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