COMMUNICATIONS TO THE EDITOR
3222
Parachor values of many constituent groups have been reported? The molecular volumes used are those at the boiling point: since molecules are thereby at the same reference state and the additivity of molar volumes of constituents can be invoked. yp values calculated for various constituent groups from eq. 3 are shown in Table I. Also tabulated are the critical surface tension values yo of various polymers which correspond to the constituent groups. Figure 1 shows the correlation of yowith yp. The generally good correlation seems to suggest amorphous or liquid-like surface structure of the polymers, at least as far as the sensitivity of the contact angle and critical surface tension measurements can determine. Figure 1 should
the National Institutes of Health, National Heart Institute, under Contract PH 43-64-84. (5) R.R.Dreisbwh, Advances in Chemistry Series, No. 29,American Chemical Society, Washington, D. C., 1961, p. 9. (6) C. W. BUM, J . Polymer Sci., 17, 323 (1956).
POLYMER SCIENCES DEPARTMENT RESEARCH INSTITUTE STANFORD MENLOPARK,CALIFORNIA94023
IRENE J. LEE WILLIAMM. MUIR DONALD J. LYUAN
RECEIVED JULY 6, 1965
Apparent Violations of Gibbs-Duhem Relations and the Limiting Laws of Dilute Solutions
1
'
Sir: Elliott, et d . , l have recently published their results on the activity of Cd, ul, in liquid Cd-Ga solutions. The purpose of this communication is (1) to show that the equations used to represent a1 as a function of composition violate the Gibbs-Duhem relations and lead to the rejection of their equation for the activity of Ga, h, even if an apparent violation of Raoult's law is accepted, and (2) to clarify the discussions concerning the laws of dilute solutions.218 The Gibbs-Duhem relations are based on (1) the axiom that the extensive thermodynamic properties, e.g., the Gibbs free energy, G, are homogeneous functions of first degree in the number of moles, (2) the existence of dG = VdP - SdT in accordance with the first and the second laws, and (3) integration of dG at constant temperature leading to the recognition of chemical potential and the definition of activity. It is therefore valid irrespective of the validity of Raoult's and Henry's laws.' The equation used for the activity of Cd by Elliott, et al., is
0
40
30
-: a
9 $0
I
- 20
$
10
rl(l - N ) 0
10
16
20
ergs/am.r.
F'igure 1. Correlation of
yo with yp calculated
from pamchor.
Acknowledgment. This study was supported by The J o u d of Phyaiccrl Chemistry
- gN
+ hN2
(1)
where N is the mole fraction of Ga, yl the activity coefficient of Cd, and g and h are the constants independent of N . The values of g and h for Cd representing the best data were shown to be 0.856 and 0.943, respectively.' Their proposed equation for Ga is
az also serve as a guide for estimating surface free energy of polymers. The apparent decrease of the slope of ya us. yp may indicate gradual departure of the critical a d a c e tension from the true surface free energy. The deviation of polyvinyl alcohol may be due to the polar hydroxyl groups which are prone to adsorb water from the air through hydrogen bonding.
= 1
=
yzN = j N
+ kN2
(2)
Substitution of these equations in the Gibbs-Duhem relation, (1 - N ) d In a1 N d In = 0, and simplification yields
+
B. Elliott, J. F. Lemons, and H. 9. Swofford, J. Phys. C h . , 69, 933 (1965). (2) 8.D.Christian and N. Fogel, ibid., 69,2135 (1965). (3) G. R.B. Elliott and J. F. Lemons, ibid., 69,2135 (1965). (4)N. A. Gokcen, ibid., 64, 401 (1960). (1) G. R.
