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Fractal-like kinetics in solid—gas reactions
Volume 186, number 1
CHEMICAL PHYSICS LEITERS
1 November 1991
Fractal-like kinetics in solid-gas reactions R.P. Rastogi CentralDrug Research Institute, Lucknow, India
Ishwar Das and Anal Pushkama Department of Chemistry, University of Gorakhpur, Gorakhpur 273009, India
Received 9 August 1991
On re-examination, the kinetics of the solid-solid reaction between mercurous chloride. and iodine iSfound to be of the fractal type. Microscopic studies of the reaction interface indicates that it may be random fraetal of the percolation cluster type.
1. Introduction In view of the current interest in fractal geometry and fractal-type kinetics [ 1,2], it was thought worthwhile to examine the kinetics of solid-solid reactions which are diffusion controlled and where the reaction surface can be of the fractal type. It is found that for one-dimensional reaction propagation, one can easily deduce the following relation which is found to be valid in many experimental Situations [ 3,4]: 7r” =MU=lF.
(1)
Here, r is the distance transversed in time t, the exponent n is 2, [is a constant independent of time, k is another constant, A is the area and D is the diffusion coefficient. However, in some cases, for onedimensional reaction propagation, it is found that k is not constant but is given by rG=ki
ev(
-Pt)
,
(2)
where k, and p are constant. The following systems [ $61 are known to obey eq. (2): (i) Hg,C12+12+2HgC11+HgC12+H&, (ii) picric acid t hydrocarbons-+picrates, (iii) picric acid+naphthols+picrates, this behaviour persisting also when surface-diffusion or grain boundary diffusion becomes important, In the case of tarnishing reactions, in many cases
eq. ( 1) with n= 3 is found to conform to the experimental data. The purpose of the present investigation is to examine whether fractal-like kinetics is obeyed in these systems, since it has been pointed out by Newhouse et al. [7] that a reaction medium does not have to be a geometrical fractal in order to exhibit fractal kinetics. It may be noted that fractal kinetics has been used as a structural tool for examining whether porous glass is a fractal, like a percolation cluster, or is it one-dimensional, and to investigate the photodimerization reaction of anthracene in a one-dimensional fractal-like system.
2. Experimental Materials. Mercurous chloride (G.R., S. Merck) and iodine (AR, BDH) were used as such. Procedure. A thin film of mercurous chloride was prepared on a microslide. It was pressed by another glass plate in order to have uniform thickness. Later on, the second plate was removed. Then, the first plate was put in an iodine chamber to facilitate diffusion of iodine vapour into mercurous chloride from all sides. Experiments were performed in an air thermostat maintained at 30.0 ? 0.1 ‘C. Photographs were taken at various time intervals using an “Olympus”
0009-2614/91/$ 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
I
Volume 186, number I
CHEMICAL PHYSICS LETTERS
1 November 1991
0.25 mm Fig. 1. Various stages of two-dimensional growth bchaviour for reaction between mercumus chloride and iodine, Plate (a) indicates the initial stage of mercurous chloride (fitm thickness 0.4 mm), plate (b)-(f) indicate patterns obtained after 15 min, 2.5,24,48 and 72 h, respectively.Temperature= 30.0&O.I “C.
microscope fitted with a camera. Results are shown in fig. 1.
