J. Phys. Chem. 1984, 88, 2655-2660
lating the two straight-line portions as shown in Figure 4. The values of concentration 1.00 X lO-l, 3.15 X lO-l, and 3.70 X lo-' M are almost the same as those obtained from conductance measurements. 4. Solvent Effect on H-Bonded Ion Pairs. The effect of the dielectric constant coupled with solvating power of the solvents on AH is illustrated in Table IV and Figure 5. At a particular constant temperature, the strengthening, weakening, and breaking of the hydrogen bond of the H-bonded ion pair 'H-
- --O,CCF,
R
in dilute solutions will mainly depend on the dielectric constant of the solvent used. Since strengthening and weakening do not produce significant change in the entropy of solvation of an H-bonded ion pair, the free energy of transfer of the H-bonded ion pair from one solvent to another will approximately be equal to the enthalpy change of the process. Therefore, AH in low dielectric constant media where the dissociation of H-bonded ion pairs is negligible', may be represented with the help of the Born equation as follows:
AH = AG
+ constant = ( - N Z e 2 / 2 r ) ( l - l / t ) + constant (7)
AG = - ( N Z e 2 / 2 r ) ( l - l / c )
Born equation
Accordingly, AH of pyridine-F,CC02H in p-dioxane, carbon
2655
tetrachloride, o-xylene, and chloroform is linearly related to the inverse of permittivity (e) of the solvent (see Figure 5 ) . The increase in the AH with the increase of dielecric constant (or decrease of 1/c) is due to weakening of H-bonding in B+H--A-. Acknowledgment. We are thankful to the Department of Science and Technology, New Delhi, India, for providing the financial support for purchasing a Tronac 450 titration calorimeter. Registry No. TFA, 76-05-1; TCA, 76-03-9; DCA, 79-43-6; MCA, 79-1 1-8; AcOH, 64-19-7; triethylamine, 121-44-8; 2,4,6-trimethylpyridine, 108-75-8;2,6-dimethylpyridine,108-48-5;3,5-dimethylpyridine, 59 1-22-0;4-methylpyridine,108-89-4;4-ferf-butylpyridine,3978-8 1-2; 3-methylpyridine, 108-99-6; pyridine, 110-86-1; isoquinoline, 119-65-3; quinoline, 9 1-22-5; 3-chloropyridine, 626-60-8; 4-cyanopyridine, 10048-1; 3-cyanopyridine, 100-54-9; 2-bromopyridine, 109-04-6; 3,5-dichloropyridine, 2457-47-8; triethylamine-TFA, 454-49-9; 2,4,6-trimethylpyridine-TFA, 57313-93-6;2,6-dimethylpyridine-TFA,7032026-2; 3,5-dimethylpyridine-TFA,8 1675-57-2;4-methylpyridine-TFA, 29885-74-3; 4-tert-butylpyridine-TFA, 30254-33-2; 3-methylpyridineTFA, 8 1675-56-1; pyridine-TFA, 464-05-1; isoquinoline-TFA, 421 541-2; quinoline-TFA, 89937-06-4;3-chloropyridine-TFA, 89937-07-5; 4-cyanopyridine-TFA, 29885-70-9; 3-cyanopyridine-TFA, 8 1675-55-0; 2-bromopyridine-TFA, 89937-08-6; 3,5-dichloropyridine-TFA,8993709-7; 2,4,6-trimethylpyridine-TCA, 53088-80-5; 2,4,6-trimethylpyridine-DCA, 573 13-89-0;2,4,6-trimethylpyridine-MCA, 24145-70-8; 2,4,6-trimethylpyridine-AcOH, 89937-10-0;pyridine-TCA, 3486-54-2; pyridine-DCA, 16983-41-8;pyridine-MCA, 933-24-4;pyridine-AcOH, 5153-63-9; 4-cyanopyridine-TCA,89937-11-1; 4-cyanopyridine-DCA, 89958-45-2; 4-cyanopyridine-MCA, 89937-12-2; 4-cyanopyridineAcOH, 89937-13-3.
Phase Diagram of a Ternary Fluid Mixture In the Vicinity of Its Critical Line in the Presence of Isotope-Exchange Reactions P. Gansen and D. Woermann* Institut fur Physikalische Chemie, Universitat Koln, Federal Republic of Germany (Received: July 26, 1983; In Final Form: November 22, 1983)
The coexistence surface of the ternary system isobutyric acid/H20, D 2 0 in which two independent isotope-exchangereactions take place is determined by chemical analysis of the two fluid phases coexisting at temperatures below the upper critical solution temperature. The acid is generated by hydrolysis of isobutyric acid anhydride in H20, D 2 0 mixtures of known composition. The form of the coexistence surface practically does not change with the equivalent fraction Y of exchangeable protons. The critical composition determined on the basis of the equal volume criterion decreases slightly with increasing values of Y (about 3%). The critical temperature is a linear function of Y. It increases by about 20 K. In the systems isobutyric acid (COOH)/D20 and isobutyric (COOD)/H20 the composition of the mixture with maximal phase separation temperature is not identical with the critical composition. This can be understood on the basis of the form of the coexistence surface of the system isobutyric acid/H20, DzO, confirming predictions of Knobler and Scott. The properties of the coexistence surface of the system isobutyric acid/H20, D 2 0 appear to be typical for critical systems in which isotope-exchange reactions take place. This is shown by studies of the temperature of phase separation vs. composition of the systems phenol (OH)/D20 and 2-butoxyethanol (OH)/D20 near the upper and lower critical point, respectively.
