J. Phys. Chem. 1996, 100, 12131-12134
12131
Nonequilibrium Partition Constants Akio Morita Department of Chemistry, Graduate School of Arts and Sciences, The UniVersity of Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan ReceiVed: March 14, 1996; In Final Form: May 14, 1996X
We have treated a nonequilibrium diffusion process where a solute is added to phases consisting of two immiscible solvents in a very long glass tube, for example. If we maintain the tube undisturbed, the diffusion will go on indefinitely. Whereas if we shake the tube in the middle, we will have the ordinary equilibrium case where the partition of the solute can be described by the partition constant, K. We have found theoretically that the long time limit of the concentration of the solute in the former nonequilibrium case becomes constant in each phase, and it will be shown that we can define a new nonequilibrium partition constant, KD, that is related by the expression KD ) xD1/D2K, where Di in phase i ) 1 or 2 is the diffusion coefficient. This predicts at least three new important implications in the text.
Introduction Although partition in liquid phases is a well-known example for the description of understanding their equilibrium properties, diffusion is another typical one to describe nonequilibrium processes.1 Since interconnections between these two processes have not been attempted, it might be worthwhile considering them in detail in this letter with the help of recent studies on Brownian motion under square-well potentials.2-4 To clarify the problem, imagine two long glass tubes where two immiscible solvents are fed and a solute is suddenly put into one of the solvent phases. If we shake the first tube and keep it still, the solute will eventually distribute uniformly, which is simply equilibrium partition described by the partition constant, K. However, if we do not shake the second tube and maintain it so for a long time, the diffusion will go on forever, because we have assumed that if the tube is very long, the partition will be different from that in the first tube, because we are now observing the nonequilibrium processes. It will be shown in this letter that even though the latter process is intrinsically nonequilibrium and goes on forever, the concentrations of the solute in the two solvent phases becomes approximately invariant at least within the experimental precision after a sufficient length of time and so we will be able to introduce the nonequilibrium partition constant KD, which is the ratio of the product of the square root of the diffusion coefficient and the equilibrium concentration of the solute in one solvent and that in the other phase. In other words, we will obtain KD ) xD1n1/xD2n2 where ni and Di represent the equilibrium concentration and the diffusion coefficient of the solute in solvent phases, respectively, and i ) 1 or 2 so that K ) n1/ n2.5,6 Implications of this new finding are (i) we can determine the diffusion coefficient, D, from a nondynamic quantitative analysis that is being determined by time-dependent dynamic measurements, (ii) we can control the extraction by using D as a parameter which has been totally neglected by keeping the system in equilibrium on shaking, and so on and finally (iii) we will be able to get fresh information on the liquid interface by comparing experimental values with the theoretical results in this letter. X
Abstract published in AdVance ACS Abstracts, July 1, 1996.
S0022-3654(96)00787-3 CCC: $12.00
Figure 1. Potential V(x) for phases 1 (0 e x < ∞) and 2 (-∞ < x < 0).
II. Theory and Discussion We start with the following diffusion equation with the square-well potential whose heights are corresponding to standard chemical potentials, µQi in Figure 1:
∂F(x,t) ∂2F(x,t) )D ∂t ∂x2
(1)
where x is the position of a solute molecule and F(x,t) dx is the probability of finding the molecule within the range of x + dx, the flux J(x,t) is defined by the expression
∂F(x,t) ∂x
J(x,t) ) -D
(2)
The reason we should introduce µQi in Figure 1 for our problem is based on the requirement that the glass tube with a finite length should lead to the equilibrium partition without the disturbance as time goes on and K in this case should be determined only by the difference in chemical potentials in light of thermodynamics. In fact, we will be able to solve eq 1 with the potential of Figure 1 in this case and find K in harmony with the requirement. The solution of eq 1 for the natural boundary condition that F(x,t) f 0 as x f ∞ can be obtained by taking the Laplace transform of both sides of eq 1 with respect to t, and it is given by © 1996 American Chemical Society
12132 J. Phys. Chem., Vol. 100, No. 30, 1996
F(x,λ) ) B0e-qx +
1 -q|x-x0| e 2qD
Letters
(3)
mentary error function in eq 14, we find for D1t . 1 that
N2(t) )
where we have used the usual initial condition of F(x,t) ) δ(x - x0) at t ) 0, B0 is a constant to be determined later:
F(x,λ) ) ∫0 F(x,t)e-λ t dt ∞
2
(4)
(5)
limN2(t) ) N∞2 ) tf∞
(6)
where C0 is a constant. We now introduce boundary conditions at x ) 0 to determine the unknown constants, B0 and C0. To this end, we assume that the flux is continuous, otherwise some of particles must be trapped at x ) 0 but F(0,λ) is discontinuous as the potential at the interface with the relation3,4
F1(0,t) ) e-AF2(0,t)
x0
xπD1t
)
+ ...
