Proportion defined

We will begin our discussion of these recipes by examining two of the main principles on which they are based the conservation of mass and the law of definite proportions. The law of definite proportions defines our ingredient list and the conservation of mass tells us how big a cake weTl bake. 

Figure 5 Ca and Ti isotopic compositions in hibonite (Ca[AiMgTi]i20,9) inciusions dispiayed a iarge range from excesses in the most neutron-rich isotopes to deficits, whiie the other isotopes are ciose to soiar proportions (defined as deviations, 5 -Ca or 5 -Ti, in %o). The most anomaious grains are both from the Murchison meteorite, 13-13 and BB-5 which have iarge excesses and iarge depietions respectiveiy in Ca and =°Ti (insets). The Ca and °Ti anomaiies are cieariy correiated and are inferred to come from neutron-rich supernova ejecta.

Porter and Momoh have suggested an approximate but simple method of calculating the total vapor rate for a sequence of simple columns. Start by rewriting Eq. (5.3) with the reflux ratio R defined as a proportion relative to the minimum reflux ratio iimin (typically R/ min = 1-D- Defining Rp to be the ratio Eq. (5.3) becomes... 

In current practice, rationalized units are not used in IR the absorption bands have long been identified in terms of wavelengths, i. e., in micrometers. The general trend now is to express energy by a scale proportional to the frequency the wave number designated by v is defined as (... 

There are several criteria used to define solvent power. Chemical analysis is ideal because it can indicate the proportion of hydrocarbons known to be good solvents in particular, the aromatics. 

Waxes are less well defined aliphatic mixtures of n-alkanes, isoalkanes and cycloalkanes in various proportions. Their average molecular weights are higher than those of the paraffins from 600 to 800. 

If the DHT is evaluated numerically from its defining expression, the time taken for long data sequences of length N is proportional to as with the DFT, because for each value of v the are N evaluations of the product /(r)cas(2nvT/ N) and there are N values of v. 

Even if K is temperature dependent, the ooexistenoe ourve oan still be defined m tonus of a reduoed temperature = 2RT/K(T), although the reduoed temperature is then no longer direotly proportional to the temperature T. 

Iterative approaches, including time-dependent methods, are especially successfiil for very large-scale calculations because they generally involve the action of a very localized operator (the Hamiltonian) on a fiinction defined on a grid. The effort increases relatively mildly with the problem size, since it is proportional to the number of points used to describe the wavefiinction (and not to the cube of the number of basis sets, as is the case for methods involving matrix diagonalization). Present computational power allows calculations... 

For each combination of atoms i.j, k, and I, c is defined by Eq. (29), where X , y,. and Zj are the coordinates of atom j in Cartesian space defined in such a way that atom i is at position (0, 0, 0), atomj lies on the positive side of the x-axis, and atom k lies on the xy-plaiic and has a positive y-coordinate. On the right-hand side of Eq. (29), the numerator represents the volume of a rectangular prism with edges % , y ., and Zi, while the denominator is proportional to the surface of the same solid. If X . y ., or 2 has a very small absolute value, the set of four atoms is deviating only slightly from an achiral situation. This is reflected in c, which would then take a small absolute value the value of c is conformation-dependent because it is a function of the 3D atomic coordinates. 

When developing the dusty gas model flux relations in Chapter 3, the thermal diffusion contributions to the flux vectors, defined by equations (3.2), were omitted. The effect of retaining these terms is to augment the final flux relations (5.4) by terms proportional to the temperature gradient. Specifically, equations (5.4) are replaced by the following generalization... 

The state of the surface is now best considered in terms of distribution of site energies, each of the minima of the kind indicated in Fig. 1.7 being regarded as an adsorption site. The distribution function is defined as the number of sites for which the interaction potential lies between and (rpo + d o)> various forms of this function have been proposed from time to time. One might expect the form ofto fio derivable from measurements of the change in the heat of adsorption with the amount adsorbed. In practice the situation is complicated by the interaction of the adsorbed molecules with each other to an extent depending on their mean distance of separation, and also by the fact that the exact proportion of the different crystal faces exposed is usually unknown. It is rarely possible, therefore, to formulate the distribution function for a given solid except very approximately. 

Equation (2.39) is the weight average molecular weight as defined in Sec. 1.8. It is important to note that this result. My = M y, applies only in the case of nonentangled chains where 17 is directly proportional to M. A more general definition of My for the case where 17 a is... 

The various elastic and viscoelastic phenomena we discuss in this chapter will be developed in stages. We begin with the simplest the case of a sample that displays a purely elastic response when deformed by simple elongation. On the basis of Hooke s law, we expect that the force of deformation—the stress—and the distortion that results-the strain-will be directly proportional, at least for small deformations. In addition, the energy spent to produce the deformation is recoverable The material snaps back when the force is released. We are interested in the molecular origin of this property for polymeric materials but, before we can get to that, we need to define the variables more quantitatively. 

Figure 2.1 served as the basis for our initial analysis of viscosity, and we return to this representation now with the stipulation that the volume of fluid sandwiched between the two plates is a unit of volume. This unit is defined by a unit of contact area with the walls and a unit of separation between the two walls. Next we consider a shearing force acting on this cube of fluid to induce a unit velocity gradient. According to Eq. (2.6), the rate of energy dissipation per unit volume from viscous forces dW/dt is proportional to the square of the velocity gradient, with t]q (pure liquid, subscript 0) the factor of proportionality ... 

There is an intimate connection at the molecular level between diffusion and random flight statistics. The diffusing particle, after all, is displaced by random collisions with the surrounding solvent molecules, travels a short distance, experiences another collision which changes its direction, and so on. Such a zigzagged path is called Brownian motion when observed microscopically, describes diffusion when considered in terms of net displacement, and defines a three-dimensional random walk in statistical language. Accordingly, we propose to describe the net displacement of the solute in, say, the x direction as the result of a r -step random walk, in which the number of steps is directly proportional to time ... 

The sensitivity equation can be developed by differentiating equation 4 with respect to J. Since the signal is proportional to J and detection is defined as to when the measured signal to noise ratio equals 3, the gas detection sensitivity (ppm) in terms of the NEP for gas detection becomes ... 

The abihty of a fiber to absorb energy during straining is measured by the area under the stress—strain curve. Within the proportional limit, ie, the linear region, this property is defined as toughness or work of mpture. For acetate and triacetate the work of mpture is essentially the same at 0.022 N/tex (0.25 gf/den). This is higher than for cotton (0.010 N/tex = 0.113 gf/den), similar to rayon and wool, but less than for nylon (0.076 N/tex = 0.86 gf/den) and silk (0.072 N/tex = 0.81 gf/den) (3). 

Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

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