$(1)$ \[ 等温だから\hspace{10mm} pV=C=p_1V_1=p_2V_2 \Longrightarrow p=\frac{C}{V}\] \[W_{12}=\int_1^2pdV=C\int_{V_1}^{V_2}\frac{dV}{V}=C\ln\frac{V_2}{V_1}=p_1V_1\ln\frac{V_2}{V_1}\] \[ここでp_1V_1=p_2V_2より\frac{V_2}{V_1}=\frac{p_1}{p_2}\] \[\therefore W_{12}=p_1V_1\ln\frac{p_1}{p_2}\hspace{15mm}\\ 証明終わり\] (2) \[Q_{12}=\Delta V+W_{12}=mc_v\Delta T+W_{12}\] \[等温だから\hspace{10mm}\Delta T=0\hspace{10mm}\therefore Q_{12}=W_{12}\hspace{10mm}\\ 証明終わり\] \[Q_{12}=\Delta H-L_{12}=mc_p\Delta T+L_{12}\] \[等温だから\hspace{10mm}\Delta T=0\hspace{10mm}\therefore L_{12}= Q_{12}=W_{12}\hspace{10mm}\\ 証明終わり\]