DISSOLUTION LIFETIME O F A “HYDRATED” SOLUTE SPHERE
2017
Dissolution Lifetime of a “Hydrated” Solute Sphere1 by Daniel E. Rosner*2and W. S. Changa Department of Engineering and Applied Science, Yale University, New Haven, Connecticut 06610 (Received January 7, 1971) Publication costs assisted by the U. S. Air Force Officeof Scientific Research
An exact method is presented for calculating the total dissolutionlifetime of a solvent-containing (e.g., hydrated) sphere in terms of the corresponding lifetime of a solvent-free (pure solute) sphere. The resulting algorithm, which is free of the common quasi-steady and/or sparing solubility approximations, is illustrated herein for the dissolution of lithium iodide and its three hydrates (LiI .zHzO, z = 1, 2, 3) in water at, 25’.
Introduction Recent theoretical studies of diffusion-limited dissolution have provided quantitative results for the total lifetime of an isolated sphere4J free of the mellknown but overly restrictive quasi-steady and sparing solubility approximations. These lifetime predictions explicitly apply to the isothermal dissolution of a stationary, pure solute sphere in an otherwise quiescent, incompressible solvent fluid. Since the sphere/solvent density ratio,po/p, may be large, of order unity, or small, formal application to problems of solid-liquid, liquid (droplet)-liquid, and gas (bubble)-liquid dissolution is possible. I n the present note we (i) demonstrate that these results may equally well be used to predict the lifetime of spheres initially containing a prescribed amount of solvent (as in the case of hydrated salts dissolving into water) ; (ii) illustrate the computational algorithm, using as an example the dissolution of lithium iodide and its three well-known hydrates, LiI.xH20 (x = 1, 2, 3); and (iii) examine the inaccuracy of the familiar quasi-steady (QS) approximation when applied to this same class of problems. Physicochemical and Mathematical Model With the exception that we now allow the sphere to contain a constant fraction of solvent, our constant property, spherically symmetric continuum model is identical with that discussed in detail in ref 4 and 5 . Thus, we visualize that a t time t = 0 a sphere of initial radius Ro R(0) is immersed in an incompressible solvent, which is undersaturated with respect to the solute (at the prevailing temperature). For definiteness, we can consider the sphere to have the composition A.xH20, where A represents the solute species, and use the phrase “hydrate,” although it is clear that our arguments would be valid regardless of the identity of the Throughout the ensuing dissolution dR/dt < 0, local saturation process, during which R is assumed a t the sphere-solvent interface, in the sense that the solute mass fraction, c, evaluated a t r = R(t), will be taken to be the constant,7 csat. Owing to our basic assumptions concerning the direction of solute
mass transfer, csat is furthermore assumed to be between the solute mass fraction, co in the sphere,8 and the value c, pertaining to the solute-containing ambient solvent. As in the cases treated in ref 4 and 5 , the following set of solute conservation equations, boundary conditions, and initial conditions suffices to determine the transient, solute concentration field c(r,t), and the associated sphere radius-time “history” R(t)
c[R(t),t] = csat
=
constant
(3)
c ( ~ , t )= c, = constant
(4)
c(r,O) = c, = constant
(5)
from which the total sphere lifetime, he, is obtained, using the definition (1) Supported by the Propulsion and Energy Conversion Division of the U. S. Air Force Office of Scientific Research under Contract F44620-70C-0026. (2) Associate Professor, Chemical Engineering Group; to whom inquiries concerning this paper should be addressed. (3) Max Planck Institut far Biophysikalische Chemie, Giittingen, West Germany. (4) D. E. Rosner, J . Phys. Chem., 74, 4001 (1970). (5) D. L. Duda and J. S. Vrentas, Int. J . Heat Mass Transfer, 14, 395 (1971). (6) Thus, for a fuel droplet evaporating into air, the “solvent” would be air and the initial fuel droplet could contain the saturated amount of air a t the prevailing pressure and droplet temperature. I n such cases the ratio of solvent to solute in the sphere is not restricted to integer (stoichiometric) values. Similarly, for the dissolution of a gas bubble into a liquid, we can now take due account of the nonzero solvent vapor pressure within the bubble. (7) Only when interface kinetic limitations to the dissolution process cause significant interfacial undersaturations would the value of c at r = R(t) itself be time dependent. For a discussion of the validity of the solute diffusion ”control” assumption see, e.g., D. E. Rosner, J . Phys. Chem., 73, 382 (1969). In what follows, D is the concentration-independent Fick diffusion coefficient for solute transport. (8) For a hydrated salt A.xHz0, co will by definition be given by M A / ( M A 1 8 . 0 2 ~ )where M A is the molecular weight of the solute A and x is the degree of hydration. The mass density of the sphere, PO, will usually depend on the degree of hydration.
