RUBINBATTINO
3408
Thermodynamics of Binary Solutione of Nonelectrolytee with 2,2,4=Trimethylpentane. I.
Volume of Mixing (25') and Vapor-Liquid
Equilibrium (35-75') with Cyclohexane'.
by Rubin Battinolb Department of Chemiatry, Illinois Institute of Technology, Chicago, Illinois 60616
(Received February 1'7, 1966)
The vapor-liquid equilibrium in the 2,2,4-trimethylpentane-cyclohexanesystem was measured at 10' intervals in the range 35-75' and over the entire range of composition. The volume of mixing for the same system was determined a t 25' and found to be extremely small (less than 0.012 cc/mole). The excess thermodynamic functions were calculated for the conditions of mixing a t constant pressure and were found to be positive over the entire range of composition and temperature and small in magnitude (at 35' the maxima are AGpE, 10.9 cal/mole, ARPM,42 cal/mole, and ASPE,0.10 cal/mole deg). The excess Gibbs function decreases slightly with increasing temperature, but the other functions are independent of temperature within the experimental error. Owing to the very small volume change on mixing, the constant volume thermodynamic functions are identical with the constant pressure functions. The results are in good agreement with a new approach proposed by Flory and co-workers.
Introduction The present work can be considered to be a continuation of the precision studies of binary solutions of nonelectrolytes which were begun by Scatchard and coworkers2 and which have been continued by Wood and c o - ~ o r k e r s . ~The recent studies by Wood and Sandus4* and Weissman and Wo0dt4b and the paper by Wood, Sandus, and Weissman5 all indicate the importance of using isooctane (2,2,4-trimethylpentane)as the central component in an investigation of orientation effects in the pure liquid components. Several workerse have suggested that the difference in the heat capacity a t constant volume of the pure substance in the liquid and ideal gas states is a good measure of the intermolecular orientation in the pure liquid. This difference has been calculated by Wood, et 4 5 for a number of solvents which are convenient to use for vapor-liquid equilibrium measurements; for isooctane this value is 1.0 cal/mole deg, for cyclohexane 4.7 cal/mole deg, for benzene 4.9 cal/mole deg, and for carbon tetrachloride 3.4 cal/mole deg. The benzene-isooctane system has been studiedj4the cyclohexane-isooctane system is the subject The Journal of Physical Chemistry
of this paper, and the carbon tetrachloride-isooctane system is currently under investigation in this laboratory. This paper reports the vapor-liquid equilibrium (35-
(1) (a) Presented in part before the Division of Physical Chemistry, 151st National Meeting of the American Chemical Society, Pittsburgh, Pa., March 1966; (b) Chemistry Department, Wright State College, Dayton, Ohio 45431. (2) G. Scatchard, C.L. Raymond, and H. H. Gilmann, J . Am. Chem. SOC.,60, 1275 (1938); G. Scatchard and C. L. Raymond, ibid., 60, 1278 (1938); G. Scatchard, S. E. Wood, and J. M. Mochel, J . Phys. Chem., 43, 119 (1939);G. Scatchard, 8.E. Wood, and J. M. Mochel, J . Am. Chem. SOC.,61,3206 (1939); G.Scatchard, S. E. Wood, and J. M. Mochel, ibid., 62, 712 (1940); G. Scatchard, S. E. Wood, and J. M. Mochel, ibid., 68, 1957 (1946). (3) S. E. Wood, ibid., 79, 1782 (1957); S. E.Wood and J. A. Gray, ibid., 74, 3733 (1952); and see ref 4, 5, 12, 13, and 16 listed in
this paper. (4) (a) S. E.Wood and 0. Sandus, J . Phys. Chem., 60, 801 (1956); (b) S. Weissman and S. E. Wood, J . Chem. Phys., 3 2 , 1153 (1960). (6) 8. E. Wood, 0. Sandus, and S. Weissman, J. Am. Chem. SOC., 79, 1777 (1957).
(6) J. A.Pople, Discussiona Faraday Sac., 15,35 (1953);L. A.Stavelay, K. R. Hart, and W. I. Tupman, ibid., 19, 130 (1953);J. Frenkel, "Kinetic Theory of Liquids," Oxford University Press, New York, N. Y., 1947.
THERMODYNAMICS OF
BINARY SOLUTIONS O F NONELECTROLYTES
75’ at 10’ intervals) and the volume of mixing (25’)’ for the cyclohexane-isooctane system over the entire range of compositim. (The phrase “volume of mixing” rather than the more cumbersome “volume change on mixing” will be used throughout.) The excess thermodynamic functions for the conditions of mixing at constant pressure and at constant volume have been calculated. These functions are positive for the temperatures investigated and are relatively small. The very small volume change on mixing results in the constant volume properties being essentially identical with those at constant pressure. The excess entropy and enthalpy of mixing are independent of the temperature, but the excess Gibbs function decreases slightly as the temperature increases. The results are compared with values calculated from the Scatchard-Hildebrand equation and a recent approach developed by Flow and coworkers.
