Paragