Due: Tuesday February 14th

Problem 1.

In many thermodynamics problems, the ideal gas state heat capacity at constant pressure, Cp, is assumed to have a constant value. Over small temperature ranges this approximation in a good one. However, when large changes in temperature occur, the dependence of Cp on temperature can significantly effect the computation of [[Delta]]H and [[Delta]]S for this process. Some compounds have more significant variations in Cp than others as can be seen in Fig. 4.1 of S&VN, p. 108. The numerical values of Cp for many substances can be calculated using Table 4.1 of S&VN, p. 109.

We will use Maple to eliminate the "number crunching" aspects of evaluating Cp(T) and integrating Cp(T)dT. This will allow us to try a number of engineering design modifications to processes with little time investment. Changing units and generating graphs is also easy. However, the biggest payoff from using Maple for these calculations is being able to easily calculate an unknown temperature in cases where using the inclusion of Cp(T) produces an equation which can not be solved explicitly for temperature.

By starting with the Maple worksheet "Cp.ms" which is in your 10.213 locker, you can see examples of how to generate many of the commands you will need. If you would like more information about a particular command like plot or fsolve, simply type ?plot or ?fsolve at the maple prompt (>). This will generate a new window with the desired information. Look at the commands under the "File" menu which help you open existing files such as Cp.ms and save changes you make. There is also a print command which allows you to get a hardcopy of the worksheet. Also look at the commands in the "Edit" menu, which can help you modify the worksheet.

After opening "Cp.ms" begin executing the worksheet line by line by simply hitting the return key. There are numerous comment lines to help you understand what is being done. Comments containing "***" indicate things are required to be handed in for this problem.

For parts a-d, only handwritten answers are required (please DO NOT hand in printouts of your Maple worksheets or your graphs)

a) Look at the graph of Cp and Cpmh vs. T for CO2 on your screen. State in 25 words or less why Cpmh < Cp at any temperature and why the difference between these two quantities increases with temperature.

b) Notice that this graph of Cp and Cpmh also contains a horizontal line at the value of

Cpmh(300, 1900, C02). Move your cursor to the intersection of this horizontal line and the curve corresponding to Cp(T, CO2). Click the left mouse button to display the temperature where this intersection occurs. Compare this value to the arithmetic mean of the initial and final temperatures, (T1+T2)/2. Repeat this comparison for N2 over the same temperature range.

c) On your screen, graph Cp(T, CO2) and Cpms(300, T, CO2) from 301 K to 1900 K. (an aside: Would there be a problem to start the plot at 300 instead of 301 K?). Determine the temperature where Cp(T, CO2)=Cpms(300,1900, CO2). You may wish to try this for some other gases as well. Is there a simple way to estimate this temperature from the endpoint temperatures of the range considered?

d) The worksheet shows a temperature rise of 420 K when one mole of methane, initially at 298 has 200,000 J of heat added at constant pressure. Calculate the temperature rise for a second isobaric heating step of the one mole of methane. For this second step, the initial temperature is 718 K and 200,000 J of heat are added. Compare the temperature rise of the two steps.

e) Generate a graph similar to Fig. 4.1 in S&VN. However, instead of the vertical axis being Cp/R have your vertical axis should be Cp (BTU/lbmol K). One way to do this is to modify the statement "R=83.14:" and executing this change by hitting return after the change. Subsequent statements employing R should also be reexecuted. Note that you can skip reexecution of commands which the new plot will not depend on.

In addition, change the title of the graph to include your name.

A hardcopy of this graph should be made using the print command under the FILE menu at the top of the window containing the graph. This will generate a postscript file. Send this postscript file to a printer using the lpr command from your xterm window.

Hand in the original printout of your graph for part e.

You may wish to explore the Maple worksheet, "eos.ms" before proceeding with

the next problem.

Problem 2

Propane is interesting to consider since it is common fuel for outdoor cooking. Assume all cooking is done at an ambient temperature of 300 K, where the vapor pressure of propane is 10 bar.

a) The propane gas tank is filled, closed and attached to a barbecue grill. The tank remains undisturbed for 4 days. What is the state of the propane inside the tank? What are the temperature and pressure inside the tank?

b) On the 5th day, a cookout is held which uses up 80% of the propane in the tank. What is the pressure inside the tank on the day after the cookout?

c) In order to study the vapor-liquid equilibrium (VLE) properties of propane in more detail, 0.1 moles of propane are added to an initially empty piston. After filling, the piston is allowed to come to thermal equilibrium with the ambient while the pressure is maintained at 10 bar. Assuming the tank is completely filled with saturated liquid, calculate the extensive volume of the piston using the cubic equation of state of your choice.

d) Heat is added to vaporize the propane in the piston. This process occurs isothermally and isobarically. Determine the extensive volume of the piston when all the propane has been vaporized and calculate the amount of extensive work associated with this expansion. Use the same EOS as chosen for part (c).