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