Intermediate Thermodynamics Assignment 5 — Second Law of Thermo  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
For a system composed of $k=1..n$ bodies each with a temperature $T_k$, and after defining the entropy as $$ {\rm d} S_k\equiv \frac{\delta Q}{T_k} - \frac{\delta W}{T_k}$$ prove that, when $\delta W =0$, the entropy of the system always increases: $$ \sum_{k=1}^n {\rm d} S_k \ge 0$$ Note: this question is awarded double the points given to the other questions.
Question #2
Start from the $T{\rm d}s$ equations and derive the following expressions for the entropy change of a perfect gas: $$ s_2-s_1=c_v \cdot \ln \left( \frac{T_2}{T_1}\right) - R \cdot \ln \left( \frac{\rho_2}{\rho_1}\right)$$ $$ s_2-s_1=c_p \cdot \ln \left( \frac{T_2}{T_1}\right) - R \cdot \ln \left( \frac{P_2}{P_1}\right)$$ with $c_v$ and $c_p$ the specific heats at constant volume and pressure and $R$ the gas constant.
Question #3
Using the expressions derived in the previous question, calculate the specific entropy change for the following changes of state for a system. Consider one mole of N$_2$ at 1 bar and 300 K as initial conditions.
(a)  Isentropic compression to 10 bar.
(b)  Constant volume heating to 2 bars
(c)  Constant pressure heating to 600 K
Question #4
(a)  One kg of water at $300$ K is brought into contact with a heat reservoir at 600 K. When the water has reached 600 K, what is the entropy change of the water? Of the heat reservoir? Of the universe?
(b)  If the water has been heated from 300 K to 600 K by first bringing it into contact with a reservoir at 400 K and then with a reservoir at 600 K, what would have been the entropy change of the universe?
(c)  Explain how the water might be heated from 300 K to 600 K with almost no change of entropy of the universe.
Question #5
Consider one kg of air ($R=0.287$ kJ/kgK, $c_p=1.0035$ kJ/kgK, $c_v=0.7165$ kJ/kgK) expanding from 2 bar and 600 K to 1 bar and $500$ K. Calculate the entropy change and verify it is the same for different paths of integration.
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Use the following paths: a constant volume and constant pressure 1-A-2, a reversible adiabatic and constant pressure 1-B-2, an isothermal and constant pressure 1-C-2.
Question #6
A rigid nonconducting tank with a volume of $120$ cubic meters is divided into two equal parts by a thin membrane. Hydrogen gas is contained on one side of the membrane at $3.5$ bar and 80$^\circ$C. The other side is a perfect vacuum. The membrane is suddenly ruptured, and the H$_2$ gas fills the tank following the polytropic process $P V^{1.2}={\rm constant}$. What is the entropy change of the hydrogen? Consider hydrogen to be a perfect gas ($R=4.124$ kJ/kgK, $c_p=14.307$ kJ/kgK, $c_v=10.183$ kJ/kgK).
Question #7
Consider three water jets entering a mixing chamber as follows:
question03.png  ./download/file.php?id=5088&sid=ac9f329976ce3c8992345601709ac3b3  ./download/file.php?id=5088&t=1&sid=ac9f329976ce3c8992345601709ac3b3
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 #8
Consider a 1 m$^3$ tank in which air is contained in three different zones separated by membranes, as follows:
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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)
6.  20.83 kJ/K.
7.  310.4 K, $-614.5$ W/K.
8.  2.1 bar, 300 K, 41.28 J/K.
Due on Monday April 22nd at 16:30. Do Questions #6, #7, and #8 only.
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