(31.8
VOl. 00
GEORGES C A T C H A R D
The Gibbs adsorption isotlierin is derived directly xiid simply from the choinicnl potentials and surface tcnsion of Gibbs’ model of a phase boundary when the derivation is not, coinplicated by the discussion of stability. The advantages of expressing surface concentrations in units of Imgth are illustrated for the vapor-liquid boiinciary of water-ethanol. A discussion of Gibbs which is reputed to be vague is shown to be very precise.
Gibbs’ treatment of surfaces2 is often considered difficult and abstract. The chief reason sccms to be that the development of the equations alternates with proofs of stability. I hope that it can be rcsolved by this more limited presentation. When Gibbs neglects surface effects (62) he considers a system “enclosed in a rigid and fixed envelope, which is impermeable to and unaltcrcd by any of the substanc,es enclosed, and perfectly nonconducting to heat.” I t is difficult t,o justify this ncglcct when the volume is madc to approach zero in his derivation of t’he Gibbs-Duhem equation (87). When he considers surface effect.s Gibbs avoids these difficulties (278) by considering a system separatcd by imaginary surfaccs from other regions with the same density of energy, of entropy, and of each component as the adjacent portion of the system. We will follow him in limiting the discussion a t first to fluids uriaffectcd by gravity. Thcn, if the system is all in one bulk phase, conditions of equilibrium are that t.he tcmpera.ture, pressure, and potential of each component are the same throughout the system. The “cxt8ensivc” propert’ies are rigorously extensive or mathematically homogeneous. Boundaries between Two Fluid Phases.-If the system includes a boundary betwecn two phases, the tcmpcrat)urct and potent.ia1 of cach component are the same in the two phases arid in the boundary (with the usual limitation for a possible, hiit not actual, component of aiiy phase that its poteiitial is not lcss than in a phase in whidh it is prescnt), but, t,he pressurc is not, the same in the two phascs if the boundary is curved. Gibbs recognizes that the boundary, t,hough very thin, is not a mathematical surface. He selects a point in (or very ncar) the bouiidary and imagincs “a geometrical siirface t o pass through this point and all other points which are similarly situat’ed with respect to tlhe condition of the adjacent matt>cr” (219). He calls this tho diiridiiig surface, S. He also imagines a closed surface with cuts off a port>ion,s, of the surface S. It, is composed of a surfacc n “such as may be generated by a moving normal to S” “as far as there is any want of perfect homogeneity in the fluid masses.” It is convenient to close the surface n by t,wo surfaces, S’ and S”, cach parallel to S. The systcm is that matter cnclosed by this closed surface. It is divided into two parts by the siirface S, with (1) Presented in p a r t before t h e Division of Colloid Chemistry, American Chemical Society, Boston, Massachusetts, April 3, 1951. (2) “The Collected Works of J. Willard Gibbs,” Longmans Green a n d Co., New York, N. Y..1928. pp. 219-331. Reference will be made t o this reprint rather than t o t h e less accessible original (1878) and page numbers will be given as (219), etc.
vohimes V‘ and V‘‘ and pressures p’ and p“, This is ilhist,rated in Fig. la. The properties of this sytem are mathematically homogeneous if changes in its volume are made by shifting tjhc surface n, changing the size but not the position of the surfaces S, S’, and S”. The surface may be made an independent variable if this change is accompanied by a movement of the planes S’ and S”’ sufficient to maintain V’ and VI‘ each constant. We then may write immcdiatdy
+ ay
G = Zinipi
(1)
in which G is the Gibbs free energy, n i thc number of units of component i, pi the chemical potential of one unit of i, Q. is the area of the surface, and y the surface free cncrgy or surfacc tcnsion. dG = Zi(pidni
By the definitions of 1 and p ,
+ nidpi) + yda + ad?
p
and
y,
dG = Zipidni
so 0 = Zinidpi
(2)
however, a t constant
+ ?de + ad?