Introduction The systems isobutyric acid (COOH)/H,O (HA/H,O), isobutyric acid (COOD)/D,O (DA/D,O), isobutyric acid (COOH)/D,O (HA/D,O), and isobutyric acid (COOD)/H,O (DA/H,O) have upper consolute points. The critical composition of all four systems has approximately the same value. However, they differ in one respect, in the systems H A / D 2 0 and DA/H20, isotope-exchange reactions take place. In each of these systems the composition with the maximal phase separation temperature is different from the critical composition determined by the equal volume The critical temperature is about 0.1 K lower 0022-3654/84/2088-2655$01.50/0
than the maximal temperature of phase separation (Figures 1 and 2). Knobler and Scott have proposed an explanation of this p h e n ~ m e n o n . ~For further study of the influence of isotope-exchange reactions on the composition dependence of the temperature of phase separation, the coexistence surface of the ternary ~~
(1) E. Gulari, B. Chu, and D. Woermann, J . Chem. Phys., 73, 2480 (1980). (2) P. Gansen, T. Jansen, W. Schan, and D. Woermann, Ber. Bunsenges. Phys. Chem., 89. 1149 (1980). (3) I. Schafke, Master Thesis, Universitat Kaln, 1983. (4) Ch. H. Knobler and R. L. Scott, J . Chem. Phys., 76, 2606 (1982).
0 1984 American Chemical Society
2656 The Journal of Physical Chemistry, Vol. 88, No. 12, 1984
LLO
calculations of the concentrations of the species HA, DA, H20, and D 2 0 in the fluid phase. The experimentally determined value of K1 in the fluid phase at a temperature of 25 "Cis K1 = 3.76.5 The satisfactory agreement between the experimental and theoretical values of K, leads us to the assumptions that in the fluid phase K2 also agrees with its statistical value within 10%. From eq 3 and 4 it can be c0nclude4~that in the homogeneous phase the ratio of the exchangeable protons and deuterons in water and acid molecules has the same value:
L 01
= (nH/nD)water (5) If the influence of isotope effects on the activity by coefficients is neglected, we conclude4 (nH/nD)acid
0 15
:A' X& c
Figure 1. Temperature of phase separation T, of the system isobutyric acid (COOH)/D,O as function of the stoichiometric mole fraction of the acid (xOHA = )IoHA/(noHA noDlo)). The critical composition = 0.1132) is determined by the equal-volume criterion of coex-
+
isting fluid phases. I
1 0
d
o
o 0
eo
0 0
0 "
0 0
275
0
0 0
0 0 0
Tp
270
0
0
O C 0 0
26 5
0
0
I
I
0.10
I
I
0.15
0.2(
X;A
Figure 2. Temperature of phase separation Tpof the system isobutyric acid (COOD)/H,O as function of the stoichiometric mole fraction XODA of the acid (xoOA= ~ O D A / ( ~ ' D A + noD2,-,). The critical compositon (X'DA,~ = 0.1 165) is determined by equal-volume criterion of coexisting fluid
phases. system isobutyric acid/H20, D 2 0 is determined. The acid is generated by hydrolysis of isobutyric acid anhydride in HzO, DzO mixtures. Equilibrium Constants of Isotope-Exchange Reactions In the system isobutyric acid/H,O, D 2 0 two independent isotope-exchange reactions take HzO DzO = 2 H D 0 (1)
+
HA
+ DzO = DA + H D O
(2)
Statistical thermodynamic calculations of the thermodynamic equilibrium constants Kl and K2 of these reactions in the gas phase, neglecting isotope effects and taking into account only the symmetry numbers q (aH20= 2, UD~O= 2, OHDO = 1, UHA = 1, UDA = l), yield K , = - = 4CHDO'
(3)
CH~OCD,O
K2 =
CHDOCDA
-=2 CD20CHA
Gansen and Woermann
(4)
It is assumed that these values can also be used for approximate
