John F. Remark1 The University o f Konsos Lawrence, 66045
Computerized Three-Dimensional
illustrations of Gas Equations
Students in chemistry have great trouble visualizing three-dimensional data in their minds without the use of models or illustrations. Until recently, a three-dimensional surface viewed from various perspectives could he drawn only by a very talented draftsman. SURFACE 11, a graphics system developed by the Kansas Geological Survey, has the capability of taking threedimensional data and plotting it to produce a perspective block diagram (also referred to as a fishnet plot). This graphics system has the capability of plotting the surface from different angles and/or different elevations by altering only two parameters in the computer program. This routine was used to plot examples of the van der Waals' and ideal gas law equations. Gas Equations Above the Critical Temperature In working with the gas laws, students very rarely are required to have a knowledge of the true shape of threedimensional data. Figure 1 illustrates the relationship between pressure, volume, and temperature in the ideal gas law above the critical temperature of the gas. From eqn. (1) P = RTIV (1) it can be seen that at constant volume the pressure doubles as the temperature is doubled. It can also he determined from eqn. (1) that as the volume is halved a t constant temperature, the pressure is again doubled. This relationship is stressed in teachine eas laws to new students and many teaching devices are i s e d to emphasize this connection to the students. In Figure 1 this relationship is illustrated pictorially in three dimensions. Without the use of a three-dimensional plot, the students' minds are trying to graph the points mentally while also trying to get a grasp of their relationship. In using a three-dimensional plot as in Figure 1 for a visual aid, the students only have to worry about comprehending the relationship and do not have to worry about sorting out the data. Figure 2 is a three-dimensional plot of the van der Waals' equation. The gas used is carbon tetrachloride which has an "a" term equal to 20.38 I2 atm/mole2 and a "b" term equal to 0.138 l/mole. In this illustration the pressure does not increase as rapidly as in the ideal gas equation. This fact can be determined by visually comparing Figures 1 and 2. This fact can also be explained by examining eqn. (2). 308
Figure 1. Plot of the ideal gas equation. The volume varies from 0.2-4.0 I and the temperature varies from 600-750%
In this equation a small volume "b" is subtracted from the total volume. This small volume "b." represents four times the area that the molecules actually displace.2 As the volume is decreased in Figure 2, the pressure increases at a much slower rate than shown in Figure 1. This occurs until the volume approaches the value for the "b" t e r n in the van der Waals' equation. At this noint the molecules would he forced into each other, thus making the pressure approach infinity. The two ~reviousillustrations deoict the shane of ~-~ how pressure, temperature, and volume are related in the ideal gas equation and the van der Waals' equation. These three-dimensional graphs still do not show the student where the differences between these two equations exist. Figure 3 is a plot of this difference between the van der Waals' and ideal gas pressures. This illustration shows graphically the volume in which van der Waals' equation should he used to approach the real gas pressure and the volume in which the ideal gas equation should he used to
.
~
Presented at the American Chemical Society's Midwest Regional Meeting, The University of Kansas, Lawrence, Kansas, October 25-26, 1973. 'Present address: Battelle Pacific Northwest Laboratories, P.O. Box 999, Richland, Washington 99352. 2 Parsonage, N. G., "Gaseous State," 1st Ed., Pergaman Press, London, England, 1966, p. 15.
600
o o2
Figure 2. Plot of the van der Waalr' equation. The volume varies from 0.2-4.0 1 and the temperature varies from 600-750%
Figure 3. Plat of the difference between the van der Wads' equation and the ideal gas equation above the critical temperature.
