Window diagram plot

Figure 2.13. Plot of the gas-liquid partition coefficients (Ka and Kb) on two stationary phases to estimate the gas-liquid partition coefficients for all solutes at any intermediate mixed-phase composition and the resulting window diagram plot of the separation factor (a) as a function of the volume fraction of the B stationary phase. (From ref. [205] Elsevier)...

A log (k) versus IIT plot very nicely demonstrates that we can predict accurate peak retention factors as a function of isothermal column temperature. However, the separation of those peaks is of more interest for optimization. We can observe from the plot that the naphthalene and dodecane peaks will merge between 100 and 110°C, but it is difficult to imagine at what temperatme or temperatures all of the peaks exhibit maximum separation. In order to better visualize these relationships, Laub and Purnell (9,10) described a window diagram plot of the peak s separation factors (a), which can be constructed from retention factor data such as presented in Figure 4.9 according to... 

FIGURE 4.10 Window diagram plot of separation factor (a) vctsus column temperature. Derived from the retention factor data in Figure 4.9 [Reprinted from LC/GC Magazine with permission of Advanstar Publication (8).]... 

Prus and Kowalska [75] dealt with the optimization of separation quality in adsorption TLC with binary mobile phases of alcohol and hydrocarbons. They used the window diagrams to show the relationships between separation selectivity a and the mobile phase eomposition (volume fraction Xj of 2-propanol) that were caleulated on the basis of equations derived using Soezewiriski and Kowalska approaehes for three solute pairs. At the same time, they eompared the efficiency of the three different approaehes for the optimization of separation selectivity in reversed-phase TLC systems, using RP-2 stationary phase and methanol and water as the binary mobile phase. The window diagrams were performed presenting plots of a vs. volume fraetion Xj derived from the retention models of Snyder, Schoen-makers, and Kowalska [76]. 

Figure 5.16a is a plot of the retention against the composition. These retention lines (surfaces) are required for the construction of the actual window diagram (figure 5.16b). In figure 5.16a the distribution coefficient (K) is shown on the vertical axis. If the total volume of the stationary phase is kept constant, then the phase ratio is constant and K is directly proportional to the capacity factor k (eqn.1.10). On the horizontal axis is the mixing ratio of the two components of the stationary phase (stationary phases 5 (left) and A (right). 

Hsu et al. [553] applied the window diagram method to the optimization of the composition of abinary mobile phase in RPLC. However, a straight line was not obtained by plotting 1 /k vs. composition and therefore more than two experimental locations were required. 

A plot of x versus a for the hypothetical four-component mixture yields a diagram like that shown in Figure 8.186, if a is always defined to keep its value greater than 1. Laub and Purnell called this type of diagram a window diagram, and that term has persisted. The best mixture of X and... 

To recap, first one needs to determine the partition coefficients for each of the analytes on each of the two liquid phases—each chosen because it is effective in separating most of the sample analytes when used alone. Second, one plots these K values versus < > by using the two K values for each analyte at 4>a- = 0 and < >a- = 1.0 and joining these two points with a straight line. Then, the window diagram is drawn using <(> values calculated from Eq. (10) at various a values. 

Figure 8.18. Plots of (a) KR versus and (b) window diagram for four hypothetical analytes. Reprinted from ref. 44 with permission.

Figure 18.14 Window diagram. The separation was performed with a linear gradient from 0 to 45% B and the optimum runtime needs to be found out. (a) Gradient in 15 min (b) gradient in 45 min with some elution orders reversed (c) window diagram calculated from the initial two experiments with a linear relationship between retention time and %B assumed the plot shows the resolution / of the peak pair which is critical under the respective conditions it is necessary to use long gradient runtimes to obtain a good resolution (d) optimized chromatogram with 0-45% B in 80 min, but separation is already finished after 45 min and 25% B.

On the following diagram, plot the steady-state current-potential curve of a system containing an inert working electrode dipped in a deaerated acidic aqueous solution pH = 0), with no Fe " ions and an amount of Fe " ions befitting the existence of a limiting current. The reference electrode is a saturated calomel electrode. It will be assumed that the electrochemical window is determined by the fast half-reactions of water. 

Window diagram Window diagrams, developed by Laub and Purnell for optimizing the composition of mixed stationary phases in gas chromatography, can be used for optimizing mobile phase composition in LC. From two initial experiments (if a linear relationship is assumed between log k and mobile phase composition) or more (in the case of a quadratic relationship), the retention models are calculated for all solutes, and a response function (selectivity between every possible pair of solutes) is calculated and plotted versus the mobile phase composition. Areas or windows in which all solutes are separated can be located graphically. No particular effort of computation is required in such a procedure. 

Single-parameter optimization employs several experiments at preselected values of the optimized parameter (such as the concentration of the strong solvent in a binary mobile phase, pH, temperature) to predict the resolution as a function of the optimized parameter using empirical or simple model-based calculations. Then, plots are constructed (the window diagrams ) in which the range of the optimized parameter is searched for the value that provides the desired resolution for all adjacent bands in the chromatogram in the shortest time. An example of a window diagram (Pig. 5) illustrates the approach adopted for the optimization of a binary mobile phase in NPLC. 

A series of thirteen pesticides were applied for the optimization of the mobile phase composition [8], A window diagram which plots the AR/vs. mobile phase composition for all seventy-eight pairs of the thirteen pesticides is shown in Figure 1. 

FIGURE 5.17 Plots of ko versus a for an arbitrary five-component mixture (a) and the resulting resolution window diagram (b). Vertical broken lines indicate a values that result in coelution of the corresponding component pairs. The value of ao gives the volume fraction of stationary phase A that will give the greatest resolution of the critical pair. 

Optimization procedures are used to determine which of these (1)a values will obtain the greatest resolution of the critical pair (36). Figure 5.17b shows a resolution window diagram where the resolution Rs of the critical pair is plotted against the volume fraction of phase A. The zero-resolution points are defined by the (t)A values that result in the crossing of the various plot pairs in Figure 5.17a. Between the zero-resolution points are windows of various amplitude, which... 

FIGURE 5.21 High-speed isothermal separation of a 20-component mixture using a pressure-tunable column ensemble. The plots of band position versus time were obtained from the retention factors for the mixture components on the two separate colimms along with the column dimensions and the inlet, outlet and junction point pressures. A window diagram was used to determine the junction point pressure for the complete separation of the first 14 components. Compounds are 1, n-pentane 2, methyl alcohol 3, 2,2-dimethylbntane 4, 1,1,1-trichloroethane 5, cyclopentane 6, w-hexane 7, w-propyl alcohol 8, cyclohexane 9, benzene 10, n-heptane 11, l,2-dichloropropane 12, toluene 13, n-bntyl alcohol 14, w-octane 15, 2-hexyl alcohol 16, n-pentyl alcohol 17, ethylbenzene 18, m-xylene 19, w-nonane 20, o-xylene. 

A window diagram is a visual optimization technique used to find the operational parameter that gives, in this case, the best resolution. Each line represents the resolution between two peaks in a chromatogram. Since there are six lines, this indicated that there are three analytes being considered and the resolution for AB, BC, and AC are plotted. Maximal resolution is at pH = 4.5 and gives an Rs of 1.6. 

The command cloop is used to find the closed-loop transfer function. The command max is used to find the maximum value of 20 logio (mag), i.e. Mp and the frequency at which it occurs i.e. tUp = uj k). A while loop is used to find the —3 dB point and hence bandwidth = ca (n). Thus, in addition to plotting the closed-loop frequency response gain diagrams,/ gd29.7 will print in the command window ... 

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

This factor corrects for the effect of flow through the baffle window, and is a function of the heat-transfer area in the window zones and the total heat-transfer area. The correction factor is shown in Figure 12.33 plotted versus Rw, the ratio of the number of tubes in the window zones to the total number in the bundle, determined from the tube layout diagram. 

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