Canceling proportions

To conclude this section let us note that already, with this very simple model, we find a variety of behaviors. There is a clear effect of the asymmetry of the ions. We have obtained a simple description of the role of the major constituents of the phenomena—coulombic interaction, ideal entropy, and specific interaction. In the Lie group invariant (78) Coulombic attraction leads to the term -cr /2. Ideal entropy yields a contribution proportional to the kinetic pressure 2 g +g ) and the specific part yields a contribution which retains the bilinear form a g +a g g + a g. At high charge densities the asymptotic behavior is determined by the opposition of the coulombic and specific non-coulombic contributions. At low charge densities the entropic contribution is important and, in the case of a totally symmetric electrolyte, the effect of the specific non-coulombic interaction is cancelled so that the behavior of the system is determined by coulombic and entropic contributions. 

For forbidden transitions in atoms and molecules this phenomenon may be experimentally observed in spectra induced by collisions. As is known, the selection rules on some transitions may be cancelled during collision. The perturbers are able to induce a dipole moment of transition having the opposite direction in successive collisions due to intercollisional correlation. Owing to this, the induced spectra do involve the gap (Fig. 1.7), the width of the latter being proportional to the gas density [46, 47], Theorists consider intercollisional correlation to be responsible for the above phenomenon [48, 49, 50]. 

The original bend and polish idea is due to Bernard Schmidt who built what has become known as a Schmidt telescope with a spherical primary and an aspheric corrector plate that introduces exactly the right amount of spherical aberration to cancel the spherical aberration that would be introduced by the spherical primary (Schmidt). The corrector has to be thicker at the edge in proportion to the radial distance in the aperture to the fourth power to provide the needed correction. 

The addition of trichloro- ortetrachloroethylene to aluminium components in dry cleaning equipments is responsible for many accidents. The effect of the carbon tetrachloride/methanol mixture in the 1/9 proportion of aluminium, magnesium or zinc causes the dissolution of these metals, whose exothermicity makes the interaction dangerous. There is a period of induction with zinc, which is cancelled out when copper dichloride, mercury dichloride or chromium tribromide is present. 

The proportionality between intensity and pressure is intuitively obvious, since at constant speed the number of collisions must be proportional to the number of molecules in the container and therefore proportional to the pressure. The inverse square root temperature dependence is a result of a partial cancellation... 

The molar rotation of the dendrimers 49-51 is proportional to the excess of (R,R)- or (S,S)-threitol units. This means that the chiroptical effects of threitol building blocks of opposite chirality cancel out each other. For the homo dendrimers an average positive molar optical rotation value of 146 for each (i, k)-threitol unit was calculated whereas a value of -185 resulted for each... 

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. 

In the binary-electrolyte experiments carried out at large, constant cell potentials, the cell current is ohmically limited. If the conductivity of the solution is proportional to the concentration of electrolyte, the current density at a given overpotential is then proportional to Cb. Under this regime, the concentration cancels out of Eq. (2.3), and the velocity is proportional to the applied potential. For this special case, the velocity can be expressed in terms of the anion drift velocity [27, 28]. For a binary solution, this is equivalent to replacing (1 — t+) by t and i by the ohmically limited current density. 

The first and third order terms in odd powers of the applied electric field are present for all materials. In the second order term, a polarization is induced proportional to the square of the applied electric field, and the. nonlinear second order optical susceptibility must, therefore, vanish in crystals that possess a center of symmetry. In addition to the noncentrosymmetric structure, efficient second harmonic generation requires crystals to possess propagation directions where the crystal birefringence cancels the natural dispersion leading to phase matching. 

The advantages described above of using waveguiding structures can cancel out if the field overlap integral of the interacting modes is small. This arises since the SHG efficiency is proportional to it. 

By contrast, in the CD spectrum of the p derivative (39), no bands were observed even at —80°C, as shown in Figure 4.19b. This is because the Cotton band of the p derivative is canceled by an oppositely signed CD band originating in an equivalent proportion of opposite screw sense helical segments (which may be referred to as masked helicity or masked circular dichro-ism ). The difference between the helical conformations of m and p derivatives may be due to the degree of steric hindrance of (S>2-methylbutoxyphenyl rings, since the p derivative may have enough space to freely rotate the... 