+
The Journal of Physical Chemistrg, Vol. 76, N o . 14, 1971
DANIELE. ROSNER AND W. S. CHANG
2018
(7)
R(tlite) E 0
The only difference between eq 1-7 and the equations of ref 4 and 5 is the appearance of co (rather than unity) in the solute conservation boundary condition at r = R ( t ) , eq 2. This latter equation follows from the fact that, relative to a control volume straddling the interface, solute A flows in per unit area at the rate poco(-&) and hence must flow out a t the equal rate: pcsat[v(R,t) - I?] - Dp(bc/br),,R, where the second contribution is due to Fick diffusion. Combining this statement with overall mass conservation to eliminate the radial mean mass fluid velocity v(R,t) gives eq 2. Analysis Because co < 1 enters only eq 2 and, by hypothesis, remains constant, the present problem can be readily transformed to the problem recently solved in ref 4 and 5. Toward this end we introduce the ratios
e(r,t) B
(C
-
cm)/(cs,,t
-
E (~siat - c ~ ) / ( c o -
cm)
sat)
(8) (9)
where, as noted in ref 7, when cm = 0 the parameter defined by eq 9 has the following simple physical interpretation: B i s numerically equal to the solubility of the “hydrated” solute, expressed in grams of hydrate per gram of solvent. When the present boundary-value problem is rewritten in terms of e and the egective solubility parameter B, it becomes identical with that already solved in ref 4 and 5; hence the dimensionless lifetime function nife(p/pO,B)introduced in ref 4 and 7 can be applied to the present problem as well, where
Uo
Numerical Example and Discussion To illustrate (i) how to apply our computational algorithm and (ii) the effect of hydration on the masstransfer-controlled lifetime, we consider the dissolution The Jou.rnal of Physical Chemistry, Vol. 76, No. 14, 1972
2
3
DEGREE OF HYDRATION, X
Figure 1. Exact and approximate diffusion-controlled total lifetime predictions for the dissolution of lithium iodide and its hydrates in water at 25”.
of lithium iodide crystal (LiI) and its well-known hydrates (LiI.xH20 with 2 = 1, 2, 3) in pure water (Le., cm = 0) at 25”. I n ref 11, the densities of LiI and LiI.3Hz0 are given as 3.494 f 0.015 g/cmS and 3.48 g/cm3, respectively.’l I n the absence of additional data we therefore simply neglect the dependence of density POon degree of hydration; i e . , we put po 3.49 g/cm3 for LiI and each of its hydrates. The molecular weight of LiI is 133.84. The saturation mass fraction, csat, of LiI in the liquid water at 25” is12 0.626. It should be noted that this same value applies in the fluid phase, no matter what the degree of hydration is in the adjacent solid phase. Thus, in this particular example, the final form for the effective solubility parameter B is B
and now B is given by eq 9. Thus, as first suggested in ref 7, the mere introduction of a more general solubility parameter, eq 9, allows us t o solve the “hydrated” sphere problem, making no approximations other than those already underlying the n i f e predictions of ref 4 and 5. I n particular, since the transient term, b c / b t , in eq 1 has been retained, and csat has not been considered negligible compared to unity, the present results are free of the restrictive “quasi-steady” and “sparing solubility” (B
