J. A. R. Renuncio, G. J. F. Breedveld, and J. M. Prausnitz
324 (15) “JANAF Thermochemical Data”, Dow Chemical Co., Midland, Mich, 1964. (16) E. K. Storms, High Temp. Sci, 1, 456 (1969). (17) L. H. Treiman, Temp. Its Meas. ControlSci Ind, Proc. Symp. Temp.,
7967, 3, 523 (1962). (18) J. K. Hulm and B. T. Matthias, Phys. Rev., 82, 273 (1951). (19) E. K. Storms, B. Caikin, and A. Yencha, High Temp. Sci., 1, 430 (1969).
Internal Pressures and Solubility Parameters for Carbon Disulfide, Benzene, and Cyclohexane J. A. R. Renuncio,t G. J. F. Breedveld,” and J. M. Prausnltr Chemical Engineering Department, University of California, Berkeley, California 94720 (Received March 29, 1976) Publication costs assisted by the National Science Foundation
Densities of carbon disulfide, benzene, and cyclohexane were measured in the region 23-65 “C at pressures to 1kbar. The data were used to calculate internal pressures and these were compared with solubility parameters obtained from liquid molar volumes at saturation and isothermal energies of vaporization to the ideal-gas state. Along the saturationline, the square root of the internal pressure is somewhat larger than the solubility parameter (square root of cohesive energy density). For benzene and cyclohexane the internal pressure decreases with temperature while the reverse holds true for carbon disulfide. As the external pressure rises, the internal pressures of benzene and cyclohexane generally increase while that of carbon disulfide tends to decrease. The new data also suggest that the previously reported internal pressures for carbon disulfide are too low. The new data show that the ratio of internal pressure to cohesive energy density is slightly larger than unity for all three liquids.
The role of internal pressure in liquid-solution thermodynamics was recognized many years ago by Hildebrandl following earlier work by van Laar.2 The use of this fundamental property has for a long time been limited to descriptive or qualitative purposes. As noted by B a r t ~ nit, ~is only recently that its usefulness has been explored for quantitative study of intermolecular forces. Since the early attempts to calculate this property by the van der Waals equation (as suggested by van Laar) yielded only qualitative results, Hildebrand and Scatchard2 proposed the concept of cohesive energy density which is the energy of isothermal vaporization from the liquid to the ideal-gas state, not per mole but per unit volume of fluid. The large advantage of the cohesive energy density is that it can be calculated easily for liquids at normal conditions using readily available data. The square root of the cohesive energy density is the well-known solubility parameter 6. This property is extensively discussed in a review presented by Barton3 In our work we present some new data on the internal pressures of three common fluids and we discuss the relation between these data and the solubility parameter. The quantitative relation between solubility parameter and internal pressure Pi has not been completely resolved. The formal definitions of these quantities are 6 = [AlJ/V]1’2
(11
where V is the molar volume of the liquid and AU is the molar energy of complete vaporization as indicated above. Pi is clearly the isothermal volume derivative of the internal energy and a detailed knowledge of U as a function
’
Current address: Dpto. de Quimica Fisica, Facultad de Quimicas, Ciudad Universitaria, Madrid-3, Spain. The Journal of Physical Chemistry, Voi. 81, No. 4, 1977
of volume would be required to relate Pi to cJ2, the latter being an integral quantity and a commonly used measure of molecular cohesion in the liquid state. To relate the two quantities, Hildebrandl proposed the empirical equation
(3) where n is a constant. Equation 3 was suggested by van der Waals’ equation where n is exactly equal to unity; data by Hildebrand and co-workers1indicated that n is a little larger than unity (within about 10%)for typical nonpolar liquids. For benzene and n-heptane Hildebrand found, respectively, n = 1.05 and 1.09. Carbon disulfide gave a surprisingly low value (0.89) which indicated either that there was something qualitatively “different” about carbon disulfide in the liquid state or else that the experimental data for carbon disulfide were not reliable.
