Variance contributions

Figure 3.8 Pareto chart showing variance contribution of each characteristic to the final assembly variance (for the paper-based analysis)...

Figure 3 Pareto chart showing the variance contribution of each design variable in the tension bar problem...

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

The dispersion of a solute band in a packed column was originally treated comprehensively by Van Deemter et al. [4] who postulated that there were four first-order effect, spreading processes that were responsible for peak dispersion. These the authors designated as multi-path dispersion, longitudinal diffusion, resistance to mass transfer in the mobile phase and resistance to mass transfer in the stationary phase. Van Deemter derived an expression for the variance contribution of each dispersion process to the overall variance per unit length of the column. Consequently, as the individual dispersion processes can be assumed to be random and non-interacting, the total variance per unit length of the column was obtained from a sum of the individual variance contributions. 

There are four major sources of extra-column dispersion which can be theoretically examined and/or experimentally measured in terms of their variance contribution to the total extra-column variance. They are as follows ... 

Example 48 The result is thus CL(/4) = 7.390 0.028 mM/g, and should be either left as given or rounded to one significant digit in C1 7.39 0.03 The %-variance contributions are given in parentheses (Eq. (4.24)). Note that the analytical method with the best precision (titrimetry), because of the particular numerical constellation, here gives rise to the largest contribution (77%). 

This is a statistical test designed by Malinowski [43] which compares the variance contributed by a structural eigenvector with that of the error eigenvectors. Let us suppose that is the variance contributed by the last structural eigen-... 

An expression can be written which lumps together RTMT in both mobile and stationary phase and sums the variance contributions which can be written as... 

Unfortunately, the magnitude of the variance contribution from each source will be different and the ultimate minimum size of each is often dictated by the limitations in the physical construction of of the different parts of the apparatus and consequently not controllable. It follows that equipartition of the permitted extra column dispersion is not possible. It will be seen later that the the maximum sample volume provides the maximum chromatographic mass and concentration sensitivity. Consequently, all other sources of dispersion must be kept to the absolute minimum to allow as large a sample volume as possible to be placed on the column without exceeding the permitted limit. At the same time it must be stressed, that all the permitted extra column dispersion can not be allotted solely to the sample volume. 

As the extra column dispersion becomes large, the column diameter must be increased, to ensure that its variance contribution remains no more than 0% of the that of the e/utedpeak. 

One of the key concerns of analytical science is how good are the numbers produced . Even with an adequately developed, optimised and collaboratively tested method which has been carried out on qualified and calibrated equipment the question remains. Recently it has become fashionable to extend the concepts of the physical metrology into analytical measurements and to quantify confidence in terms of the much more negative uncertainty.It is based on the bottom-up principle or the so called error budget approach. This approach is based on the theory that if the variance contributions of all sources of error involved in analytical processes then it is possible to calculate the overall process... 

The description of an object in the sense of environmental investigation may be the determination of the gross composition of an environmental compartment, for example the mean state of a polluted area or particular location. If this is the purpose, the number of individual samples required and the required mass or size of these increments have to be determined. The relationship between the variance of sampling and that of analysis must be known and both have to be optimized. The origin of the variance of the samples can be investigated by the study of variance contribution of the different steps of the analytical process by means of the law of error propagation (Eq. 4-21) according to Section 4.3.4. 

Since the variance contributions a1 (in length units) of longitudinal diffusion, injection, and detection are not correlated, their variances can be added... 

Band broadening arises from three principal mechanisms, one of which depends on the mean velocity of the carrier and two of which are independent of . The latter two represent nonidealities in instrument and sample. The total variance of an eluting peak is the sum of variances contributed by each bandbroadening mechanism. Expressed as plate height H, which is the total variance divided by column length L (12), the zone broadening is described by... 

The program used in this study was a modified version of Dixon s BMD08M factor analysis with varimax rotation (12), The principal components analysis was conducted using covariance matrices. Five factors were created from the data set. Examination of the individual proportion of the total variance contributed by each of the factors demonstrated that 96.3% of the total variance could be accounted for by the first three factors. These three factors were used in the following cluster analysis. 

The variance due to the detection system is additive with the other variances contributing to peak broadening. The maximum effective volume of the detection system should be no greater than K/2>/n, a limitation identical to that for sample injection. Thus, in analogy to sample injection, the effects of detector volume are most significant when a peak with small retention volume is eluted. 

In many HPLC detectors the column eluent flows through a cell within which some physicochemical interaction with the solutes takes place. Exceptions to this include mass spectrometric detectors where the eluent has to be vaporised before introduction into the vacuum system, or the evaporative mass detector, where again the eluent is heated and vaporised before undergoing analysis by light scattering. Often a number of flow cell options are offered by detector manufacturers, and this reflects the effect of the detection volume on the detected peak. The total peak variance, a, is the sum of all the variance contributions. 

In a packed column, the individual solute molecules will describe a tortuous path through the interstices between the particles, and some will randomly travel shorter routes than the average and some will travel longer routes. Consequently, those molecules taking the shorter paths will move ahead of the mean and those that take the longer paths lag behind the mean which will result in band dispersion. Van Deemter et al. derived the following function for the multipath variance contribution (oif) to the overall variance per unit length of the column (o ) ... 

Van Deemter derived the following expression for the variance contribution by the resistance to mass transfer in the mobile phase (cr ) to the overall variance per unit length of the column aff. 

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