1183
SURFACE TENSION OF BINARY LIQUIDMIXTURES
The Surface Tension of Binary Liquid Mixtures’
by J. G. Eberhart Sandia Corporation, Albuquerque, New Mexico
(Received October 91 1966) ~
An equation for the surface tension of binary liquid mixtures is presented which is based on the assumption that the surface tension, u, is a linear function of the surface layer mole fraction. The condition for equilibrium between the surface layer and the bulk liquid phase provides a relationship between the surface and the bulk compositions and leads to an equation of the form u = (SXAUA X B U B ) / ( & A X B ) , where UA and UB are the surface tensions of the two pure components, X A and XB are the bulk liquid mole fractions, and S is a function of temperature only, which gives the extent of surface layer enrichment in the component of lower surface tension. A statistical mechanical model is used to calculate the value of S for liquid hydrogen isotope and ortho-para mixtures.
+
+
Introduction For almost all of the binary liquid mixtures on which surface tension measurements have been made, it has been found that the surface tension deviates negatively from a linear function of mole fraction. This behavior is usually explained qualitatively by the fact that the surface layer of the liquid is enriched in the component of lower surface tension, thereby minimizing the Helmholtz free energy of the mixture. Although surface tension is clearly not a linear function of the bulk liquid mole fraction, it is possible that it is a linear function of the surface layer mole fraction. A model based on this assumption is explored here. The Surface Model A liquid mixture of two components, A and B, is assumed to be in equilibrium with its own vapor. The liquid-vapor interface is treated here in the manner of Bakker12*Verschaffelt,2band Guggenheimaas a thin phase separated from the bulk liquid and vapor by two dividing surfaces which enclose the region of property variation normal to the interface. The bulk liquid phase is assumed to have mole fractions X A and X B , while the bulk vapor phase has mole fractions ZA and ZB. Although the interfacial or surface phase has a composition which varies in a direction normal to the dividing surfaces, it is assigned over-all mole fractions Y A and YB which satisfy the material balance conditions for the system. For component A, for exnsyA nvzA, ample, this condition is n A = n G A where n A is the total number of moles of A in the three
+
+
“phases,” and TLL, ns, and nv are the number of moles of A and B in the liquid, surface, and vapor phases, respectively. Since, in general, ns and nv are very much smaller than n L , it will be assumed that X A is essentially equal to the over-all mole fraction of A, nA/(nA n ~ ) .The surface tension, u, is then assumed to be a linear function of the surface layer mole fraction, i.e.
+
u = YAUA
+
YBQB
(1)
where CA and U B are the surface tensions of the pure components. In order to relate the surface layer composition to the bulk liquid composition, the conditions for equilibrium between these two phases are employed, namely, that the chemical potential for each component is the same in the surface layer as it is in the bulk liquid. Then, by the usual argument
KA=
bA/aA,
KB = bs/as
(2)
where b and a are the activities in the surface layer and the bulk liquid, respectively, and K is a distribution constant which depends on temperature only. A sepa(1) This work was supported by the U. S. Atomic Energy Commission. (2) (a) G. Bakker, “Handbuch d. Experimentalphysik,” Vol. 6, Akademisohe Verlags-Gesellschaft, Leipzig, 1928; (b) J. E. Verschaffelt, Bull. Classe Sei. Acad. Roy. Belg., 2 2 , 373, 390, 402 (1936). (3) E. A. Guggenheim, Trans. Faraday Soc., 36, 398 (1940), or “Thermodynamics,” North-Holland Publishing Co., Amsterdam, 1957, p 46.
Volume 70, Number 4
April 1966
JAMESG.EBERHART
1184
s = -K=A - - - -
bA/bB
KB
~ - 6 ~ / 8YA/YB
aA/aB
XA/XB
YA/YB
(4) where S > 1 for surface enrichment in A. Using eq 1 and 4, along with the conditions X A X B = 1 and PA YB = 1, u can be expressed in terms of the bulk liquid composition of the mixture, namely
+
u =
+ +
SXACAXBQB SXA
(5)
XB
Equation 5 is of the same form as an empirical mixing rule suggested by Reshetnikov6 for a variety of properties of binary mixtures and used by Lauermann, Metzger, and Sauerwald' to fit their surface tension data on the silver-tin alloy for tin mole fractions between 0.35and l.
