Iiomwr A . PIEROTTI
1840
The LiFeOz crystals grown by this method proved to be much more suitable for order-disorder heat treatmentll than polycrystals of the same composition. Table I gives the compositions cf some of the mixtures used, together with approximate melting temperatures and initial soaking temperature. The approximate yields and compositions of the crystals obtained after dissolving the flux also are given. TABLE
Initial mixtures, molar %
Approximate melting temp., O C .
501,i20.42Bz03.8Fez03 55Li20 37B20d8Fe203 25L1~0.70R&~ 6Fe203 29Li20 43B20129Fe2O3
750 800 950 1050
1 Max Approximate soaking % yield, b y nt., of temp., "C. iron-containing phase
1100 1090 1160 1200
15y0 LiF508 870 LiFeOz 1007, glass 10% G-FenO?
Yol. 67
It should be noted that the regions with less than 15 molar % LizO have melting points greater than 1100' and therefore are not recommended for crystal growth. The cornpositions given in the table are examples of some of the mixtures used for growth of the crystals specified. The largest crystals produced were generally of about 1.3-mm. length and weighed up to 8 mg. and the smallest were 0.2 min. in length and of weight about 1 mg. They were larger where there was little volatilization and longer soaking times were used. Acknowledgment.-This research was carried out with the aid of a research grant from the Clothworkers Guild of the City of London. (11) J. C. Andersonand
>I. Scliieber, t o b r published.
THE SOLUBILITY OF GASES I N LIQUIDS' Rr ROBERTA. PIEROTTI Georgia Instiwle of Technology, Atlanta I S , Georgia Received M a r c h 19, I963
B method is developed for predicting the solubility, the heat of solution, and the partial molar volunie of simple gases in nonpolar solvents. These properties are predicted for a number of gases dissolved in benzene, carbon tetrachloride, and liquid argon. The agreement with experiment is very good. The method uses equations derived by Reiss, Frisch, Helfand, and Lebowitz for calculating the revervible work required to introdure a hard sphere into a fluid. An estrapolation procedcre based upon esperimental solubilities is developed which is a direct test of the reliabiiity of their equations. It is found that their equations are in escdlent agwement with experiment.
1. Introduction I n a series of papers Reiss, Frisch, Helfand, and Lebovitz2 have developed a statistical mechaiiical theory of fluids based upon the properties of the exact radial distribution functions 'ilhich yields an approximate expression for the reversible work required to introduce a spherical particle into a fluid of spherical particles. They consider the case of a system of ( N - 1)particles obeying a pairmise additive potential and couple one additional particle obeying the same potential to this system by the procedure of distance scaliiig. The coupling procedure is used to obtain an expression for the chemical potential of the fluid in terms of a function related to the radial distribution function for the fluid. Although the radial distribution function is not known, it emerges that for hard sphere particles the only part of the radial distribution function which contributes to the chemical potential of the fluid is that part which determines the number density of particles in contact vith the hard sphere particle. While i t would be desirable to use their method to treat soft sphere molecules, the theory rapidly becomes too complicated. Instead of the more rigorous approach, it is possible to treat the soft potential as a perturbation to the treatment of hard spheres. The present paper mill show that the expression derived by Reiss, Frisch, Helfand, and Lebowitz for the reversible m-ork required to introduce a hard sphere iiiio a fluid is an excellent approxi(1) Thia rrork i i a s snpportrd in part b y a grant from the Petroleum Research Fund of the American Chemical 3omety. ( 2 ) (a) H Rriqs h. I,. Frisch and i L Leboritz. J Chem Phws. 31, 369 ( l U i 9 ) , (b) H Pric., I3 L. Frisch E. I h l f a n d , and J. L. Lebonitr, z b d . 32, 119 (1960). (3) r. Helfand, I1 R ~ i s sH. L. Fiisch, and J . L. Lebowitr, z b d , 33, 1379 ( 1960)
mation and that it can be used along with the molecular properties of liquids and gases to predict Fvith good agreement to experiment the solubility, the heat of solution, and the partial molar volume of simple gases dissolved in numerous solveiits. Alternatively, the hard sphere approximation may be used along with a siiigle measurement of the gas solubility in order to determine the solute-solvent interaction energy. 2. Theory Thermodynamic Equations.-The chemical potential of the solute pz', in a very dilute solution of nonelectrolytes is given by4 PZ1 = - xz
+ Pa, - kT In h 3 j 2+ kT In ( N z / V )
(1)
where -xz is the potential energy of a solute molecule in the solution relative to infinite separation, P is the pressure, f i g is the partial molecular volume of the solute, VXZ3and j , are the partition functions per molecule for the translational and internal degrees of freedom for the solute, Nz is the number of solute molecules in the solution, and V is the volume of the solution. For very dilute solutions, V = N l f i l , where N I is the number of solvent molecules and fil is the partial molecular volume of the solvent and also NP,"~ = Q, the mole fraction of the solute. The sum of the first two terms on the right of eq. 1 represents the reversible n ork required to introduce one solute molecule into a solution of concentration NZ/V. If the solution is sufficiently dilute to ignore solute-solute interactions, then the reversible work of adding a solute molecule to the solution is equivalent to (4) R. H. Fowler and E. A . Guggenheim, "Statistical Thermodynamics," Cambridge, 1939, paragi aph 823.
