NOTES
1538
which is very similar to that resulting from the Forland's calculation, although dimensionally different. We give in Table I1 the values of the ratio - A j / D for the various binary systems we have investigated. Table I1 System
Alkali Alkali Alkali Alkali
nitrates chlorides bromides sulfates
- Ai/Q
dynes/cm
Ref
475 980 720 2700
2
3 2 This paper
It appears that the values are scattered in a large range; nevertheless, a very crude correlation could be seen with the average lattice energies. On the other hand, in dealing with surface energies, it is necessary to keep in mind that particular effects of displacement and polarization occur, which make the surface energy lower than that which would result in a hypothqtical surface where the ions were able to retain the same arrangement they have in the bulk.l0 Such distortion phenomena change with changing the size and polarizability of the ions, so that attempts to strictly relate bulk and surface properties are not allowed. (IO) D. T. Livey and P. Murray, J . Am. Ceram. Soc., 39,363 (1956)'
Ternary Diffusion and Frictional Coefficients for One Composition of the System
WaterUrea-Sucrose at 25'.
A Test of the
Onsager Reciprocal Relation for This System'
by H. David Ellerton and Peter J. Dunlop Department of Physicat and Inorganic Chemistry, university of Adelaide, Adelaide, South AustTahh (Received December 6 , 1966)
In a rf3cent Paper2the authors reported activity data for the ternary system watepurea-sucrose and the results, which were obtained by the isopiestic vapor pressure were interpreted in terms Of dimerization of urea, hydration of sucrose, and binding of urea to the hydrated sucrose. The experiments in this note, which reports ternary diffusion data for one composition were undertaken to see if the Of the above interaction postulated to interpret the chemical potenThe J O U T Wof~ Physical Chemist~y
tial data would lead to large cross-term diffusion coefficients. By combining the chemical potential and diffusion data it was possible to test the Onsager reciprocal relation3 (ORR) for isothermal diffusion for this composition of the system. All the equations and derivations used in this paper have been given on many previous occasions and will be freely referred to here.
Experimental Section The sucrose used in all experiments was a British Drug House "microanalytical reagent" sample and was used without further purification. The urea was part of a sample which has been described previously.* All solutions were prepared by weight using doubly distilled water and the weight fractions converted to solute concentrations in moles per 1000 cubic centimeters, et, using densities measured in triplicate in matched, single-neck, Pyrex pycnometers of approximately 30cm3 capacity. The molecular weights4 of water, urea, and sucrose were taken to be 18.015, 60.056, and 342.303, respectively. The density of water was taken to be 0.997048 g ~ m - ~ . The ternary diffusion data were measured with a Gouy diffusiometer,6 which has been previously described. The reader is referred to previous articles for the methods used to obtain the final results,6 i.e., the four ternary diffusion coefficients for the volumefixed frame of reference. The same fused-quartz diffusion cell was used for all experiments; its thickness, a, along the light path was 2.5043 cm and the optical lever arm for this cell was 305.18 cm. Each experiment was performed within 0.005' of 25" and the Stokes-Einstein relation was used to convert each reduced height:area ratio, S A , to 25.000".
Results Throughout this note, 0 is used to designate the solvent (an arbitrary choice) while 1 and 2 designate the solutes urea and sucrose, respectively. Table I summarizes the experimental results for the four diffusion experiments. The definitions of the (1) This investigation was supported in part by a Research Grant from the Colonial Sugar Refining Co. Ltd. of Australia. (2) H.D. Ellerton and P. J. Dunlop, J . P h w . C h m . , 70, 1831 (1966). (3) L. Onsager, Phys. Reo., 37, 405 (1931); 38, 2265 (1931). (4) using atomic weights compiled in International Union of Pure and Applied Chemistry, Information Bulletin No. 14b, 1961. (5) H. D. Ellerton, G . Reinfelds, D. E. Mulcahy, and P. J. Dunlop, J . Phys. chem., 68,403 (1964). (6) L. A. Woolf, D. G. Miller, and L. J. Gosting, J . Am. Chem. floc., 84,317(1962).
NOTES
1539
Table I : Ternary Diffusion Data for the System Water-Urea-Su~rose"~~ 1 2 3 4 5 6 7 8 9
Expt no.
