Intersection of straight lines

Fig. 8. Surface-area/surface-pressure isotherms for spreading 1,5, and mixtures of 1 + 5 at 0.2,0.4, 0.6, and 0.8 mole fractions of 1 on aqueous S.O mM NaCl. Areas (A,) and pressures (iq) associated with the transition to a compressed state were taken by projecting the intersection of straight lines drawn to the appropriate sections of the isotherm to the surface-area and surface-pressure axes. The collapse pressure (nc) and collapse area (Ac) were taken by treating that transition, similarly. The insert shows an expansion of the isotherms between 20-40 mN/m. Temperature = 24.0 + 0.5 °C [116]...

Find A point as intersection of straight lines EqRi and Ej Rj + ... 

The mix point, = 0.0673, falls on a straight line connecting x and The extract composition is then determined hy drawing a straight line from x,-throiigh Zm until the line intersects the extract line at the final extract composition, i/e = 0.084. The delta point is then found at the intersection of two lines. One line connects the feed and extract compositions x and y. The other line connects the raffinate and solvent compositions x,- and y. ... 

T/N2, respectively. After that, we draw the straight lines parallel to the appropriate axes through the splitting points. The points of the intersection of those lines are adopted, as usual, as the nodes which constitute... 

The graphic interpretation of the above conditions consists in selecting the branch of the cotangent curve in Eq. (A 1.83) that intersects the straight line y = ml Am at the inflection point (see Fig. A 1.3 where the inflection points are connected by a dotted line). For the Debye spectrum, we have... 

When data in the presence of an enzyme inhibitor are presented in the form of a Lineweaver-Burk plot, a series of straight lines should be obtained. The slopes of these hnes may or may not change, and the hnes may or may not intersect at a common point. The relationships between slopes, intersection points, and inhibitor mechanisms are outlined later. Further information regarding these mechanisms, including velocity equations describing data obtained in the presence of inhibitors with diverse mechanisms, can be found in (Segel, 1993). 

Plots devised by Dixon to determine K, for tight-binding inhibitors, (a) A primary plot of v versus total inhibitor present ([/Id yields a concave line. In this example, [S] = 3 x Km and thus v = 67% of Straight lines drawn from Vo (when [/It = 0) through points corresponding to Vq/2, Vq/3, etc. intersect with the x-axis at points separated by a distance /Cj app/ when inhibition is competitive. When inhibition is noncompetitive, intersection points are separated by a distance equivalent to K. The positions of lines for n = 1 and n = 0 can then be deduced and the total enzyme concentration, [EJt, can be determined from the distance between the origin and the intersection point of the n = 0 line on the x-axis. If inhibition is competitive, this experiment is repeated at several different substrate concentrations such that a value for K, app is obtained at each substrate concentration. (b) Values for app are replotted versus [S], and the y-intercept yields a value for /Cj. If inhibition is noncompetitive, this replot is not necessary (see text)... 

Emax,f = fe[Etotai] iu wMch kg is the forward rate constant for the ESA EPA conversion. Note the difference in this expression from the rapid equilibrium expression for the previous Type I scheme. In the earlier scheme, there is a i s[A] term in the denominator whereas there is a / a[S] term in this scheme. Hence, a double-reciprocal plot (1/v versus 1/[S]) will yield a series of straight lines intersecting at a common point not on the vertical axis. This intersection point will have coordinates of 1/[S] = - VKg and 1/v = 1/Emax,f- Hence, the rapid equilibrium Type II scheme should be readily distinguishable from the previous scheme. 

A] at different constant concentrations of B, a series of straight lines will be obtained having a common intersection point. The intersection point will always be in the second quadrant. A double-reciprocal plot of 1/v vs. 1/[B] at different constant concentrations of A will consist of a series of straight lines all intersecting on the vertical axis. This observation is characteristic of the rapid equilibrium ordered system but not of the steady-state scheme. 

Data for the feed and copolymer compositions for each experiment with a given feed are substituted into Eq. 6-36 and r2 is plotted as a function of various assumed values of r. Each experiment yields a straight line and the intersection of the lines for different feeds gives the best values of r and r2. Any variations observed in the points of intersection of various lines are a measure of the experimental errors in the composition data and the limitations of the mathematical treatment (see below). The composition data can also be treated by linear least-squares regression analysis instead of the graphical analysis. 

It is seen that the plot of the experimental points versus the cross-sectional area yields a curve which is virtually a straight line. Thus, it would seem that K is determined by the point of intersection of the line with the G axis, and L by the slope. This suggestion should be treated with some caution because K, though not vanishing with wire diameter, nevertheless is certainly a function of the diameter. However, the error introduced by considering K as constant is not large... 

A Simple Picture of the Symmetry Factor. In order to employ simple geometry, one now ignores the curvature of the Morse curves and considers that the potential energy barrier near the intersection point is made up of straight lines (Fig. 9.11). This simplifying analogue of the barrier is useful for a first-base discussion of the symmetry factor p. 

Interesting possibilities arise in these potential energy-distance curves if they are not drawn in the extreme simplification of straight lines (see Fig. 9.33), but with the natural curvature that potential energy-distance relations have. Thus, Fig. 9.33 shows the simplified situation where, from the formula given above for P it can be seen that with approximately equal slopes of the potential energy curves near the intersection point, equal slopes of the value of P will be about one-half, as is often observed. 

Another plot, introduced by Dixon,71 is that of 1 / v versus [ I ] at two or more fixed substrate concentrations. The student should be able to demonstrate that this plot contains a family of straight lines that intersect at a point to the left of the origin with coordinates [I] = -If and /v = /Umax. This plot may fail to distinguish certain types of inhibition discussed in the next section.72... 

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