ab and bc curves in the figure represent an isothermal and an adiabatic process respectively. Show that at the point b the ratio of the slopes of isothermal and adiabatic curves is equal to C_P/C_V.

ab and bc curves in the figure represent an isothermal and an adiabati

1.	ab and bc curves in the figure represent an isothermal and an adiabatic process respectively. Show that at the point b the ratio of the slopes of isothermal and adiabatic curves is equal to C_P/C_V.
Answer» Solution :For an adiabatic PROCESS `PV^gamma` =CONSTANT `THEREFORE P_gamma V^(gamma-1) dV+V^gamma dP=0` `therefore (P_gamma V^gamma)/V dV+V^gamma dP=0` `therefore dP=-P_gamma/V dV` `therefore ("dP"/"dV")_"adiabatic"=-gamma P/V`….(1) For an ISOTHERMAL process PV = constant, `therefore VdP+PdV=0` `therefore ((dP)/(dV))_"isothermal"=-P/V` ...(2) `("dP"/"dV")_"adiabatic"/("dP"/"dV")_"isothermal"=(-gamma P/V)(-V/P)=gamma=C_P/C_V`