First-Law Problem Solving Gavin D peckham' University of Zululand, Private Bag X1001, Kwa Dlangezwa, 3886, South Africa Ian J McNaught University of Natal, PO Box 375, Pietermaritzburg,3200, South Africa
The subtleties of physical chemistry in general and thermodynamics in particular have led to the introduction of various charts, mnemonics, algorithms, tables, etc., to help struggling students clarify their thinking (14).For many years we have used a chart like that proposed by Hamby (5).However, we have found that its usefulness was limited because students often failed to distinguish among many of the expressions. For example, the following expressions AU = n c v ~ T and
AH = hold for isochoric, isobaric, and adiabatic processes involving ideal gases. This often confuses the students. For example, students oReu believe that AU is the same for all three cases. The convention for writing the equations does not show that the AT term differs in each case. We have tried to clarify this in our scheme by using A T , AT', Al"", etc., and by asking our students to draw the appropriate P-V diagrams. Even this approach had only limited success until
we hit on the idea of combining and superimposingthe various P-Vdiagrams as shown in the figure. A Helpful Diagram In this diagram it is assumed that an ideal gas is expanded from an initial state of Pz, Vl, T3 (point 1)by various processes to a final pressure ofP1. The diagram is constructed by
qualitatively considering the work done by the gas as it expands considering the heat changes and their effect on the temperature of the system For example, consider an adiabatic expansion against a constant pressure P1 (point 1to point 3 along the dashed line). Because the process is adiabatic, no heat exchanges occur with the surroundings. Thus, the work of expansion is done a t the expense of the internal energy of the system, which must decrease. However, for an ideal gas, the internal energy is proportional to the temperature. Thus, the temperature must also decrease, leading to a final temperature and volume a t point 3. This final temperature and volume are lower than they would have been if expansion had occurred isothermally (point 1 to point 4 along the dashed line) because, in the isothermal case, the gas does not draw on its internal energy to provide the energy required to carry out the work of expansion. Instead it draws this energy from the surroundings by absorbing heat. Consequently, point 3 must lie to the left of point 4, that is, at a lower temperature. By a similar argument point 2 must lie to the left of point 3. Interestingly, for an ideal gas, the expression AU = &AT will hold for both constant-volume and constant-pressure processes because the internal energy of an ideal gas depends on temperature only. Thus,
Similarly A H = ~ processes because
C ~ both A T for isobaric
and isochoric
The case of real gases is not quite so straightforward and is adequately described elsewhere (6). Additional Exercises The figure has been of great help to our students in visualizing the relationships and differencesamong the vari-
The expansion of an ideal gas from P2 to PI by various pathways.
' Author to whom correspondence should be addressed. Volume 70 Number 8 August 1993
625
ous processes involving ideal gas expansions. Although the figure contains most of the basic processes that are typically considered in an undergraduate textbook, other processes could be added if necessary. Aparticularly useful exercise is to get students to redraw the diagram, this time assuming that the ideal gas expands from an initial state of P2, V1, T3(point 1) to a final volume of V4 by various processes. The resulting diagram looks like the figure, but the various fmal states do not lie at appropriate volumes along the PI isobar. Instead they lie at appropriate pressures along the V4 isochore.
position. The problems that result from compartmentalizine the various thermodvnamic orocesses have been indicatcd, and the use of sup&imposedp - ~ d i a ~ r a mtos cla& the issue has been sueeested. Overall., the orinriole ofcombining P-Vdiagrams has much to recommend it.
Summary P-Vdiagrams of individual processes are widely used in thermodynamic textbooks but are rarely linked by super-
4. Handbook of Ckam&try and Physics, 55th ed.;Weaat, R. C., Ed.; CRC Press: CLmland, 1974: p D84.
626
Journal of Chemical Education
-
--
.
Literature Cited 1. Laidler, K J.; Meiser, J. H.Pkyslml Ckamislry; Addhon Wesley: Menla 1982;0 121.
P d ,CA,
2. McNaught, I. J.Spctrum 1918,16(41, 1118.
3.Fox.R. F. J C h E d u e
1976,53,441-442.
5. Hamby, M. J Chem. Edue 199%67,92&924.
6. Aflons, P WPhysicol Chemiafm, 2nd eed: OUP: Oxford, 1982:pp 84-86.
1