where the prime and double prime denote the water-rich and acid-rich phases, respectively, coexisting at temperatures below the critical. Coexistence Surface of the System Isobutyric Acid/H20, D20 Isobutyric acid/H20, D 2 0 mixtures are prepared by hydrolysis of isobutyric acid anhydride in HzO, D 2 0 mixtures of known composition. The anhydride has a gas chromatographic purity of 98% (Merck) and is further purified by fractional distillation under reduced pressure in a dry nitrogen atmosphere. The hydrolysis of isobutyric acid anhydride goes to completion under properly chosen conditions (sealed ampules containing isobutyric acid anhydride, H 2 0 , and D20; reaction temperature 80-90 OC; reaction times 36 h). This procedure allows one to fix the ratio of exchangeable protons and deuterons in the mixtures by the and noDlo)in the reaction amounts of H 2 0 and D 2 0 mixture. Working with HA, H 2 0 , and D 2 0 as components makes the mapping of the coexistence difficult: The ratio of exchangeable protons and deuterons is changed by changing the amounts of HA, HzO, and D 2 0 in the mixtures. The coexistence curve of H A / H 2 0 mixtures prepared by hydrolysis of isobutyric acid anhydride in H20is identical with the corresponding curve of this system prepared from H A and H20. (The first-mentioned curve had to be shifted by 0.065 K to higher temperature to qverlap with the other.) DA is prepared by reaction of stoichiometric amounts of isobutyric acid anhydride and D20. After the reaction has gone to completion, DA is purified by fractional distillation under reduced pressure in a dry atmosphere. Five isobutyric acid/H20, D 2 0 mixtures with different H / D ratios are prepared. It is convenient to express the H / D ratio by the equivalent fraction of exchangeable protons:*
Y = n ~ / ( +n n~ ~ = ) n o ~ 2 ~ / ( n+o~ O~D , 2O ) ~
(7)
The Yvalues of the five mixtures are Yl = 0.0, Yz= 0.2176, Y3 = 0.4259, Y4 = 0.6254, and Y, = 0.8166. This corresponds to + moDl0)= 0.0, mass fractions of water of yHlOo= m0H,o/(m0H20 0.2, 0.4, 0.6, and 0.8. The mole fraction of isobutyric acid in all five mixtures after hydrolysis has a value of xa = xHA + XDA = 0.1 1, which is very close to the critical composition. The mixtures are homogenized by heating them above their temperatures of ( N T,; T, is the critical temperature). Then, phase separation each mixture is divided into 5 to 6 equal portions (in total 28 portions). Each portion is brought into thermal equilibrium in a thermostat set at a fixed temperature below its temperature of phase separation ( Tp- T I6 K; in total 28 temperatures). The two coexisting fluid phases in each sample are separated by using thermostated syringes and analyzed (in total 56 analyses). The concentration of isobutyric acid is determined by potentiometric titration of an aliquot of each phase (double determination). Another aliquot is neutralized by adding solid water-free K2C03. The composition of the H 2 0 , DzO solvent separated from the potassium isobutyrate by freeze-drying is determined by density measurements. These measurements give the equivalent fraction
Tp
( 5 ) L.
Friedman, and V. J. Shiner, J. Chem. Phys. 44, 4639 (1966).
The Journal of Physical Chemistry, Vol. 88, No. 12, 1984 2657
Coexistence Surface of Isobutyric Acid/H20, D 2 0
TABLE I: Analytically Determined Composition of Fluid Phases of Isobutyric Acid/H20, D20 Mixtures with Different Values of Equivalent Fraction Y = noHzO/(noH20 + noDzo)of Exchangeable Protons Coexisting below the Temperature of Phase Separation of These Mixtures" Tp - T , K x IHA x 'DA 'HzO XbzO X"HA x'b~ x %Iz0 xffD,O Tp = 45.977 0.8742 0.8584 0.8400 0.8163 0.7591
Yi = 0 0.1258 0.1416 0.1600 0.1837 0.2409
0.0 10 0.096 0.495 1.499 5.963
O C
0.016 0.042 0.109 0.507 1.667 5.036
0.0279 0.0286 0.0312 0.0347 0.0412 0.0496
