The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant) [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. The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant)

Smart Achievers · Accepted Answer

(2)

$∵$ $P V = n R T$ $\Rightarrow V = \frac{n R T}{P} = \frac{n R T}{P_{0} + (H - y) ρ_{ℓ} g}$

Where P is pressure of the bubble at an arbitrary location distant y from the bottom.

Substituting the value of P and T from above we get

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

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

$∴$ $Buoyancy force$

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

Solution

1 $ρ_{ℓ} n R g T_{0} \frac{{(P_{0} + ρ_{ℓ} g H)}^{2 / 5}}{{(P_{0} + ρ_{ℓ} g y)}^{7 / 5}}$

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

3 $ρ_{ℓ} n R g T_{0} \frac{{(P_{0} + ρ_{ℓ} g H)}^{3 / 5}}{{(P_{0} + ρ_{ℓ} g y)}^{8 / 5}}$

4 $\frac{ρ_{ℓ} n R g T_{0}}{{(P_{0} + ρ_{ℓ} g H)}^{3 / 5} {[P_{0} + ρ_{ℓ} g (H - y)]}^{2 / 5}}$

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Solution

1 ρℓnRgT0(P0+ρℓgH)2/5(P0+ρℓgy)7/5

2 ρℓnRgT0(P0+ρℓgH)2/5[P0+ρℓg(H-y)]3/5

3 ρℓnRgT0(P0+ρℓgH)3/5(P0+ρℓgy)8/5

4 ρℓnRgT0(P0+ρℓgH)3/5[P0+ρℓg(H-y)]2/5

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1 $ρ_{ℓ} n R g T_{0} \frac{{(P_{0} + ρ_{ℓ} g H)}^{2 / 5}}{{(P_{0} + ρ_{ℓ} g y)}^{7 / 5}}$

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

3 $ρ_{ℓ} n R g T_{0} \frac{{(P_{0} + ρ_{ℓ} g H)}^{3 / 5}}{{(P_{0} + ρ_{ℓ} g y)}^{8 / 5}}$

4 $\frac{ρ_{ℓ} n R g T_{0}}{{(P_{0} + ρ_{ℓ} g H)}^{3 / 5} {[P_{0} + ρ_{ℓ} g (H - y)]}^{2 / 5}}$