In the Laboratory
W
The Kinetics of Dissolution Revisited
Paula S. Antonel, Pablo A. Hoijemberg, Leandro M. Maiante, and M. Gabriela Lagorio* INQUIMAE/Departamento de Química Inorgánica, Analítica y Qca. Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón II, 1er piso, C1428EHA, Buenos Aires, Argentina; *
[email protected] The experiment presented here is an extension of a previously published article in this Journal: Dissolution Kinetics of Solids. Application with Spherical Candy (1). The extension was suggested by students in the physical chemistry laboratory course at the University of Buenos Aires—a good example of the creative participation of students in experimental practices. In the laboratory course a set of standard experiments are performed in the first part of the semester. In the last 20% of the course students choose the experiments. The students select material predominately from the Journal of Chemical Education under a flexible framework dictated by the course content and availability of instruments. In this exercise students are not passive receivers of knowledge and laboratory recipes but participate in the choice and design of their laboratory experiments. Autonomous thinking and manual abilities are improved from this exercise. After having chosen and performed the kinetics analysis on the dissolution of a spherical candy according to the procedure by Beauchamp (1), a group of students was interested in studying the dissolution behavior for another geometrical shape. A cylindrical candy was examined and the results are presented here. The experiment is simple, but the concepts involved (diffusion, Nernst layer, and kinetics) and the mathematical manipulation of data are very meaningful for students in a physical chemistry laboratory course. Experimental Procedure The procedure was similar to that described by Beauchamp (1). A cylindrical candy was weighed and its radius (r) and its thickness (e) were measured (Figure 1A). The candy was then fixed to a stick and submerged (time = 0) in 900 mL of distilled water in a 1-L beaker. The liquid was stirred with a magnetic bar at constant speed (the stirring speed was high enough to ensure the Nernst model validity).
At defined time intervals, the candy was taken out of the solution, dried with an absorbing paper, weighed, and its radius and thickness measured using a caliper. The temperature was kept constant (25 ⬚C) during the experiment. Theoretical Framework Solid particle dissolution follows the Noyes–Whitney differential equation (see the Supplemental MaterialW for additional theoretical background), −
dm DS = (cs − ct ) dt h
where m is the solid mass at time t, D is the mutual diffusion coefficient (2, 3) for the diffusing substance in the solution, S is the solid surface, h is the thickness of the diffusion layer (Figure 1B), cs is the solid solubility, and ct is the concentration of solute at time t (4, 5). The diffusion coefficient represents the quantity of solute that diffuses across a 1-cm2 area in 1 s under the influence of a concentration gradient of 1 g cm᎑3兾cm. For highly soluble solutes such as sugar in the presence of excess water, cs is much higher than ct and eq 1 may be written as: DS dm = cs (2) dt h For a spherical particle, S is taken as 4πr2, where r is the particle radius, and the mass is written as a function of its density (ρ) as (4/3)πr3ρ. In this case, eq 3 may be rewritten from eq 2 as, −
2 dm = km 3 dt where k is the rate constant defined as:
−
k = A
B
Nernst boundary
concentration = cs
solution
candy
h Figure 1. (A) Radius and thickness for the cylindrical candy. (B) Visualization of Nernst diffusion layer. (The diffusion layer thickness is not drawn to scale compared to the candy size.)
1042
36π
D cs 2
3
(3)
(4)
This simple case was examined in ref 1, where a linear relation between 3 m and t was found. For a cylindrical-shaped candy, however, the surface is described by, S = 2π r 2 + 2π re
(5)
where flat surfaces are assumed and the mass is described by, m = ρπ r 2 e
x
e
3
hρ
concentration = ct
r
(1)
(6)
where ρ is the candy’s density. Equation 7 is obtained by substituting eqs 5 and 6 into eq 2: −
dm 2Dcs = dt hρ
1 1 + m e r
Journal of Chemical Education • Vol. 80 No. 9 September 2003 • JChemEd.chem.wisc.edu
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In the Laboratory
Results and Discussion The candy’s thickness and radius variations with time (in seconds) are plotted on Figure 2. The sum [(1/e) + (1/r)] was fitted to a polynomial expression,
Inverse Length / cmⴚ1
3
1/e 1/r 1/r + 1/e
2
F(t) = (1.89 × 10᎑8)t3 – (1.01 × 10᎑5)t2 + (2.78 × 10᎑3)t + 1.44
cm-1
1
0 0
100
200
300
400
t/s Figure 2. Radius inverse (1/r), thickness inverse (1/e), and the sum of radius and thickness inverses [(1/r ) + (1/e)] as a function of the dissolution time. Polynomial fitting is shown as a solid line.
