$(1)$ 等温だから \[ pV=mRT \ (一定) \ \Longrightarrow \ p＝\frac{mRT}{V} \] \[ Q=W_{12}= \int_{V_1}^{V_2}pdV =mRT \int_{V_1}^{V_2} \frac{dV}{V} \\ = mRT \ln \frac{V_2}{V_1} \] $(2)$ \[ \begin{align} &\left\{\begin{array}{cc} p_1 V_1^\kappa = p_2 V_2^\kappa \\ p_1 V_1= mRT_1 \\ p_2 V_2 = mRT_2 \end{array}\right. \end{align} \] \[ mRT_1 V_1^{\kappa-1} = mRT_2 V_2^{\kappa-1} \\ \frac{T_2}{T_1} = \left( \frac{V_1}{V_2} \right) ^{\kappa-1} \]