X terms

Here, x denotes film thickness and x is that corresponding to F . An equation similar to Eq. X-42 is given by Zorin et al. [188]. Also, film pressure may be estimated from potential changes [189]. Equation X-43 has been used to calculate contact angles in dilute electrolyte solutions on quartz results are in accord with DLVO theory (see Section VI-4B) [190]. Finally, the x term may be especially important in the case of liquid-liquid-solid systems [191]. 

Referring back to the set of simultaneous equations at the beginning of this appendix, the objective is usually to solve these for the unknown x terms, lliis is where the use of matrices has a major advantage because referring to equation [E.l] we may rewrite this as... 

This equation expresses the solution to the set of simultaneous equations in that each of the unknown x terms is now given by a new matrix [A] multiplied by the known y terms. The new matrix is called the inverse of matrix [A]. The determination of the terms in the inverse matrix is beyond the scope of this brief introduction. Suffice to say that it may be obtained very quickly on a computer and hence the solution to a set of simultaneous equations is determined quickly using equation [E.4],... 

According to Eq. (27), if a metal possessed the same value of X as Hg, its point would fall on the straight line. The plot shows that all other metals lie on the left of the straight line in terms of Eq. (27) all of them have an X term more negative than that of Hg. In other words, there are no metals on whose surface the contact with water produces a lower 4> drop than on Hg (or a positive A ). 

The denotes molar, rather than weight fraction, partition coefficient. The X terms denote the cation molar fractions of Ca, Al and Mg, while the X terms denote the molar fraction of diopside and enstatite components in clinopyroxene. Note that this expression contains no explicit P-f dependence. 

The value of x decreases as the proportion of contaminant increases. Since the logarithm of a number less than one is always negative, we see how the RT In x, term on the right-hand side of Equation (5.12) is zero for a pure material (implying p, = pf ). At all other times, x, < 1, causing the term RT In x, to be negative. In other words, the value of p will always decrease from a maximum value of pf as the amount of contaminant increases. 

Under this assumption the R x term is converted to a set of constraints and our original problem is converted to the following constrained optimization ... 

We now determine the system parameters by evaluating Eq. (64). First, although it is not necessary to limit our considerations to the saturated-surface, zero-order case, we do so to simplify the analysis of high-conversion systems. [We earlier assumed in connection with Eq. (61) that the surface is saturated and that a is constant.] Equation (62) indicates that the total rate is zero order when (kj -I- k3 -I- ks) is small in comparison to the rest of the denominator. Thus, since b = k2 + k + k )/L, h = 0 in the zero-order case, and the (b/P x) term can be removed from Eq. (64). 

In the resulting equation the experimentally determined quantities are WjU and /, and for our purpose we take the unknown quantities to be (x/u), (z/x), and (y/x). We can then obtain three equations of the form of Eq. (64) [with the (bJP x) term omitted], one equation for each of the three different pairs of WjU) and / values. With the subscripts 1, 2, and 3 used for the three values of both (W/t/) and /, these three equations are solved simultaneously to give... 

The self-energy Zee >s divergent, but there is a canceling divergence in the x term, so that the net result for (V) is finite. 

Eq.( 13) and Eq.( 15) can be split into two terms, one from the t term in the t - x factor, and the other from the X term. Since the X integration is expected to converge very rapidly, the transition probability has the form ... 

By plotting 1/AOD as a function of [Q], the ratio of fcq/ y[Y ] can be derived. This ratio corresoponds to the k x term of the Stern-Volmer equation. Here r is the lifetime of the carbene in the absence of quenchers. 

What are the coefficients of the x terms which will give the best fit to the observed values of y ... 

The inner region. In this region, located around x = 0, the exponentials in (5.5.17) are negligible, compared with the N(x) term. Thus to the leading order in A in this region... 

The contributions from the short-range repulsive potential and the long-range attractive potential are shown explicitly in Fig. 12.6. Also shown are the full Stockmayer potential for three different orientations of the dipole moments. The curve listed as no dipole is for orientation angles ft = Gj = rjr — n/2. In this case the x term in Eq. 12.9 is zero. The potential has a minimum at r, - = 21/6a,y, with an attractive well depth of e,y. The curve listed as attractive dipole has orientation angles ft = 6j = 0. Thus x = 2, and this orientation has the maximum (attractive) contribution from the dipole-dipole term. The well depth in this case is almost a factor of 6 deeper due to the dipole interaction. The repulsive... 

If the tc term in Foltmann s equation is replaced by r, the intercept of the plot, x, approaches zero. This indicates that the difference between t and tc is not a lag time between enzymic action and aggregation, but is an artifact of the measurement method. The x term added to Storch and Segelcke s equation by Holter is not necessary if r is used instead of tc (Collinge and Brown 1986). 

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