Mid-point method

Compound (1) was prepared conventionally from 4-chlorobenzoyl fluoride and SbF5 and dissolved in EtN02. The solution was distributed into phials by the usual tipping device [4] the content of the phials was obtained by the mid-point method [4, 5], and the initiator concentration was determined as described [2],... 

A Tube Cutter for Use in the Mid-point Method of Filling Phials in Vacuo, S.D. Pask, P.H. Plesch, and M.DiMaina, Chemistry and Industry (London), 1981, 329-330. 

A very elegant solution to this problem is the mid-point method (Rutherford, 1962), and its subsequent developments (Pask, Plesch and DiMaina, 1981). The method is suitable for weighing small quantities (ca. 0.1-4.0 g) of liquids, solutions or solids in phials filled under vacuum and it is described in detail below, because although now quite old, it does not seem to be very widely known. 

Fig. 3.4. The mid-point method for determining the content of a breakable phial. For explanation see text.

Although the mid-point method requires practice, it is not as time-consuming as it may appear at first sight, since it is possible to fill batches of up to 20 phials in a single operation. This has the distinct advantage that the impurities are the same, at least from this source, for a particular set of experiments in which these phials are used. 

Plesch s and Sigwalt s schools have established a sound tradition in building specific devices for hi vacuum handling of precursor solutions and cationic polymerisations, Some of these are highly sophisticated, others simple but extremely useful, such as Rutherford s mid-point method for determining the amount of substance contained in a vacuum-filled breakable phial ... 

The efficiency of the methods outlined above has been tested by calculating the intermolecular Coulomb energies and forces for a series of water boxes (64,128,256, 512 and 1024) under periodic boundary conditions [15, 62], The electron density of each monomer is expanded on five sites (atomic positions and bond mid-points) using two standard ABSs, A2 and PI.These sets were used to fit QM density of a single water molecule obtained at the B3LYP/6-31G level. We have previously shown that the A1 fitted density has an 8% RMS force error with respect to the corresponding ab initio results. In the case of PI, this error is reduced to around 2% [15, 16], Table 6-1 shows the results for the 5 water boxes using both ABSs (Table 6-7). 

For routine use by unskilled personnel, the best method of measuring velocities of detonation is one due to Dautriche. The principle is illustrated in Fig. 6.5. The two ends of a length of detonating fuse are inserted in the explosive under test at a known distance apart. The mid point of the piece Mark will appear here... 

The graph is obtained by plotting Y,- against Y, results for each of the ten laboratories. The axes are drawn such that the point of intersection is at the mean values for Y, and 7/. As a single method is used in the trial, the circle represents the standard deviation of the pooled Y and Y data. The plot shows the predominance of systematic error over random error. Ideally, for bias-free data (i.e. containing no systematic error) the points would be clustered around the mid-point with approximately equal numbers in each of the four quadrants formed by the axes. In practice the points lie scattered around a 45° line. This pattern has been observed with many thousands of collaborative trials. 

Measurement of the chemical shift. When a nucleus (or set of equivalent nuclei, see below) gives rise to a single absorption peak in the spectrum it is a simple matter to determine the chemical shift from a measurement of its separation from the reference peak. In -spectra when coupling of the nucleus results in a first order multiplet (see below) measurement of the separation from the reference peak must be made to the mid-point of the multiplet. In more complex spin-spin interactions it is not possible to determine directly the chemical shift by measurement in this way, and resort must be made to the application of mathematical methods of analysis. 

The direct plot method is most commonly used to determine the end point. A titration curve is drawn by plotting the electrode potential in the Y-axis against mL titrant added in the X-axis. Near the end point, there is a sharp increase in potential. The mid-point of the steeply rising portion of the curve is taken as the end point of the titration (Figure 1.6.4). 

Discrete dipole approximation. For particles with complex shape and/or complex composition, presently the only viable method for calculating optical properties is the discrete dipole approximation (DDA). This decomposes a grain in a very big number of cubes that are ascribed the polarizability a according to the dielectric function of the dust material at the mid-point of a cube. The mutual polarization of the cubes by the external field and the induced dipoles of all other dipoles is calculated from a linear equations system and the absorption and scattering efficiencies are derived from this. The method is computationally demanding. The theoretical background and the application of the method are described in Draine (1988) and Draine Flatau (1994). 

This is a second-order Runge-Kutta method (Finlayson, 1980), sometimes called the midpoint rule. The first step is an approximation of the solution halfway between the beginning and ending time, and the second step evaluates the right-hand side at that mid-point. The error goes as (At), which is much smaller than that achieved with the Euler method. The second-order Runge-Kutta methods (there are several) also have a stability limitation. 

DCS at 20°C/min. Diverse phase diagrams Samples prepared by casting from CHClj DSC measurements at 20°C/min T < 530K T taken at half-height Samples cast from DMF into Petri dish DSC at 20°C/min to 190°C. Miscibility only for the GMA < 35.7 wt% in PMMA-GMA Samples hot cast from o-dichlorobenzene into Petri dish DSC at 20°C. Samples annealed at 200°C for 5 min THF solution precipitated by MeOH. DSC at 10°C/min T taken as mid-point in the inflection Samples either cast, precipitated from solution, or melt mixed at 250°C. DSC at 20°C/min T from second scan. Solution cast samples gave two T, whereas the other methods only one Melt mixing at 300°C. DSC at 20°C/min. T taken at onset... 

Naturally, the performance of the numerical method could be improved by dividing each bond at a point other than the mid-point. This choice could be made on the basis of the relative electronegativities of the atoms at each end of a bond, for example, so that the polyhedra are chosen to contain similar amounts of electron density. 

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