Proportionality factor

So far, the themiodynamic temperature T has appeared only as an integrating denominator, a fiinction of the empirical temperature 0. One now can show that T is, except for an arbitrary proportionality factor, the same as the empirical ideal-gas temperature 0jg introduced earlier. Equation (A2.1.15) can be rewritten in the fomi... 

This expression shows diat if die detuning Acuj is negative (i.e. red detuned from resonance), dieii die cooling force will oppose die motion and be proportional to die atomic velocity. The one-diniensional motion of die atom, subject to an opposing force proportional to its velocity, is described by a damped haniionic oscillator. The Doppler damping or friction coefficient is die proportionality factor. 

Using the molecular weight of the repeat unit Mq as the proportionality factor between n and M, we write... 

Apart from some numerical coefficients, the details of which we shall forgo, the required proportionality factor involves kT and the concentration of entanglement points. Multiplying Eq. (2.67) by these factors, we obtain... 

Note that Eqs. (6.5) and (6.12) are both first-order rate laws, although the physical significance of the proportionality factors is quite different in the two cases. The rate constants shown in Eqs. (6.5) and (6.6) show a temperature dependence described by the Arrhenius equation ... 

Another parameter that plays an important role in unifying viscosity, diffusion, and sedimentation is the friction factor. This proportionality factor between velocity and the force of frictional resistance was introduced in Chap. 2, and its role in interrelating the topics of this chapter is reflected in the title of the chapter. 

This shows that Schlieren optics provide a means for directly monitoring concentration gradients. The value of the diffusion coefficient which is consistent with the variation of dn/dx with x and t can be determined from the normal distribution function. Methods that avoid the difficulty associated with locating the inflection point have been developed, and it can be shown that the area under a Schlieren peak divided by its maximum height equals (47rDt). Since there are no unknown proportionality factors in this expression, D can be determined from Schlieren spectra measured at known times. 

Coulomb s law. This relationship poses no particular difficulties as a qualitative statement the problem arises when we attempt to calculate something with it, since the proportionality constant depends on the choice of units. In the cgs system of units, the electrostatic unit of charge is defined to produce a force of 1 dyne when two such charges are separated by a distance of 1 cm. In the cgs system the proportionality factor in Coulomb s law is unity and is dimensionless. For charges under vacuum we write... 

The fact that the proportionality factor in Eq. (10.101) is numerically equal to unity and is dimensionless makes us tend to forget that any such factor is needed. 

Thus when an electric field is appHed to a soHd material the mobile charge carriers are accelerated to an average drift velocity v, which, under steady-state conditions, is proportional to the field strength. The proportionality factor is defined as the mobility, = v/E. An absolute mobility defined as the velocity pet unit driving force acting on the particle, is given as ... 

In contrast to other spectroscopies, such as IR/Raman or VIS/UV, NMR spectroscopy is inherendy quantitadve. This means that for a given nucleus the proportionality factor relating the area of a signal to the number of nuclei giving rise to the signal is not at all sample-dependent. For this reason, NMR spectroscopy has been used extensively for absolute and relative quantitadon experiments, using chemically well-defined model compounds as standards. 

Ix is the background-corrected net intensity of the principal peak of analyte X, Kx a proportionality factor for the absolute sensitivity of the standard reference, e. g. an Ni plate, and c the concentration of X. Multielement analyses are based on known relative sensitivities S ... 

The simplest mode of IGC is the infinite dilution mode , effected when the adsorbing species is present at very low concentration in a non-adsorbing carrier gas. Under such conditions, the adsorption may be assumed to be sub-monolayer, and if one assumes in addition that the surface is energetically homogeneous with respect to the adsorption (often an acceptable assumption for dispersion-force-only adsorbates), the isotherm will be linear (Henry s Law), i.e. the amount adsorbed will be linearly dependent on the partial saturation of the gas. The proportionality factor is the adsorption equilibrium constant, which is the ratio of the volume of gas adsorbed per unit area of solid to its relative saturation in the carrier. The quantity measured experimentally is the relative retention volume, Vn, for a gas sample injected into the column. It is the volume of carrier gas required to completely elute the sample, relative to the amount required to elute a non-adsorbing probe, i.e. 