Experimental Section We have measured the volumes of benzene, cyclohexane, and carbon disulfide as a function of temperature and pressure, using a high-pressure bellows dilatometer described e l ~ e w h e r e . Measurements ~,~ extend to 1000 bars and cover the temperature region 23-65 “ C . Volumes are measured relative to those a t the same temperature a t atmospheric pressure. Pressures were measured with a precision Heise gauge with an accuracy of about fl bar. The high-pressure dilatometer was in a constant-temperature bath controlled to *0.01 OC by a Hallikainen proportional controller. Temperatures were measured with calibrated thermometers with an accuracy of f0.05 O C . The overall uncertainty in the volumetric measurements is estimated to be f0.04%. The liquids were of spectroquality; benzene and cyclohexane were obtained from Matheson Coleman and Bell (Norwood, Ohio) and carbon disulfide was obtained from
325
Internal Pressures and Solublllty Parameters
TABLE I: Constants in the Tait Equation of Statea Carbon disulfide This work Bidaman
Constant
B,
BZ B3 B4
B, B6
C D,
L
-0.303 21 0.986 87 -0.322 59 0.386 63 0 0 0,143 57 0.892 69 0.537 27
X X X
lo-' lo-'
X
10"
0.310 81 0.361 21 X -0.110 54 x 0.147 8 3 X 0 0 0.107 69 0.122 9 2 x 0.783 9 2 X
lo-' lo-' io5
X lo-' D2 Volumes in cm3/g,pressures in bars, temperatures in K.
Mallinckrodt Chemical Works (St. Louis, Mo.).
L
Results Detailed experimental results are available as supplementary material (see paragraph at end of text). We report here only a summary. The data were represented by a particular form4 of the Tait equation VIVO= 1 - C In (1 + P / F ) (4)
Cyclohexane 1.018 62 0.610 96 X -0.11010 x 0.319 34 X lo-' 0 0 0.925 98 X l o - ' 0.935 28 X l o 4 0.815 70 X lo-'
Benzene 1.692 0 -0.831 40 X 10.' 0.416 86 X -0.119 73 x 0.173 54 X lo-' -0.103 24 X lo-'' 0.978 18 X lo-' 0.108 70 X l o 5 0.801 30 X lo-'
t
"
'
I
50
60
" / /
I I .0
where V is the volume of the liquid at pressure P and temperature T and V, is the volume at P = 1 bar and temperature T. Units are cm3/g, bars, and kelvin. The constant C is independent of temperature but the constant F depends on T according to
F
= D1 exp(-D,T)
where D 1and D z are constants. For carbon disulfide and cyclohexane the volume Vowas expressed as a function of temperature by
V, = B1
+ B2T + B3T2+ B4T3
Pressure Term Included Pressure Term Neglected
(5)
(6)
while for benzene, the equation used has the form
+
V o= [ B , B2T + B3T2+ B4T3+ B S T 4 + B,T5]-' (7) To obtain the empirical B constants for carbon disulfide, we used data from the literature.6-10 For cyclohexane the constants are given by Nelson'l and for benzene by Ambrose and Lawrenson.12All empirical coefficients are given in Table I. Internal pressures were calculated from the Tait equation using standard thermodynamics. Solubility parameters were calculated for the saturated liquids as discussed by Hildebrand and Scott1using vapor-pressure data, molar liquid volumes along the saturation curve, and (small) corrections for gas-phase imperfections. We also calculated internal pressures using Bridgman's13 carbon disulfide data for pressures to 980 bars using his B constants. However, Bridgman's volumetric data for this region are extremely limited and, therefore, internal pressures calculated from these data are not highly reliable. Figure 1 shows internal pressures for carbon disulfide as a function of temperature. We compare our results with those based on Bridgman's data and note that our internal pressures are 4-9% larger than those based on his measurements. Further, we call attention to the importance a t high pressures of the second term in the definition of the Pi (eq 2). In the literature one often finds a definition of the internal pressure which neglects P. At low pressures, where P
