Logarithms, the Advent of Calculators, and Chemistry Ronald Fietkau Southeastern Oklahoma State University, Durant, OK 74701 Logarithms whether natural or common are a part of many equations used in chemistry. Examples that appear in freshman chemistry texts include the equation for pH, the integrated rate equation for a first-order reaction, the relationship between standard free energy change and the equilibrium constant, the Clausius-Clapeyron equation, Arrhenius equation, and the Nernst equation. With the exception of the equation for pH these examples employ natural loaarithms in their derivations. Tv~icallv in the .. liiit stt,p u cmvcrslon to common logarithms is performed. natural The Nernst cauauon is used a s an e x a m ~ l eThe . logarithmic fo- is shown below.
5" is the standard cell potential, R is the gas constant (8.31 J/Kmol), T i s the absolute temperature, n is the number of moles of electrons transferred, F is the Faraday Constant (9.6485 x 10%lmol). Because 1 V = 1 JIC and
cal with, natural logarithms (6, 7). The invention of common logarithms (base 10) is generally attributed to Henry Briggs (1561-1631) a n admirer of Napier. In 1624 Briggs published his Arithmetica Logarithmica wherein he acknowledged Napier's contribution (6, 7). The properties of common logarithms or their natural logarithm counterparts usually are found in a mathematical review appendix of a chemistry text. Up to the late 70's the numerical examples that were given involved use of the logarithm tables that were found in another appendix. There was no need for another appendix containing natural logarithm values due to eq 2 above. Typical examples found in a mathematical review appendix follow (81. Example I: The log of 3.540 x 10'' is to be found. log (3.540 x 1 0 ' ~ )= log 3.540 +log loL2 = 0.5490 =
Inr = 2.303 logx
a t 25 OC the Nernst equation becomes
+ 12
(6)
12.5490
(71
The factor 0.5490 (log 3.540) was obtained from the log tables, which typically contained four-place values. Example 2: The log of 2.00 x
is to be found.
lag (2.00 x 1 0 % lag 2.00 x log This conversion appears in recent new editions of general chemistry texts ( 1 3 ) but does not in a t least one new book (4) which is notably in its first edition. The question that immediately comes to mind is why make the conversion in this age of inexpensive scientific calculators. No more effort is exerted in pressing the ln key than the log key Students following the development of the Nernst equation may wonder why the extra step is taken. The authors of the texts ( 1 3 1 do not give any explanation for making the conversion. Without the conversion the equation is
The reason for the conversion is historical as any chemist educated prior to the advent of hand held calculators will attest. Prior to the age of calculators common logarithm tables and slide rules were, for many generations of chemistrv students. the standard tools for comoutations. Hy wny ol'sornr histuric;il information and persuual recollectiuns I will discuss loearithms. slide rules. and ralculntors that should provide some perspective to recent stud e n t s of c h e m i s t r y on why t h e s e conversions a r e performed. Logarithms are a mathematical device bv which manv computauons are made shorter and easier. Logarithms re,dure mulr~plicationto addition, division to iubtraction, raising to pmv~:rito multiplirntion, and extraction of roots Thc imentlon of natural logarithms I baser to divisinn 61. = 2.718. . .) is usually attributed to ~ o h Napier n (155016171, Baron of Merchiston, who published a table of logarithms in Edinburgh in 1614 under the title Mirifici Logarithmorum Canonis Descrlptio even though the functions tabulated by Napier are merely related to, but not identi932
Journal of Chemical Education
(51
= 0.301 + (-5)
(8)
(9)
Once again the log tables would be used to obtain 0.301 (log 2.00). Prior to the age of calculators this type of calculation was necessary in order to solve problems involving logarithmic relationships, such a s the Nernst equation. The index finger would track the columns and rows until the appropriate value was found, then the addition or suhtraction would be carried out to obtain the final result. Richard Feynman, 1965 Nobel Prize winner in physics, committed many logarithm values to memory, a s did many scientists, so that he could make certain calculations mentally (9). Calculators made the process of determining logarithms using tables obsolete virtually overnight. The values could be determined faster and easier with the cal, the log culator. Just "plug" in the value, 3.54 x l ~ ""hit" key and 12.54900326 is displayed. The time it takes for the computation is determined by the number of keystrokes. The new problem created by the calculator is that significant figures is now a very important concept for students to understand. As calculators made their way into the classroom logarithm tables slowly disappeared from chemistry texts. Unfortunately this omission did not make new editions of the texts shorter. The author of the chemistry text I used a s a freshman a t Simon Fraser Universitv in the Fall of 1976 converts every equation involving a natural logarithm to one involving a common logarithm (8).At this time calculators such a s t h e Texas 1nsGments SR-50 were widely available and affordable (I paid around $110.00 for mine, on sale!). These calculators had the in and log keys. I recall that everyone in my freshman chemistry class was required to have a calculator because they had become so affordable. By the
