Thermodynamics Assignment 2 — First 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
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 $$e\equiv \frac{3}{2}RT ~~~{\rm and}~~~ T\equiv \frac{\overline{q^2}}{3R}$$
Question #2
Starting from the first law of thermo for a closed system, $$d(me)+PdV=\delta Q-\delta W$$ show that: $$d(mh)-VdP=\delta Q-\delta W$$ with $h\equiv e+ P/\rho$.
Question #3
Starting from the first law of thermo for a closed system $$d(mh)-VdP=\delta Q-\delta W$$ show that for an isentropic process, the following relationship holds: $$ \frac{P}{\rho^\gamma}={\rm constant}$$ with $\gamma\equiv c_p/c_v$, $c_p= (\partial h/\partial T)_P$, $c_v= (\partial e/\partial T)_V$.
Question #4
Five kmol of oxygen occupy a volume of $(10+\xi_1)$ m$^3$ at 300 K.
(a)  Determine the work required to decrease the volume (reversibly) to 5 m$^3$ (i) at constant pressure, (ii) at constant temperature, and (iii) adiabatically.
(b)  What is the temperature at the end of process (i) and the pressure at the end of processes (ii) and (iii)?
(c)  Sketch each process on a PV diagram
(d)  At the end of process (i), how much heat must be added at constant volume to bring the gas to the state corresponding to the end of process (ii)?
Question #5
A gas is contained in a closed rigid tank fitted with a paddle wheel. The paddle wheel stirs the gas for 20 min, with the power varying with time according to $\dot{W}=-(10+\xi_1)t$, where $\dot{W}$ is in Watts, and $t$ is in minutes. Heat transfer from the gas to the surroundings takes place at a constant rate of 50 W. Determine:
(a)  the rate of change of energy of the gas at time $t=10$ min in Watts.
(b)  the net change in energy of the gas after 20 minutes, in kJ.
Question #6
A totally enclosed cylinder is perfectly insulated thermally around its side and one end and contains a frictionless thermally non-conducting piston:
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The other end of the cylinder is perfectly conducting thermally. Initially, each side of the piston contains 0.0283 m$^3$ of air and 0.0283 m$^3$ of argon, respectively at 21$^\circ$C and $(103+\xi_2)$ kN/m$^2$. The argon side is completely insulated. If by adding heat to the air side the argon is compressed until its pressure reaches 206 kN/m$^2$:
(a)  What is the internal energy change of the argon?
(b)  How much heat was transferred to the air?
Question #7
Air is compressed to 1/5 of its volume following the polytropic relation $PV^n=$constant, rising in pressure from 90 to 470 kN/m$^2$. If the initial volume is 0.113 m$^3$ and the initial temperature is $38^\circ$C, find:
(a)  The value of the exponent $n$
(b)  Work done
(c)  Gain or loss of internal energy
(d)  Heat added or removed
Due on March 26th 2014
This assignment is more difficult than usual — expect to spend a bit more time solving the problems.
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