(el)A (e2)A (P)A (e1)B
(e2)B
iP)B e -1
e2
10
,rexptl
11 12 13 14 15 16 17 18 19
ircalcd
"0
a1
x x
10' (pA)oslcd 10' Qexptl X IO4
(pA)exptl
Qcalcd
x
lo4
vo = 18.058
6 0.49722 0.483400 1.067959 0.49740 0.516474 1.072249 0.49731 0.499937 73.68 73.67 0.0009 0.3968 0.3962 5.96 7.54
5 0.48169 0.487756 1.068311 0.51795 0.512217 1.072038 0.49982 0.499986 68.48 68.52 0.2056 0.4585 0.4594 72.48 70.58 vi
=
=
0.36201
vz
44.76
RI X 10' = 8.470 Eo
4 0.43254 0.497098 1.068752 0.56776 0.502915 1.071592 0.50015 0.500006 65.47 65.48 0.8023 0.76% 0.7628 76.96 75.73
R2
2 0.41527 0.499496 1.068782 0.58368 0.499479 1.071369 0.49948 0.499488 65.46 65.38 1.0006
0.9341 0.9332 -14.71 -13.06 = 213.17
X 10' = 48.525
E1 = 18.1656
E2
= 18.9435
Units: concentrations, e , moles per 1000 cubic centimeters; reduced height-area ratios, PA,square centimeters per second; partial molar volumes, Pi, cubic centimeters per mole; refractive index derivatives, ai, 1000 cubic centimeters per mole, referred to the velocity of light in air at standard temperature and pressure and for wavelength 5460.7 x 10-8 cm in air. = water, 1 = urea, 2 = sucrose.
various quantities listed have been given p r e v i o ~ s l y . ~ ~indicate ~ that the interaction assumed to explain the In addition, the partial molar volumes of each compochemical potential data in ref 6 does not cause unusually nent were calculated6 from the density derivatives in large coupling of the solute flows. The density values, pi, in Table I were fitted by the eq 16Jand are listed in line 17 of the table. The differential refractive derivatives for sucrose and urea, method of least squares to the truncated Taylor series I% and (R2, were also computed for wavelength 5460.7 A p = 1.070163 O.O154O(e1 - 0.50) from the values of the e, and the Jexptlin Table I. 0.1296(e2 - 0.50) (1) Two-Component Systems. Two binary diffusion experiments were also performed, one with the system with an average deviation of *0.001%. The density water-urea and the other with water-sucrose. The derivatives in this equation were then used to calculate diffusion coefficients and the differential refractive the partial molar volumes in Table I for later use in increments for both experiments agreed to better than the calculation of the frictional coefficients. 0.15% with the data in the 1iteratu1-e.~The deviation Test of the Onsager Reciprocal Relation. One way of graphs6 obtained in these experiments were less than testing the ORR for isothermal ternary diffusion is to 2.0 x 10-4. calculate7J1 frictional coefficients12by combining difTuTernary System. Using methods previously outlined,6 sion data and the corresponding chemical potential the data in Table I were used to calculate the ternary derivatives with respect to the solute concentrations, diffusion coefficients, (Dfj)v, given in Table 11. It is el and e,. This method has been given in great detail believed that these coefficients apply to the volumein previous p~blications.7,~~ Table I1 lists all the prifixed frame of referencelo because the experiments were mary and intermediate datal1-la used to compute performed with small differences of concentration across the chemial potential derivatives p i $ . the diffusion boundaries. The error in each (D& value was calculated by assuming an error of =t2 X (7) P. J. Dunlop, J . Phys. C h a . , 69, 4276 (1965). in each of the areas, Qexptl, of the fringe deviation graphs (8) P. J. Dunlop and L. J. Gosting, ibid., 63, 86 (1959). in Table I. For this system the errors in the (D,,)v (9) D. F. Akeley and L. J. Gosting, J . Am. Chem. Soc., 75, 5685 were computed by assuming errors of +2 X lo-' for (1953). (10) J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting, experiments 5 and 6 and -2 X lo-' for experiments and G. Kegeles, J . Chem. Phys., 33, 1505 (1960). 2 and 4. Errors in the (D& were also calculated by (11) P. J. Dunlop, J. Phys. Chem., 68, 26 (1964). changing the signs in all these Qexptlerrors. (12) See, for example, the references at the bottom of p 4277 of The small values of (D12)~and (DZ1)v in Table I1 ref 7.
