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The Graphical Method

How many real roots does a given polynomial have Generally, in chemical problems, x represents a quantity that can only be positive, such as a concentration or an equilibrium constant. Descartes Rule of Signs states that for any polynomial F x) written in decreasing powers of x, the number of positive [Pg.193]

X term. Thus there is only one positive root of the equation. Almost always, in chemical problems, such a polynomial will have only one positive real root, and this one root will be the solution to the problem. [Pg.194]


Derive this relationship and explain the graphical method it suggests for evaluating and p. The critical temperatures for precipitation for the data shown in Fig. 8.3b are the following ... [Pg.581]

Use the graphical method outlined above to evaluate and and, from these, X for polyisobutylene in diisobutylketone. [Pg.581]

As can be seen from Figure 11, the graphical method does provide a good visual means for analysing life data and is easUy understood and explained. If used with discretion, graphical analysis can provide a useful means for data analysis. [Pg.14]

Gas-phase temperatures have been obtained by an extension of the graphical method illustrated (22). When equation 30 is divided by equation 34, the result is... [Pg.102]

Note that use of the Lee-Kesler fit [Eq. (2-78)] would give a slightly more accurate answer than the graphical method, and this fit is used for any computer applications. [Pg.402]

Figure 14-6 illustrates the graphical method for a three-theoretical-plate system. Note that in gas absorption the operating line is above the equihbrium curve, whereas in distillation this does not happen. In gas stripping, the operating line will be below the equihbrium curve. [Pg.1357]

It therefore appears that other hydrocarbons than medrane take part, but to a lesser extent. The other species which were considered in this calculation were SiCl, SiH4, CH3CI, SiHs, H, Si, Sia, Cl and CI2, all of which are gaseous at this temperature. It is difficult to predict before dris calculation what will be the major species, and attempts to limit die number of compounds which are included is to some extent subjective, but this can be conected by using the graphical method as a preliminaty. [Pg.98]

The rea.son for this choice of low pressure ratio is illustrated by an approximate analysis [12], which extends the graphical method of calculating gas turbine pierformance described in Chapter 3. If the gas turbine higher plant is assumed to ojjerate on an air standard cycle (i.e. the working fluid is a perfect gas with a constant ratio of specific heats, y), then the compressor work, the turbine work, the net work output and the heat supplied may be written as... [Pg.123]

The graphical methods of Bras provide helpful short-cuts to avoid the tedious trial-and-error solutions required of the rigorous methods. Reference 9 is the most recent and perhaps the easier to use. The results agree in general within about 10%. [Pg.144]

The graphical method of Hulden is also helpful as it is not as tedious as the arithmetic methods, and based on his comparison with the Colbum-Hougen method, the proposed results are within 1%. [Pg.144]

The graphical method of showing how the corrosion rate is dependent on the extent of the polarisation of the anodic and cathodic reactions constituting the corrosion reaction was due originally to Evanswho used... [Pg.92]

Corrosion reactions involving two simultaneous cathodic processes have already been referred to, and it is now appropriate to consider the graphical method of representing the corrosion rate. It should be noted that although the simultaneous reduction of HjO and dissolved oxygen occurs frequently this does not exhaust the possibilities, and reactions such as Fe - Fe, -> Cu, CI2 Cl may accompany either or both of the... [Pg.103]

Fig. 7.83 The graphical method for obtaining equilibrium compositions from free energy vs. composition curves at a given temperature. Points of contact give equilibrium compositions Xi for phase I and X2 for phase II... Fig. 7.83 The graphical method for obtaining equilibrium compositions from free energy vs. composition curves at a given temperature. Points of contact give equilibrium compositions Xi for phase I and X2 for phase II...
By a brilliant physical intuition B. van der Pol succeeded finally (1920) in establishing his equation (which is given in Section 6.11) but, not having any mathematical theory at his disposal, he determined the nature of Ike solution by the graphical method of isoclines. It became obvious that the problem, which was a real stumbling block for many years, had been finally solved, at least in principle. [Pg.321]

The best known application of the graphical method is that in which the variables are... [Pg.45]

The determination of k from Eq. (3-28), given a series of t, [A], pairs, requires a nonlinear least-squares program for the analysis of the data. A formidable calculation is thus made relatively simple. Otherwise, the graphical method suggested by Eq. (3-27) can be used. If the reaction is followed by an instrumental method, one can easily show that these readings, designated Y, are related to concentrations by... [Pg.51]

The reaction of nitrous acid with hydrogen peroxide1 illustrates the graphical method. Figure 4-3 shows a plot of absorbance against time, reconstructed from the reported parameters. The plot of In (Yt — Too) versus time consists of two portions. The linear segment at long times is characterized by a rate constant of 8.54 x 10"2 s-1. [Pg.74]

