In adiabatic container with adiabatic massless piston, containing 4 gm $N_{2}$ gas and 0.8 gm He gas are separated by a weakly conducting light partition. Both piston and partition are free to move. Initial temperature of $N_{2}$ and He gases are $‘ 3 T_{0}^{'}$ and $‘ T_{0}^{'}$ respectively. Initially, system is at rest and in equilibrium. Final temperature of the gas is

2 T_{0}

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\frac{3 T_{0}}{2}

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\frac{7 T_{0}}{4}

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\frac{5 T_{0}}{2}

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Solution

The correct option is A $2 T_{0}$
$\frac{(mass) N_{2}}{(atomic weight) N_{2}} \times \frac{7 R}{2} (3 T_{0} - T)$
$= \frac{{(mass)}_{H e}}{{(atomic weight)}_{H e}} \times \frac{5 R}{2} (T - T_{0})$
$\frac{4}{28} (\frac{7 R}{2}) (3 T_{0} - T) = \frac{(0.8)}{4} (\frac{5 R}{2}) (T - T_{0})$
$T = 2 T_{0}$

Why this question?

Caution: The massless piston is free to move is suggestive of the fact that heat exchange is happening at constant pressure, and not at constant volume .

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