Ratio Plots

These easily constructed diagrams consist entirely of straight lines. The mathematics was given in Chapter 5 for polyprotic acid-bases. We may choose the ratio of each species to any one of the others as a common basis. Let us illustrate with the A1(III)-F system. First with the ratios to Al(III) aquo ion, we use the P expressions directly, from set (5-2). Taking logs gives the convenient plotting form  

The log equations give lines of slopes 1-6 and intercepts at logP, log[L] = 0 (1 M F ) for plots of logR vs. log [L]. We obtain a diagram that is somewhat easier to read if we take one of the central species as the reference for our ratios. It is a simple matter to convert our set of equations above to the new ratios. Using AIF3 as the reference. 

This conversion may impress the reader with the useful point that any species in step equilibrium can be expressed as a ratio to any other in terms of the constants and only one other variable, commonly log [L] or the pH. 

Although the maxima are not clearly visible, these diagrams can serve many purposes of the a diagrams. For example, a trace of Al(III) in a natural water containing lO M F should be about half AIF3 and half AIF2 The relative amounts of the other species can be read off as fractions of the AIF3 as follows AP 10 , AIF 10 ,  

Mark M., Elementary Coordination Chemistry. Prentice-Hall, Englewood Cliffs, New Jersey, 1964. 

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Mole-ratio plots used to determine the stoichiometry of a metal-ligand complexation reaction. 

Another TSK combination (precolumn -I- PWM -I 6000 -I 5000 -I- 4000 -I-3000) was tested on differences in separation performance between individual narrow distributed samples and mixtures of several narrow distributed samples. The result is summarized in Eig. 16.31 within experimental error the summed chromatograms (theory) of four narrow distributed glucans (dextran) match perfectly with the experimentally determined chromatogram of the mixture. The (theory/experimental) ratio, plotted for quantification of the match, in-... 

X 0.75 cm) Ve i = 28 ml = 50 ml eluent 0.05 M NaCI flow rate 0.80 ml/min detection Optilab 903 interferometric differential refractometer applied sample mass/volume 200 /tl of 2-mg/ml aqueous solutions sum of individual chromatograms (theory —) and (theory/experimental) ratio (—) plotted for quantification of deviations in separation performance between narrow distributed samples and broad distributed samples. 

Using either of the above approaches we have measured the thermal rate constants for some 40 hydrogen atom and proton transfer reactions. The results are tabulated in Table II where the thermal rate constants are compared with the rate constants obtained at 10.5 volt cm.-1 (3.7 e.v. exit energy) either by the usual method of pressure variation or for concurrent reactions by the ratio-plot technique outlined in previous publications (14, 17, 36). The ion source temperature during these measurements was about 310°K. Table II also includes the thermal rate constants measured by others (12, 13, 33, 39) using similar pulsing techniques. 

Figure 14. Area ratios plotted to compare with homopolymer calibration...

Common Clinical Trial Graphs 200 Scatter Plot 200 Line Plot 201 Bar Chart 202 Box Plot 203 Odds Ratio Plot 203... 

As a part of logistic regression analysis, odds ratio plots are an excellent way to see how much more likely a condition is to exist based on the presence of another condition. Just by glancing at an odds ratio plot, you can see whether an independent variable is significant to the dependent variable. For instance, if the odds ratio confidence interval does not cross the value of 1, then the independent variable odds ratio is significant. Examine the following graph. 

The following is an example of an odds ratio plot. It shows the odds ratios for clinical therapy, race, gender, and baseline pain score with regard to the overall clinical success of a patient. 

Here is the SAS program that creates the preceding graph. It is a bit complex, because SAS/GRAPH does not provide horizontal box plots and this is typically what is desired for odds ratio plots. So, this sample program relies extensively on the Annotate facility to produce the plot. Notes follow the program. 

Program 6.7 Creating an Odds Ratio Plot Using PROC GPLOT... 

CREATE THE ODDS RATIO PLOT. THIS IS DONE PRIMARILY THROUGH THE INFORMATION IN THE ANNOTATION DATA SET. PUT A HORIZONTAL REFERENCE LINE AT 1 WHICH IS THE LINE OF SIGNIFICANCE. proc gplot... 

Fig. 3. Schematic of peak purity determination by using the ratio-plot method.

Figure 3. Monthly oxygen and deuterium isotope ratios plotted vs. temperature...

Fig. 8.28. Europium and thorium to iron ratios plotted against metallicity [Fe/H], Curves represent the model predictions as in Table 8.2 and the symbols represent observational results by different authors. After Pagel and Tautvaisiene (1995).

Fig. 8.29. Europium to iron ratios plotted against metallicity [Fe/H] according to the model of supernova-induced star formation, after Tsujimoto, Shigeyama and Yoshii (1999). Grey scales represent predicted stellar surface densities in the ([Fe/H],[Eu/Fe]) plane convolved with a Gaussian with o = 0.2dex for Eu/Fe and 0.15 dex for Fe/H, and symbols show observational data from various authors. The inset shows the unconvolved predictions.

Fig. 8.30. Main s-process element to iron ratios plotted against metallicity [Fe/H] according to the analytical model by Pagel and Tautvaisiene (1995), compared to observational data. The s-process begins to contribute, superimposed on a pure r-process contribution, already at [Fe/H] = —2.5 ( >A = 0.01 A 0.3 Gyr), followed by a more delayed s-process that sets in at [Fe/H] = -0.65 ( >A = 0.8 A 2 Gyr, compared to 1 Gyr for iron). The large scatter displayed by strontium is probably real. After Pagel and Tautvaisiene (1997).

Fig. 10.5. Logarithmic differential Th/Eu ratios plotted against stellar age. The crosses represent the average of two UMP r-process-rich stars CS 22892-052 and HD 115444 (Westin et al. 2000), and a third one BD +17° 3248 (Cowan et al. 2002) which are a kind of Rosetta stone for the r-process, assuming an age of 13 1 Gyr. Curves show predictions from various models discussed in the text.

Figure 9.11 Bivariate isotopic ratio plot of Cypriot and Anatolian ore deposits.

In a ratio-ratio plot (Figure 3.5), typically r/2 vs r/1, the ratios combine as... 

For ratios with the same denominator, e.g., Fe for Fe/ Fe and FePFe, the slope of the noise correlation line in a ratio-ratio plot is the ratio of the standard deviations [Pg.134]

Swelling profile determinations of the hydrogel films are provided elsewhere [35]. The results were given as the volume swelling ratio plotted... 

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