SOLUBILITY OF GASESIS LIQTXDE;
Sept., 1963
that of adding it to the pure solvent. It is convenient to consider the process of introducing the solute molecule into the solvent as consisting of two steps.5 Step 1.-The creation of a cavity in the solvent of suitable size to accommodate the solute molecule the reversible work or partial molecular Gibbs free energy, &, required to do this is identical mithi that required to introduce a hard sphere of the same radius as the cavity into the solution. Step 2.-The introduction into the cavity of a so'ute molecule n hich interacts mith the solvent according to some potential law, for instance a Lennard-Jones (6-12) pairwise potential. The reversible work here, &, is identical mith that of charging the hard sphere or cavity introduced in step 1 to the required potential.2b If the cavity size in step 1 were properly chosen there need be no change in the size of the cavity upon charging. Replacing (-xz Pa,)and Na/V in eq. 1 by (& &) and xa/u1, respectively, .yields for the chemical potential of the solute in the liquid phase
gi
where p? is the isothermal compressibility of the solvent. The Expression for Go.-The partial molar Gihbs free energy of creating a cavity in a fluid of hard spheres was derived by Reiss, et aLZa They obtained the expression
Q = R, +
K1a12
+ Kza1Z2 +
(11)
where the K 's are functions of the density, temperature, pressure, and hard sphere diameter of the fluid and a12 is the radius of a sphere which excludes the centers of the solvent molecules. The K's were evaluated to ke
Ko
=
R T { -In (1
- Y)
+ ( 9 / 2 ) [ ~ / ( 1- ~ 1 1 ~-) (aPa?)/6
+
+
pi =
1841
+ E,- - kT (In ( h 3 j 2 ) + IcT In
( ~ 2 / ~ i )
(2)
The chemical potential of the solute in the gas phase (assumed ideal) is pClzg==
-kT In
(XZ3j2)
+ kT In (pz/kT)
(3)
nhere p 2 is the partial pressure of the solute. If the internal degrees of freedom are not affected by the sohtion process then equating p 2 g to p$ yields In p 2
=
g,/kl'
+ &/kT + I n (kT/vl) + In xz
(4)
The Henry's 12w elonstant, K H ,is given by p2
=
KH X
(3
~2
Comparing eq. 4 n ith eq. 5 gives In K H = g,/kT
+ & / k T + In ( k T / v l )
(6)
The reversible viork for each of the solution steps given above may be written as Step 1. go
= I, -
Step 2. g,
= I;
-
+ Pz?, = h, Ts, + Pai = hi Ts,
- T i , (7a) -
TSi (7b)
The quantities I, h, S, and Z? are the partial molecular energy, enthalpy, entropy, and volume, respectively. The partial molar quaiitities will be indicated by the corresponding capital letter with a bar over it. I n terms of partial molar quantities eq. 6 becomes In K H
=
C,/RT
+ c,/RT + Iln ( R T / V l )
+
ei
(8)
The molar heat of solution is
(9)
mhere a p is the
[email protected] of thermal expansion of the solvent; the partial molar volume of the solute is
( 5 ) H. H. Uhlia, J. Phvs. Ciaem., 41, 1215 (1937). ( 6 ) D. D. Eley. "Tans. Faraday Sac., SS, 1281, 1421 (1938).
