Solution curves, algebraic

The glass transition temperature Tgc of the composite corresponds to the intersection of the first and the last part of the curve, or it is the solution of the Equation Alcl = A1o3, from which, after some algebra, one obtains ... 

Equation (9) is the basic differential equation that describes the rate of change of concentration of solute in the mobile phase in plate (p) with the volume flow of mobile phase through it. Thus, the Integration of equation (9) will provide the equation for the elution curve of a solute from any plate in the column. A simple algebraic solution to equation (9) is given below and the resulting equation for the elution curve from plate (p) is as follows -... 

The physical significance of Y2 is determined by the arbitrary placement of the mathematical surface that made T, = 0 that is, Y2 equals the algebraic difference between the overestimated and underestimated areas of the curve describing moles of solute when this curve is divided at a location jc0 that makes the surface excess of the solvent zero. 

The modification of Schreinemakers wet residue method which was used in this part of the investigation is that instead of analyzing the wet residue, a complex is prepared of known composition and the solution only is analyzed. This again gives two points on the diagram the solution point on the curve and the complex point which replaces the wet residue point. Hill and Ricci (7) claim that the complex method is as accurate or more accurate than the residue method if algebraic extrapolation of the tie-lines is used. 

From equation (23) an explicit algebraic expression for as a function of along the Hugoniot curve may be obtained. Equations (8) and (23) therefore determine the complete solution for the simplified system. 

Figure 2. Small-angle X-ray-scattering curves from silica polymers grown in alkaline solution with various ratios (R) of water to TEOS. The data are slit smeared and are plotted as log 1(h) versus log h, with the curves displaced vertically for clarity. The observed slopes are algebraically one greater than would be observed with pinhole geometry. The curve for R = 1 is consistent with a mass fractal structure with D/ = 2.84. The curves for R = 2-4 arise from colloids (Df = 3) with fractally rough surfaces with D, = 2.71, 2.51, and 2.45, respectively. (Reproduced with permission from reference 10. Copyright 1984 Elsevier.)...

Because one-dimensional EDMs require the numerical solution of a partial differential equation (as opposed to simple algebraic equations with empirical models and one-dimensional LDMs, or ordinary differential equations with hygroscopic LDMs), EDMs are more difficult to program, require somewhat more computational resources (typically many minutes on a PC), and have only recently been modified to include two-way coupled hygroscopic effects [37], For these reasons, only a few examples exist of one-dimensional EDMs being used with inhaled pharmaceutical aerosols (e.g., Ref. 11), although they have been used to aid in the development of purely empirical models (e.g., the ICRP 1994 [6] model is partly a curve fit to data from the one-dimensional EDM of Ref. 38). 

In following hand calculations of titration results we will ignore activity coefficients to avoid excessively complicating the calculations. Titration curves are typically plotted in two ways. The plot is usually either (1) of the volume of a titrant acid or base added (Vg) versus the pH of the solution or (2) of the concentration of acid or base added (Q or C ) versus the solution pH. The plots have a similar appearance. We will derive algebraic expressions for each type of plot. The advantage of or Cfi versus pH plots and mathematical expressions is that they can be used to derive the buffer... 

These values are plotted on the equilibrium-distribution diagram, as shown in Figure 3.7. The resulting curve intersects the equilibrium curve to give the interface compositions xA. - 0.231 and yA = 0.494. This solution agrees with the solution obtained previously by the algebraic method. 

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