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
(a)  Use the generalized charts to evaluate the enthalpy change and the entropy change per unit mole of $\rm CO_2$ due to a change of state from $\rm 80^\circ C$ and $(7.5+\xi_1)$ MPa to $\rm 135^\circ C$ and 15 MPa. Take $C_P=0.862$ kJ/kgK.
(b)  Calculate the heat transfer per kg when $\rm CO_2$ is compressed reversibly and isothermally from 10.35 MPa to $(14.78+0.3\times\xi_1)$ MPa at a temperature of $\rm 61.6^\circ C$ in a steady flow process. Take $C_P=0.862$ kJ/kgK.
(c)  The compressed $\rm CO_2$ at $(14.78+0.3\times\xi_1)$ MPa, $\rm 61.6^\circ C$ is now throttled to 10.35 MPa. Find the final temperature and entropy change of the process. Take $C_P=0.862$ kJ/kgK.
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