3409
I
6.5crn
i M IXlNG
BOTTLE
Figure 1. Mixing bottle.
shown in Figure 1 is a modification of Wood and Experimental Procedure and Results Brusie’s, but has some advantages over the earlier deMaterials. The isooctane used was Phillips pure sign. The mixing bottles were fabricated from 35-mm grade and was purified by the methods described Pyrey tubing and each chamber holds about 53 cc. earlier. The purified solvent was vacuum-distilled The U-tube is made of 10-mm 0.d. tubing and each from calcium hydride into ampoules where it was stored chamber is tilted 20’ from the vertical. Using male under vacuum until ready for use. The density of the joints on the bottle minimizes the possibility of wetting isooctane used in both the volume of mixing and the the joints (which were used ungreased) during the mixvapor pressure experiments was 0.68780 g ~ m at- 25’. ~ This value compares favorably with 0.68774 obtained (7) Unpublished results by R. Battino and E. L. Washington give by Weissman and and Wood and S a n d u ~ , ( ~ the volume of mixing of this system as a function of composition and temperature ix the range 10 to 80”. with 0.68775 reported by Kretschmer, et with (8) C. B. Kretschmer, J. Nowakowska, and R. J. Wiebe, J . Am. 0.68776 reported by Brooks, et U Z . , ~ and with 0.68779 Chem. Soc., 70, 1785 (1948). reported in the “Selected Values of Properties of Hy(9) D. B. Brooks, F. L. Howard, and H. C. Crafton, J . Res. Natl. drocarbons.”’” Bur. Std., 24, 33 (1940). (10) “Selected Values of Properties of Hydrocarbons,” National The cyclohexane used in both the vapor pressure and Bureau of Standards Circular C461, U. S. Government Printing volume of mixing experiments was Phillips research Office, Washington, D. C., 1947. grade (Lot No. 435 with a purity of 99.94 mole %) with (11) D. H. Everett and F. L. Swinton, Trans. Faraday Sac., 59, 2476 (1963). a density of 0.77386 g cmM3at 25’. This compares well (12) S. E. Wood and A. E. Austin, J . Am. Chem. Sac., 67, 480 with the highly purified materialof Everett and Swinton” (1945). with a density of 0.77390 g with 0.77387 re(13) S. E. Wood and J. A. Gray, ibid., 74, 3729 (1952). ported in ref 10, and with 0.7738012 and 0.77382,13 re(14) I . A. McLure and F. L. Swinton, Trans. Faraday Sac., 61, 421 ported by Wood and co-workers. McLure and S ~ i n t o n ‘ ~ (1965). (15) They found that the more highly purified material reported in report a density of 0.7738 g for cyclohexane used ref 11 gave identical volumes of mixing within their experimental for volume of mixing experiments. l5 error (i0.003 cm3 mole-’). It appears that very high purity material is not necessary for precise volume of mixing measurements. The water used to calibrate the pycnometers was obHowever, the same does not hold true for precise vapor-liquid tained by distillation from alkaline permanganate in an equilibrium measurements. all-Pyrex system followed by distillation in an all-quartz (16) The procedure of boiling the water before use (to remove dissolved air and COz) is only necessary where the possibility exists of system.16 The mercury used in the mixing bottles was outgassing during the calibration procedure. Since this possibility cleaned by dropping through both alkaline and nitric acid did not exist in the present work, the water was not boiled. The effect of dissolved air is to increase the density of water by about solutions followed by triple distillation. 3 X 10-7 at room temperature and this is entirely negligible. N. Apparatus-Volume of Mixing. The volume of mixBauer and 5. 2. Lewin in “Physical Methods of Organic Chemistry,” Vol. I, Part I, 3rd ed, Interscience Publishers, Inc., New York, N. Y., ing was determined at 25’ following the procedure of 1959, p 135. Wood and S a n d u ~with ~ ~ the use of pycnometers de(17) S. E. Wood and J. P. Brusie, J . Am. Chem. Sac., 6 5 , 1891 scribed by Wood and Brusie.17 The mixing bottle (1943). Volume 70,Number 11 November 1966‘
RUBINBATTINO
3410
ing process. The l0/l8 joint on side A was used for filling both the U-tube with mercury and side A with solvent. The T/15 joint on side B is for filling side B. The slanting chambers and the large bore of the U-tube facilitates the mixing process. The mixtures were prepared in duplicate each a t approximately one-eighth mole fraction with alternation of the substance first weighed. Results-Volume of Mixing. The densities determined for each mixture were used to calculate the relative volume of mixing, AVM/Vo,where AVMis V - Vo, V is the volume of the solution, and V ois the sum of the volumes of the components. The equation AVM
- - - &&(7.63 X lod4 - 1.74 X lo-?&) VO
(1A)
obtained by least-squares smoothing fits these data with a standard deviation of *0.23 X lo-'. Subscript 1 refers to cyclohexane, X is the mole fraction, and 2 is the volume fraction. Equation 1B gives the volume of mixing, AVM,directly in terms of the mole fraction with an average deviation of *0.003 cc/mole.
