Logarithms common, table

Values of are small for weak acids and they range very widely (Table 4.1). It is common practice to quote values as the negative logarithm to the base ten, i.e. — logjo K.. since such numbers are less cumbersome and positive when Aj < 1. The symbol for -logio is by convention "p/ fhus -logjo becomes pK,. Table 4.1 shows some typical pAg values. 

Unless otherwise stated, tables of logarithms are always tables of common logarithms. 

Mantissa of a Common Logarithm of a Number. An important consequence of the use of base 10 is that the mantissa of a number is independent of the position of the decimal point. Thus 93 600, 93.600, 0.000 936, all have the same mantissa. Hence in Tables of Common Logarithms only mantissas are given. A five-place table gives the values of the mantissa correct to five places of decimals. 

Many other mathematical operations are commonly used in analytical chemistry, including powers, roots, and logarithms. Equations for the propagation of uncertainty for some of these functions are shown in Table 4.9. 

Fig. 1. Some common curves (a) linear (b) paraboHc (c) polynomial (d) exponential (e) hyperboHc (f) logarithmic and (g) logistic. See Table 2 for...

Another solution to the problem of non-constant variance is to transform the response data. A common way of transforming data has been by taking the logarithms of both the response and amount variables ( 8-10 ). However, for all the data we looked at, the log transformation has been too strong. See Tables I and V. 

The pHs of a number of common substances are presented in Table 99.2. Because the pH and pOH scales are based on logarithms, a change in one pH or pOH unit represents a change... 

When a number is not an integral power of 10, the logarithm is not a simple integer, and assistance is needed to find it. The most common forms of assistance are electronic hand calculators and log tables. With calculators, you simply enter into the keyboard the number (A) whose log you want, press the log key (or keys), and observe the log in the lighted display. For practice, and to make sure that you know how to useyour calculator for this purpose, check that... 

The data can be evaluated using any commonly available non-linear regression program or with a linear regression, in which k,a is the slope from the plot of the natural log of the concentration difference versus time. Linearity of the logarithmic values over one decade is required for the validity of the measurement. Of course the assumptions inherent in the model must apply to the experimental system, especially in respect to completely mixed gas as well as liquid phases and reactions are negligible. Two common problems are discussed below. Other common pitfalls and problems are summarized in Table 3-3. 

These values differ from those in Table II, in that the rates are extrapolated back to sero time and do not necessarily represent the overall rate for the reaction nor do they represent anomerization. Rate constants are first order, given with time (in hours) and common logarithms. Reaction conditions Dowex-50 resin and methanol at 65c. 

The relatively rapid hydrolysis of glycofuranosides by acids was the first property used for differentiating these compounds from the more normally encountered, more stable glycopyranosides. In Tables VIII and IX are assembled data from the literature the rate constants are given in sec.-1 and common logarithms.118 The conditions of acid concentration and temperature varied in the data reported,48 48 the conditions were the same, and so the rate constants can be compared. Heidt and Purves188 used a term, fc = fc/[H], to eliminate differences in the concentration of acid used. 

Exponentials and logarithms appear in many formulas in chemistry. We have already encountered them in the definitions of prefixes in Table 1.2, which are essentially a shorthand to avoid large powers of ten (we can write 17 ps instead of 1. 7 x 10-11 s). In addition to powers of 10, we frequently use powers of e = 2.7183. .. and occasionally use powers of 2. The number e (base of natural logarithms) arises naturally in calculus, for reasons we will discuss briefly later (calculus classes explain it in great detail). Powers of e occur so often that a common notation is to write exp(x) instead of e. ... 

The set of data in Table 6.4 was generated so that the concentration of dissolved ions increases from data set 1 to set 5, but the relative abundance of the ions is preserved (check it, for example, by comparing the Mg Ca ratio in data sets 1-5, Table 6.4). This imitates dilution of a saline water by different amounts of a fresh (ideally, distilled) water, a common occurrence in nature. The data of Table 6.4 have been plotted once with a linear concentration axis (Fig. 6.2) and once with a logarithmic concentration axis (Fig. 6.3). The outcome is striking the same data plotted with a different concentration axes reveal intrinsically different patterns. On the linear plot the lines of the individual water samples differ in their gradients, whereas on the semilogarithmic plot the lines have the same gradients. The latter well reflects the dilution of saline water with fresh water, whereas the former... 

A carboxylic acid may dissociate in water to give a proton and a carboxylate ion. The equilibrium constant Ka for this reaction is called the acid-dissociation constant. The pKa of an acid is the negative logarithm of Ka, and we commonly use pKa as an indication of the relative acidities of different acids (Table 20-3). 

The pH values of many common solutions fall within a range from 0 to 14, as shown on this pH scale. The table above the pH scale relates the positive pH values to their hydronium ion concentrations and their logarithms. 

The potency of compounds derived from concentration response assays is expressed most commonly as IC50 or EC50 defined as the compound concentration that produces half maximum response. A common model is the four-parameter Hill-slope equation (Table 14.1). A three-parameter model can be used if a maximum or minimum asymptote is not available because compound potency falls outside the concentration range. One recommendation is to fit the logarithm (loglO) of IC50 or EC50 instead of the untransformed concentration because the concentration response errors are normally... 

Significant digits in logarithms are treated in Appendix 1, pages 591-93. The pH values for some common items are presented in Table 19.3. 

Particle-Size Analysis Methods for particle-size analysis are shown in Fig. 17-34, and examples of size-analysis methods are given in Table 17-1. More detailed information may be found in Lapple, Chem. Eng, 75(11), 140 (1968) L ple, Particle-Size Analysis, in Encyclopedia of Science and Technology, 5th ed., McGraw-Hill, New York, 1982 Cadle, The Measurement of Airborne Particles, Wiley, New York, 1975 Lowell, Introduction to Powder Surface Area, 2d ed., Wiley, New York, 1993 and Allen, Particle Size Measurement, 4th ed. Chapman and Hall, London, 1990. Particle-size distribution may be presented on either a frequency or a cumulative basis the various methods are discussed in the references just cited. The most common method presents a plot of particle size versus the cumulative weight percent of material larger or smaller than the indicated size, on logarithmic-probability graph paper. 

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