4119
J . Phys. Chem. 1991,95, 4119-4126 L - e x p [ a ( a - VIZ + ,B(b - Y)zl dY =
stant, leaving only the 6-dimensional integral
.\
u(x1,...xm,t) = S#iJ#i
5) 112
- b)2](
exp[ *a(+a @
(B-6)
twice to the double integral in eq B-5,we obtain J-2exPl-b!lcvl -
+ X[(xf.-
+ (-4- r',)211dJ4 d d =
Since (B-4)
where X = -1 /4t. Finally, after inserting eq B-7into eq B-5,we obtain
we have
U(xl,...,xmJ) =
m i-1
L
Caf(1 i=Zj=l k-l
+ 8bfr)-'l2
-Hill2 - x1112 1
+ 8b#
)
which is essentially eq 9 of the text. Registry No. Ar, 7440-37-1.
Applying the formula
Vapor Pressures of Aqueous 2-Butoxyethanoi Solutions at 25 'C: Transitions in Mixing Scheme Yoshikata Koga Department of Chemistry, The University of British Columbia, Vancouver, British Columbia, Canada V6T 1 Y6 (Received: September 17, 1990)
The vapor pressures of aqueous solutions of 2-butoxyethanol (BE) were determined at 25.00 OC. The partial pressures of BE and H20and hence the excess partial molar free energies, G E(i) ( i = BE or H20) were calculated by the Boissonnas method. Using the values of the excess partial molar enthalpies, H,,, E(i) (i = BE or H20),measured previously in this laboratory (Can. J . Chem. 1989, 67, 671), the excess partial molar entropies, Sme(i) (i = BE or H20), were calculated. From the concentration dependence of these excess partial molar quantities, more detailed insight into the mixing scheme became available. In particular, two boundaries were noted within the single-phase domain that separate the regions in which the thermodynamic properties and hence the mixing schemes are different. The transition in mixing scheme from the water-rich region to the intermediate region is associated with cusp or peak anomalies in the quantities that are proportional to the third derivatives of the free energy. The transition from the intermediate to the BE-rich region appears to be accompanied by jump anomalies of the fourth derivatives of the free energy.
Introduction Table I lists the thermodynamic quantities of a mixture in the @,T,n,)variable system. They are classified in terms of the order of derivative of the Gibbs free energy, G, with respect to variables p, T, and n,. The braces (()) contain the variables of differentiation.
As the order of the derivative increases, the information contained in a quantity becomes more detailed. The free energy, G, dictates the fate of a mixture a t equilibrium. The first derivative of G with respect to T separates out the entropic and enthalpic contributions; that with respect to p provides the volume that the system occupies, and that with respect to n1 gives the chemical potential of the ith component. The second derivatives with respect to purely single variables, T, p, or nt, yield the heat capacity, C,, the isothermal compressibility, Kn or the concentration fluctuation, N ( ( A ~ B )respectively. ~), The physical meanings of these quantities have been rigorously proved to be the fluctuations in enthalpy, density, and the concentration, respectively.' The partial molar quantities, H(0,S(i), and V(i)provide the real value, or the actual ( I ) Lifshitr, E. M.;Pitaevskii, L. P.Srarisrical Physics, 3rd ed.;Sykes, J. B., Kearsley, M.J., translators; Pergamon Press: Oxford, 1980.
0022-3654/91/2095-4119$02.50/0
TABLE I: Thermodynamic QwaHHea of tbe (p,?'dtf) system 0th 1st deriv deriv 2nd deriv 3rd deriv 4th deriv
a = RT( 1 - x)*/(N(ap,/an,)]:concentration fluctuation. G, = S;4r?3(gu(r) - 1 ] dr: Kirkwood-Buff parameters.
contribution, of the ith species in terms of enthalpy, entropy, and volume, respectively, at the given concentration. For example, H(i) or HmE(i),is measured as the enthalpic response of the entire system when an infinitesimal amount of the ith component is added. Figure 1 depicts such a measurement process. Its de0 1991 American Chemical Society
4120 The Journal of Physical Chemistry, Vol. 95, No. 10, 199'I
Koga by fitting a single analytic function, eq 1, for the excess (integral) CmE= -a(xB In (1 - bxw) + xw In (1 - cxe))
f o r small 6 n g
Figure 1. Measurement process of HmE(B).
