# A diatomic ideal gas is used in a Carnot engine as the working substance. If during the adiabatic expansion part of the cycle the volume of the gas increases from V to 32 V, the efficiency of the engine is Option 1) 0.25 Option 2) 0.5 Option 3) 0.75 Option 4) 0.99

As we learnt in

$PV^\gamma = constant$

$TV^{\gamma -1}= constant$

$T_{b}{V_b}^{\gamma-1}= constant = T_{c}V_{c}^{\gamma-1}\\\Rightarrow {T_b}/T_{c}= (V_{c}/V{_b})^{\gamma-1} = (\frac{32V}{V})^{7/5-1}$

$\frac{T_b}{T_c}= (25)^{2/5} \Rightarrow T_{b}=4T_{c}$

We know that $\eta = 1-\frac{T_{c}}{T_{b}} = 1-\frac{1}{4} = \frac{3}{4}=0.75$

Option 1)

0.25

This is incorrect.

Option 2)

0.5

This is incorrect.

Option 3)

0.75

This is correct.

Option 4)

0.99

This is incorrect.

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