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DONNABOGNER Hutchinson Senior High School Hutchinson. KS 67501
insights Avogadro's Number: Surprise! Thomas H. Bindel Pomona Senior High School 8101 West Pomona Drive A~ada,CO 80005 There have been many articles dealing with Avogadro's number over the past 30 years. They have dealt with determining Avogadro's number through electrochemistry (I), organic carboxylic acid monolayers on water ( 2 4 ) ,Brownian movement (5), surface tension (6),and X-ray crystallography (7).Other articles have dealt with visualizing the magnitude of the number using analogies (8-10). Grasping the Mole Concept I would like to point out that first-year high-school chemistry students struggle with the idea that a gram atomic mass (GAM)of any element has the same number of atoms as a GAM of any other element. The number is of course Avogadro's number, 6.022045(31)x 10 23 mol-'(11~).Agood grasp of the mole concept is critical when the student is faced with understanding stoichiometry Of course, one can talk about the relative masses of the individual atoms of the elements and relate this to the CAM's, but the average student just doesn't seem tograsp this idea. For this reason, I have developed the following discovery-oriented activity. The Calculation The activitv should be done rieht before anv discussion of the mole. show the class how one can arriveat the number of atoms in a CAM of an element when piven the X-ray crystallographic data (llb) using the follo&g procedu& (The calculation for aluminum is shown below.) Sample Calculation for Aluminum 4.041 A 2.702 g/cm3 26.98 g/mol
Length of side of unit cube Density Atomic mass Face-centered unit cube
Volume of Unit Cube - 6.599 x loq3 om3 (4.041 A)3(1~"cm13 cube (1 A)3
Volume of a GAM of aluminum
[
9.985cm3 26.98 Q cm3 mol ][2.702g]mol Number of unit cubes in a GAM of aluminum (NUC)
Number of atoms in 1 GAM of aluminum I ,512 x loz3cube 4 atoms
mol
cube
-
6,052 x loz3atoms mol
Calculate the volume of a unit cube of that element in cmS. Calculate the volume, in cm3,of a GAM of the element from the density (llc)and the atomic mass. Calculate the number of unit cubes (NUC)in a GAM of the element. Calculate the number of atoms in the GAM from the cubic type. In other words, determine which of the following arraneements is nresent. ' a facc-cc&rcd cubic (4 atoms per unit cube, bady-centeredcubic 12 awma per unit cube diamond cubic' 18 atoms per unit mhe. At the introductorv level of chemistrv i t is not imwrtant that the student be ible to perform tl& calculatio; in the manner resented above. Instead. after showine the calculation, k e can give the class the following eq;ation that will enable them to do the calculation. Number of atoms =
GAM x NCT x loz4 density X A ~
(1)
whereA is the edge length of the unit cube expressed in A, and NCT is the number of atoms in the cube type. The Assignment and Oral Presentations Assign each student a-different element for homework. Impress upon the students the importance of completing the calculation before the next class meeting. As you make the assignment, tell each student which cube type the element displays. Once an element has been assigned, the student should be allowed to consult a CRC Handbook (llb) for the appropriate A value. The student should also get the value for the density of the element from the CRCHandbook or from a periodic table, preferably the one mounted in the classroom. During the next class meeting, each student should present the answer to the assigneti problem orally. The students will be amazed that each element assigned gave a value of 6 x loz3! Class Work and Discussion If one has enough computers, one can have each student do all of the elements for which there is data. This can be done easily on a spreadsheet, as demonstrated in Tables 1 and 2. I t should be pointed out that the units of the A values given in the CRC Handbook are X units. One X unit is equivalent to 1.002063 f 7 A(12). For this activity, one can use 1Xunit as being approximately equal to 1 This will give only about a 0.7% error in the final answer, as can be seen in Table 2. If one does not use the above appmximation, one gets 19 of the 28 elements to fall within 0.03 (0.5%)of the accepted value for Avogadro's number. One might ask why all the values are not the same. One possible explanation is that experimental error was intmduced to the calculation by the three experimentally derived numbers that are used-GAM, density, and A. Pose the question, "If each of the numbers was in error by only
A.
'A face-centered cubic arrangement with 4 additional internal atoms.
Volume 69 Number 4 April 1992
305
Table 1. Spreadsheet Data for Various Elements Element
GAM
(sfmol) Si C (diamond) Ge
Sn ll Al Au Ca Rh
Ni AS
5.42 .3.56 5.62 6.46 5.08 4.041 4.070 5.56 3.797 3.517
28.09
Densi3
(ah
4.077 4.939 Cu 3.608 Pt 3.912 TI 4.84 Sr 6.05 lr 3.831 Pd 3.880 Na 4.29 K 5.33 W 3.157 Li 3.50 Ta 3.30 Fe 2.861 Mo 3.140 Cr 2.875 V 3.03 Ba 5.02 'The "A'value In angstroms was approximated by allowing 1 X unit to be 1 angstrom.
0.1%,what would be the ermr in Avogadro's number?"The answer is 0.5%. Acknowledgment
The author wishes to thank Kurt Haltiwanger &Ray Lab, University of ColoradwBoulder)for an enlightening discussion of X units. 1. 2. 3. 4.
White, W 0 . J Chom. Edue. 1868,43,A438. Maynihan, C. T; Goldwhite, H. J. C h m Edue. 1888,46,119. King. L. C.;Neilaan. E. K J Chem. Educ. 1958,33.198. Sloat, C . A J. Chem Edue. I-, 43, A438
306
Journal of Chemical Education
Element
NUC
loz3 _~tomg Cube
2.33
Pb
Literature Cited
Table 2. Spreadsheet Calculations of Avogadro's Number for Various Elements
Si
,757178
8
C
(diamond) Ge Sn Th Al AU Ca Rh
Ni Aa Pb Cu
Pt TI Sr lr
Pd Na K W
Li
Ta Fe MO
Cr
v Ba
126.506008 3.092091 2 6.18 (6.15) The numbers in parentheses are corrected Avogadro's Numbers. The correction involved converting X units to A by multiplying the A value by 1.002063 and then substituting the product into eq 1. K ~ g l a La , J c k m . Edue 1888.65.132. Boyko,E.R.;BeUiveau, J. F J. Chm. Edw. 1868,63.611. Hewthrone, R.M.,Jr. J C k m Edw. 1910,47,751. Posbeim, P. S; WaEmiek, J. W; lKempeQaiaa1, P ; PoakozIm, J. A. J. Chem. Edue. 1986,63,125. 9. AlexandqM. D.; Ewing,G. J.;AbboU,F.T. J Chem.Edue. 188P,61,591. 10. L u k k Henk,van J C k m Edue. 1988,66,162. 11. Hondbook of Ckmia(ryondPhyares. 61st ed,Weast. R.C., Ed.:The Chcmioal Rubber Ca.: Cleveland, OH, 1980,(a1 F-246:61E-2W, (el 8-73, 12. I n r e m t i o ~ Tablea l ofX-RoyCryalollogmphy;Lonsdale, K. Ed.; Kynrivnh Pms: Birmingham, England: 1%3.3,42. 5. 6. 1. 8.