A V =~ ~ ~ ~ ~ (-00 . 212 30~ ~ )
"0
.4
-
0
0
-.4
X
$42 Q
-1.2
0 4 ' 8
I 0
.2
I
I
4
.6
1
Zcyclohexane Figure 2. The relative volume of mixing, AVM/VO, of the cyclohexane-isooctane system at 25" us. the volume fraction of cyclohexane.
The Journal of Physical Chemistry
Mixtures a t 25" -Cyclohexan-
d,
-AVM/Vo
XI
Zl
g om-8
Obsd
0.00000 0.06327 0.11720 0.12666 0.23190 0.23328 0.32192 0.32507 0.43520 0.44733 0.56420 0.56585 0.66882 0.68028 0.77573 0.78464 0.89073 0.89627 1.00000
0.00000 0.04235 0.07988 0.08674 0.16507 0.16613 0.23715 0.23977 0.33536 0.34641 0.45880 0.46048 0.56942 0.58217 0.69373 0.70465 0.84223 0.84980 1.00000
0,68780 0.69142 0.69466 0.69523 0.70198 0.70207 0.70818 0.70839 0.71665 0.71756 0.72728 0.72741 0.73683 0.73791 0.74756 0.74850 0.76036 0.76102 0.77386
0.00 0.37 0.34 0.52 0.44 0.45 0.45 0.58 0.21 0.71 0.07 0.30 -0.31 -0.11 -0.86 -0.84 -1.09 -1.17 0.00
X 1 0 Calcd
0.00 0.28 0.46 0.49 0.66 0.66 0.64 0.63 0.40 0.37 -0.08 -0.09 -0.55 -0.60 -0.94 -0.96 -0.93 -0.91 0.00
A
0.00 0.09 -0.12 0.03 -0.22 -0.21 -0.19 -0.05 -0.19 0.34 0.15 0.39 0.25 0.49 0.08 0.12 -0.16 -0.27 0.00
(1B)
The densities of the cyclohexane-isooctane mixtures are given in Table I and the volumes of mixing as both AV"/Vo and (AV'/Vo)/ZIZz are shown in Figure 2. The volume of mixing is just larger than the experimental error and ranges from a value of 0.010 cc/mole to -0.013 cc/mole. Danusso's results18 for this system at 30' indicate that the volume of mixing is less than 0.04 cc/mole which is the experimental error in his work. The shape of the curve has been verified in ref 7. .8
Table I : Densities of Cyclohexane-Isooctane
The composition of the vapor and liquid phases for the vapor-liquid equilibrium measurements were deter mined from density measurements at 25' and eq 1A in conjunction with eq 2 and 3. The resultant error in
(3) determining the compositions from the density measurements is *0.0003 in the mole fraction, although the error in the mole fractions listed in Table I is *0.00002. Pois the molar volume of the pure component. A pparatus-Vapor-Liquid Equilibrium. The vaporliquid equilibrium measurements were made using the stills described by Weissman and following the procedures outlined in their paper. Stopcock S was replaced with one having a solid Teflon plug, and the mercury-seal joints M, L, and N were replaced with Viton O-ring joints. This system was vacuum tight and a bit easier to work with than the previous apparatus. Temperature measurement was simplified by substituting two miniature four-lead platinum resistance thermometers (manufactured by the Rosemount Engineering Co., Model No. 172A6B4) for the thermocouples previously used. These thermometers have a 2-mm o.d., are 20 mm long, and are encased in a (18) F . Danusso, Rend. Accad. A'5rE. Lincei, V I I I , 17, 109 (1954).
