Surface Tension and Avogadro's Number E. R. Boyko and J. F. Belllveau Providence College, Providence. RI 02918 Consider the following problem: Given surface tension measurements as a function of concentration for a particular system and the molar volumes of the solvent and solute, estimate Avogadro's number. This sounds virtually impossible but it will be shown that estimates within a factor of three can he obtained. In order to do this we must first present a particular derivation of the Szyskowski isotherm. Szyszkowski ( I ) measured the surface tension of dilute aqueous solutions of several organic acids and found that the data could be accurately represented by an empirical relation which was later to play an important role in Langmuir's work (2). This relation, the Szyszkowski isotherm, can be written as
where y is the surface tension of the solution, yo the surface tension of the solvent, c the concentration (molarity), and A and B are empirical constants. Derlvatlon Consider the liquid to consist of a regular array of identical cubical cells with N , internal cells andN, surface cells. In the case of binary solutions, the cell will he occupied by either a solvent or-sulute molecule. Let the number of internal cells oc(.upird hy s d u t e molecules be n,, the number of surface cells bccupied by solute molecules be n., and the enthalpy difference associated with a solute molecule going from an internal cell to a surface cell be H,.For an ideal solution the Gihbs free energy can be written as N,! NJ - kT.ln kT.In n,!(Nv - nJ! nJNs - n8)! Using Stirling's approximation, we minimize G with respect ton, and n, subiect t o the condition that n, n, is equal to a constant. This ieads to the equations
+
Eliminating A, the Lagrangian undetermined multiplier, gives "5
N, - n,
-
%
N,- n,
p J k T
---
n&
% -%
(4)
The expo~entialterm has heen written as Q. In the case of positive adsorption, Q will he greater than one (Henegative). The surface concentration, r (moles per unit area) is given by,
where S is the area and L is Avogadro's number. The maxi, is mum value of r corresponding to saturation, T
eq 4 can be rewritten, since n, >> n,,as
where X is the mole fraction of solute. Since r/T,is the mole fraction in the surface layer, eq 7 shows that a t low concentrations Q acts as a distribution coefficient. One molecule per cell, either solvent or solute, has been assumed. But if the main correction to the model arises from the difference in solute and solvent sizes in the surface layer, then eq 7 is expected to remain essentially valid. In the place of the equality, eq 5, we have T proportional to n,. The entropy term in eq 2 now becomes an approximation that should get worse with increasing concentration. We assume throughout this derivation that the solution is sufficiently dilute. The Gibbs adsorption isotherm in the form
is now applied. Setting 1- X -. 1 in eq 7 and thensubstituting for r in eq 8 gives RTr,QX dy=dlnX 1+QX which upon integration yields -,=-yo-RTC,In[l+QX]
(10)
This is the Szyszkowski isotherm, eq 1, with c replacing X and with the identifications (dilute aqueous solutions) B =RlT,
and A
-
55.61Q
(11)
(12)
Associating one solute molecule with one cell, the length of a cell edge, 1, is then Since the volume is fixed to contain L number of cells, the molar volume of the solvent, Vo, divided by P must give Avogadro's number, L
which, using eq 13 and solving for L, yields
Equation 15 may be modified to take intoaccount the difference in molecular size of the solvent and solute by
where Z is the number of cells (based on solvent) that a , from eq 11into eq solute molecule occupies. Substituting T 16 produces the final form
Using eqs 5 and 6 and fixingN,equal to Avogadro's number, Volume 63 Number 8 August 1986
671
In order to use eq 17, Zmust first be specified. Z i s taken as equal to be the ratio of the molar volume of the solute t o the molar volume of the solvent. In dilute aqueous solutions, low-molecular-weight organic com~oundsthat stronalv - .depress the surface tmsion of water should be the best systems t(1 investigate. The case of methanol is presented in detail. The co~centration/surfacetension data from ref 3 used in this investigation are shown in Table 1. Density data for converting between concentration units and obtaining molar volumes are from ref 4. With Q fixed, eq 10 shows that a plot of r versus lnll QXI . . will vield a straieht line. Q was detkrmined by $ystematically varying its va&e until the best straiaht line was found as indicated bv a minimum in the sum-of the residuals squared. ~ h r e l a n a l y s e swere performed which differed in the range of concentration data included. The ratio of molar volumes, Z, was 2.26. The intercept and slope from eq 10, along with the least-squares and L, are given in Table 2 for standard deviations (5)Q, H,, each of the three analyses. As expected, the model gives better results when the data are restricted to lower concentration ranges ( R MI. Grncrally, ir has been found that the isotherm holds at concentrations well above those expected if the assumptions made ahove are necessary to achieve its derivation. This point hasstrongly beenmade by Ross and Morrison (6). Thisexplains in part the success of the calculations. Surfacetension measurements made a t moderate concentrations produce the same result as if accurate measurements were made at extremely low concentrations where the model takes on validity. One must not restrict the data t o very low concentrations where the error increases drastically for the surface-tension lowering. This is shown in the case of acetone. Using the published data for acetone (3) in the low concentration range 0.03-0.13 M, a value for Avogadro's number of 38.9 X loz3is obtained, whereas with the concentration range 0.03-1.00 M, a more accurate value of 7.1 X loz3results. A reasonable concentration range for data with this model is 0.1-2.5 M a s shown by the results presented in Tahle - --.- 2-.
