Tangent slopes

FIGURE 6.2 Measuring the partial molar volume by dissolving ethanol in 1000 g of water at 20°C and 1 atm. 

Mols of ethanol added per 1000 g of water = molahty of ethanol in water 

FIGURE 6.3 Volume of solution plotted vs mols of ethanol added, for a constant 1000 g of water. The points represent data from Gillespie et al. [1]. See Problem 6.1. 

The corresponding ordinate must be volume of solution per 1000 g of solvent. Thus, on a plot of (Vper 1000 g solvent) vs. molality of the solute, the slope is partial molar volume of the solute, Vsolute- 

Example 6.2 Estimate the volume change on mixing for 1 mol of ethanol with KXX) g of water at 20°C. 

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We have shown (Ref. 2, Figure 6) that at surface pressures from 18 to 26 dynes/cm the partial areas of the constituents are independent of the composition of the mixed films if the molar fraction of component 2 is 0.27-0.70. In this case the error for the partial molecular areas results from the error on experimental average molecular areas equal to 10%. However, an additional error must be considered above 26 dynes/cm resulting from the tangent slope to the curve (Ref. 2, Figure 2) which can be estimated as 10%. 

Capital investment and production costs, using the techniques of Example 6.3.1 for a range of current densities, are plotted in Fig. 6.18. The optimum investment is identified by a tangent slope —3.57, which is the maximum payback time corresponding to the specified nominal plant life of 10 years with a minimum rate of return of 0.25. The optimum magnitude of capital investment is 2,550,000 with annual production costs of 580,000. 

Figure 8.8 To create the Legendre transform, a function y(x) is expressed as a tangent slope function c(x), and a tangent intercept function b(x). The tangent slopes and intercepts of points x and x" are shown here.

Comparison with the approximate expressions (135 a) and (135 b) shows, that this relation does not violate the HS upper boimd even for zirconia, as can be seen from the initial tangent slope of this relation (which is equal to -2 at = 0), although of course relation... 

If we have experimental data, or an equation that is believed to represent such experimental data for some extensive property as a function of concentration, we can compute the partial molar values by the method of tangent slopes or the method of tangent intercepts. In either case we could plot the data and make the geometric constructions. However, the mathematical procedure, which our computers can do for us is much more useful. 

The corresponding equation for the method of tangent slopes (Eq. 4.19) requires that we have the property equation in the form of an extensive property, stated as a function of the number of mols present of each species. If we have, for example, an equation for v as a function of Xj we can multiply both sides of that equation by the total number of mols present, j-. This makes the following changes... 

Example 6.5 Repeat Example 6.4, for ethanol only, using the method of tangent slopes. Multiplying both sides of Eq. 6.G by n and inserting those in Eq. 4.19 we find... 

Example 6.5 is clearly a longer and messier way to do by tangent slopes what we did more easily by tangent intercepts in Example 6.4. So why bother Because some functions are easier to do by tangent slopes, and this method appears in the historical literature, so you must understand it to understand that literature. Most often we use tangent intercepts and Eqs. 6.4 and 6.5... 

In the case of perfect symmetric peaks, the retention time can be determined directly from the peak maximum method, which is the simplest and most common. The peak maximum method is useful for determination of retention time if the skewness ratio is 0.7-1.3 [15]. The skewness ratio is defined as the ratio of tangent slope to the peak leading part and tangent slope to the peak tailing part whereas both tangents are drawn in the inflexion points. In such cases the skewness ratio is our of this interval, Ir is obtained from the first-order moment method or the Conder and Young method. Between these two methods, Conder and Young is recommended [16] ... 

As a solution must satisfy both equations, it appears as a point of intersection between the curves /i = 0 and /2 = 0. Let us modify the parameter vector 0 to move die locations of these curves in x space as shown in Figure 2.19. Initially, there are no solutions, but as the curves approach each other, they overlap to generate two solutions. Consider the particular parameter vector 0g at which the curves first touch. Clearly this is an important choice of parameter(s), as it separates the region of parameter space in which there are solutions from that in which there are none. If the curves just touch at 0 and do not cross, they must be parallel to each other at the point of contact. That is, the slopes of the tangent lines in x, JC2) space of the two curves /i = 0 and fi = 0 must be equal at the point of contact, which we note is a solution. At 0c, with solution Xs(0c), if 9 = dxi/dxi is the common tangent slope. 

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