Avogadro's of solvent molecules per cm3, n = LCs/lOOO, number and Cs, the concentration expressed in mol/dm3: dN(r,B,t$) = dr sin 0 dB dt$ (111) Notice that according to the above, when r = m, hence U = 0,
+ Y + 12 + + 41 + ... 2! 3! 4!
4!
+
* *)
-(3/4)0! +-+(1/4)1! r3/4
dY=
VI4
VI4
+
(5/4)2!
Then N( Y) = (Cs7rL4i3/1000)(~a/e2d4kT)3/4Q ( C ~ ~ L : B / l 0 o o ) b(VU) ~/~Q
The Journal of Physical Chemistry, Vol. 95, No. 14, 1991 5607
Solvation of Excess Electrons in Apolar Solvents
Q=
@A'
a:$
Q(0.5)
14.0694 13.2713 12.5291 11.8380 11.1939 10.5930 10.0319
5424 3358 2120 1365 898 603 414
9067 3985 1805 840 400 195 97
2.7560 2.7560 2.7560 2.7560 2.7560 2.7560 2.7560
14625 7315 3931 2205 1296 795 508
133.4 34.3 9.0 2.4 0.67 0.19 0.05
3.50 3.55 3.60 3.65 3.70 3.75 3.80
12.3582 11.6765 11.0412 10.4485 9.8951 9.3781 8.8939
1903 1230 812 547 377.3 266.0 191.8
1498 699 334 163 81.2 41.1 21.2
2.7560 2.7560 2.7560 2.7560 2.7560 2.7560 2.7560
3405 1928 1144 708 455.8 304.4 210.2
6.57 3.12 X 1.77 4.61 X 0.49 7.01 X 0.14 1.09 x 3.98 X 1.75 X 1.17 X 1W2 2.88 X 3.49 x 10-3 4.87 x
3.40 3.45 3.50 3.55 3.60 3.65 3.70
13.8242 4690 13.0401 2914 12.3108 1847 11.6317 1195 10.9989 789.4 10.4085 532.9 9.8571 367.8
7065 3123 1422 664 317.9 155.3 77.4
2.7560 2.7560 2.7560 2.7560 2.7560 2.7560 2.7560
11841 6057 3272 1858 1105 685.6 442.4
88.6 22.8 6.01 1.62 0.45 0.17 0.036
3.35 3.40 3.45 3.50 3.55 3.60 3.65 3.70
16.3014 15.3634 14.4920 13.6814 12.9267 12.2234 11.5673 10.9546
19081 11416 6949 4305 2717 1747 1146 766.6
79458 32576 13862 6101 2769 1291 617 301.5
2.7560 2.7560 2.7560 2.7560 2.7560 2.7560 2.7560 2.7560
102783 45 038 21 074 10473 5 501 3041 1762 1066
4244 1048 266 69.5 18.6 5.1 1.43 0.41
d, A
6
3.40 3.45 3.50 3.55 3.60 3.65 3.70
e(@
- e(0.5)
a:::
a:
Ethand 2.51 3.48 4.97 7.3 x
-AGh eV
-AH&
AS*
cal/mol
cal/mol
cal/K mol
-AGB
KO
K
0.074 33 0.077 66 0.08 108 0.08461 0.088 23 0.091 96 0.095 79
7896 3982 2123 1190 670 429 274
1979 1828 1690 1562 1445 1337 1238
0.0860 0.0194 0.0734 0.0679 0.0628 0.0581 0.0538
2353 2204 2064 1933 1811 1696 1587
-3.36 -3.38 -3.37 -3.35 -3.30 -3.23 -3.15
lo-'
0.08 1 08 0.08461 lod 0.088 24 104 0.091 96 10-' 0.095 79 lv 0.099 73 10-g 0.104
1820 1031 61 1 378 244 163 112
3105 2869 2653 2455 2213 2106 1953
0.135 0.125 0.115 0.107 0.0987 0.09 15 0.0848
2895 271 1 2538 2376 2224 208 1 1947
+1.00 +0.75 +0.55 +0.38 +0.23 +0.11 +0.03 1
n-Pentand 1.38 X 1.92 X 2.75 X 4.05 X 6.15 X 9.61 X 1.54 x