0.1001 0.1029 0.1 121 0.1248 0.1482 0.1782
Tp = 42.191 "C Yz = 0.2176 0.6822 0.0208 0.1898 0.1890 0.6795 0.0206 0.1864 0.6703 0.0189 0.1829 0.6776 0.0166 0.0139 0.1764 0.6342 0.01 10 0.1681 0.6041
0.012 0.209 0.504 1.465 6.127
0.0535 0.0621 0.0685 0.0793 0.1037
0.0722 0.0837 0.0923 0.1069 0.1398
0.3724 0.3638 0.3574 0.3466 0.3222
0.237 0.242 0.332 0.732 1.732 5.232
0.0914 0.0921 0.0947 0.1047 0.1180 0.1456
0.0547 0.0552 0.0568 0.0627 0.0707 0.0872
0.5090 0.5332 0.5306 0.5207 0.5073 0.4798
0.012 0.079 0.103 0.6 13 1.903 4.978
0.1043 0.1 148 0.1194 0.1351 0.1576 0.1976
0.0234 0.0258 0.0268 0.0304 0.0354 0.0444
0.7123 0.7017 0.6972 0.6814 0.6590 0.6189
Y3 = 0.4259
0.9007 0.9128 0.9198 0.9353 0.9525
0.0993 0.0872 0.0802 0.0647 0.0475 0.0747 0.0740 0.0679 0.0596 0.0499 0.0396
0.1969 0.1971 0.1988 0.201 1 0.2038 0.2066
0.7076 0.7083 0.7144 0.7227 0.7324 0.7428
0.0566 0.0491 0.0445 0.0372 0.028 1
0.3839 0.3895 0.3929 0.3983 0.4050
0.5175 0.5250 0.5296 0.5369 0.5460
0.0520 0.051 3 0.0500 0.0465 0.0406 0.0326
0.03 11 0.0308 0.0300 0.0278 0.0244 0.0195
0.5734 0.5740 0.5753 0.5789 0.5847 0.5928
0.3435 0.3429 0.3447 0.3468 0.3503 0.3551
30.299 OC 0.0824 0.0738 0.0730 0.0618 0.0533 0.0436
0.0185 0.0166 0.0164 0.0139 0.0120 0.0098
0.7342 0.7427 0.7436 0.7547 0.7632 0.7729
0.1649 0.1669 0.1670 0.1696 0.1715 0.1737
Tp = 38.281 "C 0.5019 0.4904 0.48 18 0.4672 0.4343
Y4 = 0.6254
0.0420 0.0364 0.0330 0.0276 0.0209
Tp = 34.670 "C 0.3049 0.3195 0.3179 0.3119 0.3040 0.2874
Ys = 0.8166
Tp 0.1600 0.1577 0.1566 0.1531 0.1480 0.1391
"The mixtures are prepared by mixing appropriate amounts of substance of isobutyric acid anhydride, H 2 0 (noH,& and D 2 0 (noDzo). In the homogeneous phase the mole fraction of isobutyric acid x, (=xHA + xDA) in all mixtures has a value of x, = 0.1 1. This corresponds to the critical composition. Tp is the temperature of phase separation.
e
-L 0O 0 I
6t
O0
7 ~ " " " ' " " " " " " " ' J 0.10
*a
1
0.20
Figure 3. Coexistence curves of isobutyric acid/HzO, D20mixtures with different equivalent fractions of exchangeable protons Y (Y = noHzo/ (noHzo + noDz0);0 5 Y 5 0.82) and the isobutyric acid (COOH)/D20 mixture (Y= O.OS92): x,, mole fraction of acid ( x , = xHA+ xDA);Tp, maximal temperature of phase separation. The data are taken from Tables I and 11. The data points indexed by numbers 2 and 3 represent 2 and 3 identical results, respectively. The black dots 0 represent the data of the HA/D20 mixture in Table 11. Y of exchangeable protons of the sample. If the equilibrium constants of the isotope-exchange reactions have indeed the statistical values, the Y values of the solvent are not changed by neutralization with K2C03(eq 5). On the basis of this assumption the mole fractions HA, DA, H 2 0 , and D 2 0 in each fluid phase
t
'n( TTD;
10-~
Figure 4. Difference between the equivalent fraction Y of exchangeable protons in coexisting fluid phases of isobutyric acid/H,O, D20 and isobutyric acid (COOH)/DzOmixtures as function of log (Tp- T)/T+ (e), Y = 0.06; (A), Y = 0.22; (X), Y = 0.43; ( 0 ) ,Y = 0.62; (*), Y = 0.82. The prime and double prime denote the water-rich and acid-rich phases, respectively.
are calculated from the analytical data. The results of the analysis are compiled in Table I. A graphical representation of the data (Figure 3) shows that the coexistence curves of all five mixtures have the same shape independent of Y. Knobler and Scott4 have argued that the coexistence curves of the systems H A / H 2 0 , HA/D20, and DA/D20 should have the same form.6 This prediction is confirmed by the experiments: Analysis of H A / D 2 0 mixtures ( 6 ) S.Greer, Ber. Bunsenges. Phys. Chew., 81, 1079 (1977).