THERMODYNAMICS OF BINARY SOLUTIONS OF NONELECTROLYTES
3411
Table II : Vapor-Liquid Equilibria Dev in Yl
YI
p, mm
0,2220 0.3006 0.4245 0.5316 0.5534 0.7350 (0.8803)
0.3580 0.4564 0.5848 0.6825 0.6999 0.8359 (0.9264)
-0.0040 -0 0024 -0.0021 -0.0022 -0.0028 -0.0011 (-0.0034)
95.57 101.70 110.93 118.67 120.19 133.07 142.72
0,0697 0,2133 0.3135 0.4387 0.5508 0.7017 0.8289 (0.8824)
0.1281 0.3370 0.4608 (E. 5907 0.6945 0.8115 0.8960 (0.9277)
-0.0007 -0.0036 -0.0051 -0.0049 0.0005 0.0010 0.0001 ( - 0.0020)
127.93 143.88 155.01 168.54 180.20 195.77 208.35 213.67
0.0834 0.1094 0.2244 0.4416 0.8245 (0.8807)
0.1464 0.1855 0.3510 0.5886 0.8905 (0.9253)
-0.0002 0.0026 -0.0003 -0.0034 -0.0008 (-0.0021)
0.0917 0.1165 0.3396 0.4454 0.7235 0.8870
0.1532 0.1913 0.4772 0.5890 0.8184 0.9299
0.2178 0.3408 0.7149 0.8131 0.8872
0.3277 0.4742 0.8119 0.8785 0.9285
XI
Dev in
AE,E,
Dev in
dP/dYi,
Dev in
P
cal/mole
AG,E
Ob8d
dP/dYi
0.0752
35" -0.34 -0.22 -0.63 -0.22 -0.25 0.02 (-0.07)
6.76 7.98 10.76 10.74 10.87 9.79 (6.35)
0.10 -0.40 0.52 -0.14 -0.02 0.62 (1.13)
0.0974 0.1101 0.1116 0.1303 (0.1285)
-0.0021 -0.0015 -0.0020 -0.0022 -0.0030 - 0.0021 (-0.0157)
45" -0.03 -0.36 -0.33 -0.26 -0.33 -0.11 -0.13 (0.02)
1.76 6.49 8.54 10.44 8.92 8.08 6.01 (5.52)
-0.34 0.68 0.74 1.03 -0.97 -0.79 -0.28 (0.83)
0.0892 0.1063 0.1227 0.1416 0.1632 0.1879 0.2005 (0.1929)
0.0006 -0.0048 -0.0044 -0.0054 -0.0010 0.0022 -0.0005 (-0.0139)
191.34 195.60 213.46 246.80 301.96 310.51
55" -0.17 -0.12 -0.72 -0.34 -0.15 (0.07)
1.87 3.54 4.29 8.92 5.29 (5.90)
-0.34 0.68 -1.12 0.46 -0.50 (1.61)
0.1292 0.1320 0.1591 0.2013 0.2746 (0.2691)
-0.0004 -0.0031 -0.0002 -0.0055 -0.0055 (-0.0201)
-0.0030 -0.0032 -0.0040 0.0004 -0.0010 -0.0003
276.94 282.46 330.09 352.14 408.64 440.18
65" -0.24 -0.24 -0.39 -0.46 0.17 -0.02
2.44 3.21 7.22 6.98 7.60 3.76
0.32 0.56 0.74 -0.53 0.73 0.09
0.1770 0.1841 0.2461 0.2824 0.3534 0.3918
-0.0059 -0.0056 -0.0073 0.0012 -0.0051 -0.0043
-0.0024 -0,0019 0.0018 -0.0013 0.0007
424.72 459.93 563.29 588.98 608.57
75" -0.74 -0.76 0.11 -0.22 -0.01
3.66 5.44 5.49 5.10 3.50
-0.33 -0.19 -0.63 0.34 0.27
0.2881 0.3350 0.4882 0.4922 0.5160
I
-
-
platinum-rhodium sheath. A 0.25-in. 0.d. brass protecting cylinder improved thermal contact. The thermometers were calibrated directly against the triple point of water and a benzoic acid cell purchased from and calibrated by the National Bureau of St,andards. I n a 15-month period, one thermometer showed a drift of O.O0lo and the other a drift of 0.003" in the temperature equivalent of the Ro value. It is believed that the measured temperatures are accurate to better than *0.01 O . The vapor pressure of pure isooctane, which was used in the reference still, was calculated froml0 log P = 6.81984 - [1262.490/(221.271 t ) ] . The vapor pres-
+
0.0849
-0.0057
- 0.0052 0.0127 -0.0145 -0.0135
sures of the pure cyclohexane at the five temperatures measured agreed with those calculated from*Olog P = 6.84498 - [1203.526/(222.863 t ) ] within * O . l l mm and thus this equation was used to calculate the vapor pressure of pure cyclohexane. Results-Vapor-Liquid Equilibriums. The experimenta.1.results for the vapor-liquid equilibria at the five temperatures investigated are given in Table 11. The first column gives the mole fraction of cyclohexane in the liquid phase, the second column the mole fraction of cyclohexane in the vapor phase, and the fourth column the total pressure. The remaining columns will be discussed later in this paper. The quantities in
+
Volume 70,Number 11 November I066
RUBINBATTINO
3412
parentheses in this table (and many others) were not used to calculate the thermodynamic functions nor to smooth the data. The differences between these values and the smoothed values were too large to be considered reliable. It should be noted in passing that although Dixon and M c G l a ~ h a n ' s(and ~ ~ others) comments about the difficulty of operating recirculating stills are true, the advantage of being able to overdetermine the system by measuring P, T, X, and Y and testing each point for thermodynamic consistency is still sufficiently important to warrant the use of this method in precision measurements. The results for 35" are shown in Figure 3. It is apparent that this system shows almost ideal behavior with very small positive deviations from the dashed Raoult's law lines. For example, at a mole fraction of 0.53, the deviation is less than 1 mm. These very small positive deviations also exist at the higher temperatures.
