Pierre Damay
3050
Surface Tension of Metal-Ammonia Solutions Pierre Damay Laboratoire de Chimie Physique, CNRS, E.R.A. 126, 59000 L i b , France (Received July 23, 1975)
A phenomenological study of surface tension at -35 and -4O’C from the Gibbs equation indicates that the minimum thickness of the interphase is very large just above the critical point (160 8, at -35OC and 360 8, at -40°C). In the interphase, the solution is much more dilute than in the bulk. These effects can be explained by the proximity of the critical point and by the large difference of surface tension between pure ammonia and concentrated solutions. Our results could help to interpret reflection spectroscopy data reported by Burrow and Lagowski.
Introduction Surface tension of sodium-ammonia solutions at -35°C is largely influenced by the proximity of the critical point (-41OC). A method is described which allows the thickness of the surface phase where gradients occur to be evaluated. It appears that the minimum thickness of the surface phase is 160 8, at -35°C and 360 8, a t -40°C at the critical concentration. The structure of the surface can be proposed as follows: there exists a sharp density gradient a t the surface corresponding to the liquid-vapor equilibrium. Inside the liquid, close to the surface, the concentration varies over a large distance. The surface phase becomes thicker as the critical point is approached. Generalities Thermodynamics permits energy balance of macroscopic systems to be studied. The simplest way to apply thermodynamic formalism to surface properties is, as did van der Waals or Guggenheim, to consider the surface of a liquid in equilibrium with its vapor as a third phase of a given thickness T. All gradients between the vapor and liquid phases take place in the surface phase. It is supposed that average thermodynamic quantities, pressure, temperature, density, and free energy can be defined in this phase. The surface phase is not isotropic and therefore pressure is a tensor. The pressure on a plane parallel to the surface is assumed to be the same as the pressure inside the vessel, but the pressure perpendicular to the surface is
P - 717
(1)
This expression can be used as a definition for surface tension y. From there the Gibbs equation can be derived quite simply for a two-component mixture as
at constant pressure and temperature. r1,2 is the number of molecules of components 1,2 in the surface phase per unit surface; a 2 is the activity of component 2, and x 1 , p are the mole fraction of components 1,2. We are going to focus all our attention on the term u = r2x1 - r l x p . It appears that this quantity can be easily determined if both the surface tension and the chemical potentials of the solution as a function of concentration are known. It must be pointed out that r1,2is closely related to the The Journal of Physical Chemistry, Vol. 79, No. 26, 1975
mole fraction x 1’)~’in the surface phase; if k is a factor taking care of the dimensionality, it is found that = rzxl- r l x z = k(x2’X1- x2xl’)
(3)
Hence, if the average concentration of the solute in the surface phase is the same as in the bulk of the solution, u vanishes, and it is shown from eq 2 that the surface tension should not vary with concentration. Conversely, any change of surface tension with concentration indicates that the concentration in the surface phase is different from that in the bulk. In all of the textbooks except Guggenheim’s,l the concentration of the solvent is supposed to be the same in the surface phase and in the bulk, and thus u/x1 in eq 2 becomes simply k(x2’ - x p ) , which indicates the excess solute in the surface phase. Some very interesting points coming from a more complete analysis of u are missed by this simplification, which is just one of the many ways to locate the surface plane. As is shown from Figure 1, this procedure consists of choosing B as the surface plane. This figure gives the number of species 1,2 per unit surface as a function of distance. The number of molecules contained between planes A and B represents the excess concentrations in the surface phase. Actually there is no need to choose a surface plane at this point, and Guggenheiml showed that u does not depend on this choice. It is seen from Figure 1 that the distance through which gradients occur must be related in some way to the value of T. The interface thickness cannot be chosen as T since the concentration gradients are not sharp but display smooth profiles, It will be shown that T is the minimum thickness of the surface phase, and that the actual thickness can be well approximated by 27. By some rearrangements, it can be seen that T is a linear function of u
V, being the volume of 1 mol of mixture. Using eq 2 and 4, can be determined directly from surface tension and activity data by T
T=-
Vm aylax2 xpRT a In aplaxp
For example for an ideal solution
Surface Tension of Metal-Ammonia Solutions B
A
DISTANCE
Flgure 1. Surface density for a two-component mixture. A and B represent the planes where the excess concentration of the solute and the solvent, respectively, is zero. The smooth curves are the actual density profiles. Dashed areas on the left and on the right of each plane are equal.
and
0
02
06
.04
CONCENTRATION
E
.08
x-*2
Figure 2. Surface tension and chemical potential of a sodium-arnmonia system at -35% as functions of concentration. A y is the difference between the surface tension of the solution and that of the Dure solvent.
For a perfect ideal mixture of two liquids for which the surface tension varies linearly with concentration between y1 and yz, the values of the pure components, aylaxz and therefore r are constants. The surface thickness depends mainly on the difference yz - 71. The absolute value of yz - y1 must be chosen because negative values of T would be meaningless. It seems very important to determine T before making any model for the structure of the surface of a two-component mixture. Most models assume that gradients occur only in a monolayer. Very often there is no quantitative basis to make this assumption. The determination of r could test the applicability of such models. If r is of the order of a few angstroms, a monolayer surface is probable; but if it happens to be of the order of a few tens of angstroms, such models must be dicarded. Application to Metal-Ammonia Solutions The Sodium-Ammonia System at -35 and -4O'C. Surface tension measurements on sodium-ammonia solutions have been performed by Holly,2 and the results have been reviewed by Sienko at Colloque Weyl Ia3It is seen from eq 2 that aylaxa must be compared to a In azlaxz in order to determine u . The surface tension and the activity of the solute a t -35°C are given on the same plot in Figure 2 as functions of the concentration. It is seen that both curves show a plateau and a point of inflection, but it is interesting to note that the minimum slope does not occur at the same concentration for the two properties. Values of r have been calculated from eq 5 and are shown in Figure 3. The minimum surface thickness r shows a maximum at the critical concentration, i.e., at the minimum slope of the activity-concentration plot. At this concentration, aylaxz is already quite large as can be seen from Figure 2. At its maximum value, r reaches 160 8, -35'C and 360 8, a t -4OOC. The surface plane can now be chosen close to plane B since it is near this plane that a large density gradient oc-
CONCENTRATION
x-
"2 n1+"2
Figure 3. The minimum surface thickness T as a function of concentration at -35% (X) and -40% (0).
