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Due: Thursday, February 11, 1999
Smith, van Ness & Abbott, problem 3.29, parts (c), (f), and (j).
Smith, van Ness & Abbott, problem 3.36 (use Table F.2).
Smith, van Ness & Abbott, problem 3.37 (use Tables E.1 - E.2).
Smith, van Ness & Abbott, problem 3.42 (use Figure 3.16).
The following problems requires you to take "measurements" using the piston-cylinder Java applet demonstrated in class (the highlighted text you just read will link you to it). It is important that you access this applet using Internet Explorer or Netscape version 4.0 (or higher). Get an early start on this, and let me know if you have difficulty getting it to work.
To measure the density, set the desired state conditions of T and P. Then hit the button to "Reset Averages". Wait a while as the system fluctuates about the average value, then record the "Average density" once it (the average) settles down to a (roughly) fixed value. Repeat this for each new measurement.
- Measure the density this way as a function of pressure and temperature. Do this for the three temperatures available on the applet (200, 400, and 600 Kelvin). At each temperature, take measurements at pressures of 50, 100, 200, 400, 600, 800, and 1000 bar (it is difficult to get exactly these pressures with the slider; just try to get data somewhere close to them).
- Make plots of your data, including pressure vs. density for the three temperatures, and compressibility factor vs. density. For each graph, put all three temperatures curves on the same figure.
- Using the appropriate construction, determine the virial coefficient from each isotherm. Does B depend on temperature for this system?
- Plot density versus temperature for all pressures. Estimate the coefficient of thermal expansion. Does it depend on temperature? On pressure?
- Fit your data to an equation of state of the form PV/RT = 1 + B/V + C/V2 (determine the coefficients B and C).
- From observing the simulation, what do you notice as the main difference in the molecular behavior between the systems at low versus high temperature?
Apply the piston-cylinder applet to to the study of adiabatic expansions. Set the system to be isothermal at T = 200 K, just to get the temperature in this range. Set the pressure to its maximum value, 1000 bar. Once the system has settled down, set it to be "Adiabatic", and reset the averages. Wait about half a minute, and note the average temperature and density (probably will be a bit different than 200 K). Then move the slider to the lowest pressure (50 bar), as quickly as possible so to simulate an irreversible expansion. Once it settles, reset the averages, wait a bit, and record the final temperature and density. Repeat this for the other two temperatures.
At no time during this experiment should you hit the "Stop piston" button, as this will take energy from the system and make it non-adiabatic.
We want to use your measurements to determine the heat capacity of this fluid. We will assume that it is independent of temperature.
a.) Following the approach presented in class, show that the equation of state from problem 5e above implies that the heat capacity of this substance is independent of density.
b.) For each expansion experiment, compute the work performed by the system, and its change in internal energy.
c.) Use your result from (b) and the measured temperature change to compute the heat capacity of the fluid. I was able to get results accurate only to within 50%, judging from the scatter in the values; I suggest (but do not require) that you repeat the expansions at each temperature to get more data.
d.) Derive an expression relating the initial and final temperatures and volumes for an adiabatic, reversible expansion of this system, based on the equation of state in problem 5e above.
e.) Repeat your expansion experiments, only this time decreasing the pressure with the slider as slowly as you can, so as to simulate a reversible expansion. Note the initial and final temperatures and densities. Compare them to the values from your newly derived formula.
f.) Just for fun, increase the pressure again slowly, and see how closely you can bring the system back to its original temperature. What happens when you try this following the irreversible expansion?
g.) Compute the work expected for a reversible expansion between the states of your experiments. Compare this to the change in internal energy computed from your previously-measured heat capacity applied to the observed temperatures.
A spreadsheet will be very helpful in doing problem 5 and 6. |
