Simple Plots

For a vector x with n entries, and a vector y with n entries, plot them in one of four ways  

Plot only one column of a matrix, here the second column. 

---

The comparison plot (cf. Section 2.10) is a simple plot of the uptake per unit mass of the experimental material against that of a reference sample at the same relative pressure. It may be regarded as an alternative to the a,-plot, in which the abscissae are actual adsorptions on a sample arbitrarily taken as a standard. 

The solubihty coefficients are more difficult to predict. Although advances are being made, the best method is probably to use a few known solubility coefficients in the polymer to predict others with a simple plot of S vs ( poiy perm Y where and are the solubility parameters of the polymer and permeant respectively. When insufficient data are available, S at 25°C can be estimated with equation 19 where k = 1 and the resulting units of cal/cm are converted to kj /mol by dividing by the polymer density and multiplying by the molecular mass of the permeant and by 4.184 (16). 

Fitting Simple Functions. In many engineering appHcations there may be no apparent theoretical basis for the relationship of two variables or the relationship may be too complex to apply. Thus the search for a correlating equation form may at first be along empirical lines. A simple plot of the data in ordinary Cartesian coordinates gives an immediate indication of the essential form of the data. 

Choice of initial conditions. To give a very obvious example, in Chapter 2 we saw that a second-order reaction A -I- B —> products could be run with the initial conditions Ca = cb, thus permitting a very simple plotting form to be used. For complex reactions, it may be possible to obtain a usable integrated rate equation if the initial concentrations are in their stoichiometric ratio. 

The models are even more helpful to the process engineer than simple plot plans. The advantages are multiplied, as with models the process engineer can study as well as solicit the advice of other engineers in visualizing a processing condition. 

Sometimes forms deviating from simple linear relations were used in this early work, particularly the relation of log A to E (31,36), which was derived theoretically (29) but later rejected (37, 38). Otherwise, the quantity log (kM ) was plotted instead of log k, M" being the sum of reciprocal masses of reacting particles according to collision theory (33), and this correction was also later abandoned (34). The two modifications mentioned, in practice, have little influence on the shape of the graph, and the simple plot of E versus log A (or AH versus AS) is now preferred. 

We have found that data interpretation other than simple plotting from texture studies requires a sophisticated array of software tools. These include least squares curve resolving, non-linear least squares fitting, 2-dimensional data smoothing, numerical quadrature, and high-speed interactive graphics, to mention only a few. 

Most hydrophobic substances have low solubilities in water, and in the case of liquids, water is also sparingly soluble in the pure substance. Some substances such as butanols and chlorophenols display relatively high mutual solubilities. As temperature increases, these mutual solubilities increase until a point of total miscibility is reached at a critical solution temperature. Above this temperature, no mutual solubilities exist. A simple plot of solubility versus temperature thus ends at this critical point. At low temperatures near freezing, the phase diagram also become complex. Example of such systems have been reported for sec-butyl alcohol (2-butanol) by Ochi et al. (1996) and for chlorophenols by Jaoui et al. (1999). 

FIGURE 3.12 Typical simple plot used to determine /M from scattering data. 

Already in this simple plot we see that are separated from each other and that ) lie far from the others. 

Figure 3.4 Radar plot to demonstrate the physicochemical characteristics of Lipitor. The simple plot demonstrates that two parameters fall within the optimal zone (logP and PSA), while three others slightly exceed the boundaries molecular weight, water solubility and number of rotatable bonds.

Generally, the energy of the olefin n -level is more sensitive to substituent effects than the 7r-level (see Table IV for example). Therefore, often a simple plot of the equilibrium constant for the metal-olefin complex vs the energy of the lowest unoccupied olefin molecular orbitai gives a reasonable straight-line graph. Such a plot reported by Tolman (220) for the equilibrium... 

A better approach to improve visualisation and decision-making is to standardise the residuals. A very simple approach is just to subtract the mean of the overall set of residuals and divide by their standard deviation. Brereton suggested that a simple plot of these standardised residuals is sufficient in... 

Thus by doing these experiments at two or more values of p, one can obtain two or more values of which can be used to obtain KAB and KA2. In this situation one must use a limiting intercept there is no simple plot available as there was from Equation 41. 

Figure 3.4B displays the data in terms of normalized i versus t 1/2. For the le and 2e cases, the two straight lines are simple plots of the Cottrell equation (Eq. 3.6) for which the slope is determined by the respective n values. The shape... 

Using the relationship between force and potential energy discussed earlier, we can represent the nuclear force in terms of a simple plot of the nuclear potential energy as a function of distance to the center (Fig. 5.1). Since low-energy particles... 

A simple plot of FJF against [H] from titration of the quenching host into a guest solution should yield a straight line of slope Kn. Common fluorescent guests such as 8-anilino-l-naphthalenesulfonate (ANS, 1.7) may be used to probe complexation ability of various hosts in this way. 

Just as Eq. (5) must be substituted into Eq. (1) or (4) to describe the variation of the with D/A separation due to varying electronic coupling, so too the dependence of k on s variation due to D/A separation should also be explicitly modeled. In fact it is not proper to assume that is independent of r (even though this is done many times). Clearly, when both HAB and s are functions of r, a simple plot of lnCi ) vs. r cannot be used to evaluate (3. 

Graphical integration requires simple plots of both (dZ/6T)P/P and (Z - 1)/P vs. P. Values of (Z - l)/P are calculated directly from the given compressibility-factor data at 360 K, The quantity (dZ/dT)P/P requires evaluation of the partial derivative... 

As mentioned before, the symmetry properties of the one-electron wave function are shown by the simple plot of the angular wave function. But, what are the symmetry properties of an orbital and how can they be described We can examine the behavior of an orbital under the different symmetry operations of a point group. This will be illustrated below via the inversion operation. 

Particle Diameter, jum Figure 2.1a Simple plot of distribution data. 

Most aerosols are polydisperse when formed, some more than others. For example, an examination of sawdust would reveal particles of various sizes, as would that of any material formed by attrition. Since raindrops could grow by condensation or by a series of collisions with other drops, they would also be expected to be polydisperse. In fact, monodisperse aerosols are very rare in nature, and when they do appear, generally they do not last very long. Some high-altitude clouds are monodisperse, as are some materials formed by condensation. Sometimes it is satisfactory to represent all the particle sizes by only a single size. Other times more information is needed about the distribution of all particle sizes. Of course, a simple plot of particle frequency versus size gives a picture of the sizes present in the aerosol, but this may not be enough for a complete quantitative analysis. 

A simple plot of the Carothers equation illustrates how functional group conversion governs step polymerization chemistry (Figure 2). Conversions of above 99% are required for high polymer to form. 

Reversible reactions are best considered as consisting of forward and reverse steps. Such step combinations will be examined in Section 5.1. Simple plots suited for evaluation exist only for first order-first order reactions A — P (see Figure 5.2). In more complex situations, the best approach often is to isolate the forward from the reverse reaction. This can be done to some extent with experiments that measure initial rates. 

---