Calculations Used in Analytical Chemistry

Chapter 3 Using Spreadsheets in Analytical Chemistry 54 Chapter 4 Calculations Used in Analytical Chemistry 71 Chapter 5 Errors in Chemical Analyses 90 Chapter 6 Random Errors in Chemical Analysis 105 Chapter 7 Statistical Data Treatment and Evaluation 142 Chapter 8 Sampling, Standardization, and Calibration 175... 

Application In Analytical And Inorganic Chemistry Knowledge about distribution coefficients is used in analytical chemistry to determine the feasibility of quantitative separation by precipitation. Therefore, D and X are also called separation factors. In order to precipitate 99.8% or more of the primary substance, X must be 3.2 x 10 or smaller. For larger values of X more than one precipitation step is necessary, and the number of steps can be calculated when X is known. 

The reactions used in analytical chemistry never result in complete conversion of reactants to products. Instead, they proceed to a state of chemical equilibrium in which the ratio of concentrations of reactants and products is constant. Equilibrium-constant expressions are algebraic equations that describe the concentration relationships existing among reactants and products at equilibrium. Among other things, equilibrium-constant expressions permit calculation of the error in an analysis resulting from the quantity of unreacted analyte that remains when equilibrium has been reached. 

In these conditions, the calculation of aM is immediate since the stability constants intervening in its expression are known. (Let s recall incidentally that the strategy consisting of adding a great excess of reagent to simplify calculations is a method frequently used in analytical chemistry. The closer we are located to the equivalence point, the better the condition [NH3] Cnh3 is since, by the definition of a satisfactory titration, the cation and ammine complexes concentrations must be very weak at diis point.)... 

Analytical chemistry is inherently a quantitative science. Whether determining the concentration of a species in a solution, evaluating an equilibrium constant, measuring a reaction rate, or drawing a correlation between a compound s structure and its reactivity, analytical chemists make measurements and perform calculations. In this section we briefly review several important topics involving the use of numbers in analytical chemistry. 

The simple-minded approach for minimizing a function is to step one variable at a time until the function has reached a minimum, and then switch to another variable. This requires only the ability to calculate the function value for a given set of variables. However, as tlie variables are not independent, several cycles through tlie whole set are necessary for finding a minimum. This is impractical for more than 5-10 variables, and may not work anyway. Essentially all optimization metliods used in computational chemistry tlius assume that at least the first derivative of the function with respect to all variables, the gradient g, can be calculated analytically (i.e. directly, and not as a numerical differentiation by stepping the variables). Some metliods also assume that tlie second derivative matrix, the Hessian H, can be calculated. 

The comparison of more than two means is a situation that often arises in analytical chemistry. It may be useful, for example, to compare (a) the mean results obtained from different spectrophotometers all using the same analytical sample (b) the performance of a number of analysts using the same titration method. In the latter example assume that three analysts, using the same solutions, each perform four replicate titrations. In this case there are two possible sources of error (a) the random error associated with replicate measurements and (b) the variation that may arise between the individual analysts. These variations may be calculated and their effects estimated by a statistical method known as the Analysis of Variance (ANOVA), where the... 

This method of using an internal standard to deal with matrix and background effects has, of course, been used for ages in analytical chemistry. Equally useful would be another analytical method, namely that of standard additions. In this latter approach, the sample is spiked with successively increasing amounts of the pure analyte, and the intercept in the response-concentration curve is used to calculate the amount of analyte in the original sample. 

The instrumental analytical techniques, developed in the last three or four decades, are almost all based on the limited signal and data processing capabilities of relatively simple analog instruments, and utilize a limited or simple theoretical basis for calculations. Apart from the rather advanced application of statistics, only a modest use of mathematical techniques in analytical chemistry has been used in these traditional analyses. 

In a modern laboratory, automated computer software for data acquisition and processing performs most of data reduction. Raw data for organic compound and trace element analyses comprise standardized calibration and quantitation reports from various instruments, mass spectra, and chromatograms. Laboratory data reduction for these instrumental analytical methods is computerized. Contrary to instrumental analyses, most general chemistry analyses and sample preparation methods are not sufficiently automated, and their data are recorded and reduced manually in laboratory notebooks and bench sheets. The SOP for every analytical method performed by the laboratory should contain a section that details calculations used in the method s data reduction. 

