J . Phys. Chem. 1988, 92, 6411-6418
6477
COMMENTS On the Irrelevance of Phase Size in Purification Sir; Recently, Reisl has suggested that it might be possible to remove a solute species completely from a small (or finely dispersed) phase by a reduction to some low but finite value of the chemical potential of that species in the medium surrounding the phase. Sciamanna and Prausnitz,* while expressing some doubts about the rigor of the theoretical approach, used similar arguments to examine the possibility of obtaining “ultrapurity” in a small dispersed phase by equilibrium purification operations such as distillation and extraction. Here we demonstrate that Reis’ original suggestion is incorrect. Furthermore, we show that, under well-defined and reasonable assumptions, the size of a phase has no influence on its purity. The system of interest is a drop (or bubble) a in contact with a much larger continuous fluid phase (3 at constant temperature T and pressure P. At a given time, the drop contains NA molecules of species A and NB molecules of species B in volume V. B is very dilute in phase a (NB > NB, which is the case of interest here. The answer to both questions is “no”. We show this using an intuitive physical argument for the first question and then via statistical mechanics for both questions. Physical Argument. If pugis finite and phase /3 is macroscopic, there must be a nonzero concentration of B in phase (3. These B molecules, through their thermal motion, must occasionally encounter the boundary with phase a . Barring an unphysical, infinitely strong barrier at the boundary, some B molecules will therefore spend a finite fraction of time in phase cy. Therefore +
-
+
Statistical-Mechanical Argument. We consider a drop (phase a ) in equilibrium with an infinite bath (phase p) at T and pB. The drop has volume V. (We are now viewing phase a as a system at constant volume instead of constant pressure; volume fluctuations will be negligible unless substance A happens to be near its critical point.) Since the A molecules are in great excess, they can be viewed as merely providing a medium for the B molecules. The B molecules in phase a can then be viewed as a pseudoone-component system with fixed T, V, and p B . Such a system is naturally analyzed in the grand canonical ensemble, for which the partition function P is4 m
where AB = exp(pB/kT) is the absolute activity of B (which depends on T and p B but not on V) and Q N B is the canonical ensemble partition function for a system with fixed T, V, and NB. The probability that the drop contains N’B B molecules is
For systems with few particles, one cannot meaningfully derive thermodynamic properties such as the entropy or pressure from the partition function. However, (2) and (3) hold for all NB 2 0.5a9b We note that our first question is again answered in the negative; P(0) < 1 (and therefore N E > 0) for all finite pB. To investigate the effect of the volume on the concentration V, we make two reasonable assumptions: ( 1 ) Interactions between B molecules are neglected. This assumption becomes exact as RE 0. (2) Surface and quantum effects are neglected. With these assumptions, each B molecule experiences the same potential as if it were the only B molecule in an infinite solvent; the problem is not too different from that of an ideal gas in an external field, where the NA solvent molecules create the field. Under these conditions, the canonical partition function QNB may be factored as4
me/
-
Q N ~=
qeNB/N~!
(4)
where qB is the partition function for a single B molecule in the phase. Furthermore, since all locations in the volume Vprovide an equivalent environment for a B molecule qB =
vfB
(5)
wherefB does not depend upon V. Substituting (4) and (5) into (2) yields (after a little algebra) In
P = XBVfB
(6)
NB is defined by m
NB =
C N’BP(N’B) NfB=O
(7)
It can then be shown that
Equation 8 tells us that the concentration N B / V = AdB, which is independent of volume. Therefore, phase size is not a relevant parameter in determining equilibrium phase concentrations.
> 0.
(4) See for example: McQuarrie, D. A. Statistical Mechanics; Harper and ( I ) Reis, J. C. R . J . Phys. Chem. 1986, 90, 6078. (2) Sciamanna, S. F.; Prausnitz, J. M. AIChE J . 1987, 33, 1315. (3) See for example: Beattie, J . A.; Oppenheim, I . Principles of Thermodynamics; Elsevier: Amsterdam, 19791
0022-3654/88/2092-6477$01.50/0
0 1988 American Chemical Society
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.
