Helium gas goes through a cycle ABCDA as shown in figure. Efficiency of this cycle is nearly _______ (Imagine He is an ideal gas)

Options

A. `9.1%`
B. `10.5%`
C. `12.5%`
D. `15.4%`

Answer

D

Explanation

Work done in complete cyclic = Area under P-V graph
(Cyclic process is clockwise so work is positive)
`therefore` W=Area of ABCD
=AB x BC
`=(2P_0-P_0)xx(2V_0-V_0)`
`therefore W=P_0V_0` …(1)
For He gas,
Specific heat at constant volume `C_v=3/2R` and
Specific heat at constant pressure `C_p=5/2R`
Heat absorbed (along path AB) at constant volume ,
`(DeltaQ)_(AB)=muC_VDeltaT=3/2muR DeltaT`
`=3/2xxDeltaP V_0 [because PV=muRT]`
`=3/2(2P_0-P_0)V_0=3/2P_0V_0`
Heat absorbed (along path BC ) at constant pressure,
`(DeltaQ)_(BC)=muC_P DeltaT =5/2muR DeltaT`
`=5/2 (2P_0)DeltaV[because muR DeltaT=PV]`
`=5/2(2P_0)(2V_0-V_0)`
`=5/2(2P_0xxV_0)`
`=5P_0V_0`
Heat is not absorbed at path CD and DA,
`therefore` Total heat absorbed `Q_1=3/2P_0V_0+5P_0V_0`
`=13/2P_0V_0` ...(2)
`therefore` Efficiency `eta=W/Q_1`
`=(P_0V_0)/(13/2P_0V_0)` (`because` From eq.(1) and (2))
`=2/13xx100%`
=15.38%
`approx` 15.4%