rivative, the third derivative, NdH(i)/an,), is the rate of change in the real enthalpic value for an infinitesimal change in n,, or it indicates whether the real enthalpic value increases or decreases on addition of an extra ith species. Therefore, N(aH(i)/ani} is a measure of the interaction among ith species in terms of ent h a l ~ y . ~Similarly, -~ N(aS(i)/ani) is a measure of the inter-ith species interaction in terms of e n t r ~ p y . ~Heple? ,~ has used a third derivative, aCp/ap, to compare the degree of structuredness between D 2 0 and H 2 0 , and also a fourth derivative, aCp(i)/ap, for discussing the structure-making or structure-breaking nature of a solute in an aqueous solution. The interpretation of the physical meanings of the third and the fourth derivatives mentioned above remains intuitive. A rigorous connection between such higher order derivatives and microscopic information is yet to be formulated. In previous papers, we have reported values of the excess partial molar enthalpies, HmE(i),for aqueous solutions of tert-butyl alcohol (TBA)2.9,'0 and 2-butoxyethanol and also those of the excess partial molar entropies, SmE(i), in aqueous TBA solution~.~.~*' Such data have provided more direct information about the solute-solvent and the solutesolute interactions. The values of HmE(BE)had not been readily available in literature, except for a single work" published independently at about the same time as O U ~ S . ~Existing '~ values of mixing enthalpies for the BE-H20 for example, are the (integral) molar enthalpies, the first derivative of the free energy. In theory, the partial molar quantities can be obtained by differentiating the data of the integral molar quantity. In practice, however, the integral molar quantities have not been determined at small enough intervals in concentration to withstand differentiation without losing accuracy. The purpose of the present paper is to obtain the values of the excess partial molar free energies, GmE(i)(i = BE or H,O), in BE-H20 mixtures. The values of the excess partial molar entropies, SmE(i)(i = BE or H20), can then be calculated using the data of HmE(i)(i = BE or H20).3 Thus we gain some additional insight into the mode of mixing in terms of entropy. Earlier, the total vapor pressures of the BE-H20 system were measured.14 However, there were only eight data points covering the entire concentration range. Moreover, the data were analyzed (2) Koga, Y. Cun. J . Chem. 1988, 66, 1187. (3) Siu, W.; Koga, Y. Can. J. Chem. 1989, 67, 671. (4) Koga, Y.; Wong, T.Y. H.; Siu, W. W. Y. IUPAC Conference in Chemical Thermodynamic and Calorimetry; Aug 25-29, 1989, Bcijing; Paper c75. (5) Koga, Y.; Wong, T. Y. H.; Siu, W. W. Y. Thermochim. Acta 1990, 169, 27. (6) Koga, Y.; Siu, W. W. Y.; Wong, T. Y. H. J . Phys. Chem. 1990, 94, 3879. (7) Koga, Y.; Siu, W. W. Y.; Wong, T. Y. H. J . Phys. Chem. 1990, 94, 7700. (8) Hepler, L. 0.Con. J . Chem. 1969, 47, 4613. (9) Koga, Y. Cun. J . Chem. 1986,64, 206. (IO) Koga, Y. Can. J . Chem. 1988, 66, 3171. (1 1) Anderson, B.; Olofsson, G. J. Solution Chem. 1988, 17, 1169. (12) Onken, U . Z . Electrochem. 1959, 63, 321. (13) Davis, M. 1.; Moliva, M. C.; Douheret, G. Thermochim. Acto 1988, 131, 153. (14) Scatchard, G.; Wilson, G. M. J . Am. Chcm. Soc. 1964, 86, 133.
(1)
molar free energy, GmE,in the entire concentration range, where xBand xw are the mole fraction in the liquid phase of BE and H20, respectively. From the concentration dependence of HmE(i) (i = BE or H20),3,6it was apparent that there were three distinct concentration regions in each of which the thermodynamic properties and hence the mixing scheme was qualitatively different from those in other regions. Therefore, a single analytic function cannot appropriately describe the thermodynamic behavior in the entire concentration range. Indeed, there are attempts at fitting different equations in four concentration regions to the data of various thermodynamic q u a n t i t i e ~ , ' ~while * ' ~ the choice of boundaries appears artificial. Thus, measurements of the vapor pressures of the BE-H20 system at closely spaced values in concentration and a data analysis without relying on any fitting function may be warranted. We have earlier reported a similar analysis for the TBA-H20 system.4,5,7
Experimental Section The total vapor pressures were measured by a static method described in detail previously for the study of the TBA-H20 Since the vapor pressure of BE is much lower than that of TBA, transfer of BE gas into the cell was facilitated by cooling the cell at the liquid nitrogen temperature. The cell was then immersed in the water bath and was allowed to stand overnight to restore the thermal equilibrium before measuring the vapor pressure.
Data Analysis At the outset, it may be useful to restate the definitions for various quantities used in the subsequent discussion. Thus, the excess partial molar enthalpy of the ith species, HmE(i),is defined as HmE(i) = (a*/ani)p,T,n,
(2)
which can be calculated by using the mole fraction, xi, as HmE(i) = (aH@/dN)p.Tj, + (1 - Xi)Ia(p/N)/aXiIp.T&
(3)
The first term on the right is rewritten as
a p / a N = PIN
= H,E
(4a)
and HmEis the excess (integral) molar enthalpy. Also HmE= CxiHmE(i) i
(4b)
The reader is requested to use caution in that there is an important difference in physical meaning between HmE,the excess (integral) molar enthalpy, and HmE(i),the excess partial molar enthalpy of the ith species, although the difference in notation does not stand out. The same expressions apply for entropy by changing H to S. The chemical potential of the ith species in the liquid solution, pi, is written as pi
= pro(pure)
+ R T In xi + GmE(i)
(5)
where the excess partial molar free energy GmE(i)can be written by using the activity coefficient Ti as GmE(i) = RT In
(6) The chemical potential of the ith species in the gas-phase mixture is given by p? = p?'(pure, 1 atm) + R T In p , (7) T,
assuming that the gas-phase mixture is ideal. pi is the partial pressure of the ith species in the gas-phase mixture. If the pure (15) Douheret, G.; Pal, A.; Davies, M. I. J . Chem. Soc., Furuduy Truns. I 1989,85, 2723.