3. Results and discussion As pointed out by Newhouse et al. [ 7 1, when a reaction obeys fractal-like kinetics, the rate constant k is a function of time of the form, k=k,t-h,
2
(3)
where O
,
(4)
where 5 is the thickness of the boundary layer at time t and E a rate constant which depends on the diffusion coefftcient D and area A of the reacting in-
CHEMICAL PHYSICS LETTERS
Volume 186, number 1
terface. Accordingly, log < was plotted against log t in fig. 2. It is found that the results can be fitted by t2
1November 1991
face is evident which approximates to a percolation cluster. The sequence of the reaction is as follows:
(5)
t =kot-5’7y
where k” is some constant independent of time. It may be noted that for a bimolecular AS A reaction on a Sierpinski gasket or percolation gasket, the exponent x 0.3 3. Fig. 1 shows the microphotographs of the reaction product taken at various time intervals at 30.0 &0.1 ‘C. The thickness of mercurous chloride on the microslide was ;2:0.4 mm. The reaction takes place when iodine vapour diffuses into the mercurous chloride. Initially, almost instantaneously yellow product is formed (fig. lb) which later on changes into the red product (figs. Id- 1f ). From fig. lc onwards, fractal geometry of the reaction inter-
It is evident that initially we start with a white continuous interface (fig. la). After about 15 min, the interface is covered with yellow HgCiI (fig. 1b), when the reaction is primarily propagated by vapour-phase diffusion. The second stage leading to the formation of HgIz and the interface of the percolation cluster type becomes apparent after 2 h as shown in figs. lc-lf. At the end of the reaction, red crystals are found embedded on the reaction interface with holes on the surface of the thin film (fig. If) facilitating surface migration and grain boundary diffusion.
Acknowledgement
l.O[
Financial support from CSIR, New Delhi is thankfully acknowledged.
References
L
-1.0
,
00
1.0
2,o
3.0
logt Fig. 2. Plots of log r versus log f for the values obtained from ref. 141.
[ 1] A. Blumen and G.H. Kohler, Proc. Roy. Sot. A 423 (1989) 189. [I!] A. Amann, L. Cederbaum and W. Gans, eds., Fractals, quasi crystals, chaos, knots and algebraic quantum mechanics, NATO ASI Series (Kluwer, Dordrecht, 1988). [3] R.P. Rastogi and B.L. Dubey, J. Am. Chem. Sot. 87 (1967) 200. [4] B.L. Dubey, Ph.D. Thesis, Gomkhpur University, Gorakhpur, India. [S] R.P. Rastogi, P.S. Bassi and S.L. Cheddha, J. Phys. Chem. 66 ( 1962) 2707; 67 (1963) 2569. [6]R.P.RastogiandN.B.Singh,J.Phys.Cltem. 70 (1966) 3315. [7] J.S. Newhouse, P. Argyrakis and R. Kopelman, Chem. Phys. Letters 107 (1984) 48.
3
CHEMICAL PHYSICS LEITERS
1 November 1991
Fractal-like kinetics in solid-gas reactions R.P. Rastogi CentralDrug Research Institute, Lucknow, India
Ishwar Das and Anal Pushkama Department of Chemistry, University of Gorakhpur, Gorakhpur 273009, India
Received 9 August 1991
On re-examination, the kinetics of the solid-solid reaction between mercurous chloride. and iodine iSfound to be of the fractal type. Microscopic studies of the reaction interface indicates that it may be random fraetal of the percolation cluster type.
1. Introduction In view of the current interest in fractal geometry and fractal-type kinetics [ 1,2], it was thought worthwhile to examine the kinetics of solid-solid reactions which are diffusion controlled and where the reaction surface can be of the fractal type. It is found that for one-dimensional reaction propagation, one can easily deduce the following relation which is found to be valid in many experimental Situations [ 3,4]: 7r” =MU=lF.
(1)
Here, r is the distance transversed in time t, the exponent n is 2, [is a constant independent of time, k is another constant, A is the area and D is the diffusion coefficient. However, in some cases, for onedimensional reaction propagation, it is found that k is not constant but is given by rG=ki
ev(
-Pt)
,
(2)
where k, and p are constant. The following systems [ $61 are known to obey eq. (2): (i) Hg,C12+12+2HgC11+HgC12+H&, (ii) picric acid t hydrocarbons-+picrates, (iii) picric acid+naphthols+picrates, this behaviour persisting also when surface-diffusion or grain boundary diffusion becomes important, In the case of tarnishing reactions, in many cases
eq. ( 1) with n= 3 is found to conform to the experimental data. The purpose of the present investigation is to examine whether fractal-like kinetics is obeyed in these systems, since it has been pointed out by Newhouse et al. [7] that a reaction medium does not have to be a geometrical fractal in order to exhibit fractal kinetics. It may be noted that fractal kinetics has been used as a structural tool for examining whether porous glass is a fractal, like a percolation cluster, or is it one-dimensional, and to investigate the photodimerization reaction of anthracene in a one-dimensional fractal-like system.
