$(1)$ ポリトロープ $(2)$ \[ p_1 V_1^n = p_2 V_2^n \\ p_1 \left( \frac{mRT_1}{p_1} \right)^n = p_2 \left( \frac{mRT_2}{p_2} \right)^n \\ p_1^{1-n} \times T_1^n= p_2^{1-n} \times T_2^n \\ \left( \frac{T_2}{T_1} \right)^n = \left( \frac{p_1}{p_2} \right)^{1-n} \\ \left( \frac{T_2}{T_1} \right)^n=\left( \frac{p_2}{p_1} \right)^{n-1} \\ \begin{align} T_2 &= T_1 \left( \frac{P_2}{P_1} \right)^{\frac{n-1}{n}} \\ &= 300\left( \frac{2.7}{0.1} \right)^{\frac{0.3}{1.3}} \\ &=641.85 \\ &=641.9 \ \rm{K} \\ &=368.9\rm{^\circ C} \end{align} \] $(3)$ \[ \begin{align} \Delta U &= m c_V \Delta T \\ &= 10 \times 0.7171 \times (T_2 - T_1) \\ &=2451.4 \,\rm{kJ} \\ &=2.451 \,\rm{MJ} \end{align} \]