Two slopes

The mathematical requirements for unique determination of the two slopes mi and ni2 are satisfied by these two measurements, provided that the second equation is not a linear combination of the first. In practice, however, because of experimental error, this is a minimum requirement and may be expected to yield the least reliable solution set for the system, just as establishing the slope of a straight line through the origin by one experimental point may be expected to yield the least reliable slope, inferior in this respect to the slope obtained from 2, 3, or p experimental points. In univariate problems, accepted practice dictates that we... 

The mole ratio of ligand-to-metal is determined from the ratio of the two slopes. 

According to the Scher-MontroU model, the dispersive current transient (Fig. 5b) can be analyzed in a double-log plot of log(i) vs log(/). The slope should be —(1 — ct) for t < and —(1 + a) for t > with a sum of the two slopes equal to 2, as shown in Figure 5c. For many years the Scher-MontroU model has been the standard model to use in analyzing dispersive charge transport in polymers. 

For a symmetrical separation of component h from c, Brian Staged Cascades in Chemical Processing, Prentice-Hall, Englewood Cliffs, N.J., 1972) reported that the ratio of wash solvent to extraction solvent W /S should be set equal to the geometric mean of the two slopes of the equilibrium lines [Eq. (15-35)]. 

FIGURE 15.8 Semilog plot of washout function showing two slopes that correspond to the two time constants in the side capacity model. 

Two slopes are compared in a similar manner as are two means the simplest case is obtained when both calibrations are carried out using identical calibration concentrations (as is usual when SOPs are followed) the average variance V u is used in a t-test ... 

The project cost of a 1,000 ton/day ammonia plant that was built in 1969 has been obtained, using Equation 9-1 with an m of 0.70 and 0.88 and the CEPI. The results appear in Table 9-5. The effect of the exponential factor is very evident for plants 1 and 2. This effect does not occur for the other plants because their rated capacity was the desired 1,000 tons/day. Exponential factors are only used when capacity extrapolations must be made. This illustrates how a difference of 0.18 in the exponential factor (m) can have a profound effect on the projected cost if the scale-up factor is large. This can be further demonstrated by drawing lines of these two slopes on log-log paper (Fig. 9-1). As the lines get farther away from the base... 

In order to estimate the relative effectiveness of the inhalation route, regression lines were drawn through dose-response values for both the oral route and the inhalation route, and the ratio of the dose values at which the two regression lines crossed the 20% performance line (an arbitrary choice) was calculated (Fig. 98). The ratio of the two slopes indicates that 3.75 times as large an aerosol dose would be needed to equal the effect of LSD given by the oral route, i.e., relative effectiveness equals 28%. 

For the sake of practical simplicity (the use of Slater rules and a two-slope space geometry [2]) one may employ the larger, just antisymmetric but non-spin-adapted space spanned by ( ) Slater determinants. 

Equations 5-92 and 5-97 allow one to obtain k, kj, and K from k pp values obtained in the absence and presence of added common ion. A plot of obtained in the absence of added common ion versus [M-]-1/2 yields a straight line whose slope and intercept are (k — feJ)X 1/2 and kj, respectively. A plot of fc pp obtained in the presence of added common ion versus [CZ]-1 yields a straight line whose slope and intercept are (k — kJ)K and kj, respectively. Figures 5-5 and 5-6 show these plots for polystyryl sodium in 3-methyl-tetrahydrofuran at 20°C. The combination of the two slopes and two intercepts allows the individual calculation of fc-, kj, and K. [Note K as well as [P-] and [P (C+)] can also be independently determined from conductivity measurements.]... 

When a drug of single compartment distribution is given intravenously, a. semilog plasma concentration time plot is obtained (Fig. 1.3.4), which has two slopes, one is due to distribution and another which is due to the drug elimination. 

The explanation for the two slopes in the plot lies in the fact that even a very pure crystal of NaCl contains some impurities, and the line corresponding to low temperatures (on the right of the plot) is due to the extrinsic vacancies. At low temperatures, the concentration of intrinsic vacancies is so small that it can be ignored because it is dominated by the defects created by the impurity. For a particular amount of impurity, the number of vacancies present will be essentially constant, jj in this extrinsic region thus depends only on the cation mobility due to these extrinsic defects, whose temperature dependence is given by Equation (5.9) ... 

It is clear that there are significant differences between the two slopes of the semi-log plots for the Cot carbon of pure PLA with the ot-helix form and the Cot carbon of PLA (ot-helix form) in the PLA/PLIL (50/50) blend sample... 

The structure of a pyrites kiln is shown in Figs. 676 and 677. In Fig. 577 the pyrites is shown as inclosed on two sides by two sloping walls, the back and front walls being vertical. The mass of pyrites rests on... 

The two lines B and D, plotted in Figure 5, were used to evaluate the constants. The ratio of the two slopes gives a value of 1.419 for Pb/Pd- It follows, by combining Equations 7 and 10 and taking the ratio of the resulting equation for the two experiments, that... 

Finally, we compute Glesjer s test statistics for the three models discussed in Section 14.3.5. We regress e2, e, and log e on 1, Xh and X2. We use the White estimator for the covariance matrix of the parameter estimates in these regressions as there is ample evidence now that the disturbances are heteroscedastic related to X2. To compute the Wald statistic, we need the two slope coefficients, which we denote q, and the 2x2 submatrix of the 3x3 covariance matrix of the coefficients, which we denote Vq. The statistic is W - q Vq 1q. For the three regressions, the values are 4.13, 6.51, and 6.60, respectively. The critical value from the chi-squared distribution with 2 degrees of freedom is 5.99, so the second and third are statistically significant while the first is not. 

The density or its reciprocal, the specific volume, is a commonly used property for polymeric materials. The specific volume is often plotted as a function of pressure and temperature in what is known as a pvT diagram. A typical pvT diagram for an unfilled and filled amorphous polymer is shown, using polycarbonate as an example, in Figs. 2.10 and 2.11 The two slopes in the curves represent the specific volume of the melt and of the glassy amorphous polycarbonate, separated by the glass transition temperature. 

In the time-dependent picture, the overall width of the total spectrum is determined by the two slopes dV/dR and dV/dr at the transition point (Re,re). 

Therefore, we plot In I vs. temperature. This should yield one straight slope (in the absence of a melting point) or two slopes separated by an area where the slope is almost equal to zero. This area characterizes the transformation of the triglycerides from a solidlike state to a liquid state. Figure 9.12 clearly shows that we are in the presence of a melting area, indicating that fusion of triglycerides is not abrupt. 

Figure 7 shows a plot of spin equilibria calculated from the data in Figure 6. The two sloping lines correspond to the temperature range where a thermal spin equilibrium exists. From the difference in height between them, the free energy difference between the equilibria in the R and T structures is calculated as 0.9-1.2 kcal/mol Fe, in agreement with the value derived from the IR measurements mentioned above, and with a recently published determination of the spin equilibrium by resonance Raman spectroscopy (46).

The Tafel slopes are given in Table 9. The first slope (bx) is the lower one and it occurs at more positive potentials the second (b2) has higher values and occurs at slightly positive potentials, i.e., at lower coverages (8). Two slopes were not observed at all temperatures. Table 9 also contains corresponding coverages and the charge transfer coefficient, a ... 

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