Intermediate Thermodynamics Assignment 6 — Non-Ideal Models  
$\xi$ 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 the Van der Waals equation of state for a non-ideal gas: $$ P=\frac{\overline{R}T}{\overline{v}-b}-\frac{a}{\overline{v}^2}$$ prove that: $$ a=\frac{27}{64} \frac{\overline{R}^2 T_{\rm c}^2}{P_{\rm c}} ~~~{\rm and}~~~ b=\frac{1}{8} \frac{\overline{R} T_{\rm c}}{P_{\rm c}}$$ with $T_{\rm c}$ and $P_{\rm c}$ the critical temperature and pressure respectively.
Question #2
Refrigerant-12 has a specific volume of $\rm 0.02638~m^3/kg$ at a pressure of 800 kPa. Estimate the temperature of the gas by the use of (a) the ideal-gas equation of state, (b) the van der Waals equation of state, (c) the Redlich-Kwong equation of state, and (d) the generalized compressibility chart.
Question #3
Can the following fluids in the specified states be treated as ideal gases? Assume that a gas can be approximated as ideal if the generalized compressibility factor differs from that of an ideal gas by less than 10%.
(a)  Air at 0.1 MPa, 20$^\circ$C
(b)  Air at 13 MPa, 900$^\circ$C
(c)  Methane at 2 MPa, 1000$^\circ$C
(d)  Water at 0.1 MPa, 20$^\circ$C
(e)  Water at 0.01 MPa, 30$^\circ$C
(f)  Refrigerant 12 at 1 MPa, 50$^\circ$C
Question #4
(a)  Calculate the specific volume of nitrogen for a pressure of 3 MPa and a temperature of $(165+\xi_2)$ K (i) assuming ideal gas behaviour, and (ii) using compressibility charts
(b)  Nitrogen has a density of $\rm 0.14~g/cm^3$ and a temperature of 150 K. Estimate the compressibility factor and the pressure of the nitrogen.
(c)  Calculate the specific volume of refrigerant-12 at a pressure of $(1.0+0.1\times\xi_1)$ MPa and a temperature of $\rm 60^\circ C$ using the generalized compressibility chart. Assume that the molecular weight is 120.91 kg/kgmol.
Question #5
Consider $\rm CO_2$ in a $\rm 1~m^3$ tank initially at a pressure 10.35 MPa and at a temperature of $\rm 61.6^\circ C$. A valve is attached to the tank and lets $\rm CO_2$ escape such that when heat is added, the pressure within the tank always remains 10.35 MPa. After the heat addition process, the temperature of the $\rm CO_2$ left in the tank is of $\rm 90^\circ C$. Calculate the heat added to the tank and the mass of $\rm CO_2$ that left the tank. Take $c_p=0.862$ kJ/kgK. Hint: you can not assume either thermally or calorically perfect gas.
Question #6
Octane (C$_{8}$H$_{18}$) is expanded isentropically from a pressure of 124.5 bars and a temperature of 683 K to a pressure of 49.8 bars. Determine the temperature after the expansion process using (i) an ideal model, and (ii) the generalized correction charts.
6.  The temperature for the second part is somewhere between 585 K and 625 K: I don't want to give you the exact number to make it more challenging.
Due on Wednesday May 8th at 16:30. Do Questions #1, #2, #5, and #6 only.
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