Problem 5. a) A free expansion from $V_i$ to $V_f > V_i$ is an irreversible adiabatic expansion ($q = 0$) against zero pressure ($w = 0$). The corresponding change of internal energy is also zero ($\Delta U = U_f - U_i = 0$) according to first law of thermodynamics. This leads to $T_f = T_i$ in the case of an ideal gas. However, $T_f = T_i$ is not true for a real gas due to particle-particle interactions. Use the Van der Waals equation of state to show that $T_f - T_i$ for one mole of gas is given by (16 points)
$T_f - T_i = (\frac{a}{C_{v,m}})(\frac{1}{V_{m,f}} - \frac{1}{V_{m,i}})$
b) Evaluate $T_f - T_i$ for one mole of $N_2$ ($C_{v,m} = 21 J \cdot K^{-1} \cdot mol^{-1}$) undergoing a free expansion from $V_i = 0.20L$ to $V_f = 20L$ (4 points). [Ans: $\Delta T = -32K$]
