10.213 Spring 1994
Due: Tuesday, February 8th
Problem 1.
(10 pts total)
The second virial coefficient, B, while independent of pressure is often a strong function of temperature (see for example, Fig 1.9 and Table 1.1, pgs. 12-13, of A&S). For ethane (C2H6), this temperature dependence between 300 and 400 K is approximately:
B = 0.86 (cm3/mol K) T - 438 (cm3/mol)
Two moles of ethane are to be compressed in a piston. Assuming the pressure explicit virial EOS truncated to one term,
Z= 1 + B/V
can be used to describe the volumetric properties of ethane, use a computer to generate graphs of pressure (P) versus extensive volume ( V ) as V decreases from
10,000 cm3 to 800 cm3 under the following conditions:
a) (4 pts) First assume the compression occurs isothermally at 300 K. On the same
P-V graph, indicate the curve for an ideal gas for comparison to the ethane behavior. Based on this graph, which will require the more work for this compression, the ethane or the ideal gas? Calculate the numerical value of the extensive work required in both cases.
b) (6 pts) A control system is programmed so that the temperature of the gas in the piston increases during the course of the expansion according to:
T = 401 - V / (99 cm3 K)
Graph the P-V curve for the ethane and for the ideal gas for this non-isothermal process and numerically calculate the work required for both gases.
Problem 2.
(15 pts total)
The synthesis of a new substance, MITol, has just been discovered. Mass spectroscopy is used to determine it has a molecular weight of 100 grams/mole.
In order to investigate the thermodynamic properties of MITol, a rigid chamber of volume 25,000 cm3 will be used. Initially, this chamber is completely empty (P=0). A pure stream of MITol is pumped into the system. The walls of the vessel conduct heat rapidly compared to the rate at which gas can be added to the chamber, so that the contents of the chamber can always be assumed to be in thermal equilibrium with their surroundings at 300 K. This chamber rests on a balance, so that the weight of the gas added can be monitored. In addition, the pressure inside the tank can be read from a gauge. The tank also has a transparent window, so that any phase change behavior can be monitored and markings on this window allow the volume of any condensed phase inside the chamber to be determined.
a) (3 pts) When 1000 grams of the new substance have been added, a tiny droplet of liquid condenses inside the container. At this point, the pressure reads 9.40 bar, corresponding to the saturation pressure of MITol at 300 K. What is the compressibility factor of the gas at this point?
b) (3 pts) The addition process is continued until liquid occupies 1/10 of the tank's total volume. Weighing the tank reveals that it now contains 4000 grams of the new substance. What is pressure in the tank at this point?
c) (7 pts) Determine the parameters a and b for the van der Waals EOS of this new compound from the measured data. What do you predict the critical temperature and pressure of MITol will be?
d) (1 pt) What experiment could you devise for the apparatus in order to measure the acentric factor?
e) (1 pt) Which of the follow two assumptions could not be used to rework the problem above:
i) the generalized correlation derived from the corresponding theory of states
ii) the pressure explicit virial EOS truncated to one term.