6418
The Journal of Physical Chemistry, Vol. 92, No. 22, 1988
Comments
Conclusions. Contrary to recent speculation, a component cannot be completely removed from a phase when that phase is in material equilibrium with another phase containing a nonzero amount of that component. Furthermore, we have shown that, with the possible exception of very small systems where surface effects become important, there is no advantage to be gained by reducing the size of a phase in an equilibrium purification operation.
#(NB+l) = In NB + 1 / ( 2 N B )for NB 2 1, it follows from eq 1 and 2 that eq 4 implies
Acknowledgment. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098. The author is grateful to D. Chandler and J. M. Prausnitz for helpful discussions.
for the number of B molecules in the open phase a. In eq 6, as well as in eq 2 and 5, NB, the total number of B molecules in the system ( a @),should be interpreted from now on as including those ones contained initially in phase @. Now Landau and Lifshitz5 state that if N o molecules of a Boltzmann gas occupy a volume Vo,then the probability P ( N ) that altogether some N molecules are in the volume V will be given by
Allan H. Harvey6
Materials and Chemical Sciences Division Lawrence Berkeley Laboratory Chemical Engineering Department Uniuersity of California Berkeley, California 94720
(6) Present address: Thermophysics Division, National Bureau of Standards, Gaithersburg, MD 20899.
Reply to "On the Irrelevance of Phase Size In Purification" Sir: There is no doubt that the neglect of fluctuations is the main limitation to the thermodynamic conditions I suggested' for isolating a pure substance. On the other hand, Harvey's criticismZ is a clear presentation of this problem from the conventional statistical-mechanical viewpoint. In this reply I shall correct my too optimistic previous claims while showing that Harvey's position is too pessimistic. Given Harvey's assumptions and notation, let us develop a pseudo-one-component in two-phase system. If we consider NB molecules of an ideal gas confined within a volume V, at T , then its molecular chemical potential is
+ 1) - In v&]
(1)
Here ( a )signifies that phase a is closed to substance B, and $(NB 1) = d In NB!/dNB. Now let phase CY be immersed into a volume V, of an immiscible phase p. This is nevertheless considered open to the actual substance B for which it provides an environment different from that of phase a . To keep with the pseudo-onecomponent analogy comprising both phases, we define an effective volume of phase /3 as KBo Vii where KBo is the Nernst partition coefficient of substance B between phases /3 and a expressed in the concentration scale. To simplify notation, we assume for the moment that initially phase p contained no B molecules. If the system a + /3 is itself closed, then, at equilibrium
+
where Vo = V , CY
+ KBoVo
(3)
We recognize that former ~ r i t e r i o n ' for , ~ ultrapurity in phase amounts in fact to the verification of the following condition
(4) Since4 $(1) = -y = -0.577 22 and, to a good approximation, ,.LB'"'(NB=O) 3
NB" = eXp(-y - 1 / ( 2 N ~ ) )
(6)
+
The pseudo-one-component in the two-phase system analogy allows applying this formula to our problem. Hence, the probability of finding zero B molecules in phase a is
Receiued: April 20, 1988
K B ' ~ ) = kT[#(NB
At equality, eq 5 gives the following mean value
/~Lg(a+')
( I ) Reis, J . C. R. J . Phys. Chern. 1986, 90, 6078-6080. (2) Harvey, A. H. J . Phys. Chern., preceding paper in this issue. (3) Sciamanna, S. F.; Prausnitz, J . M. AIChE J . 1987, 33, 1315-1321. (4) Abramowitz, H.; Stegun, I . A., Eds. Handbook of Mathematical Functions; Dover: New York, 1972; Chapter 6.
(7)
where Vois given by eq 3. Finally, combining eq 5,6, and 7 yields For large NB, eq 6 and 8 give respectively NBa = 0.561 and P(NBa=O)= 0.570. These values are thus the outcome of our earlier conditions when fluctuations are allowed for. Interestingly, one finds a mean number of fewer than one B molecule left at equilibrium in phase CY and a probability of more than 50% for obtaining an ultrapure phase CY. Equation 7 shows that the larger this probability is, the smaller V , and N B are. However, the numerical calculations made by Sciamanna and Prausnitz3 indicate that it does not seem realistic trying to improve these parameters. The validity of eq 3 and 5 of ref 1 should be restricted to closed phases. There, the prediction of a finite value for the chemical potential of a solute species at zero concentration is correct. The use of the psi or digamma function IJ in this field was first suggested by Sciamanna and P r a ~ s n i t z .Although ~ no theoretical or experimental justification was invoked by these authors but for convenience of calculation, its use nonetheless gives the correct physical picture. In summary (i) Taking into account fluctuations, conditions' previously claimed to yield ultrapurity correspond in fact to a mean number of 0.56 in impurity molecules and to a 57% probability to the phase being ultrapure at a given instant. (ii) Equations 3 and 5 of ref 1 should be applied to closed phases only. (iii) The main advantage to reducing the size of drops or bubbles in a dispersed phase is to increase the percentage of those that are ultrapure at a given moment. If these can be identified by some analytical technique, then further separation would complete purification.
Acbnowledgment. I thank Instituto Nacional de Investiga@o Cientifica for financial support (Grant QL4-LA5). Departamento de Q u h i c a Faculdade de Citncips Centro de Electroquimica e CinCtica da Uniuersidade de Lisboa 1294 Lisboa Codex, Portugal
Joslo Carlos R. Reis
Received: July 5 , 1988 (5) Landau, L. D.; Lifshitz, E. M . Statistical Physics; Pergamon: London. 1958; p 359.
