R. 6. Snadden Heriot-Watt University Riccarton, Currie Edinburgh EH14 4AS, Great Britain
Any property X of a system which is itself a function of two independent variables Y and Z, is related to the latter hy the equation X = f(Y, Z). This analytical expression can he displayed graphically as an X-Y-Z plot. One of the earliest illustrations of this technique was provided by Gihhs over 100 years ago ( I ) , an excellent up-dating of which is given by Porter ( 2 ) . The progression from X-Y-Z plots to three dimensional models was inevitable and their use as teaching aids, although widely established, is almost exclusively limited to phase models of one-component and poly-component systems (3, 4). There are, however, innumerable situations in chemistry, outwith phase equilibria, which lend themselves to an X-Y-Z model display and it is the purpose of this paper to describe a few such systems. The Models-Construction
and Use
The models to be described have two novel features, uiz.: (1) method of construction and (2) tvue of svstem studied i d consequent use by students. he modelshave all been constructed to scale, with the Wang 360 Electronic Programmable Calculator being used as the data generating system. The basic concept is, in fact, a "spin-off' from earlier work carried out on simulation studies (5). Method of Construction Each model is built up on a square Perspex base of side-length 10 cm. Slots are then made in the base at l-cm intervals. Suitable cardboard is cut to the desired shape and inserted into the slots. Cardboard that is too thin will not be sufficiently rigid, too thick, and it is difficult to cut with scissors. An optimum thickness is ahout. 0.5 mm. Finally, the model is placed on a large sheet of stiff cardboard on which is shown the orientation of the axes, together with their respective scales (Fig. 1). This method of construction is easy and quick, and reveals every bit as much detail as the plaster of paris models favored by Petrucci ( 3 ) ,the wire models used by Peretti ( 4 ) , or the plastic sheets and thread prepared by Clarke and Hepfinger ( 6 ) .
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.
, . I
Systems Studied
The type of system studied is treated either: (1) as a demonstration model, or (2) as a tutorial model. A typical example of the former is shown in Figure 1for the Maxwellian distribution as a function of both speed and temperature. In its use as a teaching aid, it impresses on the student the marked influence of temperature on the fraction of molecules exceeding a particuiar speed. I t is normally emuloved in collaboration with two other similar models which depict the fraction of molecules as a function of (1) molar mass and speed, and (2) kinetic energy and temperature (Figs. 2 and 3). The tutorial model is an extension of the demonstration model, in that it permits the student to make measurements on the model and hence carry out subsequent calculations. A typical tutorial model is depicted in Figure 4, and shows the dependence of free energy, G, with the extent of reaction a t a series of temperatures, the pressure remaining constant throughout. This model allows the student by simple inspection, initially to confirm one of the most fundamental statements applicable to all chemical reactions occurring under the conditions of constant temperature and pressure, uiz. that a spontaneous change is attended by a decrease in the free energy of the system, and that any such reaction will therefore uroceed in the direction which leads ultimatelv to the establishment of an equilibrium condition, the chaiacteristic of which is that the free energy .. is a minimum, indicated on the model by an arrow. In this respect, the model is serving no more than a useful demonstration role. More novel, however, is the fact that the student may now "interact" with the model in its tutorial mode, and proceed to perform a number of interesting calculations, the most significant being the determination of the equilibrium constant, K p . This he does by removing each card in turn from the base and measuring on the grid scale the equilihrium composition of the mixture indicated by the arrow. From this information, equilihrium
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Figure 1. The basic model: s p e d distribution as a function of temperature.
368 / Journal of Chemical Education
X-Y-Z Models
Figure 2. Speed distribution as a funmon of molar mass
t
a.S./.",-,
.
.
.._l.=..".n,ll
.
.s. .
-
, , ..,.,, n
-..
+ : A -
.
.
Figure 4 Free energy as a function of extent o f reaction (isobaric).
Model Svsternr X
system
P-V-T relation mi^ (1) (2)
z
Y pressure com-
volume
werrure
tem~erature temperature
PleS6ihilit" ~.
~.
factor
curves (1) ~ftration
121 . ~
Ellingham Diag r a m s ( 7 , 81
ml o f a d d e d bale mi of a d d e d bare
pH DH
P K-, ./ D K. ~
temperature
AGO1
elementoxide
Concentrat i o n of a c i d
mole
partial pressures of N a O d and calculated from the equation
N O 2
are determined and Kp
KO = (PN0d2/P~20d The values of Kp SO calculated may he confirmed by measurements of G a t and
Figure 5. Free energy as a function o f extent o f reaction jisothermal)
from the equation
The same N204-NO2system is also studied under conditions of variable pressure and constant temperature; in this case the Y axis is log P to ensure convenient scaling (Fig. 5). Because of the considerable amount of computation involved, the above exercises are best carried out on a group basis, with each student contributing a small part to the whole. In addition to those models already described, the table lists a selection of systems which, in the opinion of the author, lend themselves to this technique.
AGO = -RT In K,
Using the Kp values a t each of the ten temperatures makes it possihle to establish the temperature dependence of Kp expressed equationally as In K, = A
+ BIT
( A and B are constants) from which values of both AHo and AS" may he obtained. Furthermore, if the model is viewed along the X
~
x
~
~
axis, it is strikingly obvious that there exists a linear relationship between the free energj of either pure N 2 0 4 or pure N O 2 and the temperature. Measurements of the slopes made directly on the model, yield values of both AH? and ASrOfor N 2 0 4 and N O 2 .
Literature Cited 111 Oihl~~..l. W.. Trona. C m n . Arod.. 11.182 118731. (2) Pl,rter.S. K..d. C H E M EOUC.,dR,291 119711. 191 Petrucci. R. H.. J. C H E M EDUC.. 42.:129 (19681. (41 Feretti. E.A..J. CHEM. EDUC., 43,2591198fil. (51 Runquirt. O.,Oken. R..and Snadden, R. B.. J. CHEM RDUC.. 49.266 119721 (6) Clarke. P.A..and Hm1inger.N. F.. J.CHEM. EDUC.. 48. 198 (1971). lil Ellm#hnm.H.I.T.. J S o c Chsm Ind.. lLimdnnl.63. 125(19441. lJrrr. Famdoy S o e . No. 4,126 119481. (XI D8nnat.C. W..and E1lineham.H. JT..
Volume 53, Number 6.June 1976 / 369
