For curve $A B$ (isobaric),
$\frac{V_{B}-V_{A}}{T_{B}-T_{A}}$ ...........$(i)$
For curve $B C$ (adiabatic),
$T_{B}^{\top} V_{B}^{\gamma-1} =T_{C} V_{C}^{\gamma-1}$ ..........$(ii)$
$\text { Since, } V_{B} =x V_{A}$
$\text { and } V_{C} =x\left(\frac{\gamma}{\gamma-1}\right) V_{A}$
Efficiency of cycle,
$\eta=1-\frac{Q_{C A}}{Q_{A B}}$
$=1-\frac{n R T_{A} \ln \left(\frac{V_{C}}{V_{A}}\right)}{\frac{n \gamma}{\gamma-1)}\left(T_{B}-T_{A}\right)}$
$=1-\frac{\left(\frac{\gamma}{\gamma-1}\right) \ln x}{\left(\frac{\gamma}{\gamma-1}\right)(x-1)} \quad\left[\because \frac{T_{B}}{T_{A}}=x\right]$
Given, $\eta=50 \%=\frac{1}{2}$
$\Rightarrow \quad \frac{1}{2}=\frac{\ln x}{x-1} \Rightarrow x^{2}=e^{x-1}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
