Column subtraction model

T. Baczek, R. Kaliszan, K. Novotan, and R Jandera, Comarpative characteristics of HPLC Columns based on quantitative structure-retention relationships (QSRR) and hydrophobic-subtraction model, /. Chromatogr. A 1075 (2005), 109-115. 

L. R. Synder, J. W. Dolan, and P. W. Carr, Review The hydrophobic-subtraction model of reversed phase column selectivity, /. Chromatogr. A 1060 (2004), 77-116. 

To better distinguish the contributions of polar interactions to retention, the LEER model was transformed into the so-called hydrophobic subtraction model (HSM) for RPLC, where the hydrophobic contribution to retention is compensated for by relating the solute retention to a standard nonpolar reference compound. This approach was applied to characterize more than 300 stationary phases for RPLC, including silica gel supports with bonded alkyl-, cyanopropyl-, phenylalkyl-, and fluoro-substituted stationary phases and columns with embedded or end-capping polar groups. The QSRR models can be used to characterize and compare the suitabihty of columns not only for reversed-phase, but also for NP and HILIC systems. 

The present chapter describes the hydrophobic-subtraction model of RP-LC column selectivity, for characterizing columns in terms of five fundamental column properties (H, S, A, B, C). Because the hydrophobic-subtraction model can predict retention (values of k) within a few percent, coltunns with sufficiently similar values of H, S, etc. should provide equivalent separation for any sample. Similarly, columns with very different values of H, S, etc. should provide quite different separation. Several examples of the use of values of H, S, etc. for the selection of similar columns have been reported, some of which are discussed below. The present chapter also compares the selectivity of different column types (e.g., alkyl-silica columns, phenyl columns, cyano columns, etc.) and summarizes values of H, S, etc. for several commercial Cjg columns. 

The Subtraction Model of Reversed-Phase Column Selectivity... 

We have noted above that hydrophobic interaction of the solute and column represents the main contribution to RP LC retention and column selectivity. The subtraction model assumes that we first subtract the major contribution of hydrophobicity to RP LC retention, in order to better see contributions to retention from remaining solute-column interactions. The further analysis of these minor contributions then leads to a general equation for RP LC retention and column selectivity [9, 13] ... 

Mixing or splitting of flows in the plant model is handled by having flows (columns) add or subtract material in given ratios from special tanks (rows) set... 

The stochastic model applies to processes involving the stationary phase. To analyze the chromatogram, we need to subtract contributions to peak broadening from dispersion in the mobile phase and extra-column effects such as finite injection width and finite detector volume. These effects account for the width of the unretained peak. To subtract the unwanted effects, we write... 

Gas chromatograms were obtained on a twin Hewlett-Packard model 5750 Research Chromatograph. The dual columns were 6-m lengths of copper tubing 3 mm in diameter, packed with 3% OV-1 on Chromosorb G-HP (methylsilicone on calcined diatomaceous earth). Samples were introduced as about 10% solutions in carbon bisulfide. Detection was by hydrogen-flame ionization, the non-sample contribution from the idle column being subtracted from the total contribution of the active column to provide a sample chromatograph corrected for extraneous ionization. 

Mujtaba (1989) used CMH model to simulate the operations considered by Domenech and Enjalbert (1974). Since the overall stage efficiency in the experimental column was 75%, the number of theoretical plates used by Mujtaba was 3. The column was initialised at its total reflux steady state values. Soave-Redlich-Kwong (SRK) model was used for the VLE property calculations. Vapour phase enthalpies were calculated using ideal gas heat capacity values and the liquid phase enthalpies were calculated by subtracting heat of vaporisation from the... 

The B score (Brideau et al., 2003) is a robust analog of the Z score after median polish it is more resistant to outliers and also more robust to row- and column-position related systematic errors (Table 14.1). The iterative median polish procedure followed by a smoothing algorithm over nearby plates is used to compute estimates for row and column (in addition to plate) effects that are subtracted from the measured value and then divided by the median absolute deviation (MAD) of the corrected measures to robustly standardize for the plate-to-plate variability of random noise. A similar approach uses a robust linear model to obtain robust estimates of row and column effects. After adjustment, the corrected measures are standardized by the scale estimate of the robust linear model fit to generate a Z statistic referred to as the R score (Wu, Liu, and Sui, 2008). In a related approach to detect and eliminate systematic position-dependent errors, the distribution of Z score-normalized data for each well position over a screening run or subset is fitted to a statistical model as a function of the plate the resulting trend is used to correct the data (Makarenkov et al., 2007). 

The consequence for the model matrix is that the average of each squared variable, x, is subtracted fi om tbe elements of tbe square columns in tbe original model matrix. The constant term, bg, in the transformed model is related to tbe intercept term, bg, in the original model by tbe relation... 

PCA is a least square method and therefore its results depend on data scaling. The initial variance of a column variable partly determines its importance in the model. In order to avoid the problem of over- or under-representation of variables, column variables are scaled to unit variance before analysis. The column average is then subtracted from each variable, which, from a statistical point of view, corresponds to moving the multivariate system to the center of the data, which becomes the starting point of the mathematical analysis. The same auto-scaUng and centering procedures are applied in PLS discriminant analysis. 

Offsets are often handled by first centering the data and subsequently fitting the bilinear model to the centered data as also shown in the previous plots. If the data are centered by subtracting the column-average from every element in the column, this is referred to as centering across the first mode. Mathematically it can be expressed as... 

Note 17—Some gas chromatographs have an algorithm built into their operating software, which causes a mathematical model of the column bleed profile to be stored in memory. This profile is subtracted automatically from the detector signal on subsequent runs to compensate for the column bleed. 

Such measurements have indeed been performed and interpreted in terms of the model [6]. In order to correct for the impedances of the line and the electrodes, which are both in series with the system to be measured, the complex impedance of the specimen was determined by vectorial subtraction of the total impedances measured for columns of different height [6-8]. 

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