+
+
Volume 71,Number 6 April 1867
NOTES
1540
Table I1 Chemical Potential Derivatives and Diffusion and Frictional Coefficients for One Composition of the System Water-Urea-Sucrose a t 25’
el
0.500CWj
Bii X 10-8
e2
0.50000
Biz X 10-8 Bzi x lo-’
ml m2
rll r12= rZ1 rz A11 x 10-11’ (Ai2 = A n ) X 10-1” A B X 10-1’
(Dll)V
x
1@
(DdV
x
106
0.57538 0.57538 -0.06903 -0.10791 0.17646 4.1373 -0.2675 4.7459 0.9112 T0.0041 0.0020
10.0004 Rio X 10-1’
0.467 ~0.020 10.002 5.71 10.10 (bR21)l x lo-’’ (m1)2 x 10-17 10.07 AS% (exptl) = 24.0
x
10-17 (6Rio)z X lo-’’ R~~x 10-17
B22
pi1 MIZ
x
X X 10-la
pzi
x
pzt
X 10-’a
1.18034 0.14081 0.02957 1.29158 4.8755 0.2371 -0.1754 6.0921
X 106
0.0571 10.0180 (&)v x 106 0.3908 ~0.0016 RIZX lo-” 7.27 ( 6 ~ x ~10-17 ~ ) 1~2 . 2 7 ( 6 ~ x~ 10-17 ~ ) ~1 0 . 1 4 R~ x 10-17 1.390 ( 6 ~ x~ 10-17 ) ~ 10.00s ( 6 ~ x~ 10-17 ~ ) ~10.002 A% (calcd) = 39.8 (&)v
a More than the minimum number of significant digits have been retained in some of the quantities listed to minimize accumulation of errors in the Rik. The pij have units cubic centimeters erg mole-2. The Rib have units erg centimeter second mole+.
The frictional coefficients RIO, R12, R ~ I and , R20 calculated by using the above data are listed in Table 11. The errors7 (6Rm)1were calculated using the errors in the (DU)vin Table I and the errors’ (8Rtk)2 using assumed errors of 10.002 in the quantities7 I’12 and rZland 10.003 in r11 and r22. Equality of R12 and Rzl indicates validity of the ORR for a given system. It is seen that the value of A% (exptl) is less than the A% (calcd) (see eq 14 and 34 of ref 7) and thus the ORR is valid within experimental error for this composition of the system. (13) R. A. Robinson and R. H. Stokes, J . Phye. Chem., 65, 1954 (1961).
Effect of Anharmonicity on the Transfer of Energy between a Gas and a Solid’
by Hyung Kyu Shin Department of Chemistry, University of Nevadu, Reno, Nevada (Received June 9, 1966)
The calculation Of the accommodation coefficient a,associated with the energy transfer between The Journal of Phyeical Chemistry
a beam of gas particles and a solid, using the quantum mechanical perturbation theory is conventionally based on the model of harmonic oscillation of the surface atom^.^-^ Studies based on this model generally resulted in (Y significantly smaller than the observed data. However, instead of correcting such a deviation by a “realistic” model for the oscillation of the surface atoms, the calculated values were often fit to the observed data by adjusting interaction potential parameter^.^^^,^,^ When we want to improve upon the simplest possible theory of energy transfer from a beam of gaseous atoms to a solid via the oscillator states of the solids, the introduction of an anharmonic oscillator model may be a reasonable step. The purpose of the present note is to show the consequences of removing the approximation of harmonic oscillation of the surface atom from the conventional treatment. A gas atom with energy E , striking the surface of a solid, is assumed to interact with a single surface atom, which behaves as an oscillator of frequency Y, oscillating normal to the surface. Let x be the displacement of the surface atom from its equilibrium position, xe, and let r be the distance of the incident gas atom from the solid surface. The total interaction energy will be some function of r and x; we assume it by W(r,x) = U(r)Y(z). For the interaction system in which the surface atom is hit along its line of oscillation by the incident gas atom, the total interaction energy may depend on the instantaneous separation of the colliding partners, x = r - px, where p is a constant dependent on only the mass of the surface atom, m. We assume that the surface atom is bound to “inner” atoms which are at rest ( p = 1) and that the total interaction energy is the double exponential function W(z) = A [exp( -z/a) - 2 exp(-z/2a)], where A is chosen as the depth of the potential well. This function has the advanta,ge over the purely repulsive interaction in that the attractive forces of the colliding partners can play an important role in the energy-transfer process. We assume that the surface atom is bound to “inner” atoms via a Morse potential of the form VM = De(exp [ - (x ze)/bI - 1) (1) This work was supported by the Petroleum Research Fund of the American Chemical Society and was presented before the Division of Physical Chemistry at the 152nd National Meeting of the American Chemical Society, New York, N. Y., Sept 12, 1966. (2) (a) J. M. Jackson and N. F. Mott, PTOC.Roy. SOC.(London), A107, 703 (1932); (b) A. F. Devonshire, ibid., A158, 269 (1937). (3) D.M. Gilbey, J. Phy8. Chem. Solids, 23, 1453 (1962). (4) R. T. Allen and P. Feuer, J . Chem. Phys., 43, 4500 (1965). (5) For other references, see M. Kaminsky, “Atomic and Ionic Impact Phenomena on Metal Surfaces,” Academic Press Inc., New York, N. Y.,1965, Chapter 6. (6) F. 0.Goodman, J. Phya. Chem. Solide, 24, 1451 (1963),presents a classical perturbation theory.
