Sampling terms

In statistical analysis it is necessary to use a quantity called degrees of freedom, designated henceforth as d.f. This quantity allows for a mathematical correction of the data for constraints placed upon the data. In this case, in the calculation of the estimated standard deviation, the number of observations n is fixed and the estimated standard deviation is calculated from the mean. Only (n - 1) of the observations or sample terms can be varied and the last term is fixed by X and n. Thus, there are only ( — 1) degrees of freedom in estimating the standard deviation from a sample of the population data. 

The most important interaction in NMR results from the application of a large external magnetic field, Bq, to the sample. Termed the Zeeman interaction, its effect on the normally degenerate nuclear spin energy levels is to cause them to split. The Zeeman levels are perturbed by local fields generated by the motion of electrons in the vicinity... 

The spectroscopic methods, NMR and mass spectrometry for predicting cetane numbers have been established from correlations of a large number of samples. The NMR of carbon 13 or proton (see Chapter 3) can be employed. In terms of ease of operation, analysis time (15 minutes), accuracy of prediction (1.4 points average deviation from the measured number), it is... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

Since the numerator of the right-hand side of (1.81) has the dimensions of volume and the denominator those of surface area, the quantity is termed the volume-surface mean diameter of the sample. 

Whereas at the lower end of its range mercury porosimetry overlaps with the gas adsorption method, at its upper end it overlaps with photomicrography. An instructive example is provided by the work of Dullien and his associates on samples of sandstone. By stereological measurements they were able to arrive at a curve of pore size distribution, which was extremely broad and extended to very coarse macropores the size distribution from mercury porosimetry on the other hand was quite narrow and showed a sharp peak at a much lower figure, 10nm (Fig. 3.31). The apparent contradiction is readily explained in terms of wide cavities which are revealed by photomicrography, and are entered through narrower constrictions which are shown up by mercury porosimetry. 

Although the terms solute and solution are often associated with liquid samples, they can be extended to gas-phase and solid-phase samples as well. The actual units for reporting concentration depend on how the amounts of solute and solution are measured. Table 2.4 lists the most common units of concentration. 

The first important distinction we will make is among the terms analysis, determination, and measurement. An analysis provides chemical or physical information about a sample. The components of interest in the sample are called analytes, and the remainder of the sample is the matrix. In an analysis we determine the identity, concentration, or properties of the analytes. To make this determination we measure one or more of the analyte s chemical or physical properties. 

An analytical method is selective if its signal is a function of only the amount of analyte present in the sample. In the presence of an interferent, equations 3.1 and 3.2 can be expanded to include a term corresponding to the interferent s contribution to the signal. Si,... 

Finally, analytical methods can be compared in terms of their need for equipment, the time required to complete an analysis, and the cost per sample. Methods relying on instrumentation are equipment-intensive and may require significant operator training. For example, the graphite furnace atomic absorption spectroscopic method for determining lead levels in water requires a significant capital investment in the instrument and an experienced operator to obtain reliable results. Other methods, such as titrimetry, require only simple equipment and reagents and can be learned quickly. 

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