Thermodynamics

JEE Main level, single correct, medium

178 of 190

Question 178 of 190

A diatomic ideal gas is used in a Car engine as the working substance. If during the adiabatic expansion part of the cycle, volume of the gas increases from V\mathrm{V} to 32 V32 \mathrm{~V} the efficiency of the engine is

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Answer: (B) 0.750.75

The efficiency of cycle is
η=1−T2T1\eta=1-\frac{T_2}{T_1}
for adiabatic process
 TV r−1= constant \text { TV }^{r^{-1}}=\text { constant }
For diatomic gas γ=75\gamma=\frac{7}{5}
T1 V1γ−1=T2 V2γ−1 T1=T2(V2 V1)γ−1 T1=T2(32)75−1=T2(25)2/5=T2×4 T1=4 T2.η=(1−14)=34=0.75\begin{aligned} & \mathrm{T}_1 \mathrm{~V}_1^{\gamma-1}=\mathrm{T}_2 \mathrm{~V}_2^{\gamma-1} \\ & \mathrm{~T}_1=\mathrm{T}_2\left(\frac{\mathrm{V}_2}{\mathrm{~V}_1}\right)^{\gamma-1} \\ & \mathrm{~T}_1=\mathrm{T}_2(32)^{\frac{7}{5}-1} \\ &=\mathrm{T}_2\left(2^5\right)^{2 / 5} \\ &=\mathrm{T}_2 \times 4 \\ & \mathrm{~T}_1=4 \mathrm{~T}_2 . \\ & \eta=\left(1-\frac{1}{4}\right)=\frac{3}{4}=0.75 \end{aligned}

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