A Novel Approach to Brownian Movement Richard M. Neumann Polymer Science and Engineering Program & Materials Research Laboratory University of Massachusetts, Amherst, MA 01003
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The tendencv of two microsconic snherical "articles in a . liquid to wander away from each other by means of a threedimensional random walk is described from a simnle statistical mechanical perspective. A diffusional driving force responsible for the particle separation is used to derive a standard result from the classical theory of Brownian movement and t o establish a simole theorv of the meltine transition for the alkali halides and the alkalke-earth oxidis. ~
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Radial Force In order to give quantitative significance to the tendency of two particles to diffuse away from one another in three dimensions, we shall consider a fixed point and a Brownian particle in a liquid which, at a given instant, are separated by a scalar distance r. An observer, located on the particle, who maintains a fixed gaze on the point will note that r gradually increases with time as a consequence of three-dimensional Brownian motion. He may, therefore, postulate the existence of a diffusional driving force responsible for increasing r in his frame of reference. If the particle is a distance r away from the fixed point, the magnitude of this force, f,, can he calculated by considering the particle to have an entropy, S,hased on the number of configurations, W, associated with r. W is proportional to 47ir2. Using Boltzmann's equation: and the thermodynamic relationship between the radial force and the entropy ( I ) :
or r dr = 2 0 dt
(4)
D, the diffusion constant, is given (2): D = kTlfo The solution to eqn. (4) is: (5)
r2 = 4Dt
The constant of integration is zero since initial conditions require that r he zero when the time, t , is zero. The traditional equation, first derived hy Einstein (Z), is
Our result can be brought into accord with the traditional one by noting that because, for a given t, r is subject to an ensemble distrihution, the average forces should he equated in eqn. (3) or:
If the averages are computed using a "random walk" prohahility distrihution, eqn. (6) is ohtained. Proof of this assertion is left as an exercise for the reader. Ionic Salt Melting Criterion The proposed model for molten salts of the type M+"X-n where n = 1,2, or 3 is one of charged hard spheres which attract one another according to Coulomb's law: Z+Z-e2
fe = --
(8)
DorZ
a relationship between the radial force, the absolute temperature, and the separation is obtained. Brownian Motion Consider the motion of a spherical particle with respect to a fixed point in a solvent of viscosity 7. On the average, the particle undergoes no radial acceleration; thus, f, must be halanced by the viscous drag, f,, arising from the viscosity of the solvent. f,= fo drldt; fo is the friction factor for the solvent. fo = 67iR7(2); R is the radius of the particle. Equating f, with f,, one has
f, is the attractive force hetween a given cation and anion of charge Z+e and Z-e, respectively,whose centers are separated by a distance r. The ion pair under consideration is further assumed to he immersed in a dielectric medium, consisting of a l l the other ions, of average macroscopic dielectric constant Do. Above T,, the melting temperature, f, will he greater than f, when r is equal to the distance of closest approach for the given ion-cation pair. The distance of closest approach, s, is the sum of the two hard-sphere radii. Below T,, the model implies that f, < f, at r = s and the cation and anion are pre-
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Number 3
March 1982
191
vented from separating, which would correspond to the situation in a crystal. Thus, the melting temperature is defined to be that temperature at which f, = f, a t r = s. Equating f, and f, at T = T , produces:
Comparison between the calculated and observed melting temperatures for the alkali halides and the alkaline-earth oxides.
,s The model suggests that Do should be equal to the ratio of the crystal lattice energy, AHi, to the molar heat of fusion, AHi. AH1 is the energy required to separate the ions in a crystal lattice to infinity in a vacuum ( 3 ) ,where the dielectric constant is equal to one. We regard the melting process as analogous to the former one with the exception that the dielectric constant is now equal to DO. Thus, the energy required for the separation of each ion pair is reduced by Do or in total,
DO= A H I I A H ~ .
LiF LiCl LiBr Lil NaF NaCl NaBr Nal KF KC1 KBr KI RbF
I t will he left as an exercise for the reader to show that eqn. (10) is consistent with the Clapeyron equation when cons~dering the effect of pressure on T,. The table presents a comparison between the experimentally observed and calculated melting temperatures for the alkali halides and the alkaline-earth ox~des.The values of s used were taken from reference ( 4 ) ,the values of AHi from reference ( 5 ) for the alkali halides. For the alkaline-earth oxides, the following equation was used to calculate AH1 (3):
T6e agreement between the observed and calculated values is excellent with the exception of four compounds: LiI, CsF, CsCl and CaO. In the latter three salts, the heats of fusion, as shown in the table, seem to be out of place suggesting possible measurement errors. Conclusion
The novel approach to Brownian movement and the derivation of a simple criterion for melting in ionic salts are based on the existence of a radial force, f, = ZkTlr, which hears a formal similarity to the ideal gas pressure, P = nkTIV. The additional assumption used in the description of a molten salt is best expounded in reference ( 6 ) ."The molten salt is assumed to consist of sharply-delineated spherical cavities, in the dielectric continuum, outside of which the dielectric constant is everywhere D o , the macroscopic value." In our model the spatial polarization of the ions themselves as well as their orbital electron polarization contributes to D o .
192
Journal of Chemical Education
RbCl RbBr Rbl CsF CsCl CsBr Csl Be0 MgO CaO SrO BaO
(=) mole
6.47 4.78 4.22 3.50 8.03 6.69 6.24 5.64 6.75 6.34 6.10 5.74 6.15 5.87 5.57 5.27 5.19 4.84 5.64 5.64 17.(rel. 5) 18.5 12 16.7(ref.5) 14.7
Acknowledgment
I wish to thank Professor W. J. MacKnight for providing the opportunity for me to conduct this investigation. Acknowledgment is made to the Donors of the Petroleum Research Fund, administered by the American Chemical Society, for the partial support of this research. Use of the facilities of the Materials Research Laboratory a t University of Massachusetts is gratefully acknowledged. Literature Cited (11 Neumsnn,R. M..J. Chrm. Phys..66,870 (1977). (21 Einstein, A. i investigations on the Theory "rlhe Brownian Mo"ement,"Do"er, 1966, pp. 9-12. (31 Huvey, K. 8.and Porter, G. R., "Introduction toPhysicdIno=micChemiatry."Addison Wesley, 1963,pp. 27-32. (41 J a m G.,"Molten Saltr Handbook? Academic Press, 1967, pp. 1.80, and 184. 161 "Hsndbuok of Chemistry and Phyrics: 1973 Chem. Rubber Co., 1979. pp. B-244 and F-193. 161 Stiilinger, F. H., "Equilibrium Theow of Pure Fnmd Salts'' in Molten Salt Chrmiafry. Editor: (M.Blander1,1nrerarience,~.~.. 1961,pp. 21-28.