1W2 0.074 33 lW3 0.077 66 0.081 08 ol-' 0.08461 lo4 0.088 23 10-' 0.091 96 10-7 0.095 79
6310 3228 1744 990 589 365 236
5138 4744 4382 4050 3745 3465 3208
0.223 0.206 0.190 0.176 0.163 0.151 0.139
3606 3376 3162 2961 2772 2595 2428
5.18 4.63 4.13 3.69 3.30 2.94 2.64
n-Hexand 3.71 0.49 6.83 X 9.73 x 1.42 x 2.16 X 3.36 x 5.27 X
0.071 10 59289 0.074 33 25979 IC2 0.077 66 12156 6041 10-3 0.08 1 08 3173 10-3 0.084 61 1754 lo-* 0.088 24 10-5 0.091 96 1016 615 lo4 0.095 79
6508 6018 5569 5155 4774 4422 4099 3802
0.283 0.261 0.242 0.224 0.207 0.192 0.178 0.165
4587 4299 4030 3780 3546 3326 3120 2926
6.44 5.77 5.16 4.62 4.12 3.68 3.29 2.94
1.1
10-5
x 10-5
0.2 x 10-5 2.8 x lo-'
Propane
-
-
'K = (rU/1000)b3/l'Q = Kd314Q;b = Zu/cad'kT; Y = 3u/c2r'kT; Q = xi:;[fl'-(3/4))/(i(3/4))1?]lct,~. T = 111 K c = 2; = 208.15 K; t * 1.74; u 7 X cm3. d T = 295.5 K; t = 1.84; u 11.07 X lo-*' cm3. ' T = 298 K; t = 1.89; u = 13.09 X In order to correlate eq VI1 to the equilibrium constant K expressed by (I) we need to focus on the physical significance of N ( Y ) . N( Y) is the number of induced dipoles contained in a spherical shell of radius (q - d) and defined as associated to the single electron located at the center of the sphere of radius q. The ratio N( Y)/1 represents then a concentration ratio referred to the same volume Vin which both the central electron and the surrounding dipoles are contained. If order to refer numerator and denominator to the units of mol/dm3, we have to multiply both N(Y) and 1 (electron) by the conversion factor lOOO/VL which cancels out in the ratio N(Y)/l = [eS]/[e]. Then [eSI K=-=[el[Sl
MY) cs
or
K = (Ld/1000)b3/4Q = K,$'f4Q
(VIII)
It is obvious from the above that the calculation of N ( Y ) loses significance once electrons are on the average closer than the distance 2q, in other words, once the cospheres of association of two adjacent electrons overlap. In hexane (see below) for a = 13.1 X 1tYU om3, c = 1.89 at T = 298 K,it results that q = 8.01 X l P cm. The volume associated with a sphere of radius 2q is V = 4/3+7)3 = 1.72 X 10-20 cm3, which is the minimum volume that contains a single electron. Then 1/V = 5.81 X 1019 is the maximum concentration of electrons/cm3, which translates into = lOOO/VL = 0.097 mol/dm3 a concentration far above that which is generally obtainable for excess electrons in liquids.
Calculations Table I1 reports the calculations of K,according to eq VI11 for excess electrons in ethane, propane, n-pentane and n-hexane. The
u
=5X
an3. C T
cm3.