Thermodynamic Functions The change in the thermodynamic functions for mixing at constant pressure were calculated for the change of state in which X1moles of cyclohexane at 1 atm pressure and the chosen temperature T and XZmoles of isooctane at the same conditions are mixed to form one mole of solution under the identical conditions. The excess chemical potentials defined by
aplE = pl - plo - RT In Xl
(4)
were calculated by use of ~p~~ =
+ - V l W - Pl") + 6(1 - YdZP+
RT In (PY1/Pl0X1) (Pll
(71-
VlO>(l - P ) (5)
By changing the subscripts, eq 4 and 5 are valid for the second component. I n these equations, P is the total vapor pressure, Po is the vapor pressure of the pure component, X and Y are the mole fractions of the component in the liquid and vapor phases, respectively, P and Poare the partial molar volume of the component in solution and the molar volume of the pure component, respectively, Pll is the second virial coefficient, and T = 273.15 t. The term 6 is defined by 6 = 2P12 - PU - Plz. I n the derivation of eq 5 the equation of state for the vapor was the virial equation used to the second virial coefficient. The last term in eq 5 corrects the chemical potentials from the experimental pressures to 1 atm, but is entirely negligible for this system since the volume of mixing is so small. The second virial coefficients were calculated using Berthelot's equation. This procedure has been justi-
+
The Jour& of Physical Chemistry
x cyclohexane Figure 3. Vapor pressure and partial pressures vs. the mole fraction of cyclohexane.
fied by many workers.20 The values of PZ2selected for isooctane are identical with those of Weissman and Wood.4b The second virial coefficients are given by Pii
= Bit
- (A,,/T2)
(6)
where A = 27RTO2/64P,and B = 9RTc/128Pc. The values of the critical temperature and pressure used for cyclohexane were 280" and 40.0 atm, respectively, and were the values critically selected by Kobe and Lynn.z1 The resulting equation for ,811 was 011 = 80 -14.65 X 107T-2. The values calculated from this equation (Table 111) are in good agreement with the recent experimental determination by Bottomley and Coopes,2z but show a marked difference with the experimental values of Waelbroeckz3at 35" (-1592 cm3 mole-' as calculated by ScatchardZob)and are in excellent agreement with Waelbroeck's value at 75" (-1132 cm3 mole-l). (For perspective, it should be noted that a difference in B11 or PZz of 20% results in an error of about 0.5 cal/mole in AQE so that the actual choice of values for the second virial coefficient is not very critical.) The term Blz was calculated by assuming the combining laws (19) D. T.Dixon and M. L. McGlashan, Nature, 206, 710 (1965). (20) (a) J. P.Lambert, G. A. H. Roberts, J. 5. Rowlinson, and V. J. Wilkinson, Proc. Roy. SOC. (London), A196, 113 (1949); (b) G. Scatchard and F. G. Satkiewica, J. A m . Chem. SOC.,86, 130 (1964). (21) K. A. Kobe and R. E. Lynn, Jr., Chem. Rev., 52, 117 (1953). (22) G.A. Bottomley and I. H. Coopes, Nature, 193, 268 (1962). (23) F. G.Waelbroeck, J. Chem. Phys., 23, 749 (1955).
341 3
THERMODYNAMICS OF BINARY SOLUTIONS OF NONELECTROLYTES
.BIZ =
1/&1B1’8
+ B2z1’y
(74
Aiz = (AiiAzz)”’
(7b)
The resulting values of the second virial coefficients are given in Table 111. Table 111: Second Virial Coefficients, Calculated from the Berthelot Equation, in Cubic Centimeten per Mole 1, “ C
-- 811
-Ban
- 61;
6
35 45 55 65 75
1463 1:367 1280 1201 1129
2182 2039 1910 1791 1683
1786 1669 1563 1467 1377
73 68 63 58 58
ure proposed by Redlich and KisterlZ4which require that
- AkzE)dX1 = 0
[(ApiE
(12)
This is shown in Figure 4 where the data at 45” is presented. The dashed line is for the values of (AklE ApzE) determined from the smoothed values (eq 9l l ) , the circles represent the experimental points, and the solid line is a least-squares fit for the experimental points. The dashed line must satisfy eq 12 since eq
8E
40 -
I
I
I
I
I
I
I
I
-
1 m s
The molar excess change of the Gibbs free energy on mixing at constant pressure was calculated from the excess chemical potentials according to
The experimentally determined values of AdE are given in column six of Table 11. The AdEvalues were smoothed by fitting to an equation of the type
Ad’< XiXZ[a + b(X1 - Xz)]
(9) by the method of least squares. The coefficients a and b were then fitted as functions of the temperature by the method of least squares. The resulting equations are u = 166.40 - 0.3999T
(10)
b = 21.57 - 0.043T
(11)
The deviations (experimental - calculated) between the experimental and the smoothed values of AdE are given in column seven of Table 11, and the root-meansquare deviation is ~ t 0 . 6cal/mole. The values in parentheses in Table I1 were not used in the smoothing operation. The smoothed values were also checked by using them and eq 9-11 to back-calculate the total pressure P and the mole fraction in the vapor phase at each experimental point. The deviation between experimental and calculated values for Y1 is in column three and for P in column five, respectively, of Table 11. These deviations are slightly larger than those estimated from a study of the errors in the experimental method, but are considered to be satisfactory. The data were checked for thermodynamic consistency by two methods. The first followed the proced-
“;t“ -20Q
.-
1
w -40-
W
a &O
.2
P .6 cyclohexane
.8
1
Figure 4. AplE - AprzE 21s. the mole fraction of cyclohexane at 45”.
