When the gas bubble is at a height from the bottom, its temperature is [2008]

Question

A small spherical mono-atomic ideal gas bubble $(γ = 5 / 3)$ is trapped inside a liquid of density $ρ$ (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). [2008]

Q. When the gas bubble is at a height $y$ from the bottom, its temperature is

Smart Achievers · Accepted Answer

(2)

$As the bubble does not exchange any heat with the liquid.$

$So, the process is adiabatic.$

$Applying relation,$ $T^{γ} P^{1 - γ} = constant$

$T_{2} = T_{1} {[\frac{P_{1}}{P_{2}}]}^{\frac{1 - γ}{γ}}$

Here $T_{1} = T_{0}, P_{1} = P_{0} + H ρ_{ℓ} g, T_{2} = T, P_{2} = P$

$P = P_{0} + (H - y) ρ_{ℓ} g, γ = \frac{5}{3}$

$∴$ $T = T_{0} {[\frac{P_{0} + H ρ_{ℓ} g}{P_{0} + (H - y) ρ_{ℓ} g}]}^{\frac{1 - \frac{5}{3}}{\frac{5}{3}}}$

$= T_{0} {[\frac{P_{0} + H ρ_{ℓ} g}{P_{0} + (H - y) ρ_{ℓ} g}]}^{- \frac{2}{3} \times \frac{3}{5}}$

$T = T_{0} {[\frac{P_{0} + (H - y) ρ_{ℓ} g}{P_{0} + H ρ_{ℓ} g}]}^{\frac{2}{5}}$

Solution

1 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g H}{P_{0} + ρ_{ℓ} g y})}^{2 / 5}$

2 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g (H - y)}{P_{0} + ρ_{ℓ} g H})}^{2 / 5}$

3 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g H}{P_{0} + ρ_{ℓ} g y})}^{3 / 5}$

4 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g (H - y)}{P_{0} + ρ_{ℓ} g H})}^{3 / 5}$

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Solution

1 T0(P0+ρℓgHP0+ρℓgy)2/5

2 T0(P0+ρℓg(H-y)P0+ρℓgH)2/5

3 T0(P0+ρℓgHP0+ρℓgy)3/5

4 T0(P0+ρℓg(H-y)P0+ρℓgH)3/5

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1 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g H}{P_{0} + ρ_{ℓ} g y})}^{2 / 5}$

2 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g (H - y)}{P_{0} + ρ_{ℓ} g H})}^{2 / 5}$

3 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g H}{P_{0} + ρ_{ℓ} g y})}^{3 / 5}$

4 $T_{0} {(\frac{P_{0} + ρ_{ℓ} g (H - y)}{P_{0} + ρ_{ℓ} g H})}^{3 / 5}$