(15)
eA
(16)
e + xD1/D2 A
Thus in this limit, from eq 13 we also have
For phase 2, the natural condition of F(x,t) f 0 as x f -∞ and the initial one of F(x,t) ) 0 at t ) 0 by assuming that we do not feed the solute in this phase lead to
F(x,λ) ) C0eqx
eA + xD1/D2
1-
which leads in the long time limit to
that is the Laplace transform (note the difference in the exponent from the usual definition), and
q2 ) λ2/D
(
eA
N∞1 )
xD1/D2 eA + xD1/D2
(17)
It should be noted that N∞1 and N∞2 are independent of the initial condition, x0. Equations 13 and 14 indicate that even though the diffusion is still going on, N∞1 and N∞2 do not change their values appreciately once the condition of x0/xπD1t . 1 has been attained. Therefore it follows that
N∞1 N∞2
(7)
) KD )
x
D1 -A e D2
(18)
where A ) (µQ1 - µQ2 )/RT in which R and T are the gas constant and the temperature, respectively. These result in
where we have defined the nonequilibrium partition constant KD. In the mean time, the usual partition constant K where
-D2q2C0 ) D1q1B0 - 1/2e-q1x0
K ) n1/n2 ) e-A
e-AC0 ) B0 +
(8)
1 -q1x0 e 2q1D1
(9)
respectively. We now note that we have distinguished the solute molecules in phases 1 and 2 by subscripts. These lead to
[
]
A 1 -q1x0 q1D1 - q2D2e e B0 ) 2q1D1 q1D1 + q2D2eA
C0 )
eAe-q1x0 q1D1 + q2D2eA
(10)
(11)
The Laplace transform of the number of particles in phase 2, N2(λ) is thus given by
N2(λ) ) ∫-∞F2(x,λ) dx ) 0
eA e-q1x0 λ2 eA + xD1/D2
(12)
and that for phase 1 should be given by
N1(λ) )
1 - N2(λ) λ2
(13)
in view of the fact that the total number of particles must be conserved within the space -∞ < x < ∞ at all times. It follows immediately from eq 12 by taking the inverse Laplace transform that
N2(t) )
eA e + xD1/D2 A
( )
erfc
x0
2xD1t
(14)
Note that the time scale of the dynamics is governed by D1 only, despite processes in phase 2. By expanding the compli-
(19)
which is entirely determined by the difference in standard chemical potentials in both phases through A. Let us now clarify physical differences in both cases. Take the example of the glass tube again. If the tube is not long so that the solute molecules will touch both ends of the tube as time goes on, they will eventually distribute uniformly without shaking, which case corresponds to the equilibrium one whose partition is described by K in eq 19. Whereas if the tube is so long that the solute cannot distribute uniformly, in fact, eqs 3, 6, 10, and 11 lead to
F1(x,t) ) 1
[(
x4πD1t F2(x,t) )
xD1 - xD2e A xD1 + xD2e A
2xD2eA
)
2
1
2
e-1/4t(x/xD2-x0/xD1)
xD1 + xD2e x4πD2t A
]
e-(x+x0) /4D1t + e-(x-x0) /4D1t (20)
2
(21)
we see that the partition must be governed by KD in eq 18. But if we shake the tube, the partition should be given by K, because the uniform distribution will be formed in this case. In other words, shaking changes partition in two phases, which can be readily understood intuitively as well. In usual experiments in solvent extraction, the system is often shaken to get the equilibrium partition promptly. The point in this letter is rooted in the idea what will be if we do not shake it. We have seen that even though the distribution is inhomogeneous, the numbers of particles in both phases approximately become constant relatively in not a very long time. A new feature in this letter is to let the nonequilibrium process go to its full extent because we can understand it simply and clearly, instead of eliminating, and obtain more information. To visualize the short time behavior of F(x,t), we have plotted it as a function of x and t
Letters
J. Phys. Chem., Vol. 100, No. 30, 1996 12133 KD in eq 18 enables us to control the partition by using the viscosity, η of the solvent as a parameter in view of the Einstein-Stokes law stating that the diffusion coefficient is inversely proportional to η. It is interesting to note in eqs 20 and 21 that the limits of A f ∞ and A f -∞ give to the expressions for the complete absorbing and reflecting interfaces at x ) 0, respectively. Finally it is useful for calculating the mean-square displacement, 〈x2(t)〉, which indicates a degree of the progress in diffusion. For the tube of the finite length, this becomes constant as t f ∞ where the equilibrium is reached. On the other hand, our nonequilibrium case will lead to monotonical increase function of time, in fact, the contributions from phases 1 and 2 are given by
〈x2(t)〉1 ) x02 + 2D1t Figure 2. Time dependence of N1(t) and N2(t) obtained from eqs 13 and 14, respectively, for x0 ) 1, KD ) 0.01.
