Precision, 19 rounding

Figure 6.6 Plot to determine the activation energy and reaction order of a decomposition reaction. The slope indicates a second order reaction and the intercept, being Ea4>/R ( = 10°C/min), indicates that the activation energy is 111 kj/mol. The noise at the end of the trace is a result of double precision round-off error.

Now that computers are so fast, you may wonder why we don t just pick a tiny Ar once and for all. The trouble is that excessively many computations will occur, and each one carries a penalty in the form of round-off error. Computers don t have infinite accuracy—they don t distinguish between numbers that differ by some small amount 5. For numbers of order 1, typically S 10 for single precision and 5 10" for double precision. Round-off error occurs during every calculation, and will begin to accumulate in a serious way if At is too small. See Hubbard and West (1991) fora good discussion. 

In order to describe the number of primitives and contractions more directly, the notation (6s,5p) (ls,3p) or (6s,5p)/(ls,3p) is sometimes used. This example indicates that six s primitives and hve p primitives are contracted into one s contraction and three p contractions. Thus, this might be a description of the 6—311G basis set. However, this notation is not precise enough to tell whether the three p contractions consist of three, one, and one primitives or two, two, and one primitives. The notation (6,311) or (6,221) is used to distinguish these cases. Some authors use round parentheses ( ) to denote the number of primitives and square brackets [ ] to denote the number of contractions. 

More digits are retained in such presentations than are required to express the experimental precision in order that rounding errors be minimized. 

Notice that a result of this type, in order to be interpretable, must comprise three numbers the mean, the (relative) standard deviation, and the number of measurements that went into the calculation. All calculations are done using the full precision available, and only the final result is rounded to an appropriate precision. The calculator must be able to handle >4 significant digits in the standard deviation. (See file SYS SUITAB.xls.)... 

Note that a number of complicating factors have been left out for clarity For instance, in the EMF equation, activities instead of concentrations should be used. Activities are related to concentrations by a multiplicative activity coefficient that itself is sensitive to the concentrations of all ions in the solution. The reference electrode necessary to close the circuit also generates a (diffusion) potential that is a complex function of activities and ion mobilities. Furthermore, the slope S of the electrode function is an experimentally determined parameter subject to error. The essential point, though, is that the DVM-clipped voltages appear in the exponent and that cheap equipment extracts a heavy price in terms of accuracy and precision (viz. quantization noise such an instrument typically displays the result in a 1 mV, 0.1 mV, 0.01 mV, or 0.001 mV format a two-decimal instrument clips a 345.678. .. mV result to 345.67 mV, that is it does not round up ... 78 to ... 8 ). 

Example 48 The result is thus CL(/4) = 7.390 0.028 mM/g, and should be either left as given or rounded to one significant digit in C1 7.39 0.03 The %-variance contributions are given in parentheses (Eq. (4.24)). Note that the analytical method with the best precision (titrimetry), because of the particular numerical constellation, here gives rise to the largest contribution (77%). 

Comment The sequence of digits in each coefficient depends on the precision (e.g., three decimal places) and number of tabulated values (34, 50, or 64), the form of the optimization software used (Hewlett Packard HP71B Curve Fit Module), and the number of coefficients chosen (3. .. 8). Discrepancies between the approximated and the real table entries of up to j-LSD could be due either to insufficiencies of the algorithm or the rounding of table entries. The few LRR that are above 1% do not pose a risk for practical applications. 

The units match those of molality, and the result is rounded to two significant figures to match the precision of the mass percentage information. 

A C — C bond must break for isomerization to occur, and t q)ical C — C bond energies are around 350 kJ/mol (Table ). Although 270 kJ/mol is considerably smaller than this, it is a reasonable value for the activation energy because the new n bond begins to form before the C — C bond has broken completely. The value is rounded to two significant figures to match the precision of the k values. 

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