The Journal of Physical Chemistry, Vol. 88, No. 12, I984
2658
Gansen and Woermann
TABLE II: Analytically Determined Composition of Fluid Phases of Isobutyric Acid (COOH)/D20 Mixtures Coexisting below the Critical Temperature of Phase Separation T,’ Tp - T, K
0.009 0.115 0.265 0.864 2.778
X’HA
0.0075 0.0083 0.0087 0.0098 0.0121
x IDA
0.1185 0.1313 0.1369 0.1547 0.1907
x’H20
ID20
x ”HA
Y = 0.0596 T = 44.864 OC 0.0521 0.8218 0.0060 0.05 13 0.8091 0.0056 0.0509 0.8035 0.0052 0.0498 0.7857 0.0044 0.0475 0.7497 0.0036
X’bA
0.0955 0.0875 0.0816 0.0695 0.0567
Xlb20
X’;lzO
0.0536 0.0541 0.0544 0.0552 0.0560
0.8449 0.8528 0.8588 0.8709 0.8837
+
”In the homogeneous phase the stoichiometric mole fraction of the acid has a value of xom = 0.1 125 (Y@) = noHA/(noHA 2n0D20)= 0.0596). This corresponds to the critical composition.
coexisting at temperatures below the maximal temperatures of phase separation (Tp- T 5 6 K) leads to a coexistence curve whose form is idential with that of isobutyric acid/H20, DzO mixtures with varying Y values. In a ( Tp- T ) vs. xa plot the data of the system H A / D 2 0 (Table 11) coincide with that of the system isobutyric acid/H20, D 2 0 (Figure 3). The analyses demonstrate that the value of the H / D ratio in both coexisting fluid phases agrees with the H/D ratio of the initial reaction mixture within the accuracy of the measurements (Figure 4). The small differences between the Yvalues of the coexisting fluid phases (Y’, water-rich phase; Y”, acid-rich phase) may be caused by experimental errors: The amount of substance of K2C03 which has to be added to the acid-rich phase for its neutralization is bigger than that which has to be added to the water-rich phase. Traces of H20added with the solid K2CO3 could cause this effect. The results shown in Figure 4 give support to the assumption that the statistical value of the equilibrium constants K , and K2 are good approximations to the actual values. Experimental studies of the H / D isotope separation effect on T, and x, in triethylamine/H20, DzO mixtures by Linderstrom(see also ref 20) are in agreement with our findings. The effect is barely visible in that system. Three properties characterize the form of the coexistence surface of the system isobutyric acid/H20, D20: (a) The critical composition determined on the basis of the equal volume criterion of coexisting fluid phases changes slightly with the H / D ratio. It is a linear function of the equivalent fraction Y of exchangeable protons. The data are represented by Xap
= x,c(DA/D20) + D Y
(8)
with x,,(DA/D20) = 0.1132 f 0.0002, D = (3.7 f 0.3) X a n d 0 5 Y 5 1. (b) For fixed values of Y the mixture of maximal phase-separation temperature is identical with the mixture of critical composition. (c) There is a linear relation between the critical temperature and the equivalent fraction Y of exchangeable protons. The data are represented by T, = T,(DA/DzO) - C Y (9) with Tc(DA/D20) = 45.977 f 0.001 OC and C = 19.677 f 0.002
OC. Discussion The coexistence surface of the system isobutyric acid/H20, D2O can be represented in a rectangular prism diagram4J0-I6(Figure 5 ) because a double decomposition takes place (2DA + H 2 0 = 2DA D20). This representation has the advantage that the concentrations of all four relevant particle species (HA, DA, H 2 0 , and D 2 0 ) are used to construct the diagram. Furthermore, the compositions of coexisting fluid phases lie on parallel lines perpendicular to the Y axis, and the compositions of homogeneous DA/HzO mixtures (systems a) and H A / D 2 0 mixtures (system p ) , respectively, are represented by points on the diagonal of the
+
(7) C. U. Linderstrom-Lang, Acta Chem. Scand., 16, 1730 (1962). (8) C. U. Linderstrom-Lang,Pure Appl. Chem., 8, 259 (1964). (9) M. Hoffmann-Btichler,Radiochim. Acro, 16, 110 (1971). (10) D. V. Fenby, Z. S. Kooner, and J. R. Khurma, Fluid Phase Equilb., 7, 327 (1981).
