Thermodynamics

JEE Main level, single correct, medium

177 of 190

Question 177 of 190

Helium gas goes through a cycle ABCDA\mathrm{ABCDA} (consisting of two isochoric and two isobaric lines) as shown in figure. Efficiency of this cycle is nearly:
(Assume the gas to be close to ideal gas)

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Answer: (A) 15.4%15.4 \%

Work done in complete cycle == Area under P−V\mathrm{P-V} graph
=P0V0=\mathrm{P}_0 \mathrm{V}_0
from AA to BB, heat given to the gas
=nCvΔT=n32RΔT=32 V0ΔP=32P0V0=\mathrm{nC}_{\mathrm{v}} \Delta \mathrm{T}=\mathrm{n} \frac{3}{2} \mathrm{R} \Delta \mathrm{T}=\frac{3}{2} \mathrm{~V}_0 \Delta \mathrm{P}=\frac{3}{2} \mathrm{P}_0 \mathrm{V}_0
from BB to CC, heat given to the system
=nCpΔT=n(52R)ΔT=n C_{\mathrm{p}} \Delta \mathrm{T}=\mathrm{n}\left(\frac{5}{2} \mathrm{R}\right) \Delta \mathrm{T}
=52(2P0)ΔV=5P0V0=\frac{5}{2}\left(2 \mathrm{P}_0\right) \Delta \mathrm{V}=5 \mathrm{P}_0 \mathrm{V}_0
from CC to DD and DD to AA, heat is rejected.
 efficiency, η= work done by gas  heat given to the gas ×100\text { efficiency, } \eta=\frac{\text { work done by gas }}{\text { heat given to the gas }} \times 100
η=P0V032P0V0+5P0V0=15.4%\eta=\frac{P_0 V_0}{\frac{3}{2} P_0 V_0+5 P_0 V_0}=15.4 \%

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