2019 Intermediate Thermodynamics Midterm Exam  
Wednesday April 24th 2019
16:30 — 18:30
 
NO NOTES OR BOOKS; USE THERMODYNAMICS TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
04.03.19
Question #1
Starting from $F=ma$, show that the first law of thermo for a closed system corresponds to: $$d(me)+PdV=\delta Q-\delta W$$ with $$e\equiv \frac{3}{2}RT ~~~{\rm and}~~~ T\equiv \frac{m\overline{q^2}}{3k_{\rm B}}$$
Question #2
Consider three water jets entering a mixing chamber as follows:
question03.png  ./download/file.php?id=5130&sid=ad96924c479f222e9cd76696d7702b2c  ./download/file.php?id=5130&t=1&sid=ad96924c479f222e9cd76696d7702b2c
The properties of the water jets entering the chamber correspond to: $$ \begin{array}{llll} \hline ~ & \rm Jet~1 & \rm Jet~2 & \rm Jet~3 \\ \hline \dot{m} & \rm 1~kg/s & \rm 2~kg/s & \rm 3~kg/s \\ T & \rm 300~K & \rm 310~K & \rm 330~K\\ P & \rm 1~atm & \rm 1~atm & \rm 1~atm \\ \hline \end{array} $$ The chamber is sufficiently long that the 3 water jets mix completely with each other. This results in the water exiting the chamber having uniform properties. Knowing that the mixing chamber loses heat to the environment at a rate of 200 kW, determine the following:
(a)  The final temperature of the mixed water
(b)  The rate of change in entropy of the water within the chamber in W/K (that is, find the difference between the entropy of the mixed water and the sum of the entropies of the incoming 3 water jets).
Question #3
Consider a cylinder of radius $R=0.1$ m which contains helium and air gases separated by a piston as follows:
Q4.png  ./download/file.php?id=5079&sid=ad96924c479f222e9cd76696d7702b2c  ./download/file.php?id=5079&t=1&sid=ad96924c479f222e9cd76696d7702b2c
The air inside the cylinder is slowly emptied into a large dump reservoir in which the pressure remains at 0.1 atm. It is known that the initial pressure of the air inside the cylinder is of 1000 kPa, and that the initial temperature of the air and of the helium is of 27$^\circ$C. It is also known that the piston has a mass of 320.24 kg. For a final air pressure of 200 kPa, do the following (not necessarily in the order depicted):
(a)  Determine the final air temperature in the cylinder.
(b)  Determine the initial helium pressure.
(c)  Determine the final helium pressure.
(d)  Determine the final helium temperature.
Hint: the gravitational acceleration $g=9.81$ m/s$^2$.
Question #4
Consider a cylinder in which a piston made of steel separates air and helium as follows:
Q4.png  ./download/file.php?id=5066&sid=ad96924c479f222e9cd76696d7702b2c  ./download/file.php?id=5066&t=1&sid=ad96924c479f222e9cd76696d7702b2c
Knowing that the gravitational constant $g=9.81$ m/s$^2$, that the temperature of both gases is of 44$^\circ$C, that the radius of the cylinder is $R=0.1$ m, that the length of the cylinder $L=1$ m, that the mass of the piston is of 25.29 kg, that the density of the steel is of 8050 kg/m$^3$, and that the mass of the air and of the helium is of 0.0003 kg and of 0.00003 kg respectively, find $L_{\rm A}$.
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