where y = (aa13p)/6,p is the number density of fluid molecules, al is the hard sphere diameter of the fluid molecules, P is the pressure, R is Avogadro's number times the Eoltzmrmn constant, end T is the absolute temperature. The radius of the sphere (al2)which excludes the centers of solvent molecules is equal to (al a 2 ) / 2 ,nhere a2is the diameter of the cavity to be created. The Expression for (?,.--The partial molar Gibks free energy for the interaction term is given by eq. 7b. As usual, at normal pressures the PV term can be neglected when ccn-pared Kith E. It is difFcult to make a quantitative assesfment of the T S term, but it is clear that 3,mill lbe negative for the charging process and that if the interaction energy is not too great (ie.! if it is due only to dispersion forces) the magnitude of SIwill be small. We nil1 assume for the present that for the systems under consideration here TS, is negligible compared to ,E, and therefore, Bi = El. The interaction energy of a nonpolar molecule dissolved in a nonpolar solvent can be described approximately in terms of a Lennard-Jones (6-12) pairwise additive potential. The interaction energy per solute molecule is then given by = I, =
-c!c[~p-~ - r ~ l y ~ ~ p - ~ ~(13) ] [ 1P
where r p is the distance from the center of the solute molecule to the center of the Pth solvent molecule, C is the dispersion energy constant, and u12is the distance of closest approach for the solute and solvent molecules. In order to calculate I,, it is assumed that the solute molecule is immersed in the solvent. The solvent is assumed infinite in extent and uniformly distributed according to its density p around the solute molecule. The number of molecules contained in a spherical shell a distance r from the center of the solute molecule is then equal to 4 a p r 2 dr where dr is the thickness of the shell. Combining this with eq. 13, dividing by k T , and replacing the summation by an integration gives
ROBERT A. PIEKOTTI
1842
& ( R ) / k T = -(4npC/kT)
JRm
( T - ~
-
dy
cj2fir-10)
(14 where R is the distance from the center of the solute molecule to the center of the nearest solvent molecule. The integration yields
Z,(R)/kT = - (e*,’liT)[(2/R’)3- (8/3)(1/R’)’] (15) where
t*lkT = ( ~ p C ) / ( 6 k T ~ l 2 ~ ) (16) c12is the distance of closest approach for the solute and solvent molecules and R’ is equal to R/crlz. The distance of closest approach is equal to u12as defined in eq. 11.
Vol. 67
solvent and solute, respectively, and x1 and x2 are the molecular susceptibilities of the solvent and solute. Alternately, C may be evaluated in terms of the empirically determined Leimard-Jones (6-12) energy parameters9 using
CLJ =
4€i2Un6
In K H = - 5 . 3 3 ( ~ * / k T )
1
3
2
4
5
LY2.
Fig. 1.--A typical In K J us. ~ 012 curve. The filled circles are the rare gases. I
I
I
I
I
(19)
where €1 and €2 are the energy parameters for the solvent and solute, respectively, and el and cpare the distance parameters of the solvent and solute. Since the potential energy is rising very rapidly with decreasing distance at c, the values of el and u2 are also effectively the values of al and u2. 3. Results The Solubility of a Hard Sphere.-The substitution of eq. 17 for GJRT in eq. 8 yields
n
0
+
= 4 ( E 1 E ~ ) ~ ’ ~ [ ( c i c2)/2]’
+ OJRT + In (RTIV,) (20)
Since e * is related to the polarizability of the solute through eq. 18, a plot of In KH us. az for various solutes in a given solvent might be expected to give a smooth curve. Figure 1 shows such a curve for a number of gases dissolved in benzene a t 25’. Similar curves are obtained for all the solvents considered here and have been observed for many other solvents besides.1° The filled circles in Fig. 1 represent the rare gases-the only spherical, monatomic solutes. The extrapolation of this curve to zero polarizability yields a finite value of hi K H and, therefore, a nonzero solubility even though the interaction term is zero. The change from one solute to another is accompanied not only by a change in polarizability (hence, interaction energy), but also by a change in hard sphere diameter, aa. Figure 2 shows the variation of a2 with respect to a2 for the rare gases. Extrapolation of this curve to zero polarizability gives a value, azo,of 2.58 8. The extrapolation of In K H us. a2 to zero is, therefore, equivalent to determining the solubility of a hard sphere of diameter 2.58 8.in the particular solvent. This can be expressed as lini In KH orz-+o
=
111 KHo
(21)
a 2 4 2 58
0
0
1
2
3
4
5
012
Fig. 2.--The correlation of the collision diameters of the rare gases with their polarizabilities.
