Tangent lines

Figure A2.5.15. The molar Gibbs free energy of mixing versus mole fraetionxfor a simple mixture at several temperatures. Beeause of the synuuetry of equation (A2.5.15) the tangent lines indieating two-phase equilibrium are horizontal. The dashed and dotted eiirves have the same signifieanee as in previous figures.

The second important parameter is the chromatographic peak s width at the baseline, w. As shown in Figure 12.7, baseline width is determined by the intersection with the baseline of tangent lines drawn through the inflection points on either side of the chromatographic peak. Baseline width is measured in units of time or volume, depending on whether the retention time or retention volume is of interest. 

Determination of reaction rate from a tangent line at time f. 

Cyclone Separator with Integral Catch Tank (See Fig. 26-19.) The diameter of the knockout drum is calculated by the criteria given in the preceding section and Fig. 26-18. Since the liquid is also to be retained in the vessel, extend the shell height below the normal bottom tangent line to increase the total volume by an amount equal to the volume of the hquid carried over. 

Use the ethanol curve similar to Figure 8-37, or refer to the data of Reference 133 the point of tangency of the line from the distillate composition of the diagonal is xj) = 0.80 andyv = 0.80. Thus the minimum internal reflux is set by this tangent line ... 

A line which intersects a circle at only one point is a tangent to the circle at that point. Every tangent is perpendicular to the radius drawn to the point of intersection. Spheres may have tangent lines or tangent planes. 

Geometrically, the derivative of y = f(x) at any value is the slope of a tangent line T intersecting the curve at the point P(x,y). Two conditions applying to differentiation (the process of determining the derivatives of a function) are ... 

Secant modulus The secant modulus is the ratio of stress to the corresponding strain at any specific point on the stress-strain curve. As shown in Fig. 2-2(a), the secant modulus is the slope of the line joining the origin and a selected point C on the stress-strain curve this could represent a vertical line at the usual 1 % strain. The secant modulus line is plotted from the initial tangent modulus and where it intersects the stress-strain curve. The plotted line location is also based on the angle used in relation to the initial tangent line from the ab-... 

From these considerations, the general form of the concentration-temperature curve is easily deduced. It is cut in two points by a perpendicular to the T-axis and is bounded on the side of higher temperature by a tangent line also perpendicular to the temperature axis. The abscissa of this point of contact denotes a maximum temperature which is the melting-point of the hydrate, and its ordinate represents the composition. 

Hv = height, or length, between tangent lines (the length of the cylindrical section), m,... 

Loading (per linear metre) Fw = 1280 x 2.18 = 2790 N/m Bending moment at bottom tangent line ... 

Hv Height (length) of cylindrical section between tangent lines L... 

When a small quantity of a second liquid phase is present, a drawoff pot (commonly called a bootleg) is provided to make separation of the heavy liquid (frequently water) easier. The pot diameter is ordinarily determined for heavy phase velocities of 0.5 ft/min. Minimum length is 3 ft for level controller connections. Minimum pot diameter for a 4-8-ft diameter reflux drum is 16 in. For reflux drums with diameters greater than 8 ft, pot diameters of at least 24 in. are used. The pot must also be placed at a minimum distance from the tangent line that joins the head with the body of the vessel. 

Figure A 1.2 Determination of instantaneous velocity at various points in a reaction progress curve, from the slope of a tangent line drawn to a specific time point.

The method s goal is to make the residual vanish by successively improving our guess. To find an improved value we take the tangent line to the residual function at point. v(o) and project it to the zero line (Fig. 4.1). We repeat the projection from x(1) to give x(2), and so on. The process continues until we reach a value x on the (q)-d iteration that satisfies our equation to within a small... 

We can confirm that on a plot of the mole number nqtz for quartz versus time (Fig. 16.1), this value is the slope of the tangent line and hence the dissolution rate —dwqtz/dt we expect. 

A (a) The 2400-s tangent line intersects the 1200-s vertical line at 0.75 M and reaches 0 M at 3500 s. The slope of that tangent line is thus... 

In each case, we draw the tangent line to the plotted curve. 

Exothermic events, such as crystallization processes (or recrystallization processes) are characterized by their enthalpies of crystallization (AHc). This is depicted as the integrated area bounded by the interpolated baseline and the intersections with the curve. The onset is calculated as the intersection between the baseline and a tangent line drawn on the front slope of the curve. Endothermic events, such as the melting transition in Fig. 4.9, are characterized by their enthalpies of fusion (AHj), and are integrated in a similar manner as an exothermic event. The result is expressed as an enthalpy value (AH) with units of J/g and is the physical expression of the crystal lattice energy needed to break down the unit cell forming the crystal. 

The slope of the tangent (AM/At) is the instantaneous rate of the reaction at this time. To determine the rate at a different time, we would need to draw another tangent line. In most kinetic studies, we wish to know the initial rate. The initial rate comes from a tangent drawn to the curve at the very beginning of the reaction. 

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