Stability curve

A serious point is the neglect of surface tension and anisotropy in these derivations. In the experiments analyzed so far the relation VX const, seems to hold approximately, but what happens when the capillary anisotropy e goes to zero Numerically, tip-splitting occurs at lower velocities for smaller e. Most likely in a system with anisotropy e = 0 (and zero kinetic coefficient) the structures show seaweed patterns at velocities where the diffusion length is smaller than the short wavelength hmit of the neutral stability curve, as discussed in Sec. V B. 

Fig. 6 Solubility effects on drug stability curve A, drug formulated as lOmg/mL solution (q/2 = 1 year) curve B, drug formulated as a suspension with a saturated solubility of 1 mg/mL (ti/2 = 7.3 years). 

I should also point out that 25V rated capacitors from Murata have a much better voltage stability curve—only 5 to 10% fall at maximum rated voltage. That is actually close to the voltage stability figure I used in my worst-case calculations previously. 

Fig. 2. CH4 hydrate stability curves (laboratory data) showing CH4 hydrate dissociation paths (schematic) in the depressurization method.

Fig. 3. C02 hydrates stability curves (lab data) showing C02 hydrate formation paths (schematic) by injection of C02 gas.

Fig. 4. CH4 - and CO2 hydrates stability curves showing C02 enhanced CH4 hydrate dissociation zone.

A short stability curve is not necessarily a reason to reject a particular column. If the unique selectivity of a given column is essential for achieving a particular analytical separation, it can be used so long as its performance is validated to persist for a specified number of runs and a column log is maintained to document that its usage is limited accordingly. Periodic analyses to document that it is still within functional specification may also be prudent. The point is to ensure that assay performance does not fall victim to an undetected source of progressive variation. 

Now assuming a numerical value for Cq and computing / , the point S, should fall on the lowest marginal stability curve of Fig. 5. If not, Cq must be corrected appropriately. Tallin et a/. (1989) used this technique to determine the balance constant for a device similar to Fulton s, and their results are presented in Fig. 5 as the open circles. They determined the balance constant to be 0.79 with a standard deviation of 0.020, which is in good agreement with the numerical estimates of Philip et al and Sloane and Elmoursi. The data are seen to follow the marginal stability curve very well. 

Effect of pH. The pH vs. stability curves of Figure 4 are plotted using the half-lives calculated from the Arrhenius plots (Figure 3). Stability is expressed as log ti/2 to make possible to visualize the curves at the temperature range (30-60°C). Maximum stability was achieved at pH 4.5 - 5.0. The stability of the enzyme at pH 3.0 which was used for the activity assay was much lower than the stability at pH 4.5. 

Figure 5,45 P-T stability curve of phlogopite compared with the incipient melting curves of granite and basalt. Reprinted from H. S. Yoder and H. R Eugster, Geochimica et Cos-mochimica Acta, 6, 157-185, copyright 1954, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.

The P-T stability curve of muscovite intersects the incipient melting curve of granite in hydrous conditions at about 2.3 kbar total pressure and T = 650 °C. Thus, muscovite may crystallize as a primary phase from granitic melts above these P and T conditions (interstitial poikihtic crystals) or may form by reaction with pristine solid phases at lower P and T (muscovite as dispersed phase within feldspars, for instance). In this second type of occurrence, the following two equilibria are of particular importance ... 

Figure 5.47 Extrinsic stability curves for muscovite and paragonite, based on (A) equilibria 5.139 and 5.144 (quartz absent) and (B) equilibria 5.140 and 5.145 (quartz present). Incipient melting curves of granite for = 0.5 to 1, and stability curves of Al2Si05 polymorphs according to Richardson et al. (1969), are superimposed.

Figure 5.48 Extrinsic stability limits of muscovite, based on equilibrium 5.140 for variable amounts of H2O component in fluid. Also shown are incipient melting curves of granite for various h20 isopleths and stability curves of Al2Si05 polymorphs, according to Holdaway (1971) From Kerrick (1972), American Journal of Science, 212, 946-58. Reprinted with permission of American Journal of Science.

As a consequence, by breaking up a heavy nucleus (fission) to produce lighter and more stable nuclei, energy is once again released. Detailed examination of the stability curve reveals other important features of nuclear structure which are strongly correlated with observed abundances. 

