Jander equation

Komatsu [478] has put forward the hypothesis that reaction in many powder mixtures is initiated only at interparticle contact and that product formation occurs by diffusion through these contact zones. Here, one of the participating reactants is not covered with a coherent product layer. Quantitative consideration of this model leads to a modified Jander equation. 

This deceleratory reaction obeyed the parabolic law [eqn. (10)] attributed to diffusion control in one dimension, normal to the main crystal face. E and A values (92—145 kJ mole-1 and 109—10,s s-1, respectively) for reaction at 490—520 K varied significantly with prevailing water vapour pressure and a plot of rate coefficient against PH2o (most unusually) showed a double minimum. These workers [1269] also studied the decomposition of Pb2Cl2C03 at 565—615 K, which also obeyed the parabolic law at 565 K in nitrogen but at higher temperatures obeyed the Jander equation [eqn. (14)]. Values of E and A systematically increased... 

While kinetic investigations of these reactions were incomplete, it was shown that the deceleratory process obeyed the Jander equation [eqn. (14)] approximately. 

Although we have shown several kinetic models for reacting solids, none specifically applies to a reaction between two solids. A rate law that was developed many years ago to model reacting powders is known as the Jander equation, and it is written as... 

This equation has the same form as that for three-dimensional diffusion (see Table 8.2). The Jander equation was found to model the process shown in Eq. (8.27) quite well. The reaction between two solids requires the reaction to begin on the surface of the particles and progress inward. For solids in which there is no anisotropy in the structures, diffusion should take place equally in all directions, so a three-dimensional diffusion model would seem to be appropriate. 

Figure 19. Decarboxylation of inorganic carbonate in the solid state as described by the Jander equation. (Reproduced from Ref. 266 with permission.)...

The Jander equation has been applied to the degradation kinetics of various pharmaceuticals. For example, the degradation of freeze-dried thiamine diphosphate (Fig. 20) 9-270 and the degradation of propantheline bromide in the presence of aluminum hydroxide gel (Fig. 21)271 have been described by the Jander equation. 

Figure 103. Effect of water content on degradation of propantheline bromide in the presence of sodium aluminum gel. The values of the rate constant (in units of h-" ) calculated according to the Jander equation at 37°C. , Moisture equilibrium water added. (Reproduced from Ref. 454 with permission.)...

The transition of phenobarbital from forms C and E to an anhydrous form conformed to the Jander equation (Eq. 3.4), indicating a three-dimensional diffusion mechanism.587 Some results are shown in Fig. 148. 

I - a)1 3]2 - kt Three-dimensional diffusion, spherical symmetry Jander equation... 

The question arises as to what type of rate law can be used to model reactions between two solids. Probably the most widely employed rate law is that known as the Jander equation. 

The solids were suspended in dodecane, and the mixture was sonicated using a pulsed source. At specific times, samples of the soHds were separated from the reaction mixture and analyzed to determine the amount of CdS formed. The extent of reaction was found to vary with sonication time, although the data showed considerable scatter. Although the results did not represent a highly sophisticated kinetic study, they indicated that the Jander equation represents a rate law that should be considered when a model for a reaction between two soHds is being sought. 

FIGURE 7.7 The results of fitting the Jander equation to data obtained from sonicating a 1 1 mixture of Cdh and Na2S at low power. The functionis defined in Eq. (7.88). 

Fitting the data to the Jander equation gave the results shown in Figure 7.7. 

Comparing the linear correlation coefficient and standard deviation for every kinetics mechanism function in Table 3, the NO. 19 function is the most probable one. In consequence, the anti Jander equation controlled by the three-dimensional diffusion model is the most probable mechanism function for reaction of KBL anthracite coal between 40°C and 70°C. 

D, [1-(1 - ) ] = Af Three-d i mensional diffusion, spherical symmetry, Jander equation... 

Equation (2.16), referred to as the Jander equation, suffers from two oversimplifications that limit its applicability and the range over which it adequately predicts reaction rates. First, the parabolic growth law assumed for the thickness of the reaction layer is valid for one-dimension reaction across a planar boundary and not for a system witli spherical geometry. At best, it is expected to be valid only for the initial stages of the powder reaction when y r. Second, any change in molar volume between the reactant and the product is neglected. These two oversimplifications have been taken into account by Carter (37), who derived the following equation ... 

For a solid-state diffusion mechanism, the growth of the reaction product in powder systems occurs at the contact points and for nearly equal-sized spheres, the number of contact points is small. Nevertheless, for many systems, the Jander equation and the Carter equation give a good description of the reaction kinetics for at least the initial stages of the reaction. It appears that rapid surface diffusion provides a uniform supply of one of the reactants over the other. Alternatively, if the vapor pressure of one of the reactants is high enough [e.g., ZnO in Eq. (2.3)], condensation on the surface of the other reactant can also provide a uniform supply of the other reactant. In this case, the powder reaction can be better described as a gas-solid reaction rather than a solid-state reaction (32). 

Komatsu-Uemura, sometimes called anti-Jander equation)... 

Three-dimensional diffusion (Jander equation) D3 3(l-a)2 V2fl-(l-a)i l... 

Equation (5) is the well-known Jander equation relating the fraction of reaction completed to time, where kj is the rate constant. In order to determine the rate constant for an isothermal solid-state reaction, the fraction of material reacted must be determined as a function of time, and then, according to the Jander model, a plot is made of [1 — (1 — versus time. 

In the case where the reaction starts only at the contact zones between particles and the reaction proceeds by diffusion through the contact zones Jander s assumption that the surface of one component is completely and continuously covered with particles of the other component is obviously not valid. To take into account the effect of the number of contact points Komatsu ( °) introduced into the Jander equation the mixing ratio of the two components, the ratio of the radius of the two components, and a parameter which describes the packing state of the powders. 

---