Intermediate Thermodynamics Assignment 7 — Maxwell Relations  
$\xi$ 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 $T{\rm d}s$ equations, prove the following two Maxwell relations: $$ \left(\frac{\partial T}{\partial v}\right)_s=-\left( \frac{\partial P}{\partial s}\right)_v ~~~{\rm and}~~~\left(\frac{\partial T}{\partial P}\right)_s=\left( \frac{\partial v}{\partial s}\right)_P$$
Question #2
Starting from $\psi\equiv h-Ts$, $\phi\equiv e-Ts$ and the $T{\rm d}s$ equations, prove the following two Maxwell relations: $$ \left(\frac{\partial s}{\partial v}\right)_T=\left( \frac{\partial P}{\partial T}\right)_v ~~~{\rm and}~~~ \left(\frac{\partial s}{\partial P}\right)_T=-\left( \frac{\partial v}{\partial T}\right)_P$$
Question #3
If the $PvT$ behavior of a gas is given by the Berthelot equation of state, show that the change in enthalpy of the gas during an isothermal process from state 1 to state 2 can be written as: $$ \bar{h}_2-\bar{h}_1=\frac{3a}{T}\left(\frac{1}{\bar{v}_1}-\frac{1}{\bar{v}_2} \right)+ \bar{R} T \left(\frac{\bar{v}_2}{\bar{v}_2-b}-\frac{\bar{v}_1}{\bar{v}_1-b} \right)$$
Question #4
Nitrogen ($T_{\rm c}=126.2$ K, $P_{\rm c}=3.4$ MPa) expands isothermally from 125$^\circ$C and 8.5 MPa to $(3.5-0.2\times \xi_1)$ MPa. Determine the change in internal energy, enthalpy, and entropy of the gas, assuming that the nitrogen obeys the Van der Waals' equation of state. Compare your results with the values obtained assuming ideal-gas behavior.
Question #5
$(10+\xi_2)$ kilograms of argon at 20$^\circ$C fill one-half of a 0.1 m$^3$ rigid, adiabatic tank. The other half is evacuated, and the two halves are separated by a membrane. If the membrane should rupture, what final temperature and pressure would result (a) if the gas is ideal and (b) if the gas follows the van der Waals equation of state?
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