Figure 5. Schematic representation of the coexisting surface of the system isobutyric acid/H20, D20 in a rectangular prism diagram. The critical composition is chosen arbitrarily to clarify the picture. Curves CY and p represent homogeneous mixtures of the systems isobutyric acid (COOD)/H,O and isobutyric acid (COOH)/D20, respectively. The heavy lines are isothermal cuts of the coexistence surface at different T,(DA/D20) - T: (e), critical isotherm; (O),isotherm of maximal phase-separation temperature. diagram. The equivalent fractions of exchangeable protons Y and the equivalent fraction of anions Z in homogeneous mixtures of systems a and are given by
Since no isotope separation effects occur, the compositions of the conjugate fluid phases lie on straight lines of constant Y . The curves a and p represent the compositions of homogeneous mixtures of DA/HzO (system a ) and H A / D 2 0 (system p). The schematic representation of the coexistence surface of the system isobutyric acid/HzO, D 2 0 shown in Figure 5 explains the and vs. xoDAcurves of the special features of the Tpvs. xoHA systems HA/DzO and DA/H20, respectively (Figures 1 and 2). In Figure 5 the contour lines represent isothermal cuts with the coexistence surface. Each line belongs to a different temperature difference (Tc(DA/DzO) - T). The intersections of the isotherms with the curves a and /3 are of special interest. Let us turn to the system H A / D 2 0 (system 8). Following curve p from the corner D 2 0 to the corner H A in direction of increasing acid concentrations, it can be seen that the mixture with maximal phase separation temperature (point 0 in Figure 5) has a smaller value of Z@)than the critical mixture (point 0 in Figure 5 ) . The critical temperature is lower than the maximal phase separation temperature. Knobler and Scott4 have pointed out that in the system D A / H 2 0 the mole fraction of the acid xODA in the mixture with maximal phase-separation temperature should be bigger than the mole fraction of the acid in the critical mixture (Figure 5 ; intersection of curve a with the isothermal cuts). This is in agreement with the experiments (Figure 2). The special features of the temperature of phase separation vs. composition diagrams of the systems HA/DzO and D A / H 2 0 shown in Figures 1 and 2 are caused by the fact that these curves do not represent coexistence curves. The two fluid phases coexisting below the temperature of phase separation are three-component systems.2 Two concentration variables have to be specified to describe their composition.
Tp
Coexistence Surface of Isobutyric Acid/H20, D 2 0 In Figure 5 the temperature of phase separation of H A / D 2 0 mixtures as function of composition should lie on a curve resulting from a cut of the T - @ surface standing perpendicular on the base of the rectangular prism with the coexistence surface. A cut of the T - LY surface with the coexistence surface should yield a curve representing the temperature of phase separation of D A / H 2 0 mixtures as function of composition of that system. These curves can be calculated and compared with the experimental T p- xom and T p- xoDAdata of the two systems (Figures 1 and 2). The calculation is carried out in two steps: ( a ) Calculation of the Coexistence Curve of an Isobutyric Acid/H20,D 2 0 Mixture at Constant Y. The calculation is based on the observation that the form of the coexistence curve is practically independent of the value of Y of the mixture and that the critical temperature varies linearly with Y. The small dependence of the critical composition on Y is neglected. For the three-component systems isobutyric acid/H20, D 2 0 it is expected that close to the critical point its coexistence curve a t constant Y can be described by a function of the form1’ (x’: - x:) = BtB* (12) with
p* and X=
N
= p ( 1 - X C ) / ( l - &)
The Journal of Physical Chemistry, Vol. 88, No. 12, 1984 2659 I
I
TP O C
0.10
0.15 XOHA
Figure 6. Temperature of phase separation Tp vs. composition (xoHA) diagram of the system isobutyric acid (COOH)/D,O. Comparison between experimental values (data points are those shown in Figure l ) and the theoretical curve (drawn-out line; p = 0.33, xaP = 0.1123) calculated on the basis of the coexistence surface of the system isobutyric acid/H,O, D20.The calculated curve is shifted to higher temperatures by 0.135 K.
(Tc(a/H20,D20) - T,(HA/H20))/T,(HA/H20)
where t is the reduced temperature ( ( T - Tc)/Tc)2C is the critical exponent of specific heat (E-= 0.110)19 and @ is the critical exponent of order parameter (0 = 0.325);19the prime and double prime refer to the water-rich and the acid-rich phases, respectively. It has to be kept in mind that eq 12 was proposed to treat the influence of a “real” impurity. But here the impurity is D 2 0 which forms with DA mixtures, exhibiting the critical behavior of a binary mixture. It turns out that the temperature factor X i s so _small ( X 5 7 X that the bracketed correction factor of @ can be approximated by 1 (@*= 1.007). Therefore, the function (x’l, - x’,) = BtB (13)
27.5
TP
27.0
OC
and the function (x:
+ x”&/2
+
= xa,c AtP
(14) describing the diameter of the coexistence curve in a two-component system are used for the calculation. The exponent p for the diameter of the correct order parameter is related to the exponent of heat capacity by p = 1 - 6. On the other hand, an incorrect choice of the order parameter may lead to the relation p = 28.12-14 The values of the constants B and p are obtained by fitting (CURFIT) the experimentaldata shown in Figure 3 to eq 13 with the results B = 0.10 and @ = 0.330 f 0.006. The theoretically expected value of @ (neglecting the correction term X in eq 12) is @ = 0.325. In view of the fact that the composition of the coexisting fluid phases is tetermined analytically, the agreement between the two values of @ is satisfactory. This situation appears to be different from that found in a recent study of impurity effects in a near-critical binary mixture methanol/_cyclohexane with H 2 0 as impurity: The effective critical exponent @ and the mole fraction of methanol in the critical mixture increase noticeably by adding H2O.I5 The value of the constant A in eq 14 is obtained by fitting (CURFIT) this equation to the same experimental data with A and x , , ~as free parameters. Three different p values are used: 1, (1 (11) M. E. Fisher and P. E. Szesney, Phys. Rev. A , 2, 825 (1970). (12) M. Nakata, T. Dobashi, N. Kuwahara, and M. Kaneko, J . Chem. SOC.,Faraday Trans. I , 78, 1801 (1983). (13) M.S.Green, M. J. Cooper, and J. M. H.Levelt Sengers, Phys. Rev. Lett., 26, 492 (1971). (14) P. C . Hemmer and G. Stell, Phys. Rev. Lett., 24, 1284 (1970); J . Chem. Phys., 56, 4274 (1970). (15) J. L. Tveekrem and D. T. Jacobs, Phys. Rev. A , 27, 2773 (1983). (16) J. E. Ricci, “The Phase Rule and Heterogeneous Equilibrium”, Van Nostrand, New York, 1951, p 372.
