T H E DIFFUSION COEFFICIENTS O F MOLECULES AND IONS FROM MEASUREMENTS OF UNDISTURBED DIFFUSION I N A STATIONARY MEDIUM W. G. EVERSOLE AND EDw. W. DOUGHTY Department of Chemistry, The State University of Iowa, Iowa City, Iowa
Received M a y 10, 1984
Numerous measurements of the rate of penetration of dissolved substances into diffusion media have been made. The work of Stiles (2), who used an indicator for determining the rate of penetration of electrolytes into gels, illustrates a simple and direct type of experimental method for such measurements. However, the results of many such measurements have been expressed by means of empirical equations which do not permit the calculation of diffusion coefficients. The various equations which have been derived for the calculation of diffusion coefficients appear either to be inapplicable to this particular type of experimental method, or to involve a complicated and sometimes questionable method of calculation. This is particularly true of the calculation of the diffusion coefficient of an individual ion of an electrolyte in solution. The purpose of the present paper is to present a simple mathematical basis for such methods, and to point out a relationship between the distance of penetration and the diffusion coefficient which should be useful for measurements with solutions. DIFFUSION O F UNCHARGED PARTICLES
If a diffusing particle is placed in the center of a tube of motionless diffusion medium of indefhite length at zero time, then the probability curves for the position of the particle at later times are shown in figure la. These probability curves are expressed by the well-known Gaussian equation,
where ydX is the probability of the particle being in any given section, X - X + dX, in the diffusion medium and q is a function of the diffusion coefficient D and the time t . This family of curves represents also a family of concentration curves if a large number of diffusing particles are placed in the central section at 289
290
W. G. EVERSOLE AND EDW. W. DOUGHTY
zero time. Under such conditions the number of particles in any given dX, is given at any time by section, X - X
+
Thus the concentration in this section is ny = m, and
But q2, the mean square deviation from the mean, is given by i =n
where X is the mean distance traveled by the particles in time t. For uniform diffusion in both directions i =n
n
i=l
and therefore q2 =
But this is the value of
-1
i =n
n
x; i=l
G2in the equation of Einstein (1) i=n
Substituting 2Dt for q2 in equation 3, me\
n e (47rDt)*
xp --4D1
In this equation, m is the concentration of diffusing particles at any disn
tance X from the central section at any time t , - representing the (4aDt)* concentration in the central section at that time. If the concentration in the central section is kept constant at moleach ordinate in figure la will in effect be multiplied by a constant times (4nDt)i and as a result a family of concentration distance curves as in figure l b and equation 9 will be obtained.
--xz m = moe
4Dt
(9)
29 1
DIFFUSION COEFFICIENTS O F MOLECULES AND IONS
or
or for any given constant value of m (= m’) 1
D = -X2
(11)
4t In l l ~ o- In m‘
Thus the value of the diffusion coefficient may be determined, m obeing ‘;heconcentration in the central section and X being the distance from the :entral section at which the concentration is m’ at time t.
---
a
I
..
b -
4
X.
XI
+x
FIQ. 1. CONCENTRATION-DISTANCE CURVESFOR UNDISTURBED DIFFUSION DIFFUSION OF CHARGED PARTICLES
If the diffusing particles are ions, the distance of penetration due to normal diffusion X will differ from the experimentally determined distance X , by the amount X,, where X,is the distance migrated by the ions due to the potential gradient resulting from the diffusion potential E. For very small values of m’, we may assume that the diffusion potential in the system is distributed uniformly over the distance X,, and the velocity of migration is therefore
where U is the mobility of the diffusing ion. Since both this velocity and the velocity due to normal diffusion are inversely. proportional to the distance of the concentration m’ from the central section, the sum of the
292 two velocities
W. G. EVERSOLE AND EDW. W. DOUGHTY
(%)
must also be inversely proportional to this distance
and therefore
x: -=k t
Thus X i is proportional to the time, as has been observed by many investigators (2,3,4). The value of X , is given by
Therefore
Substituting the values of X and IC into equation 11
Equation 16 reduces to equation 11if the diffusing particles are uncharged or if the diffusion potential is zero. Preliminary results of colorimetric measurements of the penetration of cupric chloride into gels indicate that the equations suggested are useful and accurate. REFERENCES (1) EINSTEIN, A.: Ann. Physik. [4]17, 549 (1905);19,371 (1906). (2) STILES, W.:Biochem. J. 14,58 (1920). (3) STILES, W.,AND ADAIR,G . S.:Biochem. J. 16,620 (1921). (4) TISELIUS, A.: Nature 133,212 (1934).
