Numerical Constants

We will quote a numerical constant in some of these equations to help with actual calculations. The units can be very confusing because it is conventional to use non-SI units for several quantities. The wavenumber value, i>, is usually taken to be in cm The extinction coefficient is conveniently taken in units of 1 moH ... 

We begin by ignoring all purely numerical coefficients. In the final result we shall represent all such factors by the numerical constant K. 

The dimensional equations are usually expansions of the dimensionless expressions in which the terms are in more convenient units and in which all numerical factors are grouped together into a single numerical constant. In some instances, the combined physical properties are represented as a linear function of temperature, and the dimension equation resolves into an equation containing only one or two variables. 

The numerical constant may be adjusted to suit any particular set of data if one desires to use a certain criterion. However, surface conditions vary so greatly that deviations may be as large as 25 percent from results obtained. 

Power calculations for screw conveyors are well standardized. However, each manufaciurer has grouped numerical constants in a different fashion and assigned slightly different values on the basis of individual design variations. Thus, in comparing screw-conveyor... 

The numerical constants were obtained over the temperature range of 5°C to 45°C and a concentration range of 0 to 0.5 volume fraction of ethanol inn-hexane.The effect of temperature and solvent composition on solute retention can, again, be best displayed by the use of 3-D graphs, and curves relating both temperature and solvent composition to the retention volume of the (S) enantiomer of 4-benzyl-2-oxazolidinone are shown in Figure 23. Figure 23 shows that the volume fraction of ethanol in the solvent mixture has the major impact on solute retention. 

In considering the older literature of the subject, I have not thought it necessary to give detailed references, since these will be found in any of the standard text-books of physics. The space thus made available has been utilised in an attempt to explain in greater detail some points which usually offer difficulties to the student. The same applies to the numerical constants, which are to be found in the tables of Landolt-Bornstein, and at present are also being actively revised. 

By using the numerical constants obtained experimentally the following relations between the different rate coefficients are found... 

S where the numerical constant 3 is the m-th root of — the Bessel function of the fir t kind of order zero,... 

We introduce now two unknown numerical constants a and P and we try to check that the two conditions... 

In the case of binary solutions, Eq. (7.49) coincides with the empirical Eq. (7.14), both in form and in the value of the numerical constant k. Therefore, the empirical square-root law can be explained quantitatively on the basis of the theory of ion-ion interaction. 

The next step is to evaluate the numerical constants axi and bxi- In order to accomplish these evaluations, we must first investigate some mathematical properties of the eigenfunctions Sxi(p). 

To evaluate the numerical constant axi, which is defined in equation (6.32), we square both sides of (6.32), multiply through by p, and integrate with respect to p to obtain... 

The radius rs is sometimes called the Wigner-Seitz radius and can be interpreted to a first approximation as the average distance between two electrons in the particular system. Regions of high density are characterized by small values of rs and vice versa. From standard electrostatics it is known that the potential of a uniformly charged sphere with radius rs is proportional to l/rs, or, equivalently, to p( r,)17 3. Hence, we arrive at the following approximate expression for Ex (Cx is a numerical constant),... 

C is a numerical constant and includes the degeneracy of the vibration a. Recall from... 

Hubble constant The numerical constant, H, that relates the red shift of a galaxy to its... 

Table 3.10. Numerical constants for Eq. (3.75c) rA denotes the row of atom A and rL the row of ligand I. in the periodic table...

Here the Ac is the channel cross-section flow area, and the factor, fRe, is a numerical constant computed and tabulated for various channel geometries. The characteristic dimension has been replaced by a hydraulic diameter defined as four times the flow area divided by the channel perimeter. 

The International Units, some useful Conversion Factors, and numerical Constants... 

The constant in Equation (5.112) cannot be readily evaluated using scaling theory. Our transformation applies equally well to the radius of gyration or the root mean square end-to-end length, only the numerical constant changes. We would like to be able to apply this idea to the role of concentration in semi-dilute and concentrated polymer regimes. In order to do this we need to define a new parameter s, the number of links or segments per unit volume ... 

So by equating the two network moduli we can obtain our tube parameters. If we include all the necessary numerical constants we get... 

Valency of an ion effective number of electrons per ion Angle on a crystal surface numerical constant an angle Contact angle at the edge of a solid Surface tension of a liquid Critical surface tension"... 

Synowietz, C. in "Landolt — Bornstein Numerical Constants and Functions, New Series, Group IV, Macroscopic and Technical Properties of Matter, Vol. 1, Densities of Liquid Systems and their Heat Capacities, Part b" Springer Verlag Berlin, 1977, pp. 69-70. 

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