Significant figures calculations

The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

Using MOPAC and the MNDO Hamiltonian, calculate the energy and equilibrium bond length of N2 to 4 significant figures. The input file is... 

Significant figures are also important because they guide us in reporting the result of an analysis. When using a measurement in a calculation, the result of that calculation can never be more certain than that measurement s uncertainty. Simply put, the result of an analysis can never be more certain than the least certain measurement included in the analysis. 

Report results for the following calculations to the correct number of significant figures. 

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. 

The numerator, therefore, is 23.41 0.028 (note that we retain an extra significant figure since we will use this uncertainty in further calculations). To complete the calculation, we estimate the relative uncertainty in Ca using equation 4.7, giving... 

Evaluate the product Uj Mj for each class this is required for the calculation of both and M. Values of this quantity are listed in the third column of Table 1.4. From SjnjMj and Sjnj, = 734/0.049 = 15,000. The matter of significant figures will not be strictly adhered to in this example. 

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. Using Equations (1.11) and (1.12) calculate, to six significant figures, the wavenumbers, in cm of the first two (lowest n") members of the Balmer series of the hydrogen atom. Then convert these to wavelengths, in nm. 

Note that seven figures are retained in the calculation until the final stage, when the numbers are rounded to six significant figures. 

Question. Using Equation (1.62) calculate, to four significant figures, the rotational energy levels, in joules, for J= 0, 1 and 2 for Then convert these to units of cm. [Use a bond... 

Note that, since 5q is given, effectively, to six significant figures, the calculation has been done to seven figures and rounded to six at the final stage. 

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 ... 

Note that five significant figures have been retained for the reciprocals of the squares of the quantum numbers because, as you can see in the calculation, two of these figures are lost when the reciprocals are subtracted. 

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... 

In each example, the initial values of the factors are expressed in U.S. customary units, and the dimensionless value is calculated. Then the factors are converted to SI units, and the dimensionless value is recalculated. The two dimensionless values will be approximately the same. (Small variations occur due to the number of significant figures carried in the solution.)... 

Table 6-1 lists the experimental quantities, k, T, ct, the transformed variables x, y, and the weights w. (It is necessary, in least-squares calculations, to carry many more digits than are justified by the significant figures in the data at the conclusion, rounding may be carried out as appropriate.) The sums required for the solution of the normal equations are... 

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. 

In applying the rules governing the use of significant figures, you should keep in mind that certain numbers involved in calculations are exact rather than approximate. To illustrate this situation, consider the equation relating Fahrenheit and Celsius temperatures ... 

This conversion factor is exact the inch is defined to be exactly 2.54 cm. The other factors listed in this column are approximate, quoted to four significant figures. Additional digits are available if needed for vary accurate calculations. For example, the pound is defined to be 453.59237 g. 

Determine the number of significant figures in a calculated quantity. 

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. 

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