For lipid bilayers, equation (4) can be simplified. Above we have seen that the flat unsupported bilayer is without tension, i.e. y(0, 0) = 0, and therefore the first two terms must cancel y0 = — kcj. As argued above, JQ = 0, and thus also the third term drops out. The remaining two terms are proportional to the curvature to the power two. For a cylindrical geometry only, the term proportional to J2 is present. For spherical vesicles, the two combine into one ( kc + k)J2. The curvature energy of a homogeneously curved bilayer is found by integrating the surface tension over the available area ... 

In tliis integral only the last term is proportional to (xf - xq) and it cancels precisely the contribution of eq, (36). 

But p decreases with salt concentration with an apparent exponent of k which changes from 0 at low salt concentration to — at high salt concentrations. The N-independence of p arises from a cancellation between hydrodynamic interaction and electrostatic coupling between the polyelectrolyte and other ions in the solution. It is to be noted that the self-translational diffusion coefficient D is proportional to as in the Zimm model with full... 

N Xi t) + N2X2 t) such that N C = 0, N = [A a NbY This allows w t) to be fully independent of the original state (which is possible because of the exact cancellation of the matrix X t) in (19)). Then, N may be used to cancel the non-linear terms in the w t) dynamics and N may be used to rebuilding the unmeasured state exactly as in (4). Furthermore, in such a case, a matrix A i only left invertible is sufficient. Of course, this allows a more flexible design of the asymptotic observer. Nevertheless, in a more general case, when A t) is not proportional to the identity matrix, any... 

A mixture containing two enantiomers in equal proportions will have zero optical rotation, as the rotation due to one isomer will be cancelled by the rotation due to the other isomer. Such a mixture is known as racemic mixture or racemic modification. A racemic mixture is represented by prefixing dl or (+) before the name, for example ( ) butan-2-ol. The process of conversion of enantiomer into a racemic mixture is known as racemisation. 

An eqnimolar mixtnre of enantiomers is optically inactive, since the individnal effects from the two types of molecnle are cancelled ont. This mixtnre is called a racemic mixture or racemate, and can be referred to as the ( )-form. A mixture of enantiomers in rmeqnal proportions has a rotation nnmerically less than that of either enantiomer this measinement... 

A racemic mixture contains equal amounts of the two enantiomeric forms of the compound and has an optical rotation of zero the optical rotations arising from each of the two types of molecule are cancelled out. It follows that a mixture of enantiomers in unequal proportions will have a rotation that is numerically less than that of an enantiomer. Here, we see how to use the measured optical activity to determine the proportions of each enantiomer in the mixture, and therefore its optical purity. Optical purity is a measure of the excess of one enantiomer over the other in a sample of a compound. 

It is worth noting that the effect of temperature on ionic and polyethoxy-lated nonionic surfactants is just opposite. As temperature increases, the nonionics become more lipophilic whereas the ionics turn more hydrophilic. By mixing the two types of surfactants in a proper proportion, these effects could cancel each other out, and the mixture is said to be insensitive to temperature. This interesting feature of ionic-nonionic surfactant mixtures may be considered as a synergy, since it could be very important in practice. Analysis of this feature is not included here, because plenty of information may be found in the literature on applications of such mixtures to equihbrated and emulsified systems [10,71-74]. 

As we have seen earlier, even third-order reactions can be reduced to pseudo-first-order reactions by keeping the concentrations of all species except A constant and in great excess compared to A. This technique of using pseudo-first-order conditions is by far the most common technique for determining rate constants. Not only does it require monitoring only one species, A, as a function of time, but even absolute concentrations of A need not be measured. Because the ratio [A]/[A]0 appears in Eq. (T), the measurement of any parameter that is proportional to the concentration of A will suffice in determining k l7, since the proportionality constant between the parameter and [A] cancels out in Eq. (T). For example, if A absorbs light in a convenient... 

The mechanical properties of cancellous bone are dependent upon the bone density and porosity, and the strength and modulus are therefore much lower than those for cortical bone. The axial and compressive strength are proportional to the square of the bone density, and moduli can range from 1 to 3 GPa. 

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