edited by: RON DELORENZO Middle Georgia College Cmhran. Georgia 31014
Analogies for Avogadro's Number Paul S. Poskozlm James W. Wazorlck Permsook Tlempetpalsal Joyce Albln Poskozlm Norlheastern Illinois University Chicago, IL 60625
The utter immensity of Avogadro's number (hereafter abbreviated AN) is camouflaged in the conventional scientific notation representation 6.023 X loz3.Any true realization or comprehension of just how large AN is, is impossible. As a result, analogies are used to try to get some idea of a number that, for all practical purposes, may well be thought of as infinity. While i t is difficult to represent the truly large (or truly small) in ways fathomable by the human mind, chemistry educators should not and do not give up trying. This article reviews analogies used to try to capture the concept of the magnitude of AN and presents new ones tuned to modern technology. I t would be most difficult, short of an extensive national survey, to assess the manner in which the magnitude of AN is discussed in chemistrv classrooms today. A survey of 155 current cullege texts used in intnniucrory i n d general chemistrv courses shows that while most discuss AN and use it in appropriate calculations, only 41,or about one in four, contain additional size analogies for AN. These 41 analogies may be grouped in five categories as seen in Table 1. ~
~~
~
The most popular textbook analogies are I and 11, mathematically correct, and, depending on preference, vivid to a ereater or lesser deeree. Analow I is esoeciallveood in that it can be understood-and apprelcated b; anyoneregardless of science or mathematics background and generally uses numbers less than 100. All of the others use numbers in the millions, billions, and trillions, all magnitudes too large to have any real meaning for most people. The usefulness of analogy IV is highly suspect since it demands a strong grasp of either the size of a milliliter (in which case it might well fall under analogy I) or the size of a molecule, which itself is far too small to comprehend. Analogies of type I1 and 111 can blend reasonably well t o produce hybrid analogies appealing to those familiar with numbers of human brain cells, or with the number of individual frames of film in a reel or two of modern motionpicture film. Respectively, these can be stated as follows: Assumine the Earth has 4 billion . oeoole . and that each oerson has I 0 billion brain cells. 15 thousand planets with Earth's population would contain a n aggregate of rl.V ~ r fhrain cells. And, For an aggregate of ANof frames of film to have beenviewed, each person on Earth would have to sit through about 1.3 billion show-
ings of "Return of the Jedi". Authors and teachers seeking to capture the imagination of modern students of chemistry may well desire to employ
Table 1. Textbook Analogles Representing Avogadro's Number (AM Analogy based on.. . I. small or tiny objects
(marbles, peas, grains of sand. baseballs, etc.) 11. wuming
Ill. people
IV. water
V , money
Trnl
Premise: ANis so large mat. . .
Examples of Typical Textbook Analogles
the volume occupied by that number of smell or tiny objects would be Incredibly large.
ANof marbles. spread over the surface ol Ihe earn. would prcduce a layer of marbles about 50 mi thick. ANgrains of sand. spread over the United States. would produce a layer of sand about 3 In. deep. It would take nearly 5 million years for lhe world's entire population of 4 billion people, counting at the rate of one object per second, to count collectlvely AN of objects. 150 trillion planets, each holding 4 bllllon people like Earth, would be needed to accommodate AN of people. There are approximately two times ANof mllllliters of water on this planet.
it would take an incredibly long time to coum that number of objects even if the entire population of the world were involved in the task on a continuous basis. the total number of people on Earth now is lncredibly tiny in comparison. it can be compared to Ihe number of milliliters of water on Earth or the number of water moiecules in 18 grams, a male of water.
it would be impossible to spend that number of dollars.
If
refer en^)^^
14
12
6
6
18 g of water were spread evenly over the Earth's surface, Ihem would be approximately 100.000
molecules of water Over eaoh square centimeter. ANof dollars wuld not be spent at the rate of a billion dollars a day over a trllllon years.
3
'Anual number of surveyed texb usingmie me of analogy.
Volume 63
Number 2
February 1986
125
Table 2. Anaiqly based on. . . Vi. Modern computers and printers
New Computer-Based Analogles for Avogadro's Number (AN)
Premise: AN is so large that.
..
even the fastest computer would require an incredibly long time to process that number of instructions.
Proposed Analogy A common "mainframe" computer used by large cwporations is the IBM model 3084, which has the nominal speed of about 26 million instructiow per second. At this rate, the system would be working nonstop for 733 million years before ANof instructions would have been c a m pieted.
The CRAY S 1 super computer has a nominal speed rating of 1000 mips (millions of instructions per second) and could count the entire population of me United States in % second but would still require 1.9 million years to process AN of steps. even the fasten printen would require an im mense amount of time and consequem space to print that number of dots.
even me most modern, efflcent mlormatlon storage system lmemoryj wodlo requ re an immense amount of space to store ANof bits of information,
A common dot-matrix printer uses a 9 X 7 dot matrix, but even the most complex character (@ and #) uses perhaps 20 dots. To print AN of doh at a typical maximum of 132 characters per line, 6 lines per in.. would require approximately 600 trillion miles of printed paper. Printing this amount on a 600-lines-per-min printer would require about 700 billion years. TO store ANof o is of informatm ~ s l n gthe most advanced laser-optlca oosks ~ 0 t .ens h toes 01 a gtgaoot (1 bdloon o 1s) of informaten per squave centimeter would require about 4000 square mi of disks. roughly the size of me state of Connecticut. TO store ANof bits of information using advanced 4 megabit (million-bit) chips would require about a cubic mile of chlps.
entirely new analogies. Computers are a good source due to their high operational speed, large information-storage capacities or memories, and omnipresence. Table 2 lists five examples. The proposed analogies appeal to nearly everyone's appreciation of the speed a t which a computer operates and prints data and stores information. Even so, large numbers like millions and billions and even larger numbers must be used as part of the analogy. The more one knows about computers,-the better the analogies become. However, the examples included within the statement of the analogy help
126
Journal of Chemical Education
to bridge the gap for the novice. Analogies based on current state-of-the-art computer technology will have a finite lifetime and should be reviewed every few years. Computer analogies of A N will get better as computer technology imDroves. The analogies presentrd here may well also find use when ~ the incredihlv small size of an indiattemwinr t t deirrihe viduai atom. The authors welcome iontrihutions of analogies for AN proven effective over the years and any other derivatives conjured up by the ideas presented in this paper.
