Question

A gas is taken from $V_{i}$ to $V_{f}$ by two processes, one is isothermal and other is adiabatic.
Choose the correct option regarding the $p-V$ diagram for two processes.

• A. $(\text { Slope })_{\text {isothermal }}=(\text { Slope })_{\text {adiabatic }}$
• B. $(\text { Slope })_{\text {isothermal }}>(\text { Slope })_{\text {adiabatic }}$
• C. (Slope $)_{\text {isothermal }}<(\text { Slope })_{\text {adiabatic }}$
• D. Information is insufficient

Answer 1

Answer: (C)

Slope of isothermal curve
$=-\tan \phi=\frac{d p}{d V}=-\frac{p}{V}$
$[\because$ for isothermal process, $p V=$ constant
$\left.\Rightarrow p d V+V d p=0 \text { or } \frac{d p}{d V}=-\frac{p}{V}\right]$
Similarly, slope of adiabatic curve,
$-\tan \phi=\frac{d}{d V}\left(\frac{\text { constant }}{V^{\gamma}}\right)=-\gamma\left(\frac{p}{V}\right)$
$\left[\because\right.$ for adiabatic process, $p V^{\gamma}=$ constant $]$
As we know, $\gamma>1$
Thus, (slope) $)_{\text {adiabatic }}>(\text { slope })_{\text {isothermal }}$