Comparison with Experimental Data In order to test eq 5 with experimental data, it is convenient to define .$ and r as .$ = (u
- QA)/(QB r =
-
FA)
(6) (7)
XB/XA
i.e., t is surface tension on a dimensionless scale varying from zero for pure A to unity for pure B, while r is the bulk mole ratio. Combining eq 5-7, it is found that f = -S(f/r)
+1
(8)
Thus, if eq 5 is correct, a plot of f us. 4/r should be linear with an intercept of 1 and a slope of -S. Surface tension data on a variety of alloy, fusedsalt, organic, and inorganic liquid mixtures have been tested and found to fit eq 5 . A few examples are discussed below. A liquid alloy system is shown in The JOUTTUZ~ of Physical Chemistry
---
I
I
I
I
1600
(3)
where 6 and y are surface layer and bulk liquid activity coefficients, respectively. It has been found that for the fractionation of isotope mixtures or mixtures of components with similar properties, between either two bulk phases4 or one bulk phase and one adsorbed phase16the activity coefficient quotient in eq 3 is unity to a very good approximation. By analogy to these results, the identification ( ~ A / ~ B ) / ( Y A / Y B ) = 1 is adopted as an additional assumption of the treatment for systems where A and B are similar in properties and leads to a simple relationship between surface layer and bulk liquid composition, namely
+
.I
ism
ration factor, S, for the enrichment of the surface layer in A, the component of lower surface tension, is defined as
f 1400
I200
L
I
I
I
I
I
0.0
0.2
0.4
0.6
0.8
1.0
cu
xNI
Ni
Figure 1. The surface tension of the copper-nickel liquid alloy at 1550".
Figure 1 where the data of Fesenko, Eremenko, and Vasiliu18 on the copper-nickel system at 1550", are plotted against the mole fraction of Ni. Figure 2 shows that 5 us. t / r is linear with an intercept of 1 for this system and, from the slope, it is found that S = 1.77. This value of S was used in Figure 1 to calculate the curve drawn through the experimental points and again shows a good fit of the data to eq 5. Figure 3 shows the surface tension of the fused-salt RbNOr KNOI system at 350" as reported by Bertozzi and Sternheim.g An S value of 1.43 is required to fit this data. Figure 4 shows the surface tension data of Evans and Cleverlofor the isooctane-benzene system at 30". For this system, S is found to be 3.12. Although eq 5 accurately represents the surface tension-composition relationship for many binary liquid mixtures, there are also many binary mixtures whose surface tension cannot be accurately represented by this relationship. In general, these systems are composed of components whose properties would not be described as similar, and, in these cases, the activity coefficient ratio ( ~ A / ~ B ) / ( ? A / ~ B ) is undoubtedly a function of composition.
The Estimation of Surface Tension of Mixtures A consequence of this model is that, if the surface tension is known for a mixture of A and B and for (4) H. London, "Separation of Isotopes," George Newnes Ltd., London, 1961, pp 10, 42, 43. (5) See, for example, C. M. Cunningham, D. 5. Chapin, and H. L. Johnston, J . Am. Chem. Soc., 80, 2382 (1958), or D. White and E. N. Lassettre, J. Chem. Phys., 3 2 , 72 (1960). (6) M. A. Reshetnikov, Dokl. Akad. Nauk SSSR, 69, 45 (1949). (7) I. Lauermann, G. Metzger, and F. Sauerwald, 2. Physik. Chem. (Leipsig), 216, 42 (1961). (8) V. V. Fesenko, V. N. Eremenko, and M. I. Vasiliu, Zh. Fit Khim., 35, 1750 (1961). (9) G. Bertossi and G. Sternheim, J. Phys. C h m . , 68, 2908 (1964). (10) H. B. Evans, Jr., and H. L. Clever, ibid., 68, 3433 (1964).
1185
SURFACE TENSION OF BINARY LIQUIDMIXTURES
E
Or Figure 2. Plot for the determination of S for the liquid copper-nickel alloy at, 1550’.
IO0 0.0
I
I
I
I
1
0.2
0.4
0.6
0.8
1.0
RbN03
KN03
*K NO
Figure 3. The surface tension of the rubidium nitrate-potassium nitrate fused-salt mixture at 350’.
16
This equation was tested with some of the surface tension data of Bertozzi and Sternheimg on fused alkali nitrate systems. The values of S for the various pairs of alkali nitrates were calculated as follows: S(KNO3, NaN03) = 2.11, S(RbN03, KNOB) = 1.43, S(RbN03, NaN03) = 2.92, S(CsN03, KPU’03) = 1.43, S(CsNO3, NaN03) = 2.93, all at 350’. The product S(CsN03, KN03)S(KNOB,NaN03) = (1,43)(2.11) = 3.02 agrees with S(CsNOs NaN03) = 2.93 within 3% which is less than the precision of the S determinations for these systems. Similarly s(RbN03, KN03)S(KN03, NaN03) = (1.43)(2.11) = 3.02, which agrees equally well with S(RbN03, NaN03) = 2.92. It is possible to use eq 9 to estimate unknown surface tensions of binary mixtures. For example, Bertozzi and Sternheim measured the surface tension of all of the possible pairs of the four nitrates &NO3, RbN03, KNOa, and NaN03, except for the CsN03-RbN03 pair. The S value for this pair can be estimated, however, using eq 9. One means of estimation is S(CsN03, RbN03) = S(CsN03, KNOB)/S(RbNO,, KNO3) = 1.43/1.43 = 1.00. Another set of S values can alternatively be used, namely, S(CsN03, NaN03)/ S(RbN03, NaN03) = 2.93/2.92 = 1.00. The two approaches give consistent results and predict a surface tension which varies linearly with bulk mole fraction. I n a more recent study, Bertozzi” also reported the surface tension of various binary mixtures of the alkali bromides. Although the data for each of the five pairs investigated are well fit by eq 5, the S values did not show consistency with eq 9.
