The rectangular box shown in Fig has a partition which can slide without friction along the length of the box. Initially, each of the two chambers of the box has one mole of a mono-atomic ideal gas $(γ = 5 / 3)$ at a pressure $P_{0}$ , volume $V_{0}$ and temperature $T_{0}$ . The chamber on the left is slowly heated by an electric heater. The walls of the box and the partition are thermally insulated. Heat loss through the lead wires of the heater is negligible. The gas in the left chamber expands pushing the partition, until the final pressure in both chambers becomes $243 P_{0} / 32$ . Determine the work done by the gas in the right chamber.
[Take $R = 8.3 J/K mol$ ]

15.5 T_{0}

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- 25.8 T_{0}

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- 15.5 T_{0}

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25.8 T_{0}

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Solution

The correct option is C $- 15.5 T_{0}$
For the left chamber,
$\frac{P_{0} V_{0}}{T_{0}} = \frac{P_{0} \times 243}{32 \times T_{1}} \times V_{1}$
( $n R =$ constant)
$\Rightarrow T_{1} = \frac{243}{32} \times \frac{V_{1} T_{0}}{V_{0}}$ ......(1)
For the right chamber for adiabatic compression, we get,
$P_{0} V_{0}^{γ} = P_{0} \times \frac{243}{32} \times V_{2}^{γ}$
$\Rightarrow \frac{V_{2}}{V_{0}} = {(\frac{32}{243})}^{3 / 5} \Rightarrow V_{2} = \frac{8}{27} V_{0}$
But $V_{1} + V_{2} = 2 V_{0}$
$∴ V_{1} = 2 V_{0} - V_{2} = 2 V_{0} - \frac{8}{27} V_{0}$
$\Rightarrow V_{1} = \frac{46}{27} V_{0}$ .........(2)

From (1) and (2),
$T_{1} = \frac{243}{32} \times \frac{46 \times V_{0}}{V_{0} \times 27} \times T_{0}$
or $T_{1} = \frac{207}{16} T_{0} = 12.9 T_{0}$ (approx.)

To find the temperature in the second chamber (right), we apply
${(\frac{T_{1}}{T_{2}})}^{γ} = {(\frac{P_{2}}{P_{1}})}^{1 - γ}$
$\Rightarrow {(\frac{T_{0}}{T_{2}})}^{5 / 3} = {(\frac{243 P_{0}}{32 P_{0}})}^{1 - 5 / 3}$
$\Rightarrow T_{2} = 2.25 T_{0}$

Work done in right chamber (adiabatic process)
$W = \frac{1}{1 - γ} (P_{2} V_{2} - P_{0} V_{0})$
$= - \frac{3}{2} [(\frac{243}{32} P_{0} \times \frac{8}{27} V_{0}) - P_{0} V_{0}]$
$= - \frac{3}{2} (\frac{9}{4} - 1) P_{0} V_{0} = - \frac{15}{8} \times R T_{0}$
$\Rightarrow W = - 15.5 T_{0}$

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