summation giving Q(b) and Q(0.5) has been truncated for i = 44, as the function Q first diverges and then slowly converges to zero. In fact, Table I1 shows Q(b) dissected in four parts from i = O t o i = 11,fromi= 1 2 t o i = 2 2 , f r o m i = 2 3 t o i = 3 3 , a n d from i = 34 to i = 44. We have then calculated the free energy change associated with the scheme e + S e eS, from
AGO= -RT In K
( W
AGOis given in Table 11, expressed both in kcal/mol and in eV. We have found (Figure 1) a linear correlation between the activation energy for conduction E,, of the excess electron and the molar polarization of the solvent P. Calculating the free energy of solvation AGOon the basis of the solvent polarizability a and due to the presence of the excess electron, we have then equated E,, = AG,. This operation implies that the excess electron, in order to migrate as a quasi-free entity, has to overcome an energy bamer of the order of the solvation energy. Justification of the position E,, = AGOderives from evidence of the existence of two states for excess electrons in liquids.5a Also, in order to rationalize experimental data of electron mobility in apolar solvents and mixtures, models describing excess electrons as existing in two states (a quasi-free and a localized ~~ or solvated state) have been used in the l i t e r a t ~ r e .According to these models, the quasi-free state gives a predominant contribution to the total electron mobility of the excess electron. A recent review6 summarizes theories and results. Table 111 reports the experimental activation energies for the mobilities of excess electrons E,, at the temperatures investigated (5) (a) Itskovitch, E. M.; Kumetsov, A. M.; Ultstrup, J. In The Chemical PhyJics of Soluarion;Dogonadze, R.et al., Eds.;Elsevier: Amsterdam, 1988; Part C. Chapter 4. (b) Mindav, R. M.;Schmidt. L. D.; Davies, H.T. J. Phys. Chcm..197i,16,442; (6) Nyikos, L.; Schiller, R. The Chemical Physics of Soluarion; DogoElsevier: Amsterdam, 1988, Part C, Chapter 5. nadze, R.,et al., Us.;
5608 The Journal of Physical Chemistry, Vol. 95, No. 14, 1991 TABLE III: E x p h m t d Actimtiom hwgka E , for Mobility of Exem Eketrollrat Temperature T , S o h t b Frethwgk3,ud M U a u Apprarcb Dbtmce puwtcr d solvent T,K E,, eV 1AGd:eV 10'4 cm ethane 111 0.080(') 0.079 3.45 propane 208.2 0.1 0.11 3.65 n-pentane 295.5 0.15 (4 0.15, 3.65 n-hexane 298 0.20t5) 0.21 3.5s
Petrucci and Kunhardt TABLE Iv: Experimmtd Actimtkm hwgla (kerl/moI) for Moulltkr Of EX- Eketrolg 8t T ~ r p m t ~T, t tSolmtkm l h t b d p k 8 d hbopkq d kihi"Approach Dbbaec Panmeter d E,, A&' ASoIo IO'd, P,b solvent T,K kcal/mol kcal/mol cal/mol cm cm3/mol ethane ill 1.85 1.81 -3.30 3.60 12.52 208.2 2.6 2.54 propane +OS5 3.60 17.64 n-pentane 295.5 3.6 3.61 +5.18 3.40 27.90 n-hexane 298 4.6 k 0.8 4.59 +6.44 3.35 33.02
"Calculated according to the solvent polarization model.
for the four solvents. A match between the AG, calculated above, and the E,'s can be reached as indicated by using values of d (the minimum approach distances between the electron and the induced dipoles) between 3.45 X 10-8 and 3.65 X 10-8 cm. In the above treatment, the free energy AGO instead of the enthalpy AHo is associated with an energy of activation E,,. Recall that the process of solvation of excess electrons in hydrocarbons, according to the model used, is due to electronic and atomic polarization, not to molecular reorientation as in polar media. Therefore, it appears reasonable to ask whether AGOis mainly enthalpic or entropic, that is, whether or not the solvent is intramolecularly (rather than intermolecularly) disturbed by the presence of the neighbor electron. In other words, whatever structure the solvent has, upon the injection of the e x e s electrons, (according to thew considerations) the structure remains unaltered, only the electronic and atomic polarization being affected by the presence of the excess electrons.
'Calculated according to the solvent polarization model. is an accepted rule" to estimate the atomic polarizability as approximately 10%of the electronic polarizability. The total polarizability a will be a = a, O.la, = I.la,, and the molar polarization P = I.lP, = I.IR, with R the molar refraction.
+
Tbermodyarmic Panmeters In order to investigate the above issue, namely, the enthalpic vs the entropic nature of the AGO'S, the following analysis has been
performed. From
AGO = -RT In K = -RT In KO - Y4RT In b - RT In Q
-(y) aAGo
U o
P
(X)
= R In KO + Y4RIn b + R In Q +
a In b ab + RT a In Q ae ab
%RT-
ab aT aQ ab E As b = e2a/c2d4kT,neglecting the temperature dependence of a I
ab
-(ga/t2bkT)(
f + 2-)a In
aT
t
-b(
I
f + 2-)a In
aT
t0
+ 2%)
-
ASo= R In KO + 3/qR In b + R In Q -
We then calculate AH, from AH0
AGO+ T U 0
Therefore A H 0 = -R(-
-
1 4 + b3J4
Q
;)( + a 1
2%) In T
-
I
I
30
40
Figure 3.- Entropy change ASovs molar mass M of the hydrocarbon and vs molar polarization P for excess electrons in straight-chain hydrocarbons.
