Symbols approximately equal

The Alexander model is based on two assumptions that enable simple expressions for these two terms (1) The concentration profile of the layer is step-like. That is, the monomer volume fraction within the layer, (p Na3/d2L, is constant, independent of position (2) The chains are uniformly stretched. That is, all chain ends are positioned on a single plane at a distance L from the surface. [In this paper, we use the symbol to mean approximately equal to or equal to within a numerical factor of order one we use to mean proportional to .] The first assumption simplifies the calculation of Fin, while the second yields a simple expression for Fel. 

Figure 5.21 Tafel plots for the system illustrated in Figure 5.20, where and denote data for the 3+/2+ and 2+/1+ redox couples, respectively. The supporting electrolyte is 0.2 M TBABF4 in acetonitrile the errors on the rate constants are approximately equal to the sizes of the symbols. Reprinted with permission from R. J. Forster and T. E. Keyes, /. Phys. Chem., B, 102,10004 (1998). Copyright (1998) American Chemical Society...

In order to address this problem the concept of an estimated standard deviation (abbreviated to e.s.d.), which is related to a described previously, was introduced. It is defined by an equation analogous to Equation 10.2 and differs from the standard deviation only in that, rather than calculating the difference between a measurement and the true value, the mean value has to be used, rather than the (unknown) true value. For a large number of measurements, the standard deviation and the e.s.d are approximately equal (unless there is a systematic error), and it is common to use the symbol [Pg.391]

Because the four spin states have different energies, they also have different populations. Beeause the spin states N3 and N2 have very similar energies, we can assume that their populations are approximately equal. Now use the symbol B to represent the equihbrium populations of these two spin states. The population of spin state Ni, however, will be higher (by an amount S), and the population of spin state N4 will be reduced (also by an amount S). The intensities of the NMR hnes will be proportional to the difference in populations between the energy levels where transitions are occurring. If we compare the populations of each energy level, we can see that the intensities of the two carbon lines (X) will be equal. 

At pH = p a = 4.7, there is a 1 1 mixture of yellow and blue species, which appears green. As a crude rule of thumb, we predict that the solution will appear yellow when [Y]/[B ] 10/1 and blue when [B]/[Y] 10/1. (The symbol means is approximately equal to or greater than. ) From Equation 9-4, we predict that the solution will be yellow when pH p nin 1 (= 3.7) and blue when pH p nin + 1 (= 5.7). By comparison, Table 9-3 lists bromocresol green as yellow below pH 3.8 and blue above pH 5.4. Between pH 3.8 and 5.4, various shades of green are seen. Demonstration 9-2 illustrates indicator color changes and Box 9-2 describes an everyday application of indicators. 

A relationship between different quantities that are proportional to each otha-, such as kilometers and hours in SO kilometers per hour, is sometimes called an equivalence. This equivalence would be expressed mathematically as 80 kilometers 1 hour. The symbol means is equivalent to, is approximately equal to, or is similar to (as in similar triangles). A conversion factor derived from an equivalence is vaUd only while the conditions of the equivalence are maintained. 

Let us use a control volume approach for the fluid in the boundary layer, and recognize Newton s law of viscosity. Where gradients or derivative relationships might apply, only the dimensional form is employed to form a relationship. Moreover, the precise formulation of the control volume momentum equation is not sought, but only its approximate functional form. From Equation (3.34), we write (with the symbol implying a dimensional equality) for a unit depth in the z direction... 

Note 3 The flexural modulus has been given the same symbol as the absolute modulus in uniaxial deformation as it becomes equal to that quantity in the limit of zero amplitudes of applied force and deformation. Under real experimental conditions it is often used as an approximation to i . ... 

The joule is a rather small unit of energy. It takes approximately 10,000 joules to raise the temperature of a coffee cup of water by 10°C. Often kilojoules (kJ) are used with 1 kJ equal to 1,000 J. Another popular unit is the calorie, abbreviated with a c. A calorie is the amount of heat necessary to raise one gram of water by 1°C. One calorie is equal to 4.18 joules. The physical unit of a calorie should not be confused with a food calorie. A food calorie, symbolized with a capital C, is equivalent to 1,000 calories. When you eat a piece of cake with 400 Calories, you have actually consumed 400,000 calories. 

As the second approximation, we assume that the matrix elements coupling [Pg.258]

At least we should realize that spoken equations are at best a shorthand approximation to the meaning a verbal description of the symbols. If we were to try to speak the physical meaning, it would be something like Newton s law tells us that the net vector force acting on a body of mass m is mathematically equal to the product of its mass and its vector acceleration. In a textbook, words like that would appear in the text near the equation, at least on the first appearance of the equation. 

The approximate quantum meehanieal deseription of proton states by linear eombination of these protomerie struetures has been ealled protomerism (symbol p) [323, 324], It seems to be applieable to hydrogen bond systems in whieh a proton transfer may oeeur between two potential minima of equal depth [323, 324],... 

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