Textbook Errors, Donald E. Jones Western Maryland College Westminster
709
Significant Digits in Logarithm-Antilogarithm lnterconversions
Inspection of twenty-four freshman chemistry and quantitative analysis texts1 reveals that nearly all are in error in the use of the proper number of digits when interconverting logarithms and antilogarithms. Only one has any discussion about the number of significant digits which should be retained when finding an antilogarithm and none at all about the reverse. When thc number being converted is between 1 and 10, little error occurs in the texts. For numbers larger or smaller than these, however, most texts fail to use the proper number of digits. Since significant digit rules are meant to simplify determination of the precision of a number calculated from one or more measured numbers, and further since logarithms are used so extensively in chemical calculations (pH, pK, potentials, reaction quotients from free energy), some easily remembered rules for their use should be developed. The general rules which this writer follows are2 1 ) When convert,ing numhers to logarithms use as many decimal places in the mantissa as there are significant digits in t,he numher.
Example: log 10.35 = 1.0149 (Four significant digits, four decimal places in the mantissa.) 2) When finding the antilogarithm, keep as many significant digits as there aredecimal places in the mantissa. Example: antilogarithm 0.065 = 1.16
P dipi14i u ('l'hrce rlwimnl plure- iu the inn~,ti.-sn,~ I ~ T P-ignific~nt the I I I I I ~ I I P T . Sut(.tlmt I I I C rern of t1w mnnti--n is i,, anted.
One common occurrence of the improper use of significant digits is in the calculations of p H values (or pOH) from concentrations. For example, in the calculation of t,he pH of a 0.0300 M solution of HC1, assuming Suggestions of material suitable for this column and guest. colmnns suitable for pnhlieat,ion directly should be sent with as many details as possible, and particularly with references to modern text,books, to W. H. Eberhardt, School of Chemistry, Georgia Institute of Technology, Atlanta, Ga., 30332. ' Since t,he purpose of this column is to prevent the spread and the cont,inuation of errors and not the evaluation of individnal texts, the sonrces of errors discussed will not be cited. In order to be presented an error must occur in at least two independent, recent standard texts. W. J., AND MXMCHI:,V. W., " E l e m e n t ~ ~Quantiy BLAEDEL, tative Analysis Theory and Practice" (2nd Ed.), Harper and Row, New York, 1963, p. 639. Bnsscfi, F., ARICNTS, J., MI:ISLICH, H., AND TURK,A,, "Fundamentals of Chemistry-A Modern Introduction" (2nd Ed.), Academic Press, Now York, E. J., 1970, p. 756. MasTznsoN, W. L., AND SLOWINSKI, "hlathematical Preparation for General Chemistry," W. B. Saunder, Philadelphia, 1970, p. 49.
that the solution behaves ideally, pH = log l/[H+] or -log 3.00 -log = -log 3.00 2 = -0.4771 = 1.5229 = 1.523. I n most texts this value ~vouldbe reported as 1.52 and not 1.523. We must keep 3 dccimal places in the mantissa so that in this case the pH value calculated has four digits cven though the concentration was lmolvn to only three significant digits. The following examples illustrate t,hcvalidity of these rnles. Generally, ~vhena number such as 1.35 is reported, the reader recognizes that if significant figure rules havc been followed, the 5 is the only digit about xx-hich there is some doubt. It could be 4 or 6, or possible 3 or 7. Hon-ever, thc numher is usually knovn to within +1 of thc last digit reported or may be so regarded. That being the case, ~vhatis the variation of the logarithm if the numbcr is 1.34 to 1.36? The logarithm of 1.34 = 0.127, the logarithm of 1.36 = 0.134. Three digits in thc logarithm arc necded to convey the proper precision. Most authors ~vould properly report this logarithm since the numher of digits in each is three. Now suppose that the number is a very large number, e.g., 1.35 X loLo(or a very small one) then the variation would be 1.34 X 10IP to 1.36 X loLo with the corresponding logarithms being 10.127 and 10.134. Clearly the logarithm must carry three digits in the former case and five in this latter one even though both numbers are knovn to three significant digits. (Remember that the characteristic tells only the position of the decimal point. A three-place table is needed for a number vith three significant digits; a four-place table for four significant digits; etc). Here is xvhere most miters err. For a number ~vhichhas two significant digits, for example, 5.7 with avariation of 5.6 to 5.8, thc logarithms should be 0.75 and 0.76, respectively, while for a number with four significant digits such as 9.785 (+0.001) X with a variation 9.784 X lo6to 9.786 X lo8,the logarithm should be 6.9905 and 6.9906, respectively. Note that the variation of the logarithm occurs in the second decimal place for numbers having two significant digits and in the fourth decimal place for numbers with four significant digits. Any truncating of the decimal would yield an antilogarithm with lower precision than the original number. More examples are obvious to thc reader. The rules previously stated and examined are relatively simple and easy to follow. A more careful examination of the relation betxeen the uncertainties related logarithmically shows that in numbers larger than 5 thc uncertainty in the converted numher is larger than *1 of the last digit.