Then
= R In KO + Y4RIn b + R In Q - Y4R7'($
I
20
P(cmYmoi)
t
aT
Also, since
AS0
I
(XII)
Table I1 reports the calculated Moand ASo as a function of the distance parameter d for e x m s electrons in ethane, propane, n-pentane, and n-hexane at the temperatures where the respective activation energies for mobilities have been determined.'.'** For the calculation, the reported values9 of dc/dT = -0.16 X 10-2 (for n-pentane) and dt/dT = -0.155 X (for n-hexane) have been used. Since no values of the temperature coefficients of t are available for ethane and propane, they have been assumed to be d€/dT -0.151 X lo-'. Table IV reveals no dramatic difference in the values of d used to match the E,'s with the M < s rather than with the AGO'S, as done in Table 111. An interesting feature is that the values of the entropies seem to reverse from negative to positive by elongation of the hydrocarbon chain. If correct, this effect may reflect a reversal from (7) Doeldisacn, W.;Bakale, G.; Schmidt, W.F. Chrm. Phys. Lett. 1978, S6, 347. (8). Kmtner, N.R:; Jortner, J. J. Chem. Phys. 1973,59,26. Minday, R.; Schmidt, L.D.; Davits, H.T. Phys. Rev. Letr. 1971, 26, 360. (9) Handbook 01Physfcs and Chemistry; The Chemical Rubber Co.: Cleveland. OH.
The Journal of Physical Chemistry, Vol. 95, No. 14, 1991 5609
Solvation of Excess Electrons in Apolar Solvents
R 1P a ,
1O"m - 1 0 1 3 ~ ~
I4.1 E,,
6.72 2.67 0.29 0.24 1.13 0.785 1.63 1.13 0
11.40 4.50 0.39 0.33 1.24
16.04 6.36 0.64 0.54 2.33 1.64 3.35 2.36 2.6
0.88
1.78 1.27 1.85
20.07 8.21 0.80 0.67 2.82 1.99 4.06 2.86
25.36 10.06 0.95
+
+
0.80
3.28 2.32 4.72 3.34 3.6
30.02 11.90 1.10 0.92 3.69 2.60 5.31 3.74 4.6 f 0.8
5.5 6.0 5.5 6.0 5.5 6.0
' R = 4&H = 6.72 cm3/mol; c = 3 = 1.63 at t = -164 OC. b R = 6 k H & = 6(1.68) 1.30 = 11.40 cm3/mol. For ethane, the estimated f = 1.74 at value e s 2.0 has been uscd. e R = 8 k H 2& = 8(1.68) 2(1.30) = 16.04 cm3/mol c = bmc (4.2 X 1r2)=t 1.61 - 0.2 X 1.8. ' R = 12&H 4'& 5 25.36 cm3/mol; 1.84. f R = 1 4 R c ~+ 5& 30.02 I -65 'c. 'R IOR~H 3 R a = 20.70 Cm3/mOl; cm3/mol; (20 = 1.89. Walues for the parameter r for which m,Up and 10plwere calculated.
+
+
'
F
0
+
+
1
8
Arrhcnbus Energy of activation E l and calculated solvent molar poiarkation OWrgY IUpllOr r = 5.5 x 10-8cm and r = 8.0 x 10-Ocm lor excess electrons e Straight chain liquid hydrocarbons forming a tetrahedral trap.
+
a distance parameter d. The position AHo 5 E, implies the existence of two states for the excess electron, a view already advanced in the literat~re.~" Acknowledgment. We are grateful to the Office of Naval Research for support of this work.
020
10
040
50
P(unhol)4
Figure 4. Arrheniur, activation energy of drift mobility E,, (kcal/mol) and solvent molar larization energy lOd (kcal/mol) w solvent molar polarization P (cmP ) for exocu) electrons m straight-chain liquid hydrocarbons. The values of have been calculated with the distance parameter r = 5 3 X lo-' 6.0 X lo-' cm.
