Straight line through the origin

In Figure 4 the measured attenuation values (TT) and the corresponding estimates are plotted against each other. Ideally (with error free estimates) all sample points should lie on the straight line through the origin with unit slope. Clearly there is a strong correlation between the estimates and the true values. 

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... 

It is sufficient, as Sing has pointed out, merely to replace as normalizing factor by the amount adsorbed at some fixed relative pressure (p/p ), in practice taken as (p/p°), = 0-4. The normalized adsorption n/ o (= j). obtained from the isotherm on a reference sample of the solid, is then plotted against p/p°, to obtain a standard a,-curve rather than a t-curve. The a,-curve can then be used to construct an a,-plot from the isotherm of a test sample of the solid, just as the t-curve can be used to produce a t-plot. If a straight line through the origin results, one may infer that the isotherm under test is identical in shape with the standard the slope b, of the linear branch of the j-plot will be equal totio 4 Just as the slope b, of the t-plot was equal to nja (cf. Equation (2.34)). 

For an autocatalytic reaction some product B must be present if the reaction is to proceed. Starting with a small concentration of B, the rate rises as B is formed, and when A is used up, the rate must drop to zero. A plot of concentration versus time gives a straight line through the origin as shown in Figure 3-12. 

Because the H/e versus Q relationship is nearly linear, the HPg versus Q is essentially a straight line through the origin for a compressor with radially bladed impellers. For backswept impeller blades, the hp versus Q curve is a flat parabola, peaking near or above the maximum flow . 

Volume is directly proportional to absolute temperature. The dependence of volume (V) on the Kelvin temperature (T) is shown in Figure 5.3b (p. 106). The graph is a straight line through the origin. The equation of the line is... 

Fig. 31.17. (a) In a classical PCA biplot, data values xy can be estimated by means of perpendicular projection of the ith row-point upon a unipolar axis which represents theyth column-item of the data table X. In this case the axis is a straight line through the origin (represented by a small cross), (b) In a non-linear PCA biplot, the jth column-item traces out a curvilinear trajectory. The data value is now estimated by defining the shortest distance between the ith row point and theyth trajectory. 

Ans. The points of the second plot fall on a straight line through the origin, and so the volume and reciprocal of pressure are directly proportional to each other, making the volume inversely proportional to the pressure. 

If the shear stress versus shear rate plot is a straight line through the origin (or a straight line with a slope of unity on a log-log plot), the fluid is Newtonian ... 

A plot of the spectroscopically computed maximum ionic concentrations [SD ions]m (sum of free ions and ion-pairs) at the end of the reactions (m denotes maximum values throughout this paper, the subscript 0 denotes initial values) against the final values of the specific conductivity Km. for the equivalent runs (Tables 1 and 2) gave a straight line through the origin, showing that virtually only free ions were present in our systems if there had been important quantities of ion-pairs, this plot would have been markedly curved. 

Equations (5.34) and (5.47) have the same dependence on the concentration variables. The only difference is that for the interfacial reaction the apparent rate constants are directly proportional to the ratio S = QN. This dependency can be experimentally verified by measuring the apparent rate of extraction as a function of the interfacial area and the volume of the phases. A plot of the apparent rate constant of the forward rate of extraction vs. S must yield a straight line through the origin of the axes when case 2 holds. 

A plot of In (1 - Xp) or In CJCpjf) vs. t, as shown in Fig. 3.1, gives a straight line through the origin for this form of rate of equation. If the experimental data seems to be better fitted by a curve than by a straight line, try another rate form because the first-order reaction does not satisfactorily fit the data. 

Table E18.2 shows the details of the calculations. Plotting versus as shown in Fig. E18.2 gives a straight line through the origin, indicating a first-...

The behavior of the SrS2 salt In THP and THF looks very similar. Again only the contribution of free S ions to the propagation can be observed resulting in a straight line through the origin in a plot of kapp vs. 1/C (fig.11.Table IX). 

Eq. 12-12 implies that the logarithm of the ratio [A],/[A]0 yields a straight line through the origin with slope -k. Thus, if data from kinetic experiments are plotted as in Fig. 12.2, we can both check whether the reaction is first order in [A] and determine the rate constant k using a linear regression analysis. We note that in the case of first-order kinetics, the half-life, tm, of the compound (i.e., the time in which its concentration drops by a factor of 2) is independent of concentration and equal to ... 

Figure 12.4 Comparison of the reaction rate functions Eqs. 12-27 (curve 1) and 12-28 (curve 2). The straight line through the origin indicates the initial slope of size k. Half-saturation concentrations are at ln2 (J/k) and (J/k), respectively. Parameter values J= 1, k = 1.

Data from viscometers are often presented as a linear plot of shear stress versus shear rate, sometimes called a rheogram (Figure HI.1.2). This type of plot allows the viewer to see directly if there is Newtonian behavior because the plot will take the form of a straight line through the origin. A non-Newtonian response is, by definition, nonlinear and may or may not pass through the origin. If the sample has an apparent yield stress, then the line or curve will... 

A linear output is associated with a liquidlike or viscous response. The retardation response of a Newtonian liquid is a straight line through the origin, where the slope of the line is inversely proportional to the viscosity. A stress relaxation response for a Newtonian liquid is a narrow spike, as energy is dissipated very quickly. Typically the recoil part of a creep curve will be almost horizontal if there is little elasticity in the sample. 

Answer. A graph of A versus c is a straight line through the origin. From the slope e = 1.79 x 104 dm3 moF1 cm-1 (Figure 2.3). 

In analytical spectrometry there are many types of calibration curves which are set up by measuring spectrometric reference solutions. The measurements yield a curve of absorbance versus concentration, and the points between the data of the reference solutions are interpolated by fitting a suitable curve, which normally follows the Beer-Lambert law and which gives rise to a straight line through the origin of the coordinate system. The measurement conditions and the results of the calibration curve evaluations in the case of chromium and lead measurements by electrothermal atomic absorption spectrometry are presented in Table 1. 

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