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One mole of a diatomic ideal gas (rigid molecules) undergoes a process in which its pressure is directly proportional to its volume. The fraction of…

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One mole of a diatomic ideal gas (rigid molecules) undergoes a process in which its pressure is directly proportional to its volume. The fraction of the heat absorbed that is converted into work done by the gas is

A. $\dfrac14$
B. $\dfrac13$
C. $\dfrac17$
D. $\dfrac16$

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Answer: (D) $\dfrac16$

For

P=aV

W=\int P\,dV=\tfrac a2(V_2^2-V_1^2)=\tfrac12(P_2V_2-P_1V_1)=\tfrac12R\,\Delta T

. Also

\Delta U=C_V\Delta T=\tfrac52R\,\Delta T

. So

Q=\Delta U+W=3R\,\Delta T

(molar heat capacity

C_V+R/2=3R

). Hence

W/Q=\dfrac{R/2}{3R}=\dfrac16

. Using

W=R\Delta T

(isobaric) would wrongly give

\tfrac27

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