(3) (4)
I. Then dp
so
=
Zi(ni‘/Vf)dpi =
ZiCi‘dPi = ZiCi”dPi
(12)
April, I N 2
TTTE GIRBS
AJMORFTION 'ISOTHERM
A19 b
0
Equation 11 becomes
Plane Boundaries.-We will now limit our discussion to systems with a plane dividing surface. The position of the surface S is arbilxary as long as its normal is unchanged and it is in (or very near) the phasc boundary. Gibbs chose the position for which rl is zero and considered the case in which all dp's other than dpl and dp2 are zero. He defined rz as so determined as 1'2(1), and the Gibbs isotherm has the more familiar form r2(,)dM2 = -dr (15) For the adsorption of an unsaturated vapor, such as water, on a relativcly non-volatilc liquid, such as mercury, it is natural to make the surface concentration of the liquid zero. Making that of the vapor zero would shift the dividing surface a long distance from the phase boundary. Water-mercury was possibly the only system studied experimentally at the time of Gibbs' paper. Recently, however, emphasis has been placed on systems in which the concentration of either component may be relatively high in either phase. Then Gibbs' definition is analogous to bulk composition expressed as molality or mole ratio, and becomes increasingly inconvenient as the concentration becomes large. We may then speak equally well of I'zcl,or 1'1(2),or we may choose other conditions which keep constant the total number of moles, the total mass, or the total volume in the liquid phase. This was first noted by Guggenheim and Adam.? The last of these is the most interesting. We will call these surface concentrations I'l(v) and rzcv). It is instructive to choose as measure of the quantity of a component the volume of the pure liquid ut some temperature and pressure. The bulk concentration then becomes the volume fraction and the surface concentration is expressed as volume/surface, or length. The convenient unit is the i$ngstrom. The system water (1)-ethanol (2), vapor-liquid has been particularly well studied, and was used in (3). Figure 2 shows r?(,)and I'1c2)for this = system as dashed lines, and r2(")and -r2(")as fill1 lines. rZ(,) is nearly proportional to 9s(the volume fraction) for small values of +?, goes through a maximum and then decreases to a finite value for pure alcohol. 1'1(2) is negative. It also is nearly proportional to 41 for small values of 41 but it! becomes increasingly larger (more negative) and reaches a valuc -40 A. for pure water. The diffcrcwe between r1(2)and 1'2(1)is probably a rough measure of the thickness of the phase boiindary-a few Angstroms or a few milli= -I'l(v) is microns. For small values of 9 2 , r2(,.) cqual to for large values it is equal to -I'l(2). The maximum is at about 20 volume yo alcohol. It is so near the water end bccaiise it is much easier (3) E. A. Guggenheim and N. K. Adam, Proc. Roll Soc don). AfSS, 231 (1933).
(Lon-
+=+C
Fig. l,-at The system; b, comparison systeni; c, variations by bending.
------I
I
I
I
I
I
0
/
-10
/
/ /
f I
rA.
I
I 1
- 20
I
I I I I
I
I
I
I
-30
I
I I
I I
I I
-40
0
I
I I .o
$2.
Fig. P.---Surfnc~ concentrations, i . s / A . z = ethanol (2), vapor-liquid.
A.
Water (1)-
to obtain a positive surface concentration than a negative one of the same magnitude. The effect of pressure on the potjentials of liquids is so small that a two component liquid-liquid interface is essentially univariant with the surface tcnsion fixed by the temperature as is that a t a one caomponent gas-liquid interface. For polycomponent systems the principles are the same for liquid-liquid as for liquid-gas interfaces. The system water-phenol-sodium chloride
GEORGESCATCHARD
620
liquid-vapor gives further evidence of the small thickness of the boundary.4 The surface concentration of the phenol a t constant phenol activity appcars to 1)c independent of the salt concentration up to 25 g./lOO ml., arid the surface conccntration of the water relative to the salt is about 4 A., nearly independcnt of the phcriol activity. Curved Surfaces.-Although the experimental mcasuremcnts of surface tension depend upon the relation of the pressure difference across a surface to the curvaturc of the surface, and often upon the effect of gravity upon this surface, neither curvature nor gravity is important for surface concentrations. A surface is always compared with a bulk phase which is at very nearly the same gravitational potential, and at measurable curvatures p’ p” is very small. This was stated by Gibbs (233), and has been iterated by Guggenheim, whose digest of the thermodynamics of curved surfaces6 gives a good survey of the subject. Certain limitations should be considered. From Fig. 1 we see that the distance to the region of perfect homogeneity must be smaller than the shortest radius of curvature or there will be no such region. This limitation depcnds upon the system and not upon the position of the dividing surface. The position of the dividing surface does affect the surface free energy, however. Since it cannot affect the total free energy or the number of units or the potential of any component, it cannot by equation 1 affect the product @ y . Since it does change a, it must also change y. This process should be distinguished from that used in the derivation of the fundamental surface tension equation. This derivation considers the reversible work, energy a t constant entropy or Iklmholtz free cnergy at constant temperature, for a normal shift of the dividing surface in which the quantity of each component on each side of the dividing srirface is kept constant, and only the volumes V’ and V” and the surface @ are considered tovary. Then 6V” = -6V’
=
(c,
+
cZ)
da
(16)
in which c1 and c2 are the principal curvatures, the reciprocals of the principal radii of curvature. So Y(C1
+
Cd =
p’
- p”
(17)
(4) A. K. Goard and E. I
