\[ pV^\kappa=C=p_1V_1^\kappa=p_2V_2^\kappa \\ V=p_1^{\frac{1}{\kappa}} V_1 p^{\frac{1}{\kappa}} \] \[ \begin{align} L_{12} &= - \int_{1}^{2}Vdp \\ &= \int_{2}^{1}Vdp \\ &= p_1^{\frac{1}{\kappa}}V_1 \int_{p_2}^{p_1}p^{-\frac{1}{\kappa}}dp \\ &=p_1^{\frac{1}{\kappa}} V_1 \left[ \frac{1}{1-\frac{1}{\kappa}} \cdot p^{1-\frac{1}{\kappa}} \right]^{p_1}_{p_2} \\ &=p_1^{\frac{1}{\kappa}} V_1 \frac{V_1}{\kappa-1} \left( p_1^{\frac{\kappa-1}{\kappa}}-p_2^{\frac{\kappa-1}{\kappa}} \right) \\ &=p_1^{\frac{\kappa-1}{\kappa}} p_1^{\frac{1}{\kappa}} V_1 \frac{\kappa}{\kappa-1} \left\{1- \left( \frac{p_2}{P_1} \right)^{\frac{\kappa-1}{\kappa}} \right\} \\ &=\frac{\kappa-1}{\kappa} p_1 V_1 \left\{1- \left( \frac{p_2}{p_1} \right)^{\frac{\kappa-1}{\kappa}} \right\} \end{align} \] \[ 証明終わり\]