2061. Such decisions would require lengthy discussions, with a t least some of those participating in the discussion acting as devil's advocates. Unlike Mitchell, I would include industrial as well as academic chemists and would not assume that all college chemistry teachers are chemistry experts and all hieh school chemistrv teachers are not. 3econd, I agree with Mitchell that college and high school chemistry teachers should work together to improve instruction. However, I suggest that the average experienced high school chemistry teacher has more to offer the average experienced college chemistry teacher than vice versa. [F& some reason, college teachers seem to have one year of experience 10 times rather than 10 years of experience more often than do high school teachers, perhaps because teaching is our orimarv iob.1 The two .. erouos . toeether .. could heln. beeinnine .. high school and college chemistry teachers learn their craft. I notice that Mitchell's DaDer is "Part 1": I h o ~ that e Part 2 provides some suggestions for how to find'the time to do all this.
-. .
A repeating pattern of atoms or molecules in a crystal is ascribed to eauilibrium oositions in which the constituents of the crystai vibrate about these equilibrium positions. Usuallv. in lecture. once this has been established. solids are treated'as if the nuclei were a t rest a t their equilibrium positions. Nevertheless, our program pertains to simple inorganicmaterials such as alkali halides for which the ions are generally accepted to be on defined positions. As mentioned in our paper, line intensities are given mainI s by the square of the modulus of the structure factor. Most physical chemistry textbooks mention the additional variation of intensity due to the angle dependence of scattering factors. atomicvibrations. etc.-Our oattern simulator or& gram simulates only line positions, not intensities. As correctlv oointed out bv Goldbere. an unfortunate tvpographical error in our paper shows t h l t for a face-centered cubic lattice, the atoms are located a t (O,O, O), (a, 0, O), . . . , (aI2, a, O), . . . , (a, a, a), while the correct location is (O,O, O), (a, 0, O), , . . , (a12, al2, O), . . . ,(a, a, a).
William G. Lamb
Sllvlo Rodriguez
Meg~nEpiscopi School
University of the Pacific Stocktan. CA 95211
6300 S.W. Nicol Rd. Portland. OR 97223
On an X-Ray Diffradlon Pattern Simulator To the Editor: The recent article by Rodriguez and Rodriquez (J. Chem. Educ. 1989,66,648) describes a useful tool for those instructors who do not have facilities for the generation of X-ray powder patterns. However, the description given contains one serious. and one less serious error. The serious error reinforces the misconception often given to beginning students that in a crystal structure in the atoms (or ions) are situated a t lattice points such as 0,0,0 and translationally related ~ositions.This misconce~tiondevelops because. in the simple inorganic materials, such as the hkali halides, often discussed when introducing the concept of a crystal lattice, it is indeed true that the ions lie on special positions. However, this is the exception rather than the rule. When atoms, or ions, are locatedar general positions, the computation of the structure factors requires knowledge of those positions. It is for this reason that a previously published ( I ) powder pattern simulation program simulated only line positions hut not intensities. The less serious error is that, while it is indeed the case that the structure factors may be calculated from the scattering factors, the scattering factors themselves are functions of the scattering angle, 0, and the thermal motionof the atoms. A third error, ~ossihlvof t w o graphical origin, is that for a face-centered lattice the centering condition involves half-cell translations along two cell axes, not one. While i t is undoubtedly true that instructors with some experience in crystallography will recognize these points immediately, i t is also likely that they would not have been noticed by those with little experience in the field. Literature Cited 1. Miller. J.
S.;Goldberg,S. 2. J . Chsm. Educ. 1377 54.54 Stephen Z. Goldberg Adelphl University BOX701 Garden Clty. NY 11530
To the Editor: Given that our article "An X-ray Diffraction Pattern Simulator" was mainly concerned with the generation of a simulated X-ray diffraction pattern, we did not explicitly elaborate on basic points usually covered in most physical chemistry textbooks.
Student Understandlng To the Editor: While students' ability to solve problems may not be equivalent to their understanding molecular concepts, we must he careful in drawing conclusions from our attempts to test students' masterv of conceots. Althoueh Pickerik and Sawrey were disheartened by ;he results ;hey o b t a i k d in such an attempt (J. Chem. Educ. 1990,67,253-2551, I take a rosier view of the situation. In the gaslaw problem described in the article, nearly onethird of the students indicated the correct choice, showing essentially random positioning of dots (representingH Zmolecules) a t both 20 O C and -20 O C . An additional 50% chose an answer that indicated that the particles were randomlv arranged in a circle of smaller circu&ference a t -20 O C . his answer would be correct if the tank were a balloon (and the circle enclosing the dots were, therefore, contracted slightly). This choice, i t seems t o me, does not reflect gross misunderstandine of molecular concents hut rather the fact that balloons 6 d movable pistons are more often discussed in lectures on the eas laws than rieid containers are. T h e stoichiimetry problem discussed in the article showed three squares and eieht circles. initiallv s e ~ a r a t e d from one another, "reacting"to form three sets, each containing a square and two circles, with two circles left unattached. When asked to pick the equation which best de2Y scribes the situation, 11% chose the equation "X XY2", while 89% chose the equation "3X 8Y 3XYz
+
.
9V" u-
-
-+ -+
Sawrey indicates that the first of these equations is the correct answer. I agree that "X 2Y XYz" is the correct balanced eauation for the situation described. However. the students were not asked to choose the "correct balanced eauation": thev were asked to choose the eouation that thev feit "best described the situation". students may well have chosen the longer equation because i t conforms more closely than the balanced equation to the specifics shown in the picture-that three "molecules" were formed from three X's and six Y's and that two Y's were left over. I t is very possible to interpret the students' choice as an indication that they understand the meaning of the chemical equation, the significance of a chemical formula, and the limiting reagent concent. After all. the two eauations are aleebraicallv euuiva l e n t . ' ~ eoften ask studen& to treat chemical equa%ok as algebraic entities (in balancing redox equations, in shifting between net ionic and molecular equations, in setting up equilibrium problems); the fundamental question to be an-
+
Volume 68 Number 11 November 1991
969
swered at this point is whether the students see the relation between the two fonns of the equation and whether they are aware of the advantages and disadvantages of using each form. Toby F. Block Georgia Institute of Technoiosy Atlanta. GA 30332-0400
To the Editoc
There is, of course, merit in your assessment of the glass as being half-full while I view it as half-empty. The conceptual gas law problem is a good example of this difference. The answer chosen hy the greatest number of students may indeed reflect the fact that we discuss balloons and moveable pistons in class more often than rigid containers, but, in my admittedly random and non-quantitative poll of why students chose that particular, most common, answer, it was responses such as "the molecules were huddled together for warmth" and "the molecules are closer together in solid Hp than gaseous HZ" that convinced me that the students did not understand the kinetic molecular theory of gases whether applied to balloons or rigid containers. Regarding the conceptual stoichiometry question, I agree that answer d is partially correct but cannot go along with it being the best answer. We train our students to understand that a balanced equation represents the stoichiometry of the reaction and, like an algebraic equation, is reduced to lowest whole number ratios with like terms collected. Limiting reagent nroblems.. in . oarticular.. relv . on the students' grasp . . of the fact that nonstoichiometric quantities of reactants is a common occurrence that must be reconciled with the balanced equation. Chemists would not accept
-
&
-
The body altitudes of a tetrahedron intersect at the center of the figure, and this point is at a distance three-fourths from the vertices and one-fourth from the faces (see Proof). We are trying to find angle AMB = 8, is the bond angle of the tetrahedral molecule. By definition, t h e s i n e of this angle's supplement, i.e., cos (180 - R), is HM/BTij: = (hl 4)(3hl4) = 113. as can he seen in triangle BHM, where h is thk body altitude of the tetrahedron. Thus cos (180 - 8) = -cos R = 113 and0 = arccos (-1/3) = Proof To prove that the altitudes of an equilateral triangle intersect at a point that is at a distance two-thirds from the vertices and one-third from the sides we draw:
In equilateral triangle ABC, altitude BE is also the angle hisector, and angle DBE = 30'. BDF is a special right (30"-60° right A), where the side opposi&to the= an&e is half the hypotenuse. Thus DF = 112 BF. But BF = AF, being the sidesadjacent to the 30° angles in A ABF, therefore DF = 112 AF, which was to be proved. To prove that the body altitudes of a tetrahedron meet at a point of distance three-fourths from the vertices and onefourth from the faces we draw:
A
302+ 8Hz 6H20+ 2Hz
as the best way to express the formation of water from its elements anv more than 3A 8B = 3ABz 2B is a finished algebraic refationship. Sure, we want ourstudents to be ahle to count, and the overwhelmingly popular choice of d as an answer proves that most can do just that, hut they do so without making the connection to what chemists mean the balanced equaGon to represent.
+
+
Barbara A. Sawrey Academic Cowdinator Univwsity of California. Ssn Oiego La Jolla, CA 92093
Flndlng the Bond Angle To the Editor:
We offer an alternative method of "Findinp the Bond Angle in a Tetrahedral-Shaped Molecule" to thethree indicated bv Kawa ( I ) . This alternate method is a purely mathematicafproof that does not require the use of three-dimensional models or additional justification that a tetrahedron can be inscribed in a cube or that the center of the inscribed tetrahedral molecule is coincident with the point of intersection of the body diagonals of the cube. (This assumption was the starting point of Kawa's proof.) Consider the tetrahedron drawn below:
Assume that the edge of the tetrahedron is o k u n i t length. the face altitude, measures J3/2 units. RH, being twothirds of the face altitude (first p&t of Proof) measureLl/J3 the hodv altitude, is equal to (AB2- mZ)'", i.e., unit. (1 - (1/3'))E= Now, if HM = x , = I; being equal to also measures - x . In right triangle BHM, BM2= H M Z or:
m.
m.
+
J-m.
Jm
m- m,
m, m2
--
Therefore AH:HM = 3:1, which was to be proved. Literature Clted 1. Ksws.C. J.J Cham. Educ. 1988.65.884.
Resat Apak and izzet Tor Istanbul University Faculty of Engineering Vezneciler 34459, Istanbul. Turkey
970
Journal of Chemical Education