+
structural order making (ASo < 0 or [Sdo- (SsO S,")] < 0) to structural order breaking (ASo > 0) upon electron solvation, by elongation of the hydrocarbon chain. More data would be needed in order to render this observation less tentative. Yet,even from the above scant data, a distinct correlation emerges. First notice (Figure 3) that the ASGs are a function of the molar mass M of the solvent hydrocarbons (30,44,72, and 86 amu, respectively, for ethane, propane, n-pentane, and n-hexane). In fact, by nonlinear regression it results that ASo = -14.10 + 0.428M - 0.0022W with 9 = 0.9996 It appears then trivial, by recalling the Clausius-Mmtti expression for the molar polarizations
p=-M=%uaL c+2 p
(XIII)
that ASo ought to be also a function of P. In fact, (Figure 3) nonlinear regression between the AS,,% and the Ps (Table IV), calculated from the polarizabilities a and eq XIII, gives ASo = -16.19 + 1.238P 0.0168P with 9 = 0.9995
-
Conelurioar The reported correlation between activation energy for mobility and solvent molar polarization has lead to the calculation of the free energy and the enthalpy of solvation, using a quasi-chemical Bjerrum-like theory. The depth of the solvation well, expressed by A H 0 has been equated to the activation energy E,, by using
APpeadiX In the previous sections, a phenomenological description of electron solvation based upon a dipole-induced potential energy has been derived. Actually, a simpler and straightforward approach is feasible. In this section it is shown that the change in drift mobility of excess electrons dissolved in straight-chain hydrocarbons, in going from ethane to propane, n-pentane, and n-hexane, can be rationalized in terms of the polarizability of the solvent. Specifically, the experimental Arrhenius activation energy of the mobility, ranging from about zero in methane to -4.6 kcal/mol in n-hexane can be recalculated by equating it to the potential well created by the electrical field of the electron polarizing a tetrahedral solvent trap. Molar refractions, the Lorenz-lorentz equation, and a simple model give a satisfactory correlation between the calculated polarization energy and the activation energy for electron mobility. CalcUQtionof Actiwlioa Eaersy for Conduction. We now offer an explanation of the change of the activation energies going from methane to n-hexane in terms of charge-induced dipolar energy (due to the polarization of a tetrahedral solvent trap). The final result of this section is that we predict the order of magnitude of the Arrhenius activation energy of the mobility of excess electrons in straight-chain hydrocarbons. We begin by showing a detailed calculation for one hydrocarbon solvent, namely, propane, and collecting a tabulation of the results of this calculation for other hydrocarbons. The Lorenz-Lorentz equation10reads, for the electronic polarization
where R is the molar refraction and ar, the electronic polarizability. On the other hand, the additivity rule for the molar refraction" reads for propane, CH3-CH2-CH3,R = 8RCH 2% = 8 X 1.68 2 X 1.30 = 16.04 cm3, Hence, a, = 3R/4rL = 6.36 X cm3/molecule. It is an accepted rule" to estimate the atomic polarizability as approximately 1096 of the electronic polarizability. The total polarizability a = q 0 . 1 =~ 7~X~lWu c " / m o l ~ l e . The induced dipole moment m in propane, due to a neighbor electron of charge e at distance r is
+
+
+
(10) Loren@ H.A. ANI. Phys. Chem. 1880,9,641. Lorrnz,L.Ann. Phys. Chem. 1 8 8 O , I I , 10. (1 1) Price, H. A. Dielectric Pmpertles and Molecular Behaviour, Van Natrand Reinhold London, 1969.
J. Phys. Chem. 1991,95, 5610-5620
5610
In equation IIA, the permittivity of the solvent has been calculatedIZ at t = -65 OC from the equation12 e = 1.61 + -0.2 X 10-3(-65) = 1.74. The arbitrary parameters r = 5.5 and r = 6.0 A as the electron-solvent distances have been used. The model envisages a tetrahedral trap formed by four hydrocarbon molecules surrounding the localized electron at the center of the cm, m = 0.64 X trap. For r = 5.5 X esu cm. With this value we can now calculate the polarization energy Up due to the electron at the center of the trap as em Up = -4= -2.33 X lo-" erg/molecule = -0.146 eV
This corresponds to increasin r by a factor of (lS)'/' in order to match, after conversion to I the values of E,. In other words, the same calculated values obtained with r = 5.5 X 10% cm and r = 6.0 X lo" cm, have been obtained (for an octahedral model) cm and r = 6.64 X 10-8 cm, respectively. with r = 6.09 X In the Introduction of the paper it has been pointed out that E,, is proportional to the molar polarization of the solvent. We advance the hypothesis that lOd E,,. This implies that the barrier of energy of activation for mobility of excess electrons, subjected to a gradient of potential of 1 V/cm, is comparable to the work necessary to extract the electron from the potential well created by the polarization or the solvent. Table V reports a breakdown of the above calculation for various straight-chain hydrocarbons. Figure 4 reports again E,, vs the molar polarization of the solvent. The calculated molar polarization energy lOpl is also reported for comparison, showing a surprisingly good correlation to exist, given the simplicity of the model assumed. The correlation between E,, and 10plis, however, lost if one tries the above calculation for non-linear or branched hydrocarbons. A discussion in the literatureI3 exists involving anisotropy in the polarizability of branched hydrocarbons and possible directional effects in electron localization.