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4121
Vapor Pressures of Aqueous 2-Butoxyethanol Solutions
AhvaP(i)and HmE(i)(i = BE or H20)are the enthalpies of vaporization and the excess partial molar enthalpies,' respectively. We repeated the same analysis using the corrected values of p and arrived at the final results listed in Tables I1 and 111. As pointed out in the previous paper,' there are three concentration regions in each of which the behavior of Hme(i)(i = BE or H20) is distinctively different from that in the other regions. It is therefore expected that the ex- (integral) molar free energy, GmE,cannot be described by a single analytic function for the entire concentration range."Js The assumption that a single analytic function exists for GmEfor the entire concentration range is the prerequsite for using the common Barker's method. We thus chose to use the Boisonnas method,I7a method of numerical integration based on the Duhem-Margules relation. A small nonideality in the gas-phase mixture can be incorporated.'* However, we used the original Boissonnas method, ignoring the nonideality of the gas phase. The degree of nonideality of the liquid phase is so large that the correction for a small nonideality in the gas phase is not significant. The numerical integration in this method was possible only from either xB = 0 or xw = 0 up to the point where = xB, Le., the azeotrope, ye being the mole fraction of BE in the gas-phase mixture. This is because factor (pw, eq 16 becomes infinite at YB = xB. It turned out that = XB at XB I0.0065, as shown in Table 111 and Figure 6. We therefore applied this method from xw = 0 to xw = 0.9935 (xB = 0.0065). Namely
TABLE II: Measured Vapor Pressures temp, O C
ZB
- .
p, Torr
p-
(25.00 "C),
Torr
.
Series I
24.831 24.834 24.844 24.839 24.839 24.839 24.839 24.840 24.831 24.828 24.829 24.831 24.841 24.818 24.829 24.824 24.829 24.830
O.OO0 0.001 741 0.004105 0.006 49 1 0.009 043 0.01 1 81 (0.014 62 0.01741 0.020 35 0.023 34 0.026 46 0.029 73 0.033 01 0.038 92 0.049 41 0.064 15 0.079 64 0.095 6 1
23.519 23.537 23.551 23.576 23.571 23.562 23.541 23.544 23.529 23.517 23.512 23.508 23.514 23.474 23.487 23.466 23.446 23.429
23.756 23.769 23.770 23.802 23.797 23.788 23.768) 23.768 23.761 23.759 23.753 23.746 23.738 23.730 23.728 23.713 23.686 23.668
Series I1
24.829 24.829 24.837 24.839 24.829 24.829 24.827 24.828 24.829 24.839 24.840 24.839 24.839 24.831
1.Ooo
0.950 8 0.905 6 0.844 6 0.771 8 0.720 7 0.635 3 0.526 2 0.392 8 0.292 7 0.1995 0.1364 0.1025 0.082 21
0.848 2.188 3.582 5.840 8.629 10.446 13.649 17.278 20.556 22.108 22.920 23.245 23.394 23.471
0.859 2.213 3.619 5.899 8.720 10.555 13.793 17.459 20.769 22.323 23.141 23.470 23.620 23.709
APW = AP
vw =
With the two equilibrium conditions, pt(pure) = p?(pure) and = p,, eqs 5, 7, and 8 yield
pt
(9)
Then
SmE(i) = (HmE(i) - GmE(i)}/T
(10)
The measured quantities are listed in Table 11. zE is the mole fraction of BE present in the entire volume of the apparatus and p is the total vapor pressure. The data analysis was carried out by iteration. First, ignoring the temperature variation for each run, we assumed xB = zB. The latter assumption is reasonable since the amounts of each component in the gas phase and those adsorbed on the inner walls of the apparatus are less than 1%. We then used the method described below to obtain the partial pressures of BE and H20, pB and pw. respectively. Knowing pB and pw, we corrected for the amounts in the gas phase and those adsorbed on the walls to improve the value of xB. The procedure was repeated until xB converged within 1 X Two such iterations were sufficient. Next we corrected the measured vapor pressure for the temperature deviation from 25.00 "C for each run by the Gibbs-Konovalow relation,I6 knowing the partial pressures, pBand pw: (dP/aT)X,
(PBAhB + PWAhW)/Rp
AhB = AhV*p(BE)- HmE(BE) Ahw
AhVaP(H20)- HmE(H20)
+ PBOAXW,
(11)
(124 (1 2b)
(16) Prigogine. 1.; Defay, R. Chemical Thermodynamics: Everett, D. H., Translator; Longmans: London, 1965; p 278.
for xw = 0.0 for xw # 0.0
APW = APVW,
liquid i is in equilibrium with the pure gas at vapor pressure pi0 at the same temperature, then pt(pure) = @(pure, 1 atm) + R T In pio (8)
GmE(i)= R T In (P,/x,pio)
PB = P -PW
PW = CAP,
1/(1 - C~BXW)/C~WXB)I
(13) (14)
(15) (16)
GmE(H,O) = R T In (PW/XWPW")
(17)
GmE(BE)= R T In (PB/x@B")
(18)
Ap is the difference in the total vapor pressure, p, between the two successive data points. For the two points, XB < 0.006, it was assumed that GmE(H20)= 0.0. Namely, pw = XWPW".
