Significant figures with calculators

The values were determined at XenoTech (unpublished data). Constants are shown standard error (rounded to 2 significant figures, with standard error values rounded to the same degree of accuracy as the constant), and were calculated using GraFit software, which utilized rates of product formation (triplicate data) at 13 substrate concentrations. 

The volume was calculated and then its value was converted to milliliters. Of the given information, V2 had the fewest number of significant figures with two. Thus, the volume Z, should also have two significant figures, as it does. 

NEW BASIC MATH SKILLS APPENDIX To aid the flow of introductory chemistry material in Chapter I, a review of topics in basic mathematics skills, including scientific notation and use of significant figures, with numerous examples, now appears in Appendix A. Related exercises remain in the Measurements and Calculations section at the end of Chapter 1. 

In 1972 Hewlett-Packard (HP) introduced its Scientific Calculator (HP 32S, Fig. 15.5). This greatly extended the range to include logarithms, trigonometric functions, numbers raised to powers, and memory imits, all to over 10 significant figures with a price of 400. Soon after the HP 32S appeared, TI produced a similar scientific calculator (Fig. 15.6). 

Calculate the molar concentration of NaCl, to the correct number of significant figures, if 1.917 g of NaCl is placed in a beaker and dissolved in 50 mF of water measured with a graduated cylinder. This solution is quantitatively transferred to a 250-mF volumetric flask and diluted to volume. Calculate the concentration of this second solution to the correct number of significant figures. 

At 25°C, the Mark-Houwink exponent for poly(methyl methacrylate) has the value 0.69 in acetone and 0.83 in chloroform. Calculate (retaining more significant figures than strictly warranted) the value of that would be obtained for a sample with the following molecular weight distribution if the sample were studied by viscometry in each of these solvents ... 

Question. Calculate, to three significant figures, the wavelength of the first member of each of the series in the spectrum of atomic hydrogen with the quantum number (see Section f.2) n" = 90 and 166. In which region of the electromagnetic spectrum do these transitions appear ... 

Much of the additional material is taken up by what 1 have called Worked examples . These are sample problems, which are mostly calculations, with answers given in some detail. There are seventeen of them scattered throughout the book in positions in the text appropriate to the theory which is required. 1 believe that these will be very useful in demonstrating to the reader how problems should be tackled. In the calculations, 1 have paid particular attention to the number of significant figures retained and to the correct use of units. 1 have stressed the importance of putting in the units in a calculation. In a typical example, for the calculation of the rotational constant B for a diatomic molecule from the equation... 

Significant figures provide an indication of the precision with which a quantity is measured or known. The last digit represents, in a quantitative sense, some degree of doubt. For example, a measurement of 8.12 inches implies tliat Uie actual quantity is somewhere between 8.315 and 8.325 inches. This applies to calculated and measured quantihes quantities tliat are known exactly (e.g., pure integers) have an infinite number of significant figures. 

Most measured quantities are not end results in themselves. Instead, they are used to calculate other quantities, often by multiplication or division. The precision of any such derived result is limited by that of the measurements on which it is based. When measured quantities are multiplied or divided, the number of significant figures in the result is the same as that in the quantity with the smallest number of significant figures. 

Atomic masses calculated in this manner, using data obtained with a mass spectrometer can in principle be precise to seven or eight significant figures. The accuracy of tabulated atomic masses is limited mostly by variations in natural abundances. Sulfur is an interesting case in point. It consists largely of two isotopes, fiS and fgS. The abundance of sulfur-34 varies from about 4.18% in sulfur deposits in Texas and Louisiana to 4.34% in volcanic sulfur from Italy. This leads to an uncertainty of 0.006 amu in the atomic mass of sulfur. 

The values for K, listed here have been calculated from pK, values with more significant figures than shown so as to minimize rounding errors. Values for polyprotic acids—those capable of donating more than one proton—refer to the first deprotonation. 

We ll see how they compare in time response simulations when we come back to this problem later in Example 5.7C. A point to be made is that empirical tuning is a very imprecise science. There is no reason to worry about the third significant figure in your tuning parameters. The calculation only serves to provide us with an initial setting with which we begin to do field or computational tuning. 

In practice, in numerical calculations with a computer, both rational and imtiooal numbers are represented by a finite number of digits. In both cases, then, approximations are made and die errors introduced in the result depend on the number of significant figures carried by the computer - the machine precision. In die case of irrational numbers such errors cannot be avoided. 

Note To find e-95 66, take the inverse In of-95.66 on your calculator, inv In of-95.66 = 2.85 x 10 42. Keep one more significant figure and round off to three significant figures at the end, particularly when working with logarithms. 

B Unfortunately, we cannot use the result of Example 26-5 ( 0.0045 u = 4.2 MeV ) because it is expressed to only two significant figures, and we begin with four significant figures. But, we essentially work backwards through that calculation. The last conversion factor is from Table 2-1. 

This more detailed model is necessary for target stock calculations where productions may overlap, the lengths of quants differ significantly and quants have multiple predecessors and successors (see Figure 4.14). Calculating lot sizes with for example the formula of Andler yields completely different results and the sum of changeover costs and stock costs are much higher. 

We need to introduce a word of caution. Most modem calculators cite an answer with as many as ten significant figures, but we do not know the concentration to more than two or three significant figures. In a related way, we note how the pH of blood is routinely measured to within 0.001 of a pH unit, but most chemical applications... 

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