JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · Fluid Mechanics

A small spherical monoatomic ideal gas bubble

(γ = \frac{5}{3})

is trapped inside a liquid of density

ρ_{ℓ}

(see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains

n

moles of gas. The temperature of the gas when the bubble is at the bottom is

T_{0}

, the height of the liquid is

H

and the atmospheric pressure is

P_{0}

(Neglect surface tension).

When the gas bubble is at a height

y

from the bottom, its temperature is

(A) $T_{0} (\frac{P _{0} + ρ _{ℓ} g H}{P _{0} + ρ _{ℓ} g y})^{2/5}$
(B) $T_{0} (\frac{P _{0} + ρ _{ℓ} g ( H - y )}{P _{0} + ρ _{ℓ} g H})^{2/5}$ ✓ Correct
(C) $T_{0} (\frac{P _{0} + ρ _{ℓ} g H}{P _{0} + ρ _{ℓ} g y})^{3/5}$
(D) $T_{0} (\frac{P _{0} + ρ _{ℓ} g ( H - y )}{P _{0} + ρ _{ℓ} g H})^{3/5}$

Answer: (B)

Solution

For an adiabatic process,

P^{1 - γ} T^{γ} = constant

, which gives

P T^{- γ / (1 - γ)} = constant

. With

γ = 5/3

, this becomes

P T^{- 5/2} = constant

, or

T \propto P^{2/5}

. At the bottom,

P_{1} = P_{0} + ρ_{ℓ} g H

and

T_{1} = T_{0}

. At height

y

P_{2} = P_{0} + ρ_{ℓ} g (H - y)

. Thus

T_{2} = T_{0} (\frac{P _{2}}{P _{1}})^{2/5} = T_{0} (\frac{P _{0} + ρ _{ℓ} g ( H - y )}{P _{0} + ρ _{ℓ} g H})^{2/5}

JEE Advanced · 2008 · 13 Apr 2008 · Paper 1 · Single correct option