Polytropic processes

A polytropic process is described by the formula $pV^n=c$ , where $n\in\mathbb{R}_{\geq 0}$ . The table below shows the association of different pV curves for ideal gases with the corresponding thermodynamic processes:

Question

Why is the projection (the curve obtained by representing the process in $(p,V,T)$ space on the $p-V$ plane ) an adiabat when $n=\frac{C_p}{C_V}$ and why is $V$ a constant when $n=\infty$ ?

Answer

From eq44 and eq46, $dU=C_VdT$ and $dH=C_pdT$ for an ideal gas. For adiabatic processes, $dU=w=-pdV$ and hence, $C_VdT=-pdV$ . Furthermore, $H=U+PV\;\;\Rightarrow\;\;dH=dU+pdV+Vdp$ . Since $dU+pdV=0$ , $dH=Vdp\;\;\Rightarrow\;\;C_pdT=Vdp$ . So,

$\frac{C_p}{C_V}=-\frac{Vdp}{pdV}\;\;\Rightarrow\;\;\frac{C_p}{C_V}\int\frac{dV}{V}=-\int\frac{dp}{p}$
$\Rightarrow\;\;\frac{C_p}{C_V}(lnV+c_1)=-(lnp+c_2)$