2. Experimental Materials. Mercurous chloride (G.R., S. Merck) and iodine (AR, BDH) were used as such. Procedure. A thin film of mercurous chloride was prepared on a microslide. It was pressed by another glass plate in order to have uniform thickness. Later on, the second plate was removed. Then, the first plate was put in an iodine chamber to facilitate diffusion of iodine vapour into mercurous chloride from all sides. Experiments were performed in an air thermostat maintained at 30.0 ? 0.1 ‘C. Photographs were taken at various time intervals using an “Olympus”
0009-2614/91/$ 03.50 0 1991 Elsevier Science Publishers B.V. All rights reserved.
I
Volume 186, number I
CHEMICAL PHYSICS LETTERS
1 November 1991
0.25 mm Fig. 1. Various stages of two-dimensional growth bchaviour for reaction between mercumus chloride and iodine, Plate (a) indicates the initial stage of mercurous chloride (fitm thickness 0.4 mm), plate (b)-(f) indicate patterns obtained after 15 min, 2.5,24,48 and 72 h, respectively.Temperature= 30.0&O.I “C.
microscope fitted with a camera. Results are shown in fig. 1.
3. Results and discussion As pointed out by Newhouse et al. [ 7 1, when a reaction obeys fractal-like kinetics, the rate constant k is a function of time of the form, k=k,t-h,
2
(3)
where O
,
(4)
where 5 is the thickness of the boundary layer at time t and E a rate constant which depends on the diffusion coefftcient D and area A of the reacting in-
CHEMICAL PHYSICS LETTERS
Volume 186, number 1
terface. Accordingly, log < was plotted against log t in fig. 2. It is found that the results can be fitted by t2
1November 1991
face is evident which approximates to a percolation cluster. The sequence of the reaction is as follows:
(5)
t =kot-5’7y
where k” is some constant independent of time. It may be noted that for a bimolecular AS A reaction on a Sierpinski gasket or percolation gasket, the exponent x 0.3 3. Fig. 1 shows the microphotographs of the reaction product taken at various time intervals at 30.0 &0.1 ‘C. The thickness of mercurous chloride on the microslide was ;2:0.4 mm. The reaction takes place when iodine vapour diffuses into the mercurous chloride. Initially, almost instantaneously yellow product is formed (fig. lb) which later on changes into the red product (figs. Id- 1f ). From fig. lc onwards, fractal geometry of the reaction inter-
It is evident that initially we start with a white continuous interface (fig. la). After about 15 min, the interface is covered with yellow HgCiI (fig. 1b), when the reaction is primarily propagated by vapour-phase diffusion. The second stage leading to the formation of HgIz and the interface of the percolation cluster type becomes apparent after 2 h as shown in figs. lc-lf. At the end of the reaction, red crystals are found embedded on the reaction interface with holes on the surface of the thin film (fig. If) facilitating surface migration and grain boundary diffusion.
Acknowledgement
l.O[
Financial support from CSIR, New Delhi is thankfully acknowledged.
References
L
-1.0
,
00
1.0
2,o
3.0
logt Fig. 2. Plots of log r versus log f for the values obtained from ref. 141.
[ 1] A. Blumen and G.H. Kohler, Proc. Roy. Sot. A 423 (1989) 189. [I!] A. Amann, L. Cederbaum and W. Gans, eds., Fractals, quasi crystals, chaos, knots and algebraic quantum mechanics, NATO ASI Series (Kluwer, Dordrecht, 1988). [3] R.P. Rastogi and B.L. Dubey, J. Am. Chem. Sot. 87 (1967) 200. [4] B.L. Dubey, Ph.D. Thesis, Gomkhpur University, Gorakhpur, India. [S] R.P. Rastogi, P.S. Bassi and S.L. Cheddha, J. Phys. Chem. 66 ( 1962) 2707; 67 (1963) 2569. [6]R.P.RastogiandN.B.Singh,J.Phys.Cltem. 70 (1966) 3315. [7] J.S. Newhouse, P. Argyrakis and R. Kopelman, Chem. Phys. Letters 107 (1984) 48.
3