COMMUNICATIONS TO THE EDITOR
(2hk
3223
- hj - kg)N2 + (2k + 2hj - kg)N + j(1
-
=0
(3)
Since the coeEcients of N and N 2 in eq. 3 are independent of N and eq. 3 is zero for all values of N , it follows that the term j(l - g ) as well as the coefficients of N and N2 must each be identically zero. Substitution of the authors’ values for g and h shows that it is not possible to satisfy eq. 3 in view of the fact that j and k cannot be simultaneously zero. It is irrelevant to the present discussion to consider j = k = 0 but if this were so, then az = 0, and this would simply indicate the disappearance of Ga by a strong compound formation with Cd in which case it is necessary to choose the second component as the compound rather than Ga and again use eq. 2 so that jand k cannot both be zero. The possible sets of values satisfying eq. 3 without j = k = 0 are (1) g = 1, h = 0,k = 0, j > 0 ; (2)g = 1, h = 0.25,j = -2k; (3) g = 2,h = 1,j= 0, k > 0. The last set is of questionable validity but need not concern the present discussion. It is therefore evident that for the system Cd-Ga, eq. 1 and 2 proposed by the authors’ and the molecular reasoning supporting these equations are unacceptable, Also, the only possible value of g satisfying the set of equations al = l - gN and eq. 2, Le., their eq. 6C and 7 (or a1 = 1 - gN and az = j N ) , is g = 1, hence, the authors’ contention that Henry’s law is obeyed but Raoult’s law may not be obeyed violates mathematics and the laws of thermodynamics. Further, the restrictions imposed by eq. 3 on eq. 1 and 2 make these equations useless for the representation of activity data, There are numerous satisfactory equations6 for this purpose which satisfy the requirements of the GibbsDuhem relations and fit any set of data when a functional representation is desired. The limiting laws of solutions state that + 1 and 72 j as N 0 and j is a finite positive nonzero quantity in measurably miscible systems. In general the more accurate the measurements the smaller the composition range where the deviation from y1 = 1 is not distinguishable from the experimental errors. The deviation reported by the authors at the lowest Ga content is no larger than that in aqueous electrolytic solutions and does not constitute a violation, although it is surprising for nonelectrolytic, and in particular for metallic, solutions. The argument presented by Christian and Foge12that Ga is dissolved by substantial dimerization is plausible, but it must await direct measurements on the activity of gallium. Finally, it is suggested here that direct measurements of the vapor pressure of Cd over Cd-Ga are highly desirable before the two indirect sets of data by means of isopiestic
- -
balance‘ are accepted in order to assure that there are no unknown sources of errors. ( 5 ) See, for example, E. Hda, et aZ., “Vapor-Liquid Equilibrium,” Pergamon Press, London, 1960, p. 60.
CHEMICAL THERMODYNAMICS SECTION, LABORATORIES DIVISION AEROSPACECORPORATION EL SEQUNDO, CALIFORNIA RECEIVED JULY 16, 1965
N. A. GOKCEN
Interionic Vibrational Absorption Bands of Ion Aggregates in Benzene Solution Sir: The existence of ion pairs, triple ions, and higher aggregates in solutions of ionic salts in solvents of low
dielectric constant has long been recognized and much informaiiion has accumulated.’ Vibrational spectroscopy promises to provide additional informationthe frequency of vibration, and hence the force constant, of the ion-pair bond. The purpose of this communication is to demonstrate that absorption bands which may be assigned to interionic vibrational modes of these aggregates can be observed in the low-frequency infrared spectra of solutions in benzene. These bands were observed during the course of a study of the strongly hydrogen-bonded hydrogen dihalide ions. Spectra were recorded, using a Beckman IR-I1 instrument with polyethylene cells, of benzene solutions of some tetra-n-butylammonium and tetra-n-pentylammonium halide and hydrogen dihalide salts. The solubilities of the lower members of the tetraalkylammonium salt series in benzene were found to be too low for this study because solvent absorption restricted the usable thickness to a maximum of a few millimeters. This latter restriction also limited the concentration range of study of the more soluble salts and this in turn, coupled with the broad nature of the absorption bands which limited the accuracy of band-center determination, prevented an adequate study of the dependence of band position upon solution concentration from being made. No band shifts were observed over the concentration range 0.05 to 0.2 M . Some typical spectra are illustrated in Figure 1, which also includes the spectra of the solid materials suspended in Nujol. Table I contains the wave number values of the bands assigned to the interionic modes of aggregates in benzene solution. These assignments are (1) C. A. Kraus, J. Phys. Chem.,60, 129 (1966). (2) J. C. Evans and G. Y-S. Lo, to be published.
Volume 69,N u d e 7 9 September 1966