Results and Discussion The corrected vapor pressures for 25.00 O C are plotted against xB in Figure 2. The uncertainty in p is estimated to be *0.008 Torr. As is evident from Figure 2a, the data points from Scatchard and Wilson'4 are higher than ours for unknown reasons. At least our data points are freer from contamination by uncondensable impurities. Our data consist of two separate series of measurements, one starting from pure H20with BE added in succession and the other from pure BE. The interconnectivity between the two sets of data appears reasonable as shown in Figure 2b. The preliminary data for XB < 0.01 are shown by triangles in Figure 2c, and they fall within uncertainties. We note that the point at XB = 0,01467 appears to lie beyond the estimated error bar for some unknown reasons. We ignore this point in the subsequent discussion. As seen in Figure 2b, the p-xB curve shows the maximum at xB 0.0065 and also becomes almost flat in the range 0.05 < xB < 0.07. The former indicates the azeotrope. The latter is an indication of the critical demixing of this solution at about 49 "C. Namely, from the Duhem-Margules relation xB(d In pB/dxB) xw(d In pw/dxB) = 0
-
+
it follows that ( ~ P / ~ x B=)( ~ P B / ~ x B - Cvwx~)/Cv~xw)l )~~ Thus, at the azeotrope point, y e = xB, (dp/axB) = 0. As will be discussed in detail below, for the range 0.05 C xB < 0.07, (1 7) Boissonnas, C. G. Helu. Chim. Acta 1939, 22, 541. (1 8) Van Ness,H. C. Classical Thermodynamics of NonelectrolyfeSolurions; McMillan: N e w York, 1986; p 140.
4122 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991
XB
YE
Gme(BE)"
0.001 739 0.004 102 0.006500 0.009054 0.011 84 0.01467 0.0I7 54 0.02047 0.023 50 0.02668 0.030 00 0.033 35 0.03934 0.04990 0.06474 0.08035 0.09648 0.1012 0.1348 0.2001 0.2940 0.3950 0.529 0 0.6378 0.7224 0.7743 0.8467 0.9073 0.9522
0.002272 0.004670 0.006 374 0.007166 0.007927 0.009 424 0.009376 0.009447 0.009681 0.009827 0.009972 0.01012 0.01023 0.01026 0.01038 0.01055 0.01064 0.01088 0.01146 0.01238 0.01414 0.017 26 0.02496 0.03763 0.05638 0.07321 0.1204 0.2142 0.3696
8.893 8.552 8.135 7.604 7.188 7.084 6.627 6.264 5.981 5.703 5.448 5.220 4.839 4.254 3.636 3.138 2.704 2.637 2.041 1.217 0.505 0.093 -0.141 -0.163 -0.125 -0.119 -0.071 -0.019 -0.003
Gm'(H20)" 0.0 0.0
Koga
Hm'(H20)"
Hm'(BE)'
-16.30 -15.47 -14.45 -13.16 -1 1.52 -9.35 -6.65 -4.30 -3.32 -2.81 -2.55 -2.40 -2.20 -2.00 -1.79 -1.58 -1.45 -1.40 -1.12 -0.68 -0.24 -0.08 0 0 0 0 0 0 0
0.005 0.009 0.013 0.014 0.022 0.029 0.035 0.042 0.050 0.057 0.071 0.098 0.136 0.174 0.216 0.223 0.300 0.458 0.674 0.870 1.041 1.076 1.023 1.019 0.880 0.637 0.514
TSm'(BE)"
0.00 0.00 -0.01 -0.02 -0.05 -0.07 -0.11 -0.13 -0.15 -0.16 -0.18 -0.20 -0.22 -0.24 -0.25 -0.26 -0.27 -0.28 -0.30 -0.41 -0.52 -0.60 -0.66 -0.66 -0.66 -0.65 -0.65 -0.64 -0.64
-25.19 -24.02 -22.59 -20.76 -18.69 -16.43 -13.28 -10.56 -9.30 -8.51 -8.00 -7.62 -7.04 -6.25 -5.43 -4.72 -4.15 -4.04 -3.16 -1.90 -0.75 -0.17 0.14 0.16 0.13 0.12 0.07 0.02 0.00
TSm'(H20)" 0
0 -0.01 -0.03 -0.06 -0.08 -0.13 -0.16 -0.19 -0.20 -0.23 -0.26 -0.29 -0.34 -0.39 -0.43 -0.49 -0.50 -0.60 -0.87 -1.19 -1.37 -1.70 -1 -74 -1.68 -1.67 -1.53 -1.28 -1.15
c h ~
-3.255 -7.188 51.258 4.761 1 3.0018 2.770 2.1283 1.8395 1.6841 1 S675 1.4829 1.4211 1.3376 1.2459 1.1786 1.1390 1.1120 1.1083 1.0804 1.0528 1.0357 1.0276 1.0233 1.0227 1.0235 1.0236 1.0254 1.0286 1.0303
'kJ mol-'.