9 must satisfy the Gibbs-Duhem relation on which the Redlich-Kister method is based. The agreement between the experimental points and the line is apparently poor, but this is without taking into account the magnitude of the errors and the scale of the ordinate in Figure 4. For perspective, a similar plot presented by Weissman and Wood4bfor the benzene-isooctane system ranges from 200 to -400 cal/mole. They also indicate that a difference of 4-6% in the ratio of the areas is within the experimental error. Transposing a similar error in the area indicates that a difference of 5040% in the ratio of the areas for the system studied in the present work would be within the experimental error. Carefully drawn graphs of the type shown in Figure 4 for all five temperatures for the experimental points show the areas above and below the line to be equal within the estimated errors and thus that the data taken together are thermodynamically consistent. The more stringent point-by-point test for thermodynamic consistency used by Weissman and W00d4b compares the slopes dPldY1 for the experimental and the smoothed values. The slopes dP/dY1 are calculated from (24) 0.Redlich and A.
T.Kiater, Ind. Eng. Chem., 40,
Volume 70, Number 11
345 (1948).
November 19613
RUBINBATTINO
3414
I
d -P- d Yi (RT/P)
(Y1 - XJ[(RT/YJz) - 26Pl XlPll XZPZZ (XlY22 XZYIZ)-
+
+
+
+
40
F;l
4 0"
Pi (13)
3
The experimental values of dPldY1 are given in the eighth column of Table I1 and the deviations of the calculated values from this are in the last column. The agreement is better a t 35" than at 75", but the over-all average deviation between the observed and calculated dP/dYl is 1.6%, which is excellent when it is recalled that this sensitive test is comparing slopes. The near ideality of the cyclohexane-isooctane system presented some special difficulties. For example, the activity coefficients (the term PYl/P1OX1 in eq 5 ) were all very close to 1. This meant that the second and third terms in eq 5 , which correspond to the virial equation of state correction for the nonideality of the vapor phase, contribute significantly to the chemical potential, especially a t 75". The virial correction amounted to (on the average for the five temperatures) 36% for cyclohexane and 22% for isooctane of the experimental chemical potentials. The change of the enthalpy and the excess entropy on mixing a t constant pressure were calculated by the standard manipulation of eq 9 to 11. The results a t 40, 55, and 70" are presented in Table IV for every tenth mole fraction. Since the constants a and b in eq 9 were found to be linear functions of the temperature, both ARpM and AgPEare independent of the temperature. The values of these three functions are all positive over the entire range of composition and temperature. The results are illustrated in Figure 5. The maximum error in ASPEis estimated to be 0.015 cal/ mole deg and the maximum error in A s p M 6.3 cal/ mole. LundbergZ6directly measured the heat of mixing at 25 and 50" with an estimated error of the order of 1%, and a t XI = 0.5 his values are 43.3 and 41.3 cal/mole, respectively. Considering the maximum estimated error in A s p M the agreement with Lundberg's data is excellent. The difference between the change in the thermodynamic functions on mixing a t constant pressure and on mixing a t constant volume was calculated according to the equations developed by ScatchardZ6and was found to be entirely negligible. This is, of course, due to the extremely small volume change on mixing.
'1-
Discussion The nearly ideal behavior of the cyclohexaneisooctane system marks it as a good system for testing theories of The Journal of Physieal Chemistry
A
-
I
I
I
a
E
m
u
Xcyclohexane Figure 5. Excess thermodynamic functions a t constant pressure for the cyclohexane-isooctane system a t 40" us. the mole fraction of cyclohexane.
Table IV: Excess Thermodynamic Functions in Calories per Mole
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
3.1 5.8 8.0 9.5 10.3 10.3 9.3 7.4 4.3
2.6 4.9 6.8 8.1 8.8 8.8 8.0 6.3 3.7
2.1 4.0 5.6 6.7 7.3 7.3 6.7 5.3 3.1
13 25 33 39 42 41 37 29 17
0.033 0.060 0.080
0.094 0.100 0.098
0.088 0.068 0.039
17 32 43 51 55 55 51 40 24
Calculated from Scatchard-Hildebrand equation.
solutions. The corresponding states average potential model developed by Prigogine and his co-workersZ7has had some success with mixtures where the two components have approximately the same molecular size. However, the difference in the r* values for isooctane and cyclohexane were too large to apply this theory meaningfully. The calculated excess properties were all several orders of magnitude larger than the experimental values. The calculations reported by Benson and co-workersZ8on comparing experimental and calculated excess volumes and enthalpies of mixing for binary mixtures of hydrocarbons with benzene and (26) G. W. Lundberg, J . Chem. Eng. Data, 9 , 193 (1964). (26) G. Scatchard, Trans. Faraday SOC.,33, 160 (1937). (27) I. Prigogine, A. Bellemans, and V. Mathot, "The Molecular Theory of Solutions," Interscience Publishers, Inc., New York, N. Y . , 1957. (28) I. A. McLure, J. E. Bennett, A. E. P. Watson, and G. C. Benson, J . Phys. C h m . , 69, 2759 (1965).