Using gel filtration on columns of Sephadex G-200, Gascon and Ot-tolenghi142 discovered in Saccharomyces cerevisiae a form of /3-D-fruc-tofuranosidase of low molecular weight, and predicted correctly that this form would be found to be free from carbohydrate. This enzyme occurred within the protoplast its molecular weight (135,000) and specific activity were similar to those of the protein moiety of the external enzyme. The two /3-D-fructofuranosidases gave the same Km for sucrose, the same Km for raffinose, and the same pH optimum (3.5 to 5.5) for enzymic activity but their pH-stability curves differed, the internal enzyme being reversibly inactivated under acidic conditions, that is, below pH 5. 

Fig. 10.9. The neutral stability curve in the n n plane below the curve parametrized by tr(J) = 0 the uniform state is unstable to perturbations of appropriate spatial form...

We can think of the reactant concentration and some initial spatial distribution of the intermediate concentration and temperature profiles specifying a point on Fig. 10.9. If we choose a point above the neutral stability curve, then the first response of the system will be for spatial inhomogeneity to disappear. If the value of /r lies outside the range given by (10.79), then the system adjusts to a stable spatially uniform stationary state. If ji lies between H and n, we may find uniform oscillations. 

If, however, we start the system with a given non-uniform distribution, corresponding to n = 2 say, and a value for ji such that the initial point lies beneath the neutral stability curve, then the spatial amplitudes will not decay. Rather the positive real parts to the eigenvalues will ensure that the perturbation waveform grows. The system may move to a state which is varying both in time and position—a standing-wave solution. 

Fig. 10.10. The neutral stability curve for a system with k < 0.0279, showing curves parametrized by tr(J) = 0 and det(J) = 0. Within the latter, non-uniform profiles may be stable .

The neutral stability curve corresponding to the condition det(J) = 0 gives a closed region, within which we expect the appearance of stable (time-independent) spatially non-uniform profiles. 

As an example of how these curves should be interpreted, we consider a specific case. Let us take k = 0.02 and, for convenience, choose the size of the reaction zone such that y = 6n2 The dispersion, or neutral stability curve for this system, is shown separately in Fig. 10.12. The wave number n can only have integer values, so valid modes correspond to the horizontal lines with nn/y111 = l, etc. Only three of these horizontals intersect the... 

Fig. 10.11. The development of the neutral stability curve for stable pattern formation with the...

Fig. 10.12. Specific neutral stability curve for k = 0.02 and y = 16n2, showing the possibility of stabilizing patterns with n = 2, 3, or 4 over limit ranges of the precursor concentration fx.

Figure 11.1. pH Heat Stability Curves of Individual Cow Milks o o =... 

Sweetsur, A. W. M. and White, J. C. D. 1974. Studies on the heat stability of milk protein. I. Interconversion of type A and type B milk heat-stability curves. J. Dairy Res. 41, 349-358. 

Recently Yih (Y2) has given a detailed treatment of the stability of film flow on an inclined plane. Three cases are considered in detail small wave numbers (n), small Reynolds numbers, and large wave numbers. In the first case the results are in agreement with the results of Benjamin noted above, but for large wave numbers and zero surface tension, Benjamin s tentative conclusions are shown to be invalid. The stability curves are considered for film flows on vertical and sloped walls for liquids with and without surface tension. 

Figure 13. Neutral stability curves computed by linear analysis for the succinonitrile-acetone system as a function of acetone concentration for fixed temperature gradient of G = 67°/cm.

Figure 7.35 Mallik 2002 geothermal gradient and hydrate stability curve for pure water and water containing 40 ppt salt. Note the depths of the thermal stimulation test and the six pressure stimulation (MCT) tests. (From Wright, J.F., et al., in Scientific Results from the Mallik 2002 Gas Hydrate Production Research Well Program, Mackenzie Delta, Northwest Territories, Canada, Geological Survey of Canada Bulletin 585, including CD (2005). With permission.)...

The stability curve of the protein can be inferred from the behavior of the free enthalpy AG over the temperature T. From the other important equation for the free enthalpy AG, Eq. (17.7), the limit of stability atAG = Ois reached when K = 1 or when the concentration of native and unfolded species are equal. 

Ts is the temperature of maximum stability, at which the AG(T) versus T curve passes through its maximum, i.e., AG(TS) > 0. At Tm, the maximum temperature of stability, AG (T) = 0. By necessity, Ts < Tm. There is another special point in the temperature stability (AG(T)) curve, however as the temperature stability curve resembles a inverted parabola, there must be a second intersection point with the line of minimum stability, AG(T) = 0. This intersection point is the minimum of the range of temperature stability and is called cold denaturation temperature, Tc. The cold denaturation temperature is practically never taken into consideration in the discussion of temperature stability of biocatalysts. One reason lies in its frequent inaccessibility often, Tc is below 0 °C and thus cannot be measured in water or any other mostly aqueous medium. 

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