Factor cancellation

Since each term in the numerator and the denominator contains the factor (1 - p), this common factor cancels out ... 

When using a conversion factor, we treat the units just like algebraic quantities they are multiplied or canceled in the normal way. Thus, the units in the denominator of the conversion factor cancel the units in the original data, leaving the units in the numerator of the conversion factor. 

The permeability coefficients and molecular radii are known. The effective pore radius, R, is the only unknown and is readily calculated by successive approximation. Consequently, unknown parameters (i.e., porosity, tortuosity, path length, electrical factors) cancel, and the effective pore radius is calculated to be 12.0 1.9 A. Because the Renkin function [see Eq. (35)] is a rapidly decaying polynomial function of molecular radius, the estimation of R is more sensitive to small uncertainties in the calculated molecular radius values than it is to experimental variabilities in the permeability coefficients. The placement of the perme-ants within the molecular sieving function is shown in Figure 9 for the effective... 

Once the data have been transformed to wave vector form, they are usually multiplied by some power of k, typically, k2 or k3. Such a factor cancels the 1/k factor in Eq. (4) as well as the 1/k2 dependence of the backscattering amplitude at large values of k. Figure 9B depicts multiplication by k3. 

In other words, we simply introduce a factor in the microstate probabilities which is inversely proportional to the conventional macroscopic distribution. As a result, this factor cancels the integrated macroscopic probabilities and leaves the distribution constant - exactly the flat-histogram scenario of interest. 

Experiments 1-4 had the factor at the nominal level, A , while experiments 5-8 had the factor at the alternative level, a . Inspection of Table 4.8 shows that with this combination the effect of the other factors cancels out. The seven factors can be dealt with in a similar way by grouping the nominal level for that factor and subtracting the alternative level. Changing factor B to b is examined by calculating Ab, as follows ... 

The sample-dependent factor cancels out when a ratio of two similar intensity formulas (for two elements) is formed, leaving a simple formula connecting the ratio of measured intensities... 

Note that the factor/cancels out of this expression entirely, so that... 

We now refer to Appendix 1 to write the partition functions in terms of their translational, rotational, and vibrational components. Of the quantities appearing in the expressions for these components, only the molecular mass M, the moments of inertia /, the vibrational frequencies uu and the symmetry numbers a are different for the isotopic molecules all other factors cancel, leaving Equation A2.10. 

The second delta function on the RS of eq. (31) is zero unless K =K, when the exponential factors cancel and... 

Note Although this problem is given to us and worked in mg/mL, any volumetric concentration units could have been used in this problem, so long as the same units are used throughout because the conversion factors cancel. 

Now consider the contributions of Eq. (3.19) from intermolecular interactions. Comparing Eq. (3.17) and Eq. (3.18), this is seen to be the logarithm of a ratio of integrals. Simple proportionality factors cancel in forming the ratio. Then the denominator of that ratio is a partition function for the uncoupled N + 1)-molecule system, i.e., without interactions between the A -molecule solution and the distinguished molecule. The numerator is similarly proportional to the partition function for the physical N +1)-molecule system. We thus write... 

Strictly speaking, the result should be divided by the activity coefficient of the activated state for the reaction in any case this factor cancels out when equilibrium processes are considered. 

Multiply the given quantity by the conversion factor. Cancel units to verify that the units left are the ones you want for your answer. 

A nonlinear structure-reactivity correlation for the imidazole-catalyzed hydrolysis of a series of acetyl esters is shown in Kg. 10 (Kirsch and Jencks, 1964a). The cataljdic coefficient for the imidazole-catalyzed hydrolysis is plotted against the hydroxide term. In this way steric factors cancel out so that Aqh be used as a measure of the... 

Multiply the starting quantity and units by the conversion factor. Cancel the starting units with the units in the denominator.There are 400 cm in 4 m. 

It is interesting to note that the thermodynamic factors cancel out of Eqs. 8.8.19 and 8.8.20. The elimination of the thermodynamic factors will prove particularly useful in the estimation of transfer efficiencies in multicomponent distillation (Section 12.3). 

Still, as discussed in Section 2,8.1, normalization is in some sense still a useful concept even for such processes. As we saw in Section 2.8.1, we may think of an infinite system as a Q oo limit of a finite system of volume Q. Intuition suggests that a scattering process characterized by a short range potential should not depend on system size. On the other hand the normalization condition dx fl/(x)p = 1 implies that scattering wavefunctions will vanish everywhere like as Q CX3. We have noted (Section 2.8) that physically meaningful results are associated either with products such as A V<(x) 2 or yo V (x)p, where jV, the total number of particles, and p, the density of states, are both proportional to Q. Thus, for physical observables the volume factor cancels. 

The factor label method doesn t change the value of the physical quantity because you are multiplying that value by a factor that equals 1. You choose the factor so that when the unit you want to eliminate is multiplied by the factor, that unit and the similar unit in the factor cancel. If the unit you want to eliminate is in the numerator, choose the factor which has that unit in the denominator. Conversely, if the unit you want to eliminate is in the denominator, choose the factor which has that unit in the numerator. For example, in a chemistry lab activity, a student measured the mass and volume of a chunk of copper and calculated its density as 8.80 g/cm. Knowing that 1000 g = 1 kg and 100 cm = 1 m, the student could then use the following factor label method to express this value in the SI unit of density, kg/m. ... 

Both populations should exhibit a positive temperature dependence for the propagation rate but the more rigid sites may show a rate increase as the system density increases at lower temperature. It is possible that for the rigid sites these factors cancel. The previous observation that the center of the combined exotherm appears not to shift with temperature seems to confirm a general low or zero overall activation energy for chain propagation. 

Recall that the equations for the polarization involve ratios of currents or current densities, and therefore, the expressions are of the same form since the area factor cancels. Obviously, the numerical scale against which the plots are made will depend on the need to plot in terms of current or current density. )... 

This follows directly from the Hohenberg-Kohn theorem applied to a noninteracting system. The density n uniquely determines the Kohn-Sham potential vs (up to a constant) and therefore the also the orbitals (up to a phase factor) and eigenvalues (up to constant). The arbitrariness with respect to a constant shift and with respect to the phase factor cancels out in the energy expression and therefore the zth-order energy becomes a pure density functional. We therefore have the following series of... 

The renormalization factors cancel out, and for the physical quantities under consideration, perfectly regular expansions in powers of z,..., zN are obtained. They are given by... 

In this appendix, we deal with short-range divergences which, for d < 4, occur in the calculation of the diagrams contributing to the connected partition functions (... Slt... , S,y). The question is rather simple when there are only two-body interactions, and this case has been treated in Chapter 10, Sections 4.1.6 and 4.2.4. Then, to eliminate the divergences, we only have to renormalize the partition functions, and when we calculate the mean value of a physical quantity the renormalization factors cancel out. 

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