Numbers, Measurements, and Numerical Mathematics

The first application of mathematics to chemistry deals with various physical quantities that have numerical values. In this chapter, we introduce the correct use of numerical values to represent measured physical quantities and the use of numerical mathematics to calculate values of other quantities. Such values generally consist of a number and a unit of measurement, and both parts of the value must be manipulated correctly. We introduce the use of significant digits to communicate the probable accuracy of the measured value. We also review the fector-label method, which is a routine method of expressing a measured quantity in terms of a different unit of measurement. 

Specification of a measured quantity consists of a number and a unit. 

A unit of measurement is an arbitrarily defined quantity that people have agreed to use. 

The SI units have been officially adopted by international organizations of physicists and chemists. 

The factor-label method can be used to convert from one unit of measurement to another. 

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Both the Parameter and Reconcile cases determine (calculate) the same set of parameters. However, these cases do not get the same values for each parameter. A Parameter case has an equal number of unknowns and equations, therefore is considered "square" in mathematical jargon. In the Parameter case, there is no objective function that drives or affects the solution. There are typically the same measurements, and typically many redundant measurements in both the Parameter and Reconcile case. In the Parameter case we determine, by engineering analysis beforehand (before commissioning an online system for instance) by looking at numerous data sets, which measurements are most reliable (consistent and accurate). We "believe" these, that is, we force the model and measurements to be exactly the same at the solution. Some of these measurements may have final control elements (valves) associated with them and others do not. The former are of FIC, TIC, PIC, AIC type whereas the latter are of FI, TI, PI, AI type. How is any model value forced to be exactly equal to the measured value The "offset" between plant and model value is forced to be zero. For normally independent variables such as plant feed rate, tower... 

Once a reflectance curve is obtained spectrophotometrically, the measured information is mathematically transformed according to standard conventions, into the numbers used to describe the color of the sample. In practice, a standard observer and standard illuminant are selected based on test methods defined by the Commission International de I Eclairage (CIE). Calculation procedures involve the numerical integration of the product of the spectral power distribution S(l) of the light source and the reflectance factor R(l). Reflectance factors represent the percentage of light reflected by the sample at each wavelength. 

This section explores the mathematical basis for the statistical treatment of experimental data. Most measurements required for the completion of the experiments can be made in duplicate, triplicate, or even quadruplicate, but it would be impractical and probably a waste of time and materials to make numerous determinations of the same measurement. Rather, when you perform an experimental measurement in the laboratory, you will collect a small sample of data from the population of infinite values for that measurement. To illustrate, imagine that an infinite number of experimental measurements of the pH of a buffer solution are made, and the results are written on slips of paper and placed in a container. It is not feasible to... 

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