(apB!dxB)
i=
0 and hence (dpB/axB) = 0. Therefore, (ap/dxB)
= 0 In this range.
From eqs 13-16, the absolute uncertainty in pw and pB is about the same as that in p as long as is of the order of unity, which is the case for 0 < xw < 0.99 as shown in Table 111. It follows that the resulting uncertainty in GmE(H20)is less than fO.05 kJ mol-'. The uncertainties in GmE(BE)are estimated to be fO.05 kJ mol-' except for the first three points at the water-rich end. The error bars for these three points are shown in Figure 3a. The values of TSmE(i) (i = BE or H20) were then calculated by using the data of HmE(i)(i = BE or H20)' and listed in Table 111. All the quantities obtained are plotted against xBin Figures 3-6. As d i d in the Introduction as well as in previous papers,*' the derivative of the excess partial molar enthalpy, N(aHmE(i)/dn,), is a measure of the interaction among ith species in terms of enthalpy. Similarly, N(8Sme(i)/dn,)is that in terms of entropy. For convenience, however, they were calculated by using the data in Table 111 graphically by the right-hand expression of eqs 19 and 20 and plotted in Figure 7.
-
W H m e ( i ) / W n # , r = (1 x,)laHmE(i)/aXdN#,r
(19)
nlaT~mE(O/Wn,.p,r (1 -x,)(aTSmE(i)/dx,lN#,T (20) From Figure 3a, there may be a small wiggle in the xB-GmE(BE) curve from XB = 0.016 to 0.018, as the one seen in the xTBA-Gme(TBA)curve for aqueous TBA solutions at xTBA= 0.045.5*7The latter was reminiscent of the transition in the mixing scheme in that solution. The present case of the wiggle in the xB-GmE(BE) curve is not as apparent as that in the TBA-H20 system, being almost the same size in uncertainty, fO.05 kl mol-'. However, the transition in the mixing scheme becomes more apparent when SmE(BE)is calculated below. Figure 3a also indicates that Gme(BE) decreases as xBincreases in the water-rich region. Namely, Gme(BE) becomes less positive on addition of an extra BE molecule into the solution. This indicates that the BE-BE interaction is attractive in terms of the excess free energy. This effect is already seen at the smallest concentration measured, xB = 0.0017. Already at the concentration of one BE molecule in about 575 (=1/0.0017) molecules of H20, the BE molecule feels attraction from the other BE molecules. Thus such an attraction is of a long-range nature. This
TABLE IV: Concentration Fluctuation in tbe BE-H20 System (25.00 "C)
0.0025 0.00375 0.005 0.00625 0.0075 0.00875 0.01 0.01 125 0.0125 0.01375 0.015 0.01625 0.0175 0.01875 0.02 0.0225 0.025 a
8.75 -172.0
0.00505
-156
0.01023
-164
0.0206
-136
0.0291
-120
0.0406
-1 12
0.0601
-1 12
0.1204
-96
0.1710
8.32 7.93 7.52 7.18 6.88 6.60 6.32 5.84
kJ mol-'.
phenomena comes from a balance between the enthalpic and the entropic contributions of the excess free energy, as will be discussed below. Since BE molecules tend to attract each other, the concentration fluctuation is always larger than the value for the ideal solution, XB(1 - xB). Namely N ( ( ~ B ) ' ) RT(1
- X B ) / ( ~ P B E / ~ X=B )
- xB)/((l/RT)(dG,'(BE)/dxB)
+ (I/xB))
(21)
If GmE(BE)= 0 or aGmE(BE)/axB= 0, then N ( ( A x ~ ) ~XB(? ) Since dGmE(BE)/dxB < 0 for the present system in this concentration range, N ( ( A X ~ > ) ~xB(l ) - xB), contrary to previous indications from light-scattering m e a s ~ r e m e n t s , lthat ~ - ~the value of N ( was slightly smaller than that of the ideal solution
- xB).
(19)Ito, N.;Fujiyama, T.; Udagawa, Y. Bull. Chem. Soc. Jpn. 1983,56,
379.
(20) Kato, T. J . Phys. Chem. 1984,88, 1248.
The Journal of Physical Chemistry, Vol. 95, NO.10, 1991 4123
Vapor Pressures of Aqueous 2-Butoxyethanol Solutions
1..-... . P
* A
1
A
a .
BE-H 20, G E,,,(i)
A
25.00'C A
A
BE-H20,
Vapour Pressures
25.00'C 02
04
06
IO
08
I__ i=HzO
XB
BE-HzO,
Vapour Pressures
00
b
25.00'C
002
006
0 X0B4
008
b
.I
L
&
I-
\
BE-H20,
'1 \
23.5
W
n
GL(i)
25.00'C
-0 2 E
1
Error
'
7 W Y
23.01
t
I
0
!
"vI
I
I
01
02
W
h L
b I-
BE-H20,
v
E
a
XB
C
Vapour Pressures 25.00'C
a 23.80
0
-I!
0
I
I
02
I
I
04
I
1
06
!
