Cube root equation

For small amounts of powder, dissolution of the particulate material can often be assessed (and compared with that of other compounds) by placing the powder in a calorimeter [68] and measuring the heat evolved as a function of time. The surface area must be assessed microscopically (or by image analyzer), and the data must be plotted by a cube root equation [39] ... 

Dissolution time, tdi (for powder) Particle mass, m Mass transfer coefficient, k Solubility, S Particle radius, r Density, p Hixson-Crowell (cube root) equation t 1 — (tti 1 m0 )1/3 (kS/prJ... 

Formulation First order (equation (2a)) Square root (equation (4a)) Cube root (equation (3a)) ... 

Most often, the model used to describe the growth of pellets is the cube root equation (cf. Metz and Kossen, 1977 Pirt, 1975 Trinci, 1970). Considering the cell mass to be related to the pellet radius, R, and the number of peUets, N, according to Equ. 5.256... 

Beattie et al, 1972 Goldberg and Beattie, 1972 Moore, 1973 Addis and Moore, 1974 Moore et al, 1977). We (Moore et al, 1977) were also able to demonstrate that such a relationship between blood and water lead concentrations was nonlinear, following a cube root equation of the type ... 

Slater was one of the first to propose that one replace V m equation A 1.3.18 by a tenn that depends only on the cube root of the charge density [T7,18 and 19]. In analogy to equation A1.3.32, he suggested that V be replaced by... 

To include the volume as a dynamic variable, the equations of motion are determined in the analysis of a system in which the positions and momenta of all particles are scaled by a factor proportional to the cube root of the volume of the system. Andersen [23] originally proposed a method for constant-pressure MD that involves coupling the system to an external variable, V, the volume of the simulation box. This coupling mimics the action of a piston on a real system. The piston has a mass [which has units of (mass)(length) ]. From the Fagrangian for this extended system, the equations of motion for the particles and the volume of the cube are... 

K value is the ratio of the cube root of a boiling temperature to gravity. There are two widely used methods to calculate the K factor K, and the K, p. The equations used for calculating both factors are as follows ... 

In both equations, k and k are proportionality constants and 0 is a constant known as the critical exponent. Experimental measurements have shown that 0 has the same value for both equations and for all gases. Analytic8 equations of state, such as the Van der Waals equation, predict that 0 should have a value of i. Careful experimental measurement, however, gives a value of 0 = 0.32 0.01.h Thus, near the critical point, p or Vm varies more nearly as the cube root of temperature than as the square root predicted from classical equations of state. 

Equation (1) predicts that the rate of release can be constant only if the following parameters are constant (a) surface area, (b) diffusion coefficient, (c) diffusion layer thickness, and (d) concentration difference. These parameters, however, are not easily maintained constant, especially surface area. For spherical particles, the change in surface area can be related to the weight of the particle that is, under the assumption of sink conditions, Eq. (1) can be rewritten as the cube-root dissolution equation ... 

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. 

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 ... 

A more direct link with molecular volumes holds for alkali halides, because the lattice energy (IT) is inversely proportional to interatomic distance or the cube root of molecular volume (MV). The latter has been approximated by a logarithmic function which gives a superior data fit. Plots of AH against log(MV) are linear for alkali halides 37a). Presumably, U and AH can be equated because AH M, ) is a constant in a series, and AH (halide )) is approximately constant when the anion is referred to the dihalogen as the standard state. 

Amperometric detectors can operate over a range of conversion efficiencies from nearly 0% to nearly 100%. From a mathematical point of view, a classical amperometric determination (conversion of analyte is negligible) is one where the current output is dependent on the cube root of the linear velocity across the electrode surface as described by Levich s hydrodynamic equations for laminar flow. Conversely, the current response for a cell with 100% conversion is directly proportional to the velocity of the flowing solution. While the mathematics describing intermediate cases is quite interesting, it is beyond the scope of this chapter. 

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

L describes lightness and extends from 0 (black) to 100 (white). The a coordinate represents the redness-greenness of the sample. The b coordinate represents the yellowness-blueness. The coordinates a and b have a range of approximately [—100,100]. Notice the cube root in the above equation. The cube root was introduced in order to obtain a homogeneous, isotropic color solid (Glasser et al. 1958). Visual observations correspond closely to color differences calculated in this uniform coordinate system. A transformation of the sRGB color cube to the L a b color space is shown in Figure 5.5. 

An important conclusion to be drawn from the Stokes-Einstein equation is that the diffusion coefficient of solutes in a liquid only changes slowly with molecular weight, because the diffusion coefficient is proportional to the reciprocal of the radius, which in turn is approximately proportional to the cube root of the molecular weight. 

The Ferry-Renkin equation can be used to estimate the pore size of ultrafiltration membranes from the membrane s rejection of a solute of known radius. The rejections of globular proteins by four typical ultrafiltration membranes plotted against the cube root of the protein molecular weight (an approximate measure of the molecular radius) are shown in Figure 2.33(a). The theoretical curves... 

Equation (3.58) and Equation (3.61) are the Hixson and Crowell cube-root and the Higuchi and Hiestand two-thirds-root expressions, respectively. The cube-root and the two-thirds-root expressions are approximate solutions to the diffusional boundary layer model. The cube-root expression is valid for a system where the thickness of the diffusional boundary layer is much less than the particle radius whereas the two-thirds-root expression is useful when the thickness of the boundary layer is much larger than the particle radius. In general, Equation (3.57) is more accurate when the thickness of the boundary layer and the particle size are comparable. 

The molar volume in these equations is difficult to assign. This was found to be a problem in the case of a polar liquid. Recently Roe (29) pointed out that, in the case of polymeric liquids, the thickness of the transition layer depends not only on the size of the repeat unit but also on the degree of correlation between successive structural units, or, in other words, on the flexibility of the polymer chain. It is, therefore, not appropriate to use the cube root of the molar volume as a measure of the thickness of the monomolecular layer at the vapor-liquid interface. 

Quite a number of empirical equations describe the concentration dependence of the relative fluidity of dispersions (l/r)rei). Such equations are also to be found in a very early work by Hatschek [2], who related the fluidity to the cube root of the volume fraction of the disperse phase ... 

When the elimination half-life drops to 1 hour, as is characteristic of the rates encountered with nerve growth factor or stabilized analogs of substance P peptide or glucocerebrosidase enzyme, the treatment volume decreases to 2.7 cm, with a penetration distance of 0.9 cm. In a rapid metabolism situation, when the elimination half-life decreases to just 10 minutes, as expected for substances such as native somatostatin, enkephalin, and substance P, the treatment volume diminishes to only 0.5 cm. However, the penetration distance is still 0.5 cm and still in excess of the penetration distances encountered with modes of delivery depending on diffusional transport across tissue interfaces. Finally, it should be noted that these penetration distances, computed here for a volumetric infusion rate of 3 pL/min, will decrease with decreases in the flow rate only as the cube root of the reduction factor (cf. Equation 9.24). For example, there will be only a 30% decrease in penetration distance for a 3-fold drop in flow rate to 1 pL/ min. 

The figures in the American Table of Distances developed accdg to Marshall after the tests conducted, beginning 1910, did not agree very well with.the above equation. For small quantities of expls the distances were more nearly proportional to the cube root of the wt of expl, while for very large quantities the variation of the distance was more nearly proportional to the straight increase of the weight... 

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