THERMODYNAMICS OF BINARY SOLUTIONS OF NONELECTROLYTES
3415
Table V : Parameters for the Pure Liquids for the Flory Calculation r, t,
v,
a X 108,
O C
cc/mole
deg-1
oal/oc deg
Cyclohexane
25 40 70
108.75" 110.79' 115.25"
1,217" 1.264" 1.365"
0. 225b 0.234b 0.193*
1.2905 1.3111 1.3540
84.27 84.51 85.12
4719 4754 4834
Isooctane
25 40 70
166.06' 169.14' 175. 85c
1.197' 1.243' 1.361'
0.179' 0. 166d 0. 141d
1.2868 1.3072 1.3533
129.05 129.39 129.94
4760 4792 4845
Liquid
v*,
co/mole
T*, O K
p*, cal/oo
127 126 121 88.4 88.8 88.6
C, cc/mole
1.14 1.13 1.07 1.21 1.21 1.20
' S. E. Wood and J. A. Gray, J. Am. Chem. SOC.,74, 3729 (1952). G . A. Holder and E. Whalley, Trans.Faraday Soc., 58, 2095 (1962). ' 8. E. Wood and 0. Sandus, J.Phys. Chem., 60,801 (1956). W. A. Felsing and G. M. Watson, J. Am. Chem. Soc., 65, 780 (1943), using the values for the coefficient of compressibility ascalculated by 5. Weissman and S. E. Wood, J . Chem. Phys., 32, 1153 (1960). ' A linear extrapolation of the 40 and 70' values. toluene showed a marked sensitivity to differences in r*. The Scatchard-Hildebrand equation A E V M
=
(XiViO
+ XzVzo)(6i - 62)'$1+z
(14)
can be used to calculate the energy of mixing at constant volume. The cp's are volume fractions and the 6's are solubility parameters. The value of ARvM was calculated for each tenth mole fraction at 25, 40, 55, and 75", but since the results differed by no more than 2 cal/mole, only the average value is reported in the last column of Table IV. The solubility parameters used in the calculations for these four temperatures are for cyclohexane 8.21, 8-01, 7.82, and 7.63, and for isooctane 6.9, 6.71, 6.55, and 6.38, respectively. The agreement shown in Table IV between the calculated and experimental energies of mixing is very good. The Scatchard-Hildebrand approach can be used to calculate the volume change on mixingzgfrom the following equation
in Table V. In this table, v is the molar volume, (Y is the coefficient of thermal expansion, y is ( b P l b T ) , or the thermal pressure coefficient, calculated from y = cy/p where is the coefficient of compressibility, v' is a reduced volume, calculated from fi'h
-1=
(aT/3)/(1
+ aT)
(16)
is a reduced temperature, calculated from the reduced equation of state p = (o1h - 1)/c4/a (17)
v* is given by v/v' and is the hard-core volume per mole, T* = T / p and is the characteristic temperature, P* = yTC2 and is the characteristic pressure, and C = P*v*/RT* and is a parameter expressing the number of external degrees of freedom per molecule. The heat of mixing per mole waa calculated from
AHM = X1P,*v1*(; -
i)+
AP= aAEvM= ( ( Y ~ / ~ ~ ) ~ 5 4 p E (15) where p is the isothermal compressibility, CY is the co62)/2. efficient of thermal expansion, and 6 is (a1 Despite the many a s s ~ m p t i o n made s ~ ~ in the derivation of this equation, the calculated volume of mixing a t 25" and a mole fraction of 0.5 is 0.351 cc/mole. This result w m obtained by using the assumption in the derivation that ASVE = 0 and that AGpE = dEvM. If the experimentally determined value of AdPEis used in eq 15, then the calculated value of the volume of mixing is 0.075 cc/mole. Both of these calculated values can be considered to be in reasonable agreement with the observed value. Flory and co-workersal recently proposed a statistical thermodynamic approach using a reduced equation of state. The data used for these calculations are given
+
This is for molecules of nearly the same size. To simplify the calculations the molecular element or segment was defined (by Flory) to be in correspondence for the two species such that rl and r2 (where r is the number of elements or segments per molecule) are in the ratio of the respective molar core volumes vl* and vz* or rl/r2 = vI*/v2*. Further, SI and sz (where s is the number of intermolecular contact sites per segment) (29) J. H. Hildebrand and R. L. Scott, "Regular Solutions,'' PrenticeHall, Ino., Englewood ClitIs, N. J., 1962, p 108. (30) See, for example, p 114 in ref 29. (31) (a) P.J. Flow, J . Am. Chem. SOC.,87, 1833 (1965); (b) A. Abe and P. J. Flow, ibid., 87, 1838 (1965); (c) P. J. Flory and A. A b , ibid., 86, 3663 (1964).
