$(1)$ \[ \begin{align} L_{12} &= \frac{1.4}{0.4} \times 0.6 \times 10^6 \times 0.02 \times \left\{ 1-\left( \frac{0.1}{0.6} \right)^{\frac{0.4}{1.4}} \right\} \\ &= \frac{1.4}{0.4} \times 0.6 \times 10^6 \times 0.02 \times \left( 1-0.589337 \right) \\ &=16827.8 \\ \therefore &=16.83 \ \rm{kJ} \end{align} \] $(2)$ \[ \begin{align} W_{12} &= \frac{L_{12}}{\kappa}\\ &= \frac{16827.8}{1.4} \\ \therefore &=12.02 \ \rm{kJ} \end{align} \] $(3)$ \[ \begin{align} T_{1} &= \frac{p_1 V_1}{mR} \\ &= \frac{0.6 \times 10^6 \times 0.02}{0.1 \times 0.287 \times 1000} \\ &=418.118 \\ &=418.12 \ \rm{K} \end{align} \] \[ \begin{align} T_{2} &= T_1\left( \frac{p_2}{p_1} \right)^{\frac{\kappa-1}{\kappa}} \\ &= 418.12 \left( \frac{1}{6} \right)^{\frac{0.4}{1.4}} \\ &=250.59 \ \rm{K} \end{align} \] \[ \therefore \Delta T = T_1 - T_2 = 167.8 \ \rm{K} \]