〈x2(t)〉2 )
2xD2eA
xD1 + xD2e
A
2xD2eA
D1f(t)
D2f(t)
xD1 + xD2e
(22)
(23)
A
respectively, where
( ) ( ) ( )
f(t) ) L-1
e-q1x0 ) λ4 t+
x02 x0 erfc - x0 2D1 2xD1t
x
t -x02/4D1t e (24) πD1
The total mean-square displacement is, of course
〈x2(t)〉 ) 〈x2(t)〉1 + 〈x2(t)〉2 It therefore follows that as t f ∞
(
〈x (t)〉 ) 2 2
Figure 3. Time development of F(x,t) calculated from eqs 20 and 21 for x0 ) 5.0, D1 ) 1, D2 ) 4, and A ) 3.
for typical two kinds of initial conditions in Figures 3 and 4. The former is for x0 ) 5, and the latter for the uniform distribution in the range of 0 e x e 80 at t ) 0.
)
D1xD1 + D2xD2eA
xD1 + xD2e
A
t
(26)
which is different from Einstein’s relation, 2D1t that is valid for the particular case of D1) D2. In other words, the difference in the diffusion coefficients in two phases can increase or decrease the degree of the diffusion, because the probability of the solute molecules to phase 2 crossing the interface is dependent upon Di. It should be noted that eq 26 is the direct indication that our system is essentially nonequilibrium whose time development will never cease and the solute in both phases flows without interruptions. However, we should note that after long time, there is a region of a pseudoequilibrium state where the partition law holds. Let us clarify this point to avoid a possible misunderstanding. The boundary condition in eq 7 leads to the assumption that the partition law in eq 19 always holds for the interface and F1(0,t) is the mirror image of F2(0,t) with the magnification ratio of e-A, i.e., as soon as the solute touches the interface in phase 1, it also moves to the interface in solvent 2 instantly. Furthermore, it follows from eq 20 and 21 that for a given t and for the region satisfying the conditions
|x ( x0| . x4D1t Figure 4. Time development of F(x,t) calculated from eqs 20 and 21 assuming the initial uniform distribution for 0 e x e 80. The remaining parameters are the same as those in Figure 3.
(25)
|x - x0xD2/D1| . x4D2t (27)
the solute distributes uniformly and the concentration decreases by being inversely proportional to xt, because it is allowed to set exponential terms to be 1. In this case, we see that [F1(x,t)/ F2(x,t)] ≈ e-A so that if we take the same volume of the tiny amount of solution from each phase at this region in the unshaken tube at the same time and measure the concentration,
12134 J. Phys. Chem., Vol. 100, No. 30, 1996
Letters
we would find just the partition law. But we must be careful here that this is not the equilibrium state, although the distribution is uniform, but the process is still runing with Fi(x,t) ≈ 1/xt. In fact, in the region where
|x ( x0| ≈ x4D1t
|x - x0xD2/D1| ≈ x4D2t (28)
Fi(x,t) is no longer homogeneous. In other words, in the region near the diffusion front, the distribution is always imhomogeneous and the front moves with the conditions in eq 28. Our nonequilibrium partition relation includes both forms of distribution through the total number of the solute molecules, Ni(t) in phase i which is different from Fi(x,t) as seen from eq 12. So to measure N∞i , we should take the whole solution in each phase, not a part of the solution as in the previous case, and shake the container of the solution whose portion of a known amount of volume will be used for a quantitative analysis. Here it is important to note from eq 14 that our nonequilibrium partition relation holds in a time scale of
t . x02/4D1
(29)
This is a slightly weaker condition than the first one in eq 27, which means that in shorter than the time scale in eq 27, our nonequilibrium partition will be attained. It is hoped that predictions in this letter will be confirmed experimentally. Acknowledgment. I would like to express my sincere gratitude to Professor Toshihiro Tominaga, Okayama University of Science, for useful advice on experimental aspects and for providing with appropriate references. Professor Hitoshi Watarai, Osaka University, was also helpful for widening my scope on partition constants. References and Notes (1) Crank, J. The Mathematics of Diffusion; Oxford University Press: London, 1975. (2) Morita, A. Phys. ReV. E 1994, 49, 3697. (3) Morita, A. J. Mol. Liquids 1995, 65/66, 75. (4) Morita, A. J. Mol. Liquids, in press. (5) Tominaga, T.; Matsumoto, S., Bull. Chem. Soc. Jpn. 1990, 63, 533. (6) Watarai, H.; Tanaka M.; Suzuki, N. Anal. Chem. 1982, 54, 702.
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