The minimum in ei(R’)/kT occurs when R‘ is unity, therefore Zi/kT = -5.33(t*/kT)
(1’7)
The value of C in eq. 16 may be evaluated by means of the Kirkwood-Muller formula7 C ~ a r=
+ ( a z / x z )\
ala2
~ U Z C ~
L X 1 )
(18)
where nz is the mass of an electron, c is the velocity of light, a1 and a2 are the molecular polarizabilities of the (7) A. &fuller, Proc. Roy. Roc. (London), A154, 624 (1936). (8) R.A. Pierotti and G . D. HaLey, J . Phys. Chsm., 63,680 (1959).
where KRO is the Henry law constant for hard spheres of diameter 2.58 8. It is possible to compare the experimental values for In KHOdirectly with the theoretical value obtained using the expression derived by Reiss, et al., for the reversible work required to introduce a hard sphere of diameter 2.58 A.into various fluids. In order to make this comparison, it is necessary to know the hard sphere diameter and the density of the solvent. Table I contains a number of physical properties of the solvents considered here. Table I1 contains values of the hard sphere diameters and the energy parameters for a number of solvents. Included in Table 11are three values for a1 and a corresponding value for el. The first and fourth columns were determined from viscosity measurements. The second and fifth column mere determined semiempirically by Kobatake and Alder1’ from the entropy and the equation of state of the liquid using a cavity model (9) J. 0.Hlrschfelder, C. F, Curtiss, and R. B.Bird, “ilIolecular Theory of Gases a n d Liquids,” John Wiley and Sons, New York, E.Y.,1954, Ch 13. (10) J . H. Saylor a n d R. Battino, J. Phye. Chem., 62,1334 (1958). (11) Y. Xobatake a n d B. J. Alder, %b%d.,66, 645 (1962).
Sept., 1963
SOLUBILITY O F
GASESIX LIQVIDS
TABLEI PHYSICAL PR.OPERTIES OF SEVERAL LIQUIDS Temp.,
Liquid
O K ,
rrp X 103 dag,-’
@t X
104, atm.-l
VI, cc./mole
298 3..240a 0.953a 89.40b 8090* 298 I .296‘ ... 131.60 7540b C-COHIB 298 1.214d 1.113‘ 108.76’ 7895* CC1, 298 1.22Bd 1.096“ 97.08‘ 7830’ CS2 298 1 .190° 0.9338 61.0’ 6700’ 81 4.37e 2.03” 28.20e 1580e Ar Ar 87 4.49” 2.27e 28.66“ 1557” a N. A. Lange, “Handbook of Chemistry,” 6th Ed., Handbook Publishers, Inc., Sanduskg, Ohio, 1956. American Petroleum Institute, “Selec1;ed Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Carnegie Press, Pittsburgh, Pa., 1961. “International Critical Tables,” Vol. 111, McGraw-Hill Book Co., Inc., New York, N. Y., 1926-1928. I. Prigogine, “Molecular Theory of Solutions,” Interscience Publishers, Inc., Xew York, N. Y., 1957. e J. S. Rowlinson, “Liquids and Liquid Mixtures,” Academic Press, Inc., New York, N. Y., 1959. J. H. Hildebrand and R. L. Scotb,, “Regular Solutions,” Prentice-Hall, Inc., Ihglewood Cliffs, S . J., 1962. 0 E. A. Moelwyn-Hughes, “Physical Chemistry,” Pergamon Press, New York, iY.Y., 1957.
TABLE I1 MOLECULAR PARAMETERS FOR SEVERAL LIQUIDS a1,O A. C6H6 5.22 n-CeHid . C-CeHiz .. CCla 5 35 CS2 4 53 Ar 3 42 a Reference 11.
Substance
~
aIsb A.
5.27 5.91 6.09 5.88 4 44 3.42 Hirschfelder,
al,c A.
eI/k,’
OK.
TABLE 111 PROPERTIES AND PARAMETERS FOR THE SOLUTES x 1024,~ - x Z x
AH”,,, cal./mole
C& n-CsHii
el/k,b
OK.
5.24 504 440 5.90 ... 413 5.59 324 5.37 490 327 4.55 466 488 3.42 126 124 Curtisr3, and Bird, rpf. 9.
...
(their results a,re in excellent agreeinent with those obtained by Salsburg and 1I. Derfer, K. W. Greenlee, and C. E. Boord, J . A m . Chem. Sac., 71, 175 (1549); H. Pines, W. E. Huntsman, and 1'. N. Ipatieff, ibid., 7 5 , 2315 (1953); E . 9. Kazanskii, M. Yu. Lukina, and L. A. Nakhapetayan, Dokl. Ahad. Nauh S S S R , 101, 683 (1555). (6) J. M. Derfer, E. E. Pickett, and C. E. Boord, J. Am. Chem. Soc., '71, 2482 (1949). (7) E. R. Johnson and TT. D . Kalters, i h i d . , 76, 6266 (1964); C. .A. Wellington and 1%'. D. Walters, ibid., 85, 4888 (1961).