The proportionality factor /i, is the dynamic viscosity of the fluid, its units being force X time/length and is expressed as N s rrr or Pa s. 

In the first approach, a vapor cloud s potential explosive power is proportionally related to the total quantity of fuel present in the cloud, whether or not it is within flammable limits. This approach is the basis of conventional TNT-equivalency methods, in which the explosive power of a vapor cloud is expressed as an energetically equivalent charge of TNT located in the cloud s center. The value of the proportionality factor, that is, TNT equivalency, is deduced from damage patterns observed in a large number of vapor cloud explosion incidents. Consequently, vapor cloud explosion-blast hazard assessment on the basis of TNT equivalency may have limited utility. 

Conventional TNT-equivalency methods state a proportional relationship between the total quantity of flammable material released or present in the cloud (whether or not mixed within flammability limits) and an equivalent weight of TNT expressing the cloud s explosive power. The value of the proportionality factor—called TNT equivalency, yield factor, or efficiency factor—is directly deduced from damage patterns observed in a large number of major vapor cloud explosion incidents. Over the years, many authorities and companies have developed their own practices for estimating the quantity of flammable material in a cloud, as well as for prescribing values for equivalency, or yield factor. Hence, a survey of the literature reveals a variety of methods. 

The number of structurally isomeric hydrocarbons with formula n 2n+2-2At o symptoticdlly proportional to p-" (3M-5)/2 paraffins p is equal to zero, that is, the number is asymptotically proportional to p. For m 1, the proportionality factor is 1/4. 

These compounds are really "i-fold substituted paraffins" the radicals have to be different from each other and from alkyls. In case of i 0 and i 1, the mentioned result gives the asymptotic behavior of p and R respectively. The proportionality factor is of the form X, where L and X are independent of i. 

The number of isomeric homologues of benzene with formula Cg n 6-f2n asymptotically proportional to the number of isomeric alcohols with proportionality factor [r(p ) + (p)r(p ) J/2. 

Similarly, the increase in the number of isomers in other homologous series (e.g., in the series starting with naphthalene and anthazene) is asymptotically proportional to the number of isomers of the alcohol series. The proportionality factor can easily be derived from the cycle index of the permutation group of the replaceable bonds of the basic compound. 

Reactor productivity was obtained by dividing final ethanol concentration with respect to sugar concentration at a fixed retention time. It was found that the rates of 1.3, 2.3 and 2.8 g 1 1 h 1 for 25, 35 and 50 g 1 1 glucose concentrations were optimal. Ethanol productivities with various substrate concentrations were linearly dependent on retention time (Figure 8.12). The proportionality factor may have increased while the substrate... 

Kfne = Proportionality factor in Stokes-Cunningham correction factor, dimensionless k = Constant for wire mesh separators 1 = Wire mesh thickness, ft L = Length of vessel from hydrocarbon inlet to hydrocarbon outlet, or length of decanter, ft L[ = Liquid entering Webre separator, lbs pel- minute per square foot of inlet pipe cross-section L, = EnLrainment from Webre unit, lb liquid per minute per square foot of inlet pipe cross section... 

In this equation, % is a proportionality factor known as the bead-solvent friction coefficient which purports to account in some kind of average way for the complex molecular interactions as the polymer segments (schematized by the bead) move about in the solvent. Following Stokes law of drag resistance, this friction coefficient is usually given as = 67trisa, with a equal to the bead radius. 

Two units which have never been popular in the last two systems of units (Sections 1.2.3 and 1.2.4) are the poundal (for force) and the slug (for mass). As a result, many writers, particularly in America, use both the pound mass and pound force as basic units in the same equation because they are the units in common use. This is an essentially incoherent system and requires great care in its use. In this system a proportionality factor between force and mass is defined as gc given by ... 

R is the shear stress in the fluid and divelocity gradient or the rate of shear. It may be noted that R corresponds to r used by many authors to denote shear stress similarly, shear rate may be denoted by either dw,/dy or y. The proportionality sign may be replaced by the introduction of the proportionality factor n, which is the coefficient of viscosity, to give ... 

The relative reflected microwave power is (S is a proportionality factor) ... 

---