A single extrapolation of the ratio D/s measured at a series of very low concentrations would then suffice. The graphical method advocated above for handling osmotic and turbidity data could be applied here also for the purpose of obtaining (D/s)o = RT/M, Again, if a poor solvent is used, so that T2 is very small, D/s should be nearly independent of c over the dilute range. [Pg.306]

Equations 11.21 and 11.22 and the equilibrium relationship are conveniently solved by the graphical method developed by McCabe and Thiele (1925). The method is discussed fully in Volume 2. A simple procedure for the construction of the diagram is given below and illustrated in Example 11.2. [Pg.505]

Flaving minimized the enzyme concentration as described above, one can attempt to measure the IC50 of the compound at several enzyme concentrations, at and above the minimum, and then attempt to define A app by use of Equation (7.12) and the graphical methods described in Section 7.2. We have already discussed the limitations of this approach. Nevertheless, when the enzyme concentration can be varied over an appropriate range, relative to the Apapp, this approach can work well. [Pg.195]

As with the case for water minimization, the graphical methods used for effluent treatment and regeneration have some severe limitations. As before, multiple contaminants are difficult to handle, constraints, piping and sewer costs, multiple treatment processes and retrofit are all difficult to handle. To include all of these complications requires an approach based on the optimization of the superstructure. [Pg.616]

When more than two data points are available, the graphical method is much better to use than common averaging techniques. It gives one a visual picture of the fit of the data to 3.3.55. If one has several data points and estimates of the uncertainty in each point, a weighted least squares fit of the data would be appropriate. [Pg.63]

Step 4 Separation of Distortions of 1st and 2nd Kind. From Eq. (8.27) the graphical method for the separation of small lattice distortions of the first and the second kind is obvious. It is sketched in Fig. 8.5. In a plot of fo (r/L) vs. r/L the amount of lattice distortions of the second kind is determined from the intercept. Lattice distortions of the first kind are computed from the slope of the observed... [Pg.128]

The graphical method has the advantage of smoothing the data if the other methods are used it may be necessary to smooth the data first. Application of the difference method directly to the data in the above table will show good agreement in the lower part of the range but oscillations will be observed towards the top of the range. [Pg.107]

To determine the concentration using the graphical method draw a horizontal line from A — 0.443 to the least-squares line. Then construct a vertical line from that point to the concentration axis and read the value of the unknown. In this case it appears to be slightly less than 6.0 ppm. Using the least-squares equation we get... [Pg.133]

How the graphical method is most useful in determining the order of a reaction ... [Pg.44]

Bistability and switching are crucial concepts of cellular regulation [80, 98] and can often be detected using the graphical methods outlined above. See also [98,287] for several illustrative examples. For a number of pathways, transitions between different states were observed, either in silico, in vivo, or both. Examples include the glycolytic pathway [273, 294], the Calvin cycle [113, 125], and models of the human erythrocytes [295, 296],... [Pg.168]

Chan (Chapter 6) presents a simple graphical method for estimating the free energy of EDL formation at the oxide-water interface with an amphoteric model for the acidity of surface groups. Subject to the assumptions of the EDL model, the graphical method allows a comparison of the magnitudes of the chemical and coulombic components of surface reactions. The analysis also illustrates the relationship between model parameter values and the deviation of surface potential from the Nernst equation. [Pg.5]

Using a simple amphoteric model for the mineral surface, we have demonstrated the role specific chemical binding reactions of potential determining Ions In determining the electrical properties and thermodynamics of the oxide/solution interfaces. A by-product of our study Is that under appropriate conditions, an amphoteric surface can show marked deviations from ideal Nernstlan behaviour. The graphical method also serves to Illustrate the... [Pg.112]

This example will be solved using the graphical method. [Pg.536]

The Systematic Approach. The systematic approach for deriving rate equations was first devised by Fromm based on certain concepts advanced by Volkenstein and Goldstein. Its underlying principles, however, are more akin to the graphic method of King and Altman. The procedure to be described here is a modified method that includes the contributions from the aforementioned workers and from Wong and Hanes. ... [Pg.256]

Fit a straight line to the data by the graphical method previously described. [Pg.84]

It was concluded that within its limitations the relative pressure method is simpler and easier to apply than the graphical method, is less subject to errors in interpretation of graphical data, and extracts the max info from data points... [Pg.754]

EV from positive impulse data, the graphical method should be used in cases where the impulse vs distance curves for the expls being compared have the same slope... [Pg.755]


See other pages where The Graphical Method is mentioned: [Pg.85]    [Pg.86]    [Pg.120]    [Pg.426]    [Pg.280]    [Pg.501]    [Pg.383]    [Pg.49]    [Pg.125]    [Pg.129]    [Pg.141]    [Pg.111]    [Pg.202]    [Pg.55]    [Pg.76]    [Pg.230]   


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A Graphical Method for the Estimation of k and

Graphic methods

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