26 5
I
:‘A,
0
c 0.15
0 10
0.20
0 DA
Figure 7. Temperature of phase separation Tp vs. composition (xoDA) diagram of the system isobutyric acid (COOD)/H,O. Comparison between experimental values (data points are those shown in Figure 2) and the theoretical curve (drawn-out line; x ~=,0.1123) ~ calculated on the basis of the coexistence of the system isobutyric acid/HzO, DzO. The calculated curve is shifted ‘,o highe_r temperature by 0.205 K. Curves 1, 2, and 3 represent p - 2a,8, and p + 26, where p = 0.330 and a is the standard deviation.
- C), and 2p. The best fit is obtained with p = 2p, resulting in xa,c= 0.1 123 and A = 0.454. With these values of x , , ~A , B, @,
and p = 28 the coexisting curves of isobutyric acid/H20, D 2 0 mixtures of known values of Y close to the critical point can be calculated by using eq 15, neglecting the Ydependence of xa,,. x , = x,,,
+ AtP f (B/2)tB
(15)
The critical temperature T, of the mixtures is calculated from eq 9. ( b ) Calculation of Temperature of Phase Separation of H A / D 2 0and D A / H 2 0Mixtures of Known Composition. From the known value fl@)of a given homogeneous H A / D 2 0 mixture, the critical temperature of the corresponding isobutyric acid/H20, D 2 0mixture (Y= Y@)) is calculated by using eq 9. This T, value, the known value of x,(= xoHA), and the value of the critical
2660
J. Phys. Chem. 1984,88, 2660-2669
composition x,,, = 0.1 123 are put into eq 15, and the temperature T satisfying eq 15 is calculated by an iterative procedure. This temperature is the temperature of phase separation of the HA/D,O mixture under consideration. The calculation is repeated for mixtures of different compositions (different values of xoHA and Y)using the same value of xaB. The slight change of xapwith Y is neglected. Figure 6 shows the result of these calculations. The calculated curve is shifted by 0.135 K to higher temperatures to overlap with the experimental data. The experimental data are the same as that shown in Figure 1. The same calculation is repeated for the system D A / H 2 0 with x,,, = 0.1 123. The result is shown in Figure 7. Here, the calculated curve is shifted to higher temperatures by 0.205 K. The experimental data are the same as in Figure 2. The agreement between the form of the calculated and experimentally determined curves in Figures 6 and 7 is satisfactory. The temperature shifts of the calculated curves are assumed to reflect the influence of impurities. The properties of the coexisting surface of the system isobutyric acid/H20, D 2 0 appear to be typical for three-component systems in which H / D isotope-exchange reactions take place. This can be concluded from measurements of the temperature of phase separation as function of composition in the system phenol ( O H ) / D 2 0 with an upper critical point” and the system 2-but-
oxyethanol ( O H ) / D 2 0 with a lower critical point.’* In both systems the mixture of critical composition has a lower D 2 0 content than the mixture of maximal and minimal phase-separation temperatures, respectively.2’ For phenol (OH)/D20, T , = 78.445 “C, XOHP,~= 0.095, T,,,,, = 78.495 “C, and x ~ H ~ = 0.085. For 2-butoxyethanol (OH)/D20, T, = 42.385 “C,xoHBp = 0.061, and Tp,min = 42.350 OC. Tables of experimental data shown in Figures 1 and 2 and data leading to eq 9 and 10 can be found in ref 2 1. Acknowledgment. We thank G . Rottger, P. Harnisch, I. Schafke, and Dr. L. Belkoura for helping us during different stages of this study. The financial support of the Deutsche Forschungsgemeinschaft is also gratefully acknowledged. Registry No. Hydrogen, 1333-74-0; isobutyric acid, 79-3 1-2. (17) J. Timmermans and G. Poppe, C.R. Hebd. Seances Acad. Sci., 201, 524 (1935). (18) C. M. Ellis, J . Chem. Educ., 44, 405 (1967). (19) J. Zinn-Justin in “Phase Transitions”, M. Levy, J. LeGuillou, and J. Zinn-Justin, Eds., Plenum Press, New York and London, 1982, p 349. (20) W. J. Green, J . Chem. Eng. Data, 24, 92 (1979). (21) P. Gansen, Doctoral Thesis, University of Koln, 1984.