Statistical Mechanics of Hydrogen Isotope Mixtures Surface tension measurements have recently been made on the various liquid mixtures of H2, HD, and D2,12J3as well as the ortho-para mixtures of H2.I4 These systems have sufficient simplicity that a statistical mechanical surface model can be used to calculate S for comparison with the values found empirically. For these mass or nuclear spin isotope mixtures, S can be calculated by replacing the mole fraction terms of eq 4 with partition functions, thus 0.0
%HI8
0.2
0.6
0.4
‘C6H6
0.8
1.0
C6”6
Figure 4. The surface tension of the isooctane-benzene liquid mixture a t 30’.
another mixture of B and C at the same temperature, then the surface tension for a mixture of A and C can can be calculated at this temperature. This follows from eq 3, where it can be seen that S(A, B)S(B, C)
=
S(A, C)
(9)
=
(&A/&B)surf/(&A/&B)
liq
( 10)
where Q is the total partition function for A or B in the surface layer or the liquid phase, and B is the heavier of the two isotopes. The total partition function is the product of partition functions for transla(11) G. Bertozzi, J. Phys. Chem., 69, 2606 (1965). (12) V. N.Grigor’ev and N. S. Rudenko, Zh. Eksperim. i Teor. Fiz., 47, 92 (1964). (13) V. N. Grigor’ev, Zh. Tekhn. Fiz., 35, 332 (1965). (14) V. N. Grigor’ev, Zh. Eksperim. i Teor. Fiz., 47, 484 (1964).
Volume YO, Number 4 April 1966
JAMES G. EBERHART
1186
box, and fl is the area of the rectangle. Substituting (11) and (12) into (10) then yields, after cancellation
Using the result of eq 15, the values of S for the binary mixtures Hz-D2, HZ-HD, and HD-D2 are predicted to be
S(H2, Dz) = 1.41 S(H2, HD) = 1.22 S(HD, Dz) = 1.15 I
I
I
I
I
I
0.0
0.2
0.4
0.6
0.8
1.0
Q2
H2
XD2
Figure 5. The surface tension of the hydrogen-deuterium liquid mixture at 20.4%.
tional, rotational, vibrational, and electronic contributions. The rotational, vibrational, and electronic behavior of the hydrogen isotope molecules are assumed to be the same whether the molecule is in the liquid phase or the surface layer, and thus these partition functions cancel, leaving only the transitional partition functions in eq 10. The translational energy levels for a molecule in the liquid phase are assumed to be those of a particle in a three-dimensional box, while the translational energy levels for a molecule in the surface layer are assumed to be those of a particle in a twodimensional rectangle. The translational partition functions are, to a good approximation Qliq
=
Qeurf
(8K M k T )"'V h3 =
87rMkTfl h2
(12)
where M is the molecular mass, h is the Planck constant, k is the Boltzmann constant, V is the volume of the
The Journal.of Physical Chemistry
(14)
while, for the ortho-para mixtures of H2 and of DP, the masses M A and M B are equal and
1
S(p-Hz, o-H~)= 1 S(o-Dz, p-Dz) = 1
(15)
The H2-D2 system has been studied by Grigor'ev and Rudenko,12 and the surface tension concentration dependence at 20.4"K is plotted in Figure 5 using their original data tabulation. The data was linearized as prescribed by eq 8 and was found to be fit by an S value in the range of 1.88 f 0.20, a result which is not in close agreement with the prediction of eq 14. Grigor'ev13has also studied the HrHD and the HDDz systems and presents his results on a very small graph. Although his surface tension measurements cannot be read to their original accuracy, the concentration dependence is reproduced by S values which agree with those predicted above within *0.05. The experimental results on the three mixtures are furthermore ranked by Grigor'ev13 in order of decreasing deviation from linearity as Hz-D2 > Hz-HD > HD-D2, which agrees with the ranking of eq 14. The surface tension of three different ortho-para hydrogen mixtures has also been determined by Grigor'ev'q and was found to vary linearly from 1.90 dynes/cm for 2% o-Hz to 1.95 dynes/cni for 75% o-Hz. A linear relationship implies that S = 1, which is in agreement with eq 15. The author is not aware of any surface tension measurements on D2 at various ortho-para compositions.
