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Suggestions from the CUPS project
(CUPS = Consortium for Upper Level Physics Software, published by John Wiley).
[In the CUPS Thermal and Statistical Physics Simulations materials there is a program "Phase Diagram of Fluids" which allows the user to explore the representation of thermodynamic paths in various diagrams (for example p-T, p-v, v-T, s-T, s-v, ..... diagrams). The following short exercise uses this program for the purpose of giving the student a little exposure to the meaning of the Clausius-Clapeyron equation.]
Suppose you have a liquid and its vapor in equilibrium in the same container at a certain temperature and pressure. If the volume of the container is changed, either more vapor becomes liquid (compression) or more liquid becomes vapor (expansion), but the pressure and temperature do not change until the system is converted entirely to a liquid or to a gas. In the (expansion) process of converting one mole from 100% liquid to 100% gas, there is heat absorbed from the surroundings; the amount of heat is called the latent heat of vaporization L = T delta-s. The constant delta-s is the entropy change per mole associated with the conversion of a liquid to a gas at a certain temperature and is equal to L/T.
Let's now consider again the case where both liquid and gas are present in the container. If the temperature is changed by a small amount delta-T there will be a corresponding change in pressure delta-p, but if the changes are small (and not close to the critical point) there will still be liquid and gas in the container. One can plot a line on a p-T diagram which represents the "coexistence" line, i.e. the p-T values for which both gas and liquid coexist. It turns out there is a relationship between the slope of this line dp/dT at a certain point on the line and the entropy change per unit volume change delta-s/ delta-v as the liquid is converted to gas at the same temperature and pressure. This relationship, the Clausius-Clapeyron equation, is:
dp/dT = delta-s/ delta-v
In the following exercise the calculations are based on the van der Waals model, and the pressure, volume, and temperature values used are given as fractions of the critical values. [ The critical temperature, for example, is the temperature above which the system will not separate into a "liquid" and a "vapor" part which coexist.] The van der Waals model is only an approximation to the real world when applied to the liquid-gas part of the phase diagrams.
(1) Call up the appropriate CUPS program (cups\CUPSTP\cupstp.exe, "Phase Diagram of Fluids"). Choose the options "van der Waals" fluid and v-T, p-T, p-v, s-v diagrams
(2) Try experimenting with the diagrams a little. You will find that you can use the mouse to plot points on some of the diagrams and the points will show up at appropriate spots on the other diagrams. Values can be obtained for the last point plotted. You can also draw lines representing processes on some graphs and have the corresponding lines show up on the other graphs.
(3) The p-T line represents the values of pressure and temperature for which the gas coexists with the liquid. Go to the upper part of the line (above 0.5) and put two points on the line, recording the temperatures and pressures. (These points should be fairly close together, but far enough apart to give an appreciable pressure difference.) Calculate the slope dp/dT of the coexistence line.
(4) In about the same position on the p-T graph (about halfway between your previous two points), plot points on either side of the coexistence line but very close to it. (These points should represent 100% liquid and 100% gas.) Note the rather large difference in the entropy of these two points, corresponding to a significant amount of heat absorbed to change the system from a liquid to a gas. Record the entropies, the volumes, and the differences as the liquid is converted to gas. Compare delta-s/ delta-v with dp/dT.
(5) Repeat several times for different parts of the p-T curve, and try to convince yourself that your results bear some resemblence to the Clausius-Clapeyron prediction. [Don't expect high accuracy; you won't have many significant figures to work with and the van der Waals model is only an approximation.]
Exercise with a diesel engine.
Under 'Engines' , select Diesel. Slow down the animation and carefully follow the cycle through its stages. Pay attention to the opening and closing of the valves.
1) Be able to discuss in detail what the engine is doing when T is rising with no change of S.
2) What is supposed to be going on where the pressure is constant and the volume changes? Comment on how realistic this is.
Build a carnot engine.
Under 'Engines' select 'Design Your Own Engine'. Stay with the default choices (nitrogen, reversible cycle T=300K).
a) Do an isothermal expansion at 300K first. Record the final volume and pressure.
b) Carry out the next phase of the carnot cycle and reach 600K. Record the volume and pressure.
c) Use the p and V values from part a) and the pressure from part b) to correctly predict the pressure from part b) [pV^(1.4)=constant along an adiabat]
d) Complete the third part of the carnot cycle, guided by the T-s diagram. [The complete T-s diagram should be a rectangle. Let the isothermal compression go to the point where a vertical line can be drawn back to the starting position on the T-s diagram.]
e) Close the carnot cycle and record the efficiency as given by the program. Compare this to your calculated efficiency. [Both should be right at 50% efficiency.]
f) Did we run the cycle in the correct sense for a heat engine? [Yes. Going in the other direction, it would be a refrigerator.]
[Here is one possible cycle. (1.0,0.02462,300)->(4.5, 0.00552,300)-> ((50.4,0.00098,600)-> (11.3,0.00434,600)-> (1.0,0.02462,300).]
Partial Summary of CUPS Thermal Physics Simulations
The program is designed to allow the user to explore the representation of thermodynamic paths in various diagrams (P-v, s-v, s-T, u-v, u-T diagrams). The diagrams are phase diagrams showing regions where there is vapor only or liquid only and showing the coexistence lines beyween phases. The user can plot a path on one diagram and observe the process on other diagrams. Data for pressure , temperature, (specific) volume, energy, and entropy for a given plotted point are available. The calculations are based on an expression for Helmholtz free energy which comes either from a van der Waals model or, in the case of water, from experimental data. One part that might be useful to us is the investigation of the Clausius-Clapeyron equation; comparisons may be made between the slope dP/dT of the coexistance line at a certain place on the P-T graph and the change in entropy per unit change in volume s/ v as this line is crossed (changing phase from liquid to gas and vice-versa).
This program allows the user to choose among engine (ideal) gases (helium, argon, nitrogen, or steam), decide on whether cycle is to be reversible or irreversible, choose an intial temperature and pressure, and combine adiabatic, isothermal, isobaric, and isochoric processes to create a cycle. The program shows T-s and P-v plots and gives a summary of final pressure, volume, temperature, and entropy for each process. It also keeps track of work done, heat in or out, and the changes in internal energy and entropy for each process and displays them when the cycle is complete. It is a nice program for visualizing processes and cycles on T-s and P-v diagrams, but for most cycles you must do some pre-calculation in order to know how to bring the gas back to the starting point (the Diesel, Otto, and Wankel engine models are exceptions). It is possible to use the program for refrigeration cycles also.
PROBABILITY AND STATISTISTICS:
This program uses the Galton board to illustrate binomial and normal distributions. It also illustrates Poisson and exponential distributions as well as simulations of a two dimensional random walk, the Kac ring, and the Stadium model.
STATISTICAL PROPERTIES OF GASES, LIQUIDS, AND SOLIDS:
This program simulates a two dimensional system of Lennard-Jones particles and hard disks using molecular dynamics and Monte Carlo methods. The simulation places a number of moving particles in a box and, as time goes on, shows the paths of colliding particles, graphs of pressure and temperature as a function of time, the speed distribution function, radial distribution function, mean square displacement, and velocity component distribution function. It can be used for such things as observing the return to equilibrium after a change in box size (sudden expansion), oserving qualitatively some properties of a solid (reduced box size, lower temperature) such as CV vs T, etc .
QUANTUM IDEAL GAS:
This program calculates and plots, for quantum ideal gases containing N particles, such things as density of states, number of particles per state, and number of particles per unit energy interval d , for specified temperature, and for B-E, F-D, or M-B statistics. It also plots chemical potential, specific heat, and energy per particle vs temperature. Another part of the program shows a time sequence simulation of the changes in the state occupancy diagram in phase space along with a plot of the instantaneous energy per particle. This simulation allows the user to change temperature while it is running.
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