Ionic Atmosphere of Rodlike Polyelectrolytes. A Hypernetted Chain Study Russell Bacquet and Peter J. Rossky*t
Department of Chemistry, University of Texas at Austin, Austin, Texas 78712 (Received: August 12, 1983; In Final Form: November 16, 1983)
Numerical solutions to the hypernetted chain (HNC) integral equation have been obtained for a model system representing an infinitely dilute rodlike polyelectrolyte in an aqueous 1-1 electrolyte solution. Bulk salt concentrations, cST, of lo-], and M, and reduced polyion charges, 6, ranging from 0.5 to 5.0, have been studied. The distribution functions are analyzed in terms of various structural parameters and compared with the counterion condensation (CC) formalism of Manning and with available Poisson-Boltzman (PB) data. Although HNC and CC are in general qualitative agreement, several quantitative aspects of CC theory, including the special nature off = 1.0 and the csT independence of the Manning radius, are not reproduced by HNC.
I. Introduction It is widely appreciated that the behavior of polyelectrolytes in solution is a sensitive function of the ionic environment of the polymer. Of special importance is the influence of solution composition on nuclei acid conformation’ and on the mode of interaction of nucleic acids with other solution components such as proteinsa2 A detailed understanding of the spatial distribution of small ions in the vicinity of a polyelectrolyte is fundamental to a microscopic interpretation of such phenomena. However, experimental studies of these distributions3 are, at present, only capable of elucidating the general character of the structure and frequently require substantial assumptions for their interpretation! Complementary theoretical descriptions are therefore of great interest. Theoretical treatments have been based, with few exception^,^^^ on either the counterion condensation (CC) formulation, developed primarily by M a n n i ~ ~ g , ~or- ’the ~ Poisson-Boltzmann (PB) e q ~ a t i o n . ’ ~ ,These ’~ approaches vary in the type and detail of information provided. C C theory is developed from a free energy minimization using a simple two-state model. The resulting description has the substantial advantage that results for a variety of phenomena depend on only a few physical parameters and can be obtained without the need for complex calculations. PB theory Alfred P. Sloan Foundation Fellow.
0022-3654/84/2088-2660$01.50/0
provides a more detailed description in that the small-ion distribution functions are obtained. However, the distributions obtained from PB theory account only for the mean electrostatic field due to the small-ion environment and inherently neglect other, short-ranged, forces among the small ions. Theoretical analysis15 indicates that, at least for highly charged polyelectrolytes in dilute (1) H. Drew, T. Takano, S. Tanaka, K. Itakura, and R. E. Dickerson, Nature (London), 286, 567 (1980). (2) A. Rich, N. C. Seeman, and J. M. Rosenberg in “Nucleic acid-Protein
Recognition”, H. J. Vogel, Ed., Academic Press, New York, 1977. (3) See, for example: H. Magelhat, P. Turr, P. Tivant, M. Chemla, R. Menez, and M. Drifford, Biopolymers, 18, 187 (1979); J. L. Leroy and M. Gubron, ibid., 16, 2429 (1977); R. W. Wilson and V. A. Bloomfield, Eiochemistry, 18,2192 (1979); P. C. Karenzi, B. Meurer, P. Spegt, and G. Weill, Biophys. Chem., 9, 181 (1979). (4) C. F. Anderson, M. T. Record, Jr., and P. A. Hart, Biophys. Chem., 7, 301 (1978). (5) E. Clementi and G. Corongiu, Biopolymers, 21, 763 (1982). (6) D. Bratko and V. Vlachy, Chem. Phys. Lett., 90, 434 (1982); M. LeBret and B. H. Zimm, Biopolymers, 23, 274 (1984). (7) G. S. Manning, J . Chem. Phys., 51, 924 (1969). (8) G. S . Manning, J . Chem. Phys., 51, 934 (1969). (9) G. S . Manning, J . Chem. Phys., 51, 3249 (1969). (10) G. S. Manning, Biophys. Chem., 7, 95 (1977). (11) G. S. Manning, Biophys. Chem., 9, 65 (1978). (12) G. S. Manning, Q. Reu. Biophys., 11, 179 (1978). (13) A. Katchalsky, Pure Appl. Chem., 26, 327 (1971). (14) D. Stigter, J. Colloid Interface Sci., 53, 296 (1975). (15) M. Fixman, J . Chem. Phys., 70, 4995 (1979).
0 1984 American Chemical Society
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