and
Therefore,
Similarly,
Diffusion from Gels
To the Editov: Recently J. C. Dennis [J. CHEM.EDUC.,45, 432 (1968)] proposed that the solution for the problem of diffusion of electrolytes from a gel column, of height a (0 < x < a ) , and initial concentration unity, to a wellstirred fluid a t x = 0 (zero initial concentration and height a) is
where
and
Hence
Each of the eqns. (1) and (2) is one of Maxwell's relat,ions. The Maxwell rclations may be seen to possess a certain symmetry which suggests the following method to remember them. The convention used here is to treat S and T, V and P, and N and p as pairs of conjugate variables. The extensive parameters (X) are S, V , and N; the corresponding intensive parameters (I) are T, P, and p. Then
+
where p = (2n 1 ) r / 2 a , X = Dp2, D = diffusion coefficient (I) f: D(x, C ) ) , and u ( x , t ) = concentration in gel. However this solution appears to be in error. The derivation presented by Dennis requires the solution to the following differential equation
where ru (x, t ) = u (x, t ) - v(t),and v(t) is the concentration in the well-stirred fluid. The expression given for w ( z , t ) is not a correct solution to the differential eqn (2). The correct solution for the concentration profilc in the gel is
where p cot p
= -
1.
Thc general solut,ion (dcrivntion of t,his and other solutions t,o be published elsewhere) for the case of gcl height a , and fluid height vr, and initial conccntratio~~s Cl and Cz in the gel and well-stirrcd fluid respectively, can be shown t,o be I n the above relations, one may notice that The conjugate variables, of the numerator and the denominator on t,heleft, appear, respectively, a4 t,hedenominator and numerator on the right. On d h e r side of the equation, the variable held constant is that which is the eonj~gatevariable of the nwnerxtor. R represents any other specified vaiahles, and is common on both sides. The nnmbers in parentheses index t,hc eollpled extensive and intensive variables. The relations eqns. ( 4 ) and (;)) are basically the same.
With further reference to Maxwell's relations, choose the upper sign if N or p does not appear as a differential; choose the lower sign if N or p does appear a~ a differential; but if, in addition to the presence of N or p as a differential, P or V appears as a differential, choose the upper sign. It is of interest to note that this formalism, with the exception of the sign convention, may be extended to include additional sets of variables. This set includes the electro-magnetic extensive and intensive variables, as well as the stress and strain variables.
536 / Journal of Chemical Education
u(z,t) =
wherc
+ a l m + C1
CP- CI
-
1
p cot p
= -
a/m.
We wish to thank the U.S. Air Forcc Ctmbridge Rescarch Laborat,ories for partial financial support of this work.
