Question

Find the ratio of the relative rise in pressure in an adiabatic compression to that in an isothermal compression.

Answer 1

For Adiabatic Process,

${PV}^{\gamma }=Constant\therefore V^{\gamma }dP+\gamma V^{\gamma -1}.P.dV=0\therefore dP=-\frac{\gamma V^{\gamma -1}.P.dV}{V^{\gamma }}$

For Compression$dV=-ve\therefore d{P}_a=\gamma .\frac{P}{V}dV$

For Isothermal Compression

$PV=Constant\therefore PdV+VdP=0dP=-\frac{P}{V}dV(asdVis-ve)\therefore \frac{{dP}_a}{d{P}_i}=\gamma$