Volume YO,Number 11 November 19136
RUBINBATTINO
3416
~~
Table VI: Comparison of Observed and Calculated Excess Quantities for the Flory Calculation for an Equimolar Mixture
25 40 70
0.653 0.653 0.655
fj&d
s1/82
V,
V,
calcd
obad
mole, calcd
42 42 42
1.2902 1.3105 1.3548
1.2883 1.3087 1.3536
0.204 0.184 0.125
e2
1.4 1.2 0.8
0.57 0.57 0.57
37 33 25
x12,
rl/rZ
cal/ mole, obsd
cal/cc
t, oc
AHM
cal/ mole, oalcd
1.15 1.15 1.15
TSR,
vE, cc/ mole, obsd
AHM
VE, eo/
oal/ mole, oalcd"
TSR, cal/ mole, oalcdb
11 9 7
-2 -3 0
0.013'
From Oobd and A&omb = A s i d . From 5calcd and A s o o m b = - R [ X i In From DoaiCd and d S c o r n b= A s i d . and AScarnb = - R [ X t In M X Z In e].e At a mole fraction of 0.25, essentially zero at X1 = 0.5.
+
are in the ratio of the molecular surface areas of contact per segment. For the calculations performed here, both molecules were taken to be spherical such that the ~ (u1*/u2*) -'Ia could be equation s1/s2 = ( T ~ / T ~ ) - ' / = used. A better approximation could be made, but the results (see below) with this one certainly justify its use in this case. The segment fraction (02 (per mole) is cp2
=
1 -
cp1
and the site fraction
e2 =
1 -
el
=
=
rzXz/(r1X1
+ r2X2)
(19)
e2 is defined as
+
w ~ / ( ( ~ ~ C0zs2> s~ = x2(u2*)g'*/ [X,(u1*)*/a
+
(20)
The interaction parameter XI2was calculated from
XI2 = P1* [l -
(S1/SZ)'/~(P2*/P1*)'/~]2 (21)
ti for the mixture was calculated as indicated by Abe
and F l ~ r y . ~ The ~ b results for the calculation of AHM are given in Table VI along with values of the parameters used in the calculation. The agreement is quite good although the calculated heats of mixing show a marked decrease with increasing temperature which was not found in the present work. Lundberg's25 data for this system show a slight decrease for the heat of mixing of about 2 cal/mole over a range of 25". The residual entropy of mixing SR was calculated from
SRvalues were calculated using both ccalcd (column 8 in Table VI) and &bad for the mixture and are presented in Table VI as TSR. %,bad is calculated from the experimental molar volume %bed from gobad
=
uobad/(Xlul*
The Journal of Physical Chemistry
+
x2u2*)
(23)
31 30 30
-/-
30 31 34
xz In M].
18 18 23 From
The excess entropy, SE is related to the residual entropy, SR,according to
SE = SR
+
Ascomb
-
(24)
a i d
+
where A S i d = -R [XI In XI XZIn X,] and A s c o m b is the combinatorial entropy appropriate to the system concerned. Since there is a fairly large disparity in the molar volumes for this system, the following formulation for AScomb has been used although it has been criticized for over-correcting. Ascomb
x2(212*)p/a]
pi
TSE, T S E , T S E , cal/ eal/ cal/ mole, mole, mole, oalodC obsd oalodd
=
-R [XI In
(01
+ X2 In
~2
I
(25)
The difference between Scomb and A S i d is 0.067 cal/mole deg, and values of TSEusing this factor for calculations based on Gcealcd and &b,ad are presented in Table VI. Comparison with the experimental value of TSEshows excellent agreement with the excess entropy calculated using eq 25 and &lcd. According to eq 22, the excess entropy does not vanish when the volume change on mixing is zero, which makes it particularly applicable to the present system and a significant advance over regular solution theory. The excess molar volume vE may be calculated from
vE
+ XZVZ*)(d (26) and Bo is pldl + p2d2 or the ideal re-
=
(XIS*
50)
where 5 is ccalcd duced volume. The results of this calculation are given in Table VI where it is seen that vE is both small and decreases with increasing temperature. The agreement is good. The carbon tetrachloride-isooctane system is currently under investigation, and a discussion of orient% tion effects and further testing of the approaches discussed in this section will be reported at a later date.
Acknowledgment. The author is deeply indebted to Professor Scott E. Wood and Dr. Stanley Weissman for many helpful discussions. The support of the Petroleum Research Fund v i a Grant No. 975-A3 is gratefully acknowledged.
