March, 1958
NOTES
rigorous form by a suitable selection of variables. We hope also t o illuminate the significance of the buoyant force on a macromolecule in solution. The equilibrium distribution of n components in an external field of acceleration g is determined by v p i = g, i = 1, 2, ..., n, where p i is the chemical potential (in units of energy/mass). Let the concentrations (in mass/volume), of the n components be pi. Then
moved explicitly from eq. 5 by setting Z(l &flk)Pk = 0, where the sum goes over solvent components. Thus Q = PJQ. (6) where ps is the total density of solvent components, and vS is the total volume fraction of solvent components. The equation Q = l / f l nfor the single solvent case forms an example of eq. 6.
375
n VPi =
(1)
(aPi)laPk)l*g
k=l
where we suppress a subscript indicating constant temperature. I n the derivative all p except p k are held constant. We introduce the partial specific volumes fli by means of the equation Dh(bPi/aPUk)p
(2)
= KPi
k=l
where is the compressibility. Equation 2 is derived readily from the Gibbs-Duhem equation and fli = K(bP/dpi)p, where P is the pressure. We multiply eq. 2 by an arbitrary constant Q and subtract the result from eq. 1
v
In
Pi
(1
=g
- & ~ (ad
+
~ i / d ~ ) p QK
[kIl
1
(3)
We will select a convenient value of Q later. Now we use an expression from Kirkwood and Buff R T ( 3 In
pi/&m)fi =
M k h f NpkGik
(4)
where M k is the molecular weight of component k, R is the gas constant, N is Avogadro’s number, 6 is the Kronecker 6, and Giro is a cluster integral
.f [gik(R) - 11 dR
THE SOLUBILITY OF XENON IN SOME HYDROCARBONS1 BY H. LAWRENCE CLEVER^ Contribution from lhe Department of Chemislry, Duke Universily, Durham, N . C. Received October 1. 1967
The solubility of xenon has been determined a t a total pressure of one atmosphere and temperatures of about 16, 25, 34.5 and 43” in benzene, cyclohexane, n-hexane, isooctane and n-dodecane. Experimental The solubility apparatus, procedure and solvents were those used and described before.8 The pure xenon was furnished by the Linde Air Products Co., Tonawanda, N. Y.
Results and Discussion The solubilities of xenon in the hydrocarbons corrected to one atmosphere of xenon by Henry’s law are given in Table I expressed as the Ostwald coefficient and mole fraction. Included are least square constants for the equation log solubility =
T
+b
in both solubility units. The slope intercept equations reproduce the experimental solubilities with where g i k is a radial distribution function. Equation 4 is valid for non-electrolytes, or for electro- an average deviation of 2.8% in benzene, 2.270 in lytes if applied to individual ion species. If i and k cyclohexane, 1.1% in n-hexane, 1.0% in isooctane refer t o macromolecules, G i k reduces to an osmotic and 0.7% in n-dodecane. Much of the departure second virial coefficient G i b o in the limit of low from linearity is in the low temperature determinamacromolecule concentration. Thus ?T = RT. tion where temperature control was difficult. Entropy of Solution.-Entropies for the transfer [Z(pi/kfi)- 2 Z ( p i p k / 2 M i M k ) G 0 i k . . .], where the sums go over macromolecular components i, k , of one mole of xenon from the gas phase t o the hypothetical unit mole fraction (Fig. l a , curve 1) ... are slightly more negative than values found beI n terms of the G i k , eq. 3 becomes fore for the other rare gases in the same soln v e n t ~ . However, ~ they are not as negative as the RTvln pi = g (1 - QUi)Mi N entropy calculated from either compressing and k=l condensing the xenon (Fig. la, curve 2 ) or the cor(1 - QZik)PkGik RTQK] (5) entropy of condensation of the solvent I n the subsequent discussion we take i to be a responding (Fig. la, curve 3). macromolecule. In the particular case where there Solubility and Surface. Tension.-The is only one solvent component, say n, all macro- Uhlig plot4 of theSolvent logarithm of the Ostwald coefmolecule-solvent interactions may be made to ficient against the solvent surface tension (Fig. 1b) vanish identically from eq. 5 by taking Q = l/fln. shows more scattering of the points than the same If several solvent components are present, macrofor the other rare gases in the same solvent^.^ molecule-solvent interactions cannot be rigorously plot Hildebrand critical temeliminated from eq. 5. If, however, the solvent perature 16.7”, Equation.-Xenon, is a gas with physically conmolecules are much smaller than the macro- stants from which t o evaluate solubilityreal parammolecules, and do not interact with macromolecules (1) Presented before the Division of Physical and Inorganic Chemisstrongly or specifically, we may assume that G i k 127th National Meeting of the American Chemical Society, Cinis independent of which solvent component k is try, cinnati, Ohio, April, 1955. chosen. (Under these Circumstances, G i k will (2) Department of Chemistry, Emory University, Emory Univerequal the “volumie” of macromolecule i). Then sity, Georgia. (3) H. L. Clever, R. Battino, J. H. Saylor and P. M . Gross, THIB macromolecule-solvent interactions may be re- JOURNAL, 61, 1078 (1957). Gik =
+
[
+
+
(2) J. G. Kirkwood and F. Buff, J . Chem. Phys., 19,774 (1951).
