Question

The slopes of isothermal and adiabatic curves are related as

• A. Isothermal curve slope $=$ adiabatic curve slope
• B. Isothermal curve slope $=\frac{1}{2}\times$ adiabatic curve slope
• C. Isothermal curve slope $=\frac{1}{\gamma }\times$ adiabatic curve slope
• D. Adiabatic curve slope $=\frac{1}{\gamma }\times$ isothermal curve slope

Answer 1

Answer: (C)

For adiabatic,
$P{V}^{\prime }=$ constant
$\ln P+\gamma \ln V=$ constant
$\frac{dP}{P}+\frac{\gamma dV}{V}=0$
$\Rightarrow {\left(\frac{dP}{dV}\right)}_{adb}=-\gamma \left(\frac{P}{V}\right)$
For isothermal, $PV=$ constant
${\left(\frac{dP}{dV}\right)}_{\text{iso }}=\frac{-P}{V}\Rightarrow {\left(\frac{dP}{dV}\right)}_{\text{adb }}=\gamma \cdot {\left(\frac{dP}{dV}\right)}_{\text{iso }}$
$\Rightarrow {\left(\frac{dP}{dV}\right)}_{\text{iso }}=\frac{1}{\gamma }{\left(\frac{dP}{dV}\right)}_{\text{adb }}$