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IIT-JEE 2008 · Paper 1, Physics question 39

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Paragraph for Question Nos. 38 to 40

A small spherical monoatomic ideal gas bubble (γ=53)\left(\gamma=\frac53\right) is trapped inside a liquid of density ρℓ\rho_\ell (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains nn moles of gas. The temperature of the gas when the bubble is at the bottom is T0T_0, the height of the liquid is HH and the atmospheric pressure is P0P_0 (Neglect surface tension).

A container of liquid of height H open to the atmosphere (pressure P₀ at the top); a small gas bubble is at height y above the bottom.

When the gas bubble is at a height yy from the bottom, its temperature is

Choose one option
Answer and worked solution

Official answer: B

Pressure at height y. P=P0+ρℓg(H−y)P=P_0+\rho_\ell g(H-y); at the bottom Pb=P0+ρℓgHP_b=P_0+\rho_\ell gH.

Adiabatic change (no heat exchange): P1−γTγP^{1-\gamma}T^\gamma is constant, so T∝P(γ−1)/γ=P2/5T\propto P^{(\gamma-1)/\gamma}=P^{2/5}:
T=T0(P0+ρℓg(H−y)P0+ρℓgH)2/5.T=T_0\left(\frac{P_0+\rho_\ell g(H-y)}{P_0+\rho_\ell gH}\right)^{2/5}.

(B)

Topics:Thermodynamic processesPressure and Pascal's law
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