Vanishing fractions

Note that in the equilibrium limit PilPi dX 1 isotopic fractionations vanish but the elemental abundances in the gas will still be fractionated in proportion to their relative saturation vapor pressures. The degree of equilibrium elemental fractionation of the condensed phase will depend on the volume of gas being sufficiently large that a substantial fraction of the elements of interest are in the gas phase. 

If the denominator of this fraction vanishes, this solution is not valid. 

Finally, we always assumed implidtely that the gel is formed continuously, i.e. the gel fraction vanishes continuously at the gel point. In solutions, as a function of chemical potential the gel fraction may also jump to zero discontinuously if the system jumps over the miscibility gap. In the language of phase transition theory, this would be called a first-order transition, and we ignore its properties in this review which deals with continuous transitions for gels (cf. Chap. D.). 

Before attempting to answer this question, let us first summarize the procedure of section 11.3 in a slightly modified form. Equations (11.20) and (11.21) provide a set of simultaneous ordinary differential equations to determine the pressure and the composition, represented by mole fractions Xi,..,Xn in terms of the dummy variable. If at least one of the x s varies monotonically with X, so that its derivative never vanishes, we may use this x in place of X as an Independent variable. Without loss of generality this x may be labelled x, so we may divide equation (11.20) and each equation (11.21) for r = 2,...,n-l, by equation (11.21)... 

A comparison of the left-hand side (LHS) and the right-hand side (RHS) of Eq. (81) is given in Table IV. The comparison is made at three different values of x2, including the critical point. In order to assess their relative importance, values of all the individual terms in Eq. (81) are reported in the table. It is apparent that all the terms contribute significantly and that none may be neglected (except that In Kt must necessarily vanish at the critical mole fraction). 

In agreement with this expectation Sjogren (16) found that when bombarding C02 with Ne+ ions (RE 21.6 e.v.) of low velocity and at low pressure, vanishing fractions of 0+ ions were obtained. This result indicates that when using electron or photon impact, O + (4S) is formed at 19.1 e.v. after preionizing a highly excited triplet state of neutral C02. 

Rn, and the gel point, Figure 16 concerns the sol fraction and Figure 17 the number of EANC s per monomer. The differences are most pronounced near the gel points and they vanish with completion of the reaction. 

The weight fraction of o -mer calculated according to Eq. (36) for a trifunctional monomer is shown in Fig. 69 for several degrees of reaction. The curves resemble those for the polymers of an A—R—B2 monomer shown in Fig. 66, with the important difference that they do not vanish into the axis at a = ac. This is simply a consequence of the fact that condensation is far short of completion at the critical point. 

Commenting on above we should mention that initial expressions (1.59) - (1.63) are valid for disordered systems with exponentially broad spectrum of local values of electric conductivity. Due to existing dependence of 0 on over long times in our case the broad preadsorption spread in can grow narrow. At specific ratios between parameters of the absorbate-adsorbent system it can either vanish at all or there is a notable concentration of leveled-off barriers being formed with the fraction higher than the threshold one Xe- The straightforward analysis of each specific case characterized by a certain relationships between parameters of the system enables one easily obtain conditions... 

Figure 8.10 According to Darken s theory, the sign of the diffusion coefficient changes where the second derivative of the Gibbs function relative to the molar fraction Xt vanishes (spinodal).

An admixture of normal quark matter is found below // = 388.8 MeV. The fractions of the superconducting phases other than the 2SC phase then rapidly become smaller and vanish at fi = 388.6 MeV, while the fraction of normal matter strongly increases. Nevertheless the 2SC phase stays the dominant component for // > 360 MeV. At /i = 340.9 MeV we finally reach the vacuum. 

Obviously, vanishes, unless Up + Uq + rir + tig = 2. Sufficient though not necessary for Eq. (176) is that the Up are equal to 0,, or 1 that is, now open-shell states with fractional NSO occupation numbers are also possible. This implies via Eq. (174) that the only nonvanishing elements of >-2 are those... 

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