(4) H.H. Uhlig, ibid., 41, 1215 (1937).
NOTES
370
1’01. 62
TABLE I TIIE SOLUBILITY OF XENON IN SEVERAL HYDROCARBONS UNDER ONE ATMOSPHERE PRESSURE Solubility Mole Solvent
Benzene
Cyclohexane
n-Hexane
Isooctane
n-Dodecane
Temp., ‘C.
16.0 25.0 34.45 43.1 16.0 26.0 34.45 43.1 16.0 25.3 34.4 43.1 16.0 25.5 34.4 43.0 16.0 25.0 34.4 43.0
Ostwald
fraction
log sol.
5
90
Solvent molar vol., cm.J/mole. 130 170 210
+b
(1) Ostwald (2) Mole fraction a b
3.59 0.0132 3.08 .0111 (1) 297.4 2.98 ,0106 (2) 372.6 2.90 .0101 5.26 .0233 4.42 .0192 (1) 385.1 4.25 ,0181 (2) 463.8 4.00 .0168 5.60 .0298 4.84 .0254 (1) 491.6 4.45 ,0229 (2) 558.7 3.95 .0201 4.68 .0313 4.08 .0264 (1) 475.0 3.70 .0241 (2) 537.0 3.36 .0214 3.78 .0348 3.35 .0304 (1) 414.0 3.08 .0274 (2) 496.3 2.84 .0247
-0.4878 -3.1825
-0.6326 -3.2483
,
-0.9552 -3.4610
-0.9720 -3.3674
-0.8561 -3.1753
eter 8z7 molar volume VZ,and ideal solubility XZ’, for use in the Hildebrand and Gjaldbaek equation.6 -log XI = -log
Xz‘ +
V2 log VI
+ 0.434
Values with physical significance for V Zinclude (1) the volume at the xenon b.p., 43, (2) van der Waals b, 108.4, and (3) the critical volume, 113.7 cm.a/mole. Values for the solubility parameter, dZ, include (1) 8.0, the value at the normal b.p., (2) 5.4, calculated from the normal b.p. value t o 25” bya
14 17 20 23 26 Solvent surface tension, dynes/cm. Fig. la.-(Top) entropy of solution us. solvent molar volume. 1b.-(Bot,tom) logarithm of Ostwald coefficient us. solvent surface tension: 0 , benzene; e, cyclohexane; a), methylcyclohexane; 0, n-hexane; 6,isooctane; a, perfluoromethylcyclohexane; 0, n-dodecane.
where a,the coefficient of thermal expansion is assumed temperature independent and equal t o the corresponding state value of 0.68 Tc7and (3) 7.0, the maximum in a plot of mole fraction solubility against solvent solubility parameter (Fig. 2) which would correspond t o the gas solubility parameter if the volume of mixing terms cancelled or were 0.0 I ‘ 1 I I negligible. Included in Fig. 2 are the solubilities 6 7 8 9 in methylcyclohexane and perfluoromethylcycloSolvent solubility parameter. hexane given before.* Fig. 2.-Xenon solubility us. solvent solubility parameter, The Hildebrand and Gjaldbaek equation was same symbols for solvents as Flg. 1. solved for the ideal solubility using the experimental solubility in seven solvents and all possible a2 = 7.0 and Ve = 108.4 which reproduced the hycombinations of 8 2 and VZ. The combination giv- drocarbon solubilities to within 5% but was 54% ing the most constant “ideal solubility’’ value was in error for the fluorocarbon solubility. The calculated ideal solubility was 0.0257 0.0008 which ( 5 ) J. Chr. Gjaldbaek and J. H. Hildebrand, J . Am. Chsm. Soc., is 50% higher than the value calculated from xenon 71, 3147 (1949). (6) J. H. Hildebrand and R. L. Scott, “Solubility of Nonelectrovapor pressure. lytes,” 3rd ed., Reinhold Publ. Corp., New York, N. Y.,1950. Acknowledgment.-The author is indebted to (7) E. A. Guggenheim, J . Chem. Phya., la, 253 (1945). Professors P. M. Gross and J. H. Saylor for helpful ( 8 ) H. L. Clever, J. H. Saylor and P. M. Gross, THIEJOURNAL, 62, 89 (1958). discussions.
*
I d ! I
