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A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside a liquid of density $\rho_1$ (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).
Question:
The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)
Options:
A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside a liquid of density $\rho_1$ (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).
Question:
The buoyancy force acting on the gas bubble is (Assume $R$ is the universal gas constant)
Solution:
2388 Upvotes
Verified Answer
The correct answer is:
$\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}$
$\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}$
Buoyancy force
Here, and
$$
\begin{aligned}
F & =(\text { volume of bubble })\left(\rho_l\right) g \\
& =\left(\frac{n R T_2}{p_2}\right) p_l g \\
T_2 & =T_0\left[\frac{p_0+p_l g(H-y)}{p_0+\rho_l g H}\right] \\
p_2 & =p_0+p_l(H-y)
\end{aligned}
$$
Substituting the values we get,
$$
F=\frac{\rho_n n g T_0}{\left(p_0+\rho_l g H\right)^{2 / 5}\left[p_0+\rho_l g(H-y)\right]^{3 / 5}}
$$
$\therefore$ correct option is (b).
Here, and
$$
\begin{aligned}
F & =(\text { volume of bubble })\left(\rho_l\right) g \\
& =\left(\frac{n R T_2}{p_2}\right) p_l g \\
T_2 & =T_0\left[\frac{p_0+p_l g(H-y)}{p_0+\rho_l g H}\right] \\
p_2 & =p_0+p_l(H-y)
\end{aligned}
$$
Substituting the values we get,
$$
F=\frac{\rho_n n g T_0}{\left(p_0+\rho_l g H\right)^{2 / 5}\left[p_0+\rho_l g(H-y)\right]^{3 / 5}}
$$
$\therefore$ correct option is (b).
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