2012 Thermodynamics Midterm Exam  
Midterm Quiz
May 5th 2012
10:00 — 12:00

Question #1
Starting from $F=ma$ applied on a gas particule, show that $$ P=\rho R T=N k_{\rm B} T= \frac{n k_{\rm B} T}{V} $$ with $$ T\equiv \frac{m \overline{q^2}}{3 k_{\rm B}}~~~{\rm and}~~~R\equiv \frac{k_{\rm B}}{m}$$ Outline the definition of the pressure $P$, the density $\rho$, and the number density $N$.
Question #2
Two adiabatic tanks are interconnected through a valve. Tank A contains 0.2 ${\rm m}^3$ of air at 40 bar and 90$^\circ$C. Tank B contains 2 ${\rm m}^3$ of air at 1 bar and 30$^\circ$C. The valve is opened until the pressure in A drops to $12.5$ bar. At this instant, determine (a) the temperatures and pressures in both tanks and (b) the amount of mass that has left tank A.
Question #3
Consider a 1 m$^3$ tank in which air is contained in three different zones separated by membranes, as follows:
question03.png  ./download/file.php?id=1417&sid=b03695733b1314c87ff20f75ea41eb3b  ./download/file.php?id=1417&t=1&sid=b03695733b1314c87ff20f75ea41eb3b
Initially, the air within the three zones has the following properties: $$ \begin{array}{llll} \hline ~ & \rm Zone~A & \rm Zone~B & \rm Zone~C \\ \hline P & \rm 1~bar & \rm 2~bar & \rm 3~bar \\ T & \rm 300~K & \rm 300~K & \rm 300~K\\ V & \rm 0.2~m^3 & \rm 0.5~m^3 & \rm 0.3~m^3 \\ \hline \end{array} $$ The membranes are suddenly ruptured, mixing occurs between the zones, and after a large amount of time the properties of the air become uniform throughout the tank. Assuming no heat transfer from the air to the tank walls, calculate:
(a)  The final temperature and pressure of the mixed air
(b)  The change in entropy of the air within the tank in J/K (that is, find the difference between the entropy of the mixed air and the sum of the entropies of the air within the 3 zones)
Question #4
Bottles of compressed gases are commonly found in physics and engineering laboratories. They present a serious safety hazard unless they are properly handled and stored. Oxygen cylinders are particularly dangerous. Pressure regulators for oxygen must be kept scrupulously clean, and no oil or grease should ever be applied to any threads or on moving parts within the regulator. The rationale for this rule comes from the fact that if oil were present — and if it were to ignite in the oxygen gas — this hot spot could lead to ignition of the metal tubing and regulator and cause a disastrous fire and failure of the pressure container. Yet it is hard to see how a trace of heavy oil or grease could become ignited even in pure, compressed oxygen since ignition points probably are over 800 K if so-called “nonflammable” synthetic greases are employed.

Let us model the simple act of opening an oxygen cylinder that is connected to a closed regulator as shown below:
question04.png  ./download/file.php?id=1418&sid=b03695733b1314c87ff20f75ea41eb3b  ./download/file.php?id=1418&t=1&sid=b03695733b1314c87ff20f75ea41eb3b
Assume that the sum of the volumes of the connecting line and the interior of the regulator is $V_{\rm R}$. $V_{\rm R}$ is negligible compared to the bottle volume. Opening valve A pressurizes $V_{\rm R}$ from some initial pressure to full bottle pressure. Presumably, the temperature in $V_{\rm R}$ also changes.

The oxygen cylinder is at 15.17 MPa and 311 K. The connecting line to the regular and the regulator interior ($V_{\rm R}$) are initially at 0.101 MPa, 311 K and contain pure oxygen. Assume no heat transfer to the metal tubing or regulator during the operation, independent of pressure or temperature.

The question we have is the following: Can the temperature in $V_{\rm R}$ ever rise to a sufficiently high value to ignite any traces of oil or grease in the line or in the regulator? Specifically, do the following:
(a)  If the gas entering $V_{\rm R}$ mixes completely with the initial gas, what is the final temperature in $V_{\rm R}$?
(b)  An alternative model assumes that there is no mixing between the gas originally in $V_{\rm R}$ and that which enters from the bottle. In this case, after the pressures are equalized, we would have two separate zones which are at different temperature. Assuming no heat transfer between the two zones, what is the final temperature of each zone?
(c)  From the temperatures obtained in part (a) and (b), comment on the hazard of this simple operation of bottle opening: Do you think the models in (a) and (b) are realistic (how close do you expect them to be to the actual temperature)?
2.  260 K, 4.35 kg, 3.75 bar, 393 K.
3.  2.1 bar, 300 K, 41.28 J/K.
4.  434 K, 1302 K.
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