Cube-root law

At the critical time, there is a change in slope in the cube root law plot [37,39]. 

The simplest case for modeling particle dissolution is to assume that the particles are monodisperse. Under these conditions, only one initial radius is required in the derivation of the model. Further simplification is possible if the assumption is made that mass transport from a sphere can be approximated by a flat surface or a slab, as was the case for the derivation for the Hixson-Crowell cube root law [70], Using the Nernstian expression for uniaxial flux from a slab (ignoring radial geometry or mass balance), one can derive the expression... 

Since m is the mass of solid remaining at time t, the quantity m/m0 is the fraction undissolved at time t. The time to total dissolution (m/m0 = 0) of all the particles is easily derived. Equation (49) is the classic cube root law still presented in most pharmaceutics textbooks. The reader should note that the cube root law derivation begins with misapplication of the expression for flux from a slab (Cartesian coordinates) to describe flux from a sphere. The error that results is insignificant as long as r0 8. 

In the early days of water radiolysis, it was empirically established in several instances that the reduction of molecular yield by a scavenger was proportional to the cube root of its concentration (Mahlman and Sworski, 1967). Despite attempts by the Russian school to derive the so-called cube root law from the diffusion model (Byakov, 1963 Nichiporov and Byakov, 1975), more rigorous treatments failed to obtain that (Kuppermann, 1961 Mozumder, 1977). In fact, it has been shown that in the limit of small concentration, the reduction of molecular yield by a scavenger should be given by a square root law in the orthodox... 

The diffusion layer model satisfactorily accounts for the dissolution rates of most pharmaceutical solids. Equation (43) has even been used to predict the dissolution rates of drugs in powder form by assuming approximate values of D (e.g., 10 5 cm2/sec), and h (e.g., 50 pm) and by deriving a mean value of A from the mean particle size of the powder [107,108]. However, as the particles dissolve, the wetted surface area, A, decreases in proportion to the 2/3 power of the volume of the powder. With this assumption, integration of Eq. (38) leads to the following relation, known as the Hixon-Crowell [109] cube root law ... 

Anderberg, E.K., and Nystrom, C. (1990), Physicochemical Aspects of Drug Release X. Investigation of the Applicability of the Cube Root Law for Characterization of the Dissolution Rate of Fine Particulate Materials, Intemat. J. of Pharma., 62, 143-151. 

Under conditions leading to a porous shell of magnetite, the kinetic curve displayed an induction period corresponding to formation of nuclei and the subsequent reaction followed the cube root law. Diffusion of the reducing gas to the reactant/ product interface took place readily with a porous product. Whether chemical or diffusion control predominated depended on reaction conditions. With small crystals... 

Figure 4. Recovery of the cube-root law activity coefficients calculated from polarized sphere model using coulombic and induced dipole terms.

Another way of expressing QD values is to state them as the cube root of the expl wt because certain detonation phenomena scale according to a cube root law. One of these is the instantaneous peak overpressure with distance (Ref 11). Damage can be related to overpressure by the cube root law except with respect to damage within inhabited structures and with respect to flying debris, for both of which a square root law is more nearly correct. 

The cube root law equation is obtained wlMgiis extremely small, and the negative two-thirds law is obtained in the case wh flti = Ms. 

For dissolution of solid particles, the Hixson-Crowell cube-root law (Eq. 5.3) assumes that the thickness of the diffusion layer h is constant during dissolution. However, this is not necessarily true. In addition, most drug particles are nonspherical and nonuniform in size. Therefore, very often the dissolution mechanism of solid drug particles is actually much more complicated. Nevertheless, the Hixson-Crowell cube-root law provides the first approximation to model powder dissolution. 

Solid Dissolution. The dissolution rate of a solid, whether it be a nondisintegrating compact or a powder, generally decreases with time because of the reduction in surface area as the dissolution proceeds. The familiar cube-root law for dissolution of solids was derived by Hixson and Crowell (1 on the basis of diffusion away from the surface of a spherically-shaped solid. The convex surface of a sphere decreases in area as solid mass is lost from the surface so that the dissolution rate decreases in proportion to the decrease in area until the solid is completely dissolved. By including shape factors, this model has been extended to describe the dissolution of various prismatic forms (2). As in the case of spherical particles, the dissolution rates decrease with time as the dissolution process progresses because of the decrease in area. 

Dissolution processes of multiparticulate systems where the specific surface area decreases during the dissolution, may be described by the Hixson and Crowell cube root law in Eq. (2) ... 

Fig. 1 Dissolution data of a hypothetical solid plotted as cumulative amount released (right axis, ) and after cube-root-law data treatment (left axis, ).

These test series and data are used to determine an additional explosion parameter—the maximum rate of pressure rise normalized to a volume of 1 m (KJ. The explosion parameter is expressed in the form of the cube root law ... 

The cube root law results from a combination of exponential growth with mass transport limitations in the solid phase, which is expressed in the concept of dp as shown by Pirt (1975). 

The rate of penetration of pits in aluminium has been shown to decrease rapidly with time. Aziz and Godard found that in field test coupons the pitting rate curve follows a cube root law [2.14] ... 

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