late 70's every chemistry text could have done away with the conversions. To be fair the Nernst equation seems to be the last hold out in the recent editions cited (13).The other equations mentioned earlier have not been converted from natural logarithms to common logarithms. However, the changes did occur slowly over the past decade. The most likely reason that the Nernst equation is still found in the common loearithm form is the familiaritv amone the authors of these Gxts with the factor 0.0592 in the tion (see ea 3). The new factor of 0.0257 V in the natural logarithm ~O&I of the equation (eq4) necessitates displacement of the old value which for many chemists was memorized decades ago. Convention dies a long and hard death. Calculators have been present in classrooms for more than 20 years! I n all other equations 2.303, the constant that relates natural logarithm values to common logarithm values, is not incorporated into a factor with other constants but is explicit in the calculation. That is, there are no wellremembered factors to supplant. When that first calculator was purchased the slide rule that had been a steady companion for many chemists was relegated to a drawer. For me that time came in January 1976. I could not, however, put away my slide rule too quickly. My high school chemistry teacher required a demonstrated uroficiencv with a slide rule urior to beine uermitted to ;se the c a h l a t o r to solve cgemistry p r o & n s . My high school physics teacher had no such requirement. He encouraged its use to solve physics problems. Slide rules were made of various kinds of wood or plastic and could be found in the familiar linear shape or as a circular device. The number of scales would v a n from one model to another. Stories abound of engineers who attached the cases of their slide rules to their belts so that they hung down the side of their legs like six shooters. (The u ~ d a t e dversion is the eneineer with calculator haneing &om his belt., Many ol'tod&'s students have never seen slide rule and those u,ho have have little idea of how one works. Only people for whom it was necessary to use slide rules on a regular basis can really appreciate the advent of calculators and their impact. Today calculators are taken for granted. The history of the slide rule h e ~ n with s the invention of logarithms hy Napier mentioned uhove. In 1620 Edmund ~ G n t e r~ , r o f e s s o rof Astronomy a t Gresham College, in London, conceived the idea of using logarithm scales that were constructed with antilogarithm markings for use in simple mathematical operations. In 1630 the Rev. William Oughtred, who lived near London, produced a linear twoscale rule in sliding combination, thus inventing the slide rule. Later he also produced a circular version of the slide rule (l&ll). I n 1657 Seth Partridge made a three-strip rule with the two outer s t r i ~ held s bv cleats and the inner strip sliding between them marked with numerical and trieonometrical scales on both sides (11).This is the versi& with which many of us are familiar. In its time (which ended around 1975) it was a quick, though with limited accuracy, method of calculation. For the chemistry student
Gus-
a
slide rules were primarily used for multiplication and division. I t must he noted that addition and subtraction had to be accomplished using longhand. The typical 10- or 12inch slide rule is capable of a n accuracy of three digits. To increase the level of accuracy a larger slide rule is required. Slide rules today are nostalgic objects that may become items for collectors in the future. The advent of calculators had another affect on chemistry instruction. In the day of slide rules and logarithm tables care was given by authors to the values of variables given in textbook example problems and professors making up examination questions. For example, concentrations such a s 0.01,0.1, or 0.2 M would be given so that the auantities would cancel or that the calculation could be performed mentally resulting in a number such as 10,100, 0.1, or 0.01. Determining the logarithms of these numbers becomes trivial. Now co&entrations in examples or examination questions take on any value, for example 0.06,0.27, or 0.35 M. Calculators have made values of variables a non-issue. Since their introduction in the late 60'5, calculators themselves have gone through considerable changes. In comparison to modem calculators, the first generation were bulky due in large part to the battery packs that needed recharging a t regular intervals. I recall always charging up my calculator the night before an exam. The fear was that it would "die" during the exam. The display consisted of red light emitting diodes. The price of calculators seemed to drop on almost a monthly basis until the new model came out. The new model was not necessarily more expensive but had many more functions it could perform. The slim design appeared in the early 80's sporting a liquid crystal display and a battery that would last a t least a year. Now solar-powered calculators are available. Calculators with graphics displays cost less than $100.00. Gone are the days of limited memory. Even inexpensive calculators can store programs and can perform a staggering array of functions. The slide rule was used by many generations of chemists and was made obsolete virtually overnight. Chemists can date themselves by reminiscing about that favorite slide rule and the days before calcul&ors. For students today it is diff~cultto imaf$ne performing calculations with the use of a slide rule aid logarithm tables. Literature Cited 1. Zurndahl, S. S. Chemistry. 3rd ed.; Heath:k i n g t o n , MA. 1993. 2. Petrueci. R. H.;Harwood, W. S. G m w d Chemistry, 6th ed.; Macrnillan:New York. 1993. 3. Ebbing, D. D. Dpmml Chamislry. 4U1 ed.: Hovghton Mifnin: Baston, 1993. 4. Urnland, J. B. Gneml Chemistry; West: St. Paul. MN, 1993. 5. Newman, J. R.,Ed. TheHarperEncyelopedioofSeience: Harper andRow: NewYark, 1963 Vol. 2. 6. McDmlu-Hill Eneyeloppdio a,fScienrp and lkhndogv, 7th ed.: MeCraw-Hill: New York, 1992:Vol. 10. 7. Cajo", EAHistory ofMaIhomotics. 2nd ed.; Mamillsn: New York, 1926. 8. Monirner, Charles E. Chemistry, 3rd ed.; Van Nostrand: New York, 1975. 9. Fewman. R. P. Surely You're Jokinp. Mr Feynmoni; Norton:New York. 1985. 10. Beakley. G. C.: Leach, H. W. The Slide Rubandits USOin Pmblem Soluing, 2nd ed.: Collier-Macrnillan: hronto. 1969. 11. Thewlis, J.. Ed. EnwclopmdicDicLim~ryofPhyrks: Maernillan:New Ymk. 1962.
Volume 71 Number 11 November 1994
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