Straight line and

The straight line and consequently its coefficients are determined starting from two known viscosity values. 

A plot of P/n versus P should give a straight line, and the two constants and b may be evaluated from the slope and intercept. In turn, n may be related to the area of the solid ... 

In Eq. (C.2a), we derive the solution by solving it along the path E characterized by two straight lines and three points (see Fig. 16a) ... 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

A plot against Hammett s cr-constants of the logarithms of the rate constants for the solvolysis of a series of Mz-substituted dimethylphenylcarbinyl chlorides, in which compounds direct resonance interaction with the substituent is not possible, yielded a reasonably straight line and gave a value for the reaction constant (p) of — 4 54. Using this value of the reaction constant, and with the data for the rates of solvolysis, a new set of substituent parameters (cr+) was defined. The procedure described above for the definition of cr+, was adopted for... 

Examples of (a) straight-line and (b) curved normal calibration curves. 

In a vacuum (a) and under the effect of a potential difference of V volts between two electrodes (A,B), an ion (mass m and charge ze) will travel in a straight line and reach a velocity v governed by the equation, mv = 2zeV. At atmospheric pressure (b), the motion of the ion is chaotic as it suffers many collisions. There is still a driving force of V volts, but the ions cannot attain the full velocity gained in a vacuum. Instead, the movement (drift) of the ion between the electrodes is described by a new term, the mobility. At low pressures, the ion has a long mean free path between collisions, and these may be sufficient to deflect the ion from its initial trajectory so that it does not reach the electrode B. 

This is the equation of a straight line and indicates that a plot of Hc2/r versus C2 has the following properties ... 

A plot of a k /T) versus 1 /T is then a straight line, and its slope is —jR. Once is determined in this manner, is available from the relationship... 

It is seen that the Van Deemter equation predicts that the total resistance to mass transfer term must also be linearly related to the reciprocal of the solute diffusivity, either in the mobile phase or the stationary phase. Furthermore, it is seen that if the value of (C) is plotted against 1/Dni, the result will be a straight line and if there is a... 

Few other reactions of series of substituted pyridines have been investigated extensively. Dondoni, Modena, and Todesco have measured the rate of N-oxidation of a limited series of pyridines and found a good correlation with normal u-values with a p-value of — 2.23. The A-alkylation of pyridines with alkyl iodides in nitrobenzene has been studied by Brown and Cahn and by Clarke and Rothwell. Unfortunately, the only data available are for the parent compound and for alkyl derivatives, and, since the a-values for the various alkyl groups in a given position are substantially constant, this leaves a correlation of only three independent points. However, the rates of A-alkylation of the j8- and y-alkyl derivatives are so nearly equal that it appears as if no correlation existed. Clarke and Rothwell have also studied the alkylation with allyl bromide in nitromethane at various temperatures, and in this case a more extensive series is available. The authors state that no overall Hammett correlation is obtained however, the j8-substituted derivatives fall on one straight line and the y-derivatives on another one with a different slope. The data are shown in Fig. 2. The line for the j8-compounds, p = — 2.53 0.31, r = 0.95, is seen not to be very good the line for the y-derivatives, p = — 1.42 0.06, r = 0.99, is much more satisfactory. It does not seem likely that the discrepancy is due to the intervention of resonance effects, since in this case one would expect the correlation for the y-derivatives to be poorer than that for the j8-analogs. More extensive studies with a wider variety of substituents would seem very desirable. 

Kinematics is based on one-dimensional differential equations of motion. Suppose a particle is moving along a straight line, and its distance from some reference point is S (see Figure 2-6a). Then its linear velocity and linear acceleration are defined by the differential equations given in the top half of Column 1, Table 2-5. The solutions... 

The creep modulus, also known as the apparent modulus or viscous modulus when graphed on log-log paper, is normally a straight line and lends itself to extrapolation for longer periods of time. 

Second, the creep modulus, also known as the apparent modulus or viscous modulus when graphed on log-log paper, is normally a straight line and lends itself to extrapolation for longer periods of time. The apparent modulus should be differentiated from the modulus given in the data sheets, which is an instantaneous or static value derived from the testing machine, per ASTM D 638. 

Creep test data when plotted on log-log paper usually form a straight line and tend themselves to extrapolation. Tlie slope of the straight line, which indicates a decreasing modulus, depends on the nature of the material (principally its rigidity and temperature of heat deflection), the temperature of the environment in which the product is used, and the amount of stress in relation to tensile strength. 

If we join the points representing two such phases by a straight line, and if we imagine 1 mol of a heterogeneous complex formed of p mols of the first and q mols of the second phase, the complex will be represented by a point on the straight line with the co-ordinates. 

The values used in plotting Figs. 2-1 and 2-2 can be used to illustrate the method for first-order and second-order data. Plots of t/E versus time are shown in Fig. 2-9. The second-order data define a precise straight line, and those for n = 1 are linear to E < 0.4. The latter graph has a slope of 0.6, giving n = 1.2. 

As we have seen for first- and second-order rate laws, each integrated rate law can be rearranged into an equation that, when plotted, gives a straight line and the rate constant can then be obtained from the slope of the plot. Table 13.2 summarizes the relationships to use. 

Figure 16. Nonisokinetic dependence of calorimetric AH on AG° for ionization of substituted acetic acids (75), actual regression line (full straight line) and lines transferred from the aH° versus AS° plot (broken lines).

From the image sequences, information on the velocities of nano-particles can be extracted. The statistical effect of Brownian motion on the flowing speed of the mixed liquid is found small enough to be ignored as shown in Fig. 37 where most of the particles trajectories in the liquid are straight lines and parallel with the wall basically. Therefore, Brownian diffusive motion is ignorable. 

C, the fourth parameter, represents the relationship between the first cumulant and the particlescattering factor. For values of 1/F( ) < 10, the double logarithmic plot of the first cumulant against the reciprocal particle-scattering factor yields a straight line, and the exponent v is related to the initial slope C oiF/q D, against by the equation... 

The ko was obtained using the measured P and Eq. (15) with the given values of Da and kt. Fig. 1 shows plots of QoUeo/ko against Cbo, and the plots satisfied straight lines, and kj and k2/ki were obtained firem the slope and intercept of the straight line of Eq.(13). 

A linear relationship of free energies is extrathermodynamic, and such a correlation is hardly a theoretical corollary which directly results from the axioms of thermodynamics alone. However, the slope of the straight line and departures from linearity can often suggest something physically meaningful concerning the chemical reactivity, as seen in Fig. 2. 

If there are more than two nuclei exerting relaxation effects on one another, then it is convenient to consider them in pairs and to arrive at the overall effect by adding together the effects of various possible pairs. In the case of a three-spin system, we can consider two different situations (1) the nuclei and He are arranged in a straight line, and (2) they... 

Let us first consider the nuclei //a, Hys, and as lying in a straight line and equidistant from one another (Fig. 4.5). The central proton, can relax by interactions with two neighbors, //a and He, while //a and He can relax by interaction with only one neighbor, so //b can relax twice as quickly as //a or Hq. If we assume that relaxation can occur only through dipole-dipole relaxation, and that an equilibrium steady state has been... 

Thus, plotting log[R] versus time provides a convenient way to verify that the reaction follows first-order kinetics. Such a plot should yield a straight line and the rate constant follows from the slope of the line. 

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