curs. It has been said that concentration profiles were not taken into account when calculating r. The minimum requirement for concentration profiles is that x~ must not vanish between planes B and A. Thus in Figure 1 the solute profile must be extended from A to B. Assuming symmetry on both sides of plane A, it can be concluded that a value of 27 is a good approximation for the thickness of the interface. For metal-ammonia solutions, the surface phase is furthermore much more dilute than the bulk phase. The thickness of the surface phase increases as the critical point is approached. It probably diverges near the critiThe Journal of Physical Chemistry, Vol. 79, No. 26, 1975
3052
Pierre Damay
0
,400
,200
0 DISTANCE Y
200
400
(8)
F l W e 4. A tentatlve concentration profile for a sodlum-ammonla solution at -40% at the crltlcal concentratlon, Posltlons of planes A and In Figure 1 are indicated by the arrows.
cal temperature. At the critical point itself, both adax2 and a In azlax:! vanish and r is not defined. Discussion If we assume that sodium-ammonia solutions consist of a mixture of free ammonia and solvated sodium, Na(NH3),, it is evident from Figure 2 that the surface tension of the solvated metal is much larger than that of pure ammonia; it is probably of the order of 100 or 200 dyn/cm as for most liquid metals. In order to minimize the surface energy, ammonia tends to replace the solute a t surface sites, but energy is required to make the solution more dilute. This energy can be calculated from the difference in free energy coming from different concentrations in the surface phase and in the bulk. Damay and Schettler4 showed that the energy required to make large concentration fluctuations in the intermediate range is small because of the proximity of the critical point. A minimization is operated between these two opposite energies. Very often profiles are represented by hyperbolic tangents in order to take into account the exponential gradient of concentration which occurs on both sides of the midpoint (Figure 1).Figure 4 represents a tentative concentration profile for a sodium-ammonia system at -36OC at the critical concentration. The concentration gradient at plane A is taken as xz/r. The surface plane is chosen very close to B because it is a t this plane that a large density gradient occurs. This density gradient is very sharp due to the fact that the critical temperature of pure ammonia is far from -35oc.
Hence the surface structure in the intermediate range of concentration can be summarized as follows: a concentration gradient exists through several hundred angstroms, the surface itself presents a sharp density gradient. The
The Journal of Physical Chemistry, Vol. 79, No. 26, 1975
concentration at the surface is much more dilute than in the bulk. Semiquantative arguments (hyperbolic tangent profile and axdaz = x2/r at plane A) allows the concentration t o be calculated in the few angstroms close to the surface. It appears that the molar fraction of the solute at the surface is of the order of lom3and is surprisingly quite independent of concentration. Those results could help to interpret reflection spectroscopy data obtained by Burrow and Lagowski.5 Their results show that the,vibrational spectra of ammonia for all concentration8 is the same as for the pure solvent. These spectra could very well be attributed to the surface phase which is always much more dilute than the bulk. The same argument could be applied to the data of Beckman and Pitzere who found the reflection spectrum of solvated electrons in the intermediate range. It can also be suggested that the work function of electrons should be strongly affected by these surface effects. Conclusion The surface tension of metal-ammonia solutions has been studied from a thermodynamic point of view. It has been shown that concentration gradients occur through large distances in the intermediate range of concentration. A semiquantitative concentration profile has been suggested. It indicates that the concentration in the few angstroms close to the surface remains nearly constant and is more dilute than in the bulk. More work has to be done in order to define a more probable structure of the interface. Widom' showed that the surface and interfacial tension could be interpreted in terms of concentration and density fluctuations. In the light of recent results obtained by Chieux8 on neutron diffraction near the critical point and by Damay and Schettler4 on the microscopic structure of these solutions in the intermediate range, it should be possible to make a more quantitative model for both surface and interface structure (the interface separating the two liquid phases below the critical point).
Acknowledgments. The author is grateful to Professeur
M. J. Sienko for having extended to him the hospitality of the Chemistry Department of Cornell University, where this work was completed, and to the National Science Foundation for the award of a fellowship which made possible his stay a t Cornell. References a n d Notes (1) E. A. Guggenheim, "Thermodynamique", Dunod, Paris, 1965. (2) F. Holly, Ph.D. Dissertation, Cornell Universlty, 1962. (3) M. J. Sienko in "Metal-Ammonia Solutions", G,Lepoutre and M. J. Sienko, Ed., W. A. Benjamin, New York, N.Y., 1964. (4) P. Damay and P. Schettler. J. Phys. Chem., this Issue. (5) D.F. Burrow and J. J. Lagowski, J. Phys. Chem., 72, 169 (1968). (6) T. A. Beckman and K. S.Pitzer, J. Phys. Chem., 65, 1527 (1961). (7) 6 . Widom, J. Chern. Phys., 43, 3892 (1965). (8) P. Chieux, Phys. Lett A, 48, 493 (1974).