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Volume 49, Number 7 I , November 1972
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753
If one starts with the relation A = log B, the variation in A associated with a corresponding variation in B will be
where 2.3 is In 10, and represents the conversion between logarithms to the base e and to the base 10. The variation in A is therefore proportional to thefractional variation in B rather than to its absolute valuc. If the value of a quantity is represented as a number between 1 and 10 multiplied by an appropriate pover of 10, i.e., in "scientific notation," only the fractional variation in the number will be important. For number near thc lover end of this range, 6A = (1/2.3)'SB; for a number near the high end of the range, SA = (1/23)6B. The fact that this relation varies appreciably over the range of the number spoils the simple relation between the number of significant figures needed to represent A and B to the same precision and suggests that near the low end of the range the rules stated carlier are perfectly valid, hut that at the high end, i.e., numbers larger than 5, one additional'decimal place is needed in the log term. Below are the values for numbers near the worst possible case (near 10) and their corresponding logs. B
log B or A 0.996 0.996 0.997 0.997 0.997 0.998 0.998 0.999
9.90 9.91 9.92 9.93 9.94 9.95 9.96 9.97
*
If the antilogarithm of 0.997 0.001 is to he determined, the number could vary between 9.90 (the antilogarithm of 0.997 - 0.001) and 9.96 (the antilogarithm of 0.997 0.001) yielding a precision of +0.03 (-3/ 1000) in the antilogarithm whereas the precision in the mantissa is *1/1000. However even in this worst case the variation in the antilogarithm is in the third digit, and is the same digit in which the variation occurs in the mantissa. Therefore the rules stated previously still hold. If one desires to maintain the exact precision throughout thesc interconversions then one must abandon significant figure rules and use the more laborious methods of computing the precision. The discussion to this point has dealt only with the interconversion of logarithms-antilogarithms and not with the interpretation of them. I t seems appropriate that a word of caution bc added when interpreting measured numbers which are related logarithmically. For example, when dealing with the conversion of
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Journal of Chemical Education
[H+]to pH, we assumed for simplicity that the solution was ideal. In fact, this assumption is rarely valid and it is necessary to be very careful to use the activity in calculating the pH from a concentration. For example, a 0.100 M solution of HC1 will not have a pH of 1.000. Since the mean activity coefficient of HCl in 0.100 M solution at 25°C is 0.809, the activity of H + is 0.809 and the pH is 1.092. This is not to say that a pH meter does not give accurate values when properly standardized, however. Most pH meters in laboratory use have an accuracy of zt0.05 and an expanded scale of +0.01 pH units when properly standardized. A similar situation exists when calculating potentials from concentration terms. The Kernst equation xas derived using activity in the logarithmic term, not concentrations, so that accurate work requires the use of activities. This problem carries over in the calculation of k(,,,,., using a potential measurement to determine K. The K so determined is in actual fact an equilibrium constant which incorporates the activities of the species involved. To attempt to calculate equilibrium conccntrations of the various species with a high level of accuracy is unwise, regardless of the precision of thc original potential measurement, unless the relation between activity and concentration is taken into account. It is also important to recall that the ionic strength will be important in determining the activity of any particular electrolyte species, and that the ionic strength is determined by all ions present, not just those of particular interest. Finally, the use of the quantity measured must also be considered. If an equilibrium constant K is to be used to calculate the AGo of a reaction, it is seldom necessary to know AGO to an accuracy greater than a few hundred calories. Consequently, substantial uncertainty in the equilibrium constant for a reaction \ d l seldom lead to any serious misinterpretation of the thermodynamic values. It u.ould seem sufficient for most purposes if log K were lcnown to only three total digits, including the characteristic, even through the experimental data might justify a larger number of significant digits. However, if the K value is to he used to calculate equilibrium concentrations, then the rules stated earlier should be observed carefully, and with proper consideration to the limits imposed by non-ideality of real solutions. Acknowledgment
The author wishes to exprcss his appreciation to the editor of this column for his comments and suggestions which have added much to it.