I
08
I
I
IO
XE
Figure 3. Excess partial molar free energies, Gme(i)( i = BE or H20), at 25 OC. 23.75
23 70
I
8
0 01
1
0.02 XB
Figure 2. Vapor pressures of the BE-H20 system at 25 OC: (a) 0 , this work; A, ref 13; (b) 0 , series 1 from pure H20: X series 11 from pure B E (c) see text.
a t xB= 0.01 and 21 O C . We note that the light scattering is not very accurate when the values of N ( are small, while those determined by eq 21 using our data of GmE(BE)are more accurate in the water-rich region where Gme(BE) varies sharply. For X B < 0.02, the values of N( were calculated within 50% and listed in Table IV. For the range 0.03 < XB < 0.1 5, the absolute values of (I/RT)(KmE(BE)/axB)and 1/xB are very close to each other, and the resulting denominator of the right-hand side of eq 21 cannot be determined with confidence. It is in this latter concentration range that the values of N((AxBY)determined by ~cattering'~320 are more accurate. Another noticeable feature in Figure 3b is that for the range xB> 0.5, the values of Gme(BE) are almost 0, less than 0.15 kJ mol-'. This indicates that there is virtually no difference when
a BE molecule is in the pure liquid or in the solution in this concentration range. This point is discussed in more detail using Hme(BE) and SmE(BE)below. Water-Rich Region, X, C 0.0175. From Figure 4a, the first molecule of BE dissolves into water with a large enthalpy gain (-17 kJ mol-') and a larger entropy loss (-25 kJ mol-') a t XB@ c 0.0. This is consistent with the notion of the "iceberg formation" or of the "structure enhancement of watern by the solute BE.2'-24 As the concentration increases, both the enthalpy gain and the entropy loss become smaller. Namely, the BE-BE interaction is repulsive in terms of enthalpy but attractive in terms of entropy. Therefore, in terms of enthalpy, it is favorable for the second BE molecule to settle at an infinite distance from the first, where the effect of the structure enhancement by the first is negligible. Hence the second BE molecule would dissolve with almost an equal enthalpy gain. In terms of entropy, on the other hand, the second BE molecule would settle as close as possible to the first with a negligible entropy loss, where the structure of water is being enhanced already by the first. If the enthalpic and the entropic (21) Uedaira, H.; Kida, J. Nippon Kagaku Kaishi 1982, 539. (22) Bcndcr. T. M.;Pecora, R. J . Phys. Chem. 1986, 90, 1700. (23) Nakanishi, K.; Ikari, K.; Ozaki, S.;Touhara, H. J . Chem. Phys. 1984, 80, 1656. (24) Tanaka. H.; Nakamishi. K.; Touhara, H.J . Chem. Phys. 1984,81, 4065.
4124 The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 0
h
w
D
w
Ln
c
1
L
0 h
w
m
/r BE-HzO,
*ooi
a.
-
BE-H~O,G,~
25.00C
600
I-
*
I
7
c3
HmE(3E) and TSmE(BE)
-
Koga
200
*
*
t
1 i
W
t
E -20-
I
0
r
,
t
02
,
04
c
r
b ,
*
06
I
0' 8
IO
XB
Figure 5. Excess (integral) free energy, GmE. 1
002
0
'
004 x g 0'06
BE-H20,
vs x 25.00'C . X B =o-0.01
'
008
.
XB
A X B
001
01
IO
= Error
=o- 0.1 =ow 1
1
I
/
YB
BE-HzO,
W
m
H,'(BE)
V
I
.
4
0 005 005 0 5
25.00'C
W
E m i-
b
and TSmE(BE)
g
-44
1
0
0
0
0 0 0
05 0 05 0 005
I O 01
001
XB
Figure 6. x B vs ye in three different scales. XB
BE-H20,
HmE(H20) and TSmE(H20)
25.00'C
1
0-
V h
0 N I V
W
E
g
-I-
b h
0, c
I
I
I
0
02
04
XE
#
I
I
06
08
10
Figure 4. Excess partial molar enthalpies, HmE(i), and entropies, Sme(i) ( i = BE or H20), at 25 OC. 0, this work;-, ref 3.
terms cancel each other, (Le., GmE(BE)= O.O), the BE molecules would distribute randomly. Since the entropy effect is actually stronger, BE molecules tend to settle closer to each other than in the random distribution, with the structurally enhanced water around each of them. This is consistent with the notion of the "hydrophobic attraction".24a As mentioned above, such attraction (25) Franks, F. Faraday Symp. Chem. Soc. 1982, 17, 7.
is apparently of long range and is entropy driven. As the concentration increases, the entropic BE-BE interaction shown in Figure 7a increases sharlpy to xB= 0.0175, whereupon it suddenly diminishes, in the same manner as the enthalpic BEB E intera~tion.',~This is an indication of the transition in the mixing scheme. Namely, the mixing scheme discussed above no longer operates, and a new mixing scheme sets in at this concentration. We pointed out that this transition is associated with cusp or peak anomalies in the third derivatives of the free energy. We argued by an analogy to the glass transition that this is the transition of a short-to-medium range The boundary of this transition in the mixing scheme is reproduced in Figure 9 with an additional point (A)of the anomaly of P@TSme(BE)/t3nBe) in Figure 7a. Above this boundary, xB = 0.0175 a t 25 O C , a different mixing scheme operates. Intermediate Region. Figure 8 shows the chemical potential of BE, L(BE, taking the value of pB0 = 0, for the range 0.02 < XB C 0.14. The uncertainty in this region is estimated to be 10.03 kJ mol-I. As is evident from the figure, the slope ~ L ( B E / ~ X B becomes very small with an apparent inflection point at xB= 0.06. This must be a manifestation of the critical demixing a t 49.4 OC and xB= 0.0719 (LCST, the lower critical solution temperature). Various interpretations are probable including an attempta to map the model for a solution with a LCST to the Ising model. While this attempt does not invoke the presence of molecular clusters, we are in favor of the interpretation that the solution in this concentration range consists of the two kinds of molecular clusters, (26) Wheeler, J. C. J . Chrm. Phys. 1975, 62, 433.
The Journal of Physical Chemistry, Vol. 95, No. 10, 1991 4125
Vapor Pressures of Aqueous 2-Butoxyethanol Solutions a
-254
interaction (Entropic) 25.00’C
BE -BE
-35 i
0
002
0
,
006
004
!
b
BE-BE
interaction
$0
gj W
5-
0
c
0
e
=
c
0-
W
?
1 1
w m
0
,
1
02
I
I
04
06
1
I
I
08
IO
xB
C
V
C F 0
e .-C c
I
-
-2-
H20-HzO
interaction
25.00’C
Entropic
0 , -4-
II 0, I
_
-61
0
I
0 15
I
008
XB
E
XE
01
Figure 8. Chemical potential of BE in the intermediate region.
I
.-
1
0 05
02
04
06
!
I
08
I
XB
Fipn 7. (a) Entropic BE-BE interactions in the H20-rich region. (b)
BE-BE interactions.
(c) H20-H20
interactions.
-
precursory to the phase separation at 49.4 O C , x B 0.07. This interpretation is consistent with the conclusions of earlier studies by light scattering19” and by heat capacity mea~urement~’.~* that (27) Roux, G.; Perron, G.; Desnoyers, J. E.J . Phys. Chem. 1978,82,966. (28) Roux, G.; Perron, G.; Desnoyers, J. E. J . Solution Chem. 1978, 7 , 639.
in this concentration range BE molecules tend to cluster. The same interpretation was used for the TBA-H20 system in the intermediate concentration range, for which the angle dependence of the small-angle X-ray s ~ a t t e r i n g ~supported ~”~ the existence of clusters of about 15 A in size. More detailed studies of the BE-H20 solution system by means of small-angle X-ray scattering, particularly for extracting correlation lengths, appear warranted. BE-Rich Region. As seen in Figure 4b, it is striking that HmE(BE)is exactly zero and TSmE(BE)is very nearly so in the range xB > 0.5. In other words, a BE molecule sees its environment to be almost exactly the same as in the pure BE liquid. This implies that BE molecules form a cluster of its own kind. An additional BE molecule settles inside the cluster and does not feel the presence of H 2 0 directly. As for H 2 0 , Figure 4c indicates that the values of HmE(H20) stay constant, while those of TSmE(H20)decrease as the concentration of H 2 0 increases. These facts may suggest that H 2 0 molecules attach to sites on the surfaces of the BE clusters, with a constant but small negative enthalpy and with a decreasing entropy of “adsorption”, as the “coverage” of H 2 0 increases. Obviously, the thermodynamic quantities presented here alone cannot lead to unequivocal conclusions. It would appear important to study the solution in this concentration region also by smallangle X-ray scattering and other techniques. Thus our tentative conclusion is that in the BE-rich region, the solution consists of clusters of pure BE with H 2 0 molecules “adsorbed” on them, while in the intermediate region the solution consists of two kinds of clusters of mixed BE and H 2 0molecules with unknown composition. The crossover between the two regions may be estimated to be xB = 0.46 from the concentration dependence of N(dHme(i)/&Jand l@TSmE(i)/t3nJ(i = BE or H20), Figure 7b,c at 25.00 OC. For 35.00 O C , the crossover may also be estimated to be xB= 0.38 from the concentration dependence of HmE(BE)as reported previ~usly.~ We noted that the boundary from the H20-rich region to the intermediate region is marked by cusp or peak anomalies in the third derivatives of the free en erg^.^^^ On the other hand, for the crossover from the intermediate to the BE-rich region, the third derivatives do not show such anomalies. Instead, the derivatives once more with respect to concentration of the enthalpic BE-BE interaction and the entropic H20-H20 interaction, the fourth derivatives, would seem to show weak jump anomalies, as seen in Figure 7b,c. Mixing Scheme Diagram. Figure 9 shows all the boundaries discussed above, including that separating the two-phase region from the single phase region. Thus boundary a in Figure 9 is a (29) Koga, Y . Chem. Phys. Lett. 1984, I l l , 176. (30) Nishikawa, K.;Hayashi, H.;Iijima, T. J. Phys. Chem. 1989,93,6559.
J . Phys. Chem. 1991, 95, 4126-4130
4126
0 ‘
b
’(
011
I
I
03
02
I
04
I
05
I
06
XB
Figure 9. Mixing scheme diagram.
locus of a macroscopic phase change, a change in a long-range order. On crossing this boundary at the LCST, the heat capacity, the second derivative of the free energy, would show a cusp anomaly (a X transition). When crossing boundary a at other
points than the LCST, the transition would be the first order with a jump anomaly in the heat capacity. Boundary b is marked by cusp or peak anomalies in the third derivatives of the free en erg^.^,^ We argued6 that this transition in the mixing scheme is the transition of the short-teintermediate range order. Boundary c, on the other hand, appears to be associated with jump anomalies in the fourth derivatives of the free energy. We suggest that this transition involves a change in the composition in molecular clusters that appear to be present. In summary, we point out that there are transitions of more subtle kinds than those between macroscopic phases and that they are accompanied by various anomalies in the third and the higher order derivatives of the free energy. Thus a theoretical work appears important that would rigorously relate the following two: (1) changes in such structural characteristics as a pair correlation function; (2) various anomalies in the quantities that are proportional to the third or higher order derivative of the free energy.
Acknowledgment. I thank Drs. Iijima and Nishikawa for making ref 30 available prior to publication. Various technical assistance by William Siu, Terrance Wong, and Lisa Chao is greatly appreciated. This research was supported by Natural Sciences and Engineering Research Council of Canada. Registry No. BE, I 1 1-76-2.
A Semicontinuum Model for the Hydrated Electron Stanislas Pommerett and Yam Cauduel* Laboratoire d’Optique AppliquPe CNRS URA 1406 et INSERM U 275, Ecole Polytechnique-ENS Techniques Avancgees, Batterie de I’Yvette, 91 120 Palaiseau, France (Received: September 21 1990) I
Kevan’s structural model for the solvated electron has been studied in association with the spherical part of a continuum potential via an imaginary time splitting operator method (SOM).The study focuses on the influence of the continuum potential representing the second solvation shell interaction with the electron. The continuum potential is computed in the self-consistent approximation. The temperature dependence of the continuum potential allows us to extend our study from 77 to 300 K. The gyration radius of the solvated electron is found to be of the same order as computed by using simulation methods, and for the best value of the cavity radius more than 60% of the charge is inside the cavity. The results of the present computation are then compared to those obtained by using a full simulation approach or the semicontinuum theory. In the present work, it is interesting to note that at 300 K we have found results that are in good agreement with those computed by using a molecular dynamics approach of the medium.
1. Introduction The structure and the optical properties of the solvated electron in aqueous media have been extensively studied in the past few y e a r ~ . l - ~In recent theoretical studies2+ the solvent is treated classically and the electron is ranked as a quantum particle under the influence of the sum of the pseudopotential of interaction with each one of the classical m ~ l e c u l e s and ~ ~ Jeventually ~ a many-body polarization ~ t e n t i a l . 2The ~ computation of the induced structure around the solvated electron has been performed using the Monte Carlo (MC)*s3 or molecular dynamics (MD)@ method for the water molecules and using quantum path integral MC (QPIMC),r’ quantum path integral MD (QPIMD),c7 or splitting operator method for the electron. The main results of these simulations are (i) the water molecules show a preferential 0 - H orientation around the solvated electron in neat water, and (ii) the absorption spectrum of the electron is mainly explained by transition from the fundamental (s-like state) to nondegenerated states (plike, d-like). These two results are in contradiction with Author to whom correspondence should be addressed. ‘From February 1989 to June 1990 at the Radiation Laboratory, University of Notre Dame, Notre Dame, IN 46556.
0022-3654/91/2095-4126$02.50/0
the continuum theory which predicts a dipole orientation, a pure s state for the fundamental, and fully degenerated states for excited levels.1° Up to now it has been impossible to examine directly the induced structure of the solvated electron in water at room temperature. The only experimental investigations of the solvated (1) For a review on the solvated electron see: Dayton, F. S.Chem. Soc. Rev. 1975,4,323. Brodsky, A. M.; Tsarevsky, A. V.Ado. Chem. fhys. 1980. 44,483. Feng, D.-F.; Kevan, L. Chem. Rev. 1980,80, 1. For recent experimental investigation on the hydrated electron see: Gauduel, Y.; Pommeret, S.;Migus, A.; Yamada, N.; Antonctti, A. J. Upf.Soc. Am. E 1990,7, 1528. (2) Wallqvist, A.; Martina, G.; Berne, B. J. J . fhys. Chem. 1!?83,92, 1721. (3) Wallqvist, A.; Thirumalai, D.; Berne, B. J. J . Chem. fhys. 1987.86, 6404. (4) (5)
Romero, C.; Jonah, C. D. J . Chem. fhys. 1989, 90,1877. Jonah, C. D.;Romero, C.; Rahman, A. Chem. Phys. Len. 1986,123,
209. (6) Schnitker, J.; Rossky. P. J. J . Chem. fhys. 1987,86, 3471. (7) Rossky, P.J.; Schnitker, J. J . fhys. Chem. 1988, 92, 4277. (8) Schnitker, J.; Motakabir, K.; Rossky, P. J.; Friesner, R. Phys. Rev. Len. 1988, 60, 456. (9) Schnitker, J.; Rossky, P. J. J . fhys. Chem. 1989, 93, 6965. (IO) Jortner, J. Radiaf. Res. Suppl. 1964, 4, 24.
0 1991 American Chemical Society
