A diatomic ideal gas is used in a Carnot engi | vedclass

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(D) For a diatomic gas,the adiabatic index $\gamma = 1.4 = \frac{7}{5}$.In a Carnot cycle,the adiabatic expansion occurs from state $C$ to state $D$.

Vedclass · Accepted Answer

$0.75$

Vedclass · Answer

(D) For a diatomic gas,the adiabatic index $\gamma = 1.4 = \frac{7}{5}$.In a Carnot cycle,the adiabatic expansion occurs from state $C$ to state $D$. The relation between temperature and volume for an adiabatic process is $T V^{\gamma-1} = 	ext{constant}$.Thus,$T_C V_C^{\gamma-1} = T_D V_D^{\gamma-1}$.Given $V_C = V$ and $V_D = 32 V$,we have:$\frac{T_C}{T_D} = \left(\frac{V_D}{V_C}ight)^{\gamma-1} = \left(\frac{32 V}{V}ight)^{\frac{7}{5}-1} = (32)^{\frac{2}{5}}$.Since $32 = 2^5$,we get:$\frac{T_C}{T_D} = (2^5)^{\frac{2}{5}} = 2^2 = 4$.The efficiency of a Carnot engine is given by $\eta = 1 - \frac{T_D}{T_C}$.Substituting the value,$\eta = 1 - \frac{1}{4} = \frac{3}{4} = 0.75$.

$A$ diatomic ideal gas is used in a Carnot engine as the working substance. During the adiabatic expansion of the cycle,if the volume of the gas increases from $V$ to $32 V$,then the efficiency of the engine is

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