Cube root

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... 

The hazard posed can be limited by maintaining a zone free of people and property around a storage area of explosive material. The minimum radius of the zone depends on the type and quantity of explosive, the extent and type of barrica ding, and the magnitude of loss that would be encountered if an explosive incident occurred. The maximum distance to which hazardous explosive effects propagate depends on the blast overpressure created, which as a first approximation is a function of the cube root of the explosive weight, W. This is termed the quantity distance and is defined as... 

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... 

Figure 1. Graph of the Reciprocal of the Diffusivity against the Product of the Cube Root of the Molar volume and the Square Root of the Molecular Weight...

It is seen that if the diffusivity is to be correlated with the molecular weight, then a knowledge of the density of the solute is also necessary. The result of the correlation of the reciprocal of the diffusivity of the 69 different compounds to the product of the cube root of the molecular volume and the square root of the molecular weight is shown in Figure 1. A summary of the errors involved is shown in Figures 2 and 3... 

The vast majority of protein/peptides will have densities lying between 0.85 an 1.25 and, thus, the cube root of their density will tend to unity. Consequently, the following approximation can be made. 

The JKR model predicts that the contact radius varies with the reciprocal of the cube root of the Young s modulus. As previously discussed, the 2/3 and — 1/3 power-law dependencies of the zero-load contact radius on particle radius and Young s modulus are characteristics of adhesion theories that assume elastic behavior. 

Fullwood and Erdman, 1983 circumvent this problem by comparing risk as cubes in which linear dimensions are the cube root of the volume/risk. Figure 1.4.3-5 compares the risks associated with nuclear fuel reprocessing, refabrication and waste disposal with nonnuclear risks. 

Kubik-gehalt, m. solid content, volume, -inhalt, m. cubic contents, -mass, n. cubic measure, -wurzel, /. cube root, -zahl, /. cube. 

Except in simple cases (square and cube roots) radical signs are replaced by fractional exponents. If n is odd,... 

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 ... 

Roots of exponentials are obtained in a similar way. To extract a square root, use the Vx key. For other roots, use they key. To obtain a cube root, enter the number, press they key, enter 1/3 (0.333333333) and read the answer ... 

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. 

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

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] ... 

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 ... 

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. 

Returning to my original fanciful example, the first thing to note is that reductive explanations may fail to be relevant or appropriate even when they do introduce factors that are at least preconditions for the phenomenon in question. (Perhaps there is always some kind of reductive explanation that does this much.) This possibility is obvious for evolutionary psychology. To take an extreme (in the absurdity of the evolutionary explanation) and notorious example, there is no doubt that men have evolved both the capacity to commit rape, and a disposition, under some circumstances, to do so. This much is true for any behaviour that actually occurs, although the disposition to play the sousaphone or to extract cube roots is something that requires extremely special circumstances. 

The film thickness is seen to be proportional to the cube root of the flow rate and the fluid viscosity. The shear stress exerted on the plate is... 

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 ... 

---