Volume 52, Number 7 , January 1975
/
61
approach the real gas pressure. In Figure 3 i t can be seen that at volumes larger than 1.4 1, the ideal gas equation can be used to approximate the real gas. However, a t volumes smaller than 1.4 1 the van der Waals' equation must be used to approximate the pressure of the real gas. (1.4 1 is an arbitrary value and any value may be chosen depending on the error which one thinks is not excessive.) Note i.n Figure 3 that a t large volumes the pressure of the van der Waals' equation is smaller than the pressure of the ideal gas equation. At small volumes the van der Waals' pressure is larger than the ideal pressure. The crossover point, the point where the van der Waals' pressure becomes larger, occurs where the volume approaches the "b" term. The table lists some values for the pressure a t the isotherm of 625°K a t various volwnes around the crossover point. The crossover point occurs a t approximately 0.22 1 for carhon tetrachloride. This volume varies, depending on the particular gas used in the calculation. I t appears a t first glance that Figure 3 is incorrect due to the inconsistency at the crossover point a t low volumes. However, from the table it can he determined that the illustration is indeed correct and the inconsistency a t the crossover point is real. Gas Equations Below the Critical Temperature The critical temperature of carhon tetrachloride is 556°K. The previous illustrations are calculated and plotted for temperatures ahove the critical temperature of carhon tetrachloride. How do the relationships of the pressure, volume, and temperature of the ideal gas and van der Waals' equations change if the illustrations are plotted over a temperature range below the critical temperature of carhon tetrachloride? Figure 4 is a plot of the relationship between pressure, volume, and temperature of the ideal gas equation. Figures 1 and 4 are very similar; indeed the only noticeable difference is that the pressure in Figure 4 is lower than in Figure 1 because of the correspondingly lower temperature. Figures 1 and 4 could be used by an instructor to display the relationship between temperature and pressure. The temperature scale has been chosen so the range of the temperatures for Figures 1 and 4 is consistent from 450-750°K. This range was selected to illustrate the pressure-temperature relationship both ahove and below the critical temperature of carbon tetrachloride. How is the van der Waals' Dressure. volume. and temperature relationship altered at temperatures below the cr~t:ral temoeraturt. d carbon tetrachloride'.' As ran he observed from Figure 5, the pressure increases gradually as the volume is decreased. This gradual increase in pressure occurs until low volumes are reached. At these low volumes, as can be seen in Figure 5, the pressure actually decreases according to the van der Waals' equation. This is the area in which the van der Waals' equation breaks down. What is actually occurring in thm area is liquefaction of the gas; a two-phase solution exists in this area.3 The van der Waals' equation cannot represent the discontinuities arising during the liquefaction of a gas. Rather than the experimental straight line achieved in the laboratory during the liquefaction of a gas, the van der Waals' equation has a maximum and minimum within this two-phase regioa4 As the critical temperature of the gas is approached, the maximum and minimum points approach one another and a t the critical temperature they form an inflection point in the isotherm. Figure 6 is a plot of the difference between the ideal gas pressure and the van der Waals' pressure. This graph, together with Figure 3, illustrates that at high volumes the aparsonage, N. G., "Gaseous State," 1st Ed., Pergamon Press, London, England, 1966. p. 20. 'Moore, W. J., "Physical Chemistry," 3rd Ed., Prentice-Hall, Ine., Englewood Cliffs,New Jersey, 1962, p. 22. 62
/
Journal of Chemical Education
Values for Pressure at the Isotherm of 625°K at Various Volumes volvme
AP (vander waels'-idealgas)
0.204
61.3 -59.9 -31.1
0.404 0.6W
difference between the van der Waals' equation and the ideal gas equation is very small. When the volume approaches the "b" term of the van der Waals' equation, the pressure difference becomes large. As the pressure difference becomes larger, the van der Waals' equation must be used in order to obtain values of the pressure simulating the pressure of the real gas. With the use of these three-dimensional illustrations, it is hoped that students will obtain a better understanding of the relationship between temperature, volume, and pressure in both the van der Waals' equation and the ideal gas equation. These illustrations are not meant to replace any teaching aids currently in use but rather as a much needed visual supplement in explaining the pressure, volume, and temperature relationship to students. Each matrix illustrated in this paper consists of 620 data points. Two computer programs were used to create
0.2
V
4.0
Figure 4. Plot of the ideai gas equation. The volume varies from 0.2-4.0 I and the temperature variesfram 450-600'K.
02
v
Figure 5. Plot of the van der Waals' equation. The volume varies from 0.2-4.0 i and the temperature varies horn 450-600°K.
Figure 6. Plat of the difference between the ideai gas pressure and the
van der Waals' pressure below the critical temperature.
the illustrations. The first program, GGAS, calculated the theoretical points. The second program, SURFACE 11, calculated the plotter instructions. The illustrations were drawn on a Gerher flatbed plotter, Series 600, Model 622. Program GGAS is available upon request. SURFACE I1 is available under a controlled distribution plan to nonprofit
organizations. The author will supply additional information upon request. Acknowledgment
The author would like to thank the Kansas Geological Survey for providing the computer and plotter time neces-. sary for completion of this work.
Volume 52, Number I , January
1975
/
63