It mav a m e a r that Z has been taken arhitrarilv as the ratio of the'molar volume of the solute to the mola;volume of the solvent, but this is the only definition that the model will allow. If a mole of pure solute and a mole of pure solvent are considered as cell constructious, then the ratio of the cell volumes (solute to solvent) is Z. Without further information, it must be assumed that the number of solvent cells occupied on the surface layer is Z times the number of solute molecules on the surface layer. This model is supportive of the concept that carboxylic acids dimerize on the surface due t o their ability to form two strong hydrogen bonds. If one treats formic acid data in
672
Surface Tenslon Data for Methanol In Water (30 OC)
Table 1.
Test of Theory
Journal of Chemical Education
Molarity
Weight %
Mole Fractton
y (observed)*
0.314 0.774 1.541 3.056 4.549 6.012
1.011 2.500 4.997 9.994 15.00 20.00 25.00 29.98
0.005710 0.01421
66.44 65.32 60.96 54.60 49.89 46.05
7.464
8.903
Table 2.
0.05876 0.09027 0.1232
0.1578 0.1940
68.44 65.32 60.97 54.35 49.29 45.17 41.66 36.59
43.00
40.27
Three Analvses of Methanol-Water Data In Table 1
Molarity range Of
0.02673
r (calcla 3poim analysis
3-polnt analysis
analysis
5-poim analysis
0.314-1.514
0.31C3.056
0.314-4.549
4-point
data
2.263 70.66 i 0.01
Z
lmercept (dynelcm) Slope. -0 (dynelcm)
-16.26 f 0.01 25.06
0
4 (cal/mol)
Avogadra's number,
-1940 6.92 X 10"
L
Table 3.
Summary tor Small Molecules In Aqueous Solutlonr
Compound MBthanol Ethanol
2-propanal AllylalcOhol l-Prapanol Acetaldehyde Acetone
Temp
range
(OC)
of data
30 30 15 15 15 22 15
0.31-1.54 0.21-2.21 0.03-1.00 0.03-1.00 0.03-1.00 0.26-2.27 0.03-1.00
Q
B
Z
25.1 16.26i0.01 2.26 124.6 13.80i 0.02 3.26 427.8 12.17f 0.08 4.21 163.4 15.72 f 0.06 3.75 359.9 14.61 f 0.09 4.1 1 62.1 12.80 i 0.01 3.11 269.8 9.61 i0.03 4.05
LX1Oz3
6.9 5.4 3.1
2.0 1.9 7.2 7.1
terms of monomers on the surface, a value for Avogadro's number of 71.6 X 1023 is obtained whereas treating the data in terms of dimers results in a value of 9.7 X loz3. In going through the simplified model given above, students should obtain a clearer insight into surface phenomena and the meaning of the Szyszkowski isotherm. Literature Clted (1) von Szyszkowaki. B. 2. Phyaik. Chom. 1908.64.385. (2) Langmuir. I. J. Amer. Chem. Soc. 1917,39,1848. (31 "lnlernsfional C~itiralTablea";McCrsw-Hill: New York, 1928: Vol IV, p 467. (41 "International CriticalTsbln": Mecraw-Hill:NewYork, 1928: Vot In, p 111. ( 5 ) Bauer, E. L. "Astatistical Manual For Chemists";Aeademie Preaa: New York, 1971; p
114.
(6) Ross,S.:Morrison, I. D.J. CalloidInl~rfoeeSei.
1983,91.244.