%
f
69
which on a molar scale becomes (2.33 X 10-20)(6.02X loz3) = 3.35 kcal/md lop1 = 4.184 Repeating the calculation with r = 6.0 X 10" cm, 10pl= 2.36 kcal/mol, whereas the reported activation energy for electrical mobilityz E,, = 2.6 kcal/mol for excess electrons in propane. Notice that altering the arbitrary model of a tetrahedral trap to, for example an octahedral trap, changes Upto Up = -6em/e#. (12) Data from Handbook of Chemistry and Physics, 44th 4.; The Chemical Rubber Co.: Cleveland, OH, 1963.
(13) Dodelet, J.-P.; Freeman, G. R. Can. J . Chem. 1972, 50, 2667.
Absolute pK, Cakulattons with Continuum Dleiectric Methods Carmay Lim, Don Bashford, and Martin Karplus* Department of Chemistry, Haruard University, 12 Oxford Street, Cambridge, Massachusetts 02138 (Received: July 16, 1990)
Solvation free energies and pK, values of models for ionizable side chains of amino acids are calculated by using continuum dielectric methods, integral equation techniques are also investigated. The dependence of the solvation free energies on the parameters (charges and van der Waals interactions used to describe the model compounds) is explored by comparing different sets that are being used in protein and liquid simulations. The solvation free energies calculated with both continuum and integral equation methods and various parameter sets agree qualitatively with experiment but are not accurate enough to yield absolute pK, values. To obtain the experimental solvation free energies and pK, values of the model compounds with the continuum dielectric method, an adjusted parameter set is introduced; only very small changes from the standard parameter values are required. The set of calibrated parameters is tested on some bifunctional compounds and yields pK, changes in reasonable agreement with experiment. However, the pK, changes are very sensitive to the solution conformation. This may result in large pK, errors if conformational changes (e.g., between the ionized and neutral species) are not taken into account.
1. Introduction Knowledge of the pK, values of ionizable groups is important for an understanding of many a r m of chemistry, both in the gas phase and solution. They are of particular interest for elucidating reaction mechanisms, especially those involving proton transfers, and for interpreting the binding of substrates or inhibitors to enzymes. However, experimental determinations of individual pK, values are difficult in complex systems. One case in point concerns the direct measurement of titrating group of catalytically important residues or of substrates in enzyme-substrate complexes. Kinetic assignments of pK,'s are often complicated by uncertainties in interpreting the pH dependence of the measured parameters.' It is useful, therefore, to have reliable and accurate means of calculating relative and/or absolute pK, values and to have an understanding of the factors involved. Such an understanding is essential for interpreting the measured effective pK, values in proteins and other complex polyions. Both microscopic and macroscopic theoretical methods are now available for the estimation of solvation free energie~.~JThis ( I ) Knowlea, J.
SCHEME I BH(g)
A%
~AGw)
BH(s)
We)
AGa
+
JAGW*)
AW)
B(s)
H*(g)
+ H*(s)
makes it possible, in principle, to determine theoretical relative or absolute pK, values from the thermodynamic cycle (Scheme I). In Scheme I, AG, and AGa are the gas-phase and solution free energy of ionization and the AG,'s are solvation free energies. Given these results, the pK, of BH(s) is
1
pK, = 2.303RTAGa
=- 1 {AG8 + AG,(B) - AG,(BH) 2.303R T
+ AG,(H+))
(2) Jorgensen, W. L. Acc. Chem. Res. 1989, 22, 184. (3) Gilwn, M. K.; Honig, B. H. Proteins 1988, I , 7 .
R. CRC Crfr. Reu. Blochem. 1975, 3, 165. 0022-3654/91/2095-5610$02.50/0
-1
Q
1991 American Chemical Society
(1.1)