0.0
Conclusions The experiment allowed students: (i) to fit experimental data to mathematical functions, (ii) to successfully validate the proposed model for the dissolution of a solid in a liquid, (iii) to search the literature for diffusion coefficient values and decide which should be used under the experimental conditions, and (iv) to obtain kinetic parameters for this process (ks) and an estimation of the thickness for the Nernst diffusion layer.
-0.1
correlation coefficient = .9996
ln
m mo
-0.2 -0.3 -0.4 -0.5 -0.6
Hazards There are no significant hazards in this experiment.
-0.7 0
200
400
600
using Microsoft Excel. Both experimental and fitted data are also presented in this figure. The linear dependence between ln(m兾m0) and f (t) is shown in Figure 3. The slope, k´, is ᎑(7.9 ± 0.1) × 10᎑4 cm兾s. These results show an excellent correlation between experimental data and the model proposed to describe the dissolution process. The candy density (ρ = 1.4 g兾cm3) was calculated from its initial mass and volume. As an approximation the candy was considered to be pure sucrose and its solubility was taken as 0.67 g兾cm3 (6). From these values, ks was calculated to be: (1.6 ± 0.1) × 10᎑3cm兾s. A value of D = 6.1 × 10᎑6 cm2兾s was used to estimate the thickness of the Nernst diffusion layer (6, 7): a value of 7.4 × 10᎑3 cm for h was obtained in our experimental conditions. Typical values for the Nernst layer thickness in a well-stirred solution are about 10᎑2–10᎑3 cm (2, 8).
800
f (t)
Acknowledgments We want to acknowledge F. Molina and E. San Román for discussion of some aspects of the present manuscript.
Figure 3. ln(m/m0) as a function of the integral f (t ).
Supplemental Material Instructions for the students and notes for the instructor are available in this issue of JCE Online. W
Finding the dependence of m with time is complicated by the time dependence of both e and r. To overcome this, the experimental values of [(1/e) + (1/r)] as a function of time were fitted to a polynomial function, F(t), using the linear least-squares method. The integration of eq 7 leads to,
m ln n m0
Dc = −2 s f (t ) hρ
(8)
where m0 and m are the masses at time 0 and t, respectively and f(t) is the integral of the polynomial function F(t). A linear relation between ln(m兾m0) and f (t) will support the proposed model. Additionally from the slope of this plot, the D兾h value may be obtained provided the candy density and solubility are both known. As shown in the Supplemental Material,W D兾h is the rate constant ks. In this particular case, it is also convenient to redefine the rate constant k´ as k´ = 2
Dcs hρ
(9)
to simplify calculations when determining mass values as a function of time.
Literature Cited 1. Beauchamp, G. J. Chem. Educ. 2001, 78, 523–524. 2. Levine, I. N. Physical Chemistry; Mc Graw-Hill, Inc.: New York, 1988. 3. Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Academic Press Inc.: New York, 1959. 4. Berry, R. Stephen; Rice, Stuart A.; Ross, John. Physical Chemistry; John Willey & Sons: New York, 1980. 5. Atkins, P. W. Physical Chemistry; Oxford University Press: Oxford, England, 1994. 6. Bubnik, Z.; Kadlek, P.; Urban, D.; Bruhns, M. Sugar Technologists Manual: Chemical and Physical Data for Sugar Manufactures and Users; Bartens: Berlin, Germany, 1995. 7. Linden, P. W.; Nassimbeni, L. R.; Polson, A.; Rodgers, A. L. J. Chem. Educ. 1976, 53, 330–332. 8. Cárcamo, E. C. Cinética de Disolución de Medicamentos; The General Secretariat of the Organization of American States: Washington, D.C., 1981.
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