An ideal mono-atomic gas is confined in a cylinder with the help of a piston that can slide inside the cylinder without friction. The cylinder is placed on a horizontal platform in a vacuum chamber. The walls of the cylinder and the piston are made of heat insulating materials. Initially the temperature of the gas is $T_{0}$ and the piston stays in equilibrium as shown in the figure. The temperature of the gas is quickly increased $η$ times and then the gas is allowed to expand. If the piston rises and again acquires an equilibrium state, find temperature of the gas in this new equilibrium state. (in kelvin) (Take, $η = 2, T_{0} = 273 K$ )

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answer is 436.8.

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Detailed Solution

$\underset{I}{P_{0}, V_{0}, T_{0}} \to \underset{I I}{η P_{0}, V_{0}, η T_{0}} \to \underset{I I I}{P_{0}, V_{f}, T_{f}} P_{0} = \frac{m g}{A} W_{g} = - W_{e x t} = P_{0} Δ V = P_{0} (V_{f} - V_{0}) = n R (T_{f} - T_{0}) A p p l y i n g 1 s t l a w o n g a s f o r I I \to I I I Δ Q^{°} = Δ U + Δ W \Rightarrow O = \frac{3}{2} n R (T_{f} - η T_{0}) + n R (T_{f} - T_{0}) \Rightarrow T_{f} = (\frac{3 η + 2}{5}) T_{0} = \frac{8}{5} \times 273 = 436.80$

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