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,
$P V'=$ constant
$\ln P+\gamma \ln V=$ constant
$\frac{d P}{P}+\frac{\gamma d V}{V}=0$
$\Rightarrow \left(\frac{d P}{d V}\right)_{adb}=-\gamma\left(\frac{P}{V}\right)$
For isothermal, $PV=$ constant
$\left(\frac{d P}{d V}\right)_{\text {iso }}=\frac{-P}{V} \Rightarrow \quad\left(\frac{d P}{d V}\right)_{\text {adb }}=\gamma \cdot\left(\frac{d P}{d V}\right)_{\text {iso }}$
$\Rightarrow \left(\frac{d P}{d V}\right)_{\text {iso }}=\frac{1}{\gamma}\left(\frac{d P}{d V}\right)_{\text {adb }}$