断熱変化　ポアソンの公式

$\displaystyle \Delta U$	$\textstyle =$	$\displaystyle Q - W'$
$\displaystyle Q$	$\textstyle =$	$\displaystyle 0　Q=\Delta U +W'$
$\displaystyle 0$	$\textstyle =$	$\displaystyle dU+dW'= nC_vdT+PdV$	(33)
$\displaystyle PV$	$\textstyle =$	$\displaystyle nRT　で両辺を割る$
$\displaystyle \frac{{C_v }}{R}\frac{{dT}}{T}$	$\textstyle +$	$\displaystyle \frac{{dV}}{V} = 0 　微分方程式を解く$	(34)
$\displaystyle \frac{{dT}}{T}$	$\textstyle =$	$\displaystyle - \frac{R}{{C_v }}\frac{{dV}}{V}$	(35)
$\displaystyle \int {\frac{{dT}}{T}}$	$\textstyle =$	$\displaystyle - \frac{R}{{C_v }}\int {\frac{{dV}}{V}}$
$\displaystyle \log \left\vert T \right\vert$	$\textstyle =$	$\displaystyle - \frac{R}{{C_v }}\log \left\vert V \right\vert + C$
$\displaystyle \log \left\vert T \right\vert$	$\textstyle +$	$\displaystyle \frac{R}{{C_v }}\log \left\vert V \right\vert = C$
$\displaystyle \log \left( {TV^{\frac{R}{{C_v }}} } \right)$	$\textstyle =$	$\displaystyle Const.$	(36)
$\displaystyle TV^{\frac{R}{{C_v }}}$	$\textstyle =$	$\displaystyle Const. 　PV=nRT　を用い$	(37)
$\displaystyle \frac{{PV}}{{nR}}V^{\frac{R}{{C_v }}}$	$\textstyle =$	$\displaystyle Const.$
$\displaystyle PV^{1 + \frac{R}{{C_v }}}$	$\textstyle =$	$\displaystyle PV^{\frac{{C_{v + } R}}{{C_v }}}$
	$\textstyle =$	$\displaystyle PV^{\frac{{C_p }}{{C_v }}} = PV^\gamma = TV^{\gamma - 1} = Const.$	(38)
		$\displaystyle \frac{R}{{C_v }} = \gamma - 1$