V. A. Kropotov, Calculations in Analytical Chemistry Using Programmable Microcalculators Textbook, Simferopolskii Gos. Un-T., Simferopol, USSR, 1985. 

The ability to perform the same analytical measurements to provide precise and accurate results is critical in analytical chemistry. The quality of the data can be determined by calculating the precision and accuracy of the data. Various bodies have attempted to define precision. One commonly cited definition is from the International Union of Pure and Applied Chemistry (IUPAC), which defines precision as relating to the variations between variates, i.e., the scatter between variates. [l] Accuracy can be defined as the ability of the measured results to match the true value for the data. From this point of view, the standard deviation is a measure of precision and the mean is a measure of the accuracy of the collected data. In an ideal situation, the data would have both high accuracy and precision (i.e., very close to the true value and with a very small spread). The four common scenarios that relate to accuracy and precision are illustrated in Figure 2.1. In many cases, it is not possible to obtain high precision and accuracy simultaneously, so common practice is to be more concerned with the precision of the data rather than the accuracy. Accuracy, or the lack of it, can be compensated in other ways, for example by using aliquots of a reference material, but low precision cannot be corrected once the data has been collected. 

The assumption of heteroskedasticity is frequently rejected in analytical chemistry. Weighted regression can be used to correct for heteroskedasticity. The equations to calculate the detection decision and LoD in case of weighted linear regression models (WLRM) are given by ... 

For most purposes in analytical chemistry an expanded uncertainty (U) should be used when reporting a result, i.e. x U. U is calculated using the following equation ... 

In analytical chemistry, several calibration methods have been developed and introduced into laboratory practice [1], Most of them can be recommended for use in trace analysis. The methods differ from each other in terms of (a) preparation of the calibration solutions, (b) interpretation of the measurement results and construction of calibration graphs, and (c) calculation of the final analytical results. We present the calibration methods exploited most often in chemical analysis and point out their advantages and drawbacks, which are especially significant from the point of view of the specific conditions characterizing trace analysis. 

In this chapter, we have begun to explore the use of spreadsheets in analytical chemistry. We have examined many of the basic operations of spreadsheet use, including data entry, data import, string handling, and basic calculations. In other spreadsheets in this book and in Applications of Microsoft Excel in Analytical Chemistry, we will build on the techniques that we have acquired and learn much more about Excel that will be useful in our study of analytical chemistry and related fields. 

Spreadsheet Summary In the first exercise in Chapter 3 of Applications of Microsoft Excel in Analytical Chemistry, we use Excel to perform the t test for comparing two means assuming equal variances of the two data sets. We first manually calculate the value of t and compare it with the critical value obtained from Excel s function TINV(). We obtain the probability from Excel s TDIST() function. Then, we use Excel s built-in function TTEST() for the same test. Finally, we employ Excel s Analysis ToolPak to automate the t test with equal variances. 

Spreadsheet Summary In Chapter 3 of Applications of Microsoft Excel in Analytical Chemistry, the use of Excel to perform ANOVA procedures is described. There are several ways to do ANOVA with Excel. First, the equations from this section are entered manually into a worksheet, and Excel is invoked to do the calculations. Second, the Analysis ToolPak is used to carry out the entire ANOVA procedure automatically. The results of the five analysts from Example 7-9 are analyzed by both these methods. 

Spreadsheet Summary In the first three exercises in Chapter 5 of Applications of Microsoft Excel in Analytical Chemistry, we explore the solution to the types of equations found in chemical equilibria. A general purpose quadratic equation solver is developed and used for equilibrium problems. Then, Excel is used to find iterative solutions by successive approximations. Excel s Solver is next employed to solve quadratic, cubic, and quartic equations of the type encountered in equilibrium calculations. 

Spreadsheet Summary In Chapter 5 of Applications of Microsoft Excel in Analytical Chemistry, we explore the solubility of a salt in the presence of an electrolyte that changes the ionic strength of the solution. The solubility also changes the ionic strength. An iterative solution is first found, in which the solubility is determined by assuming that activity coefficients are unity. The ionic strength is then calculated and used to find the activity coefficients, which in turn are used to obtain a new solubility. The iteration process is continued until the results reach a steady value. Excel s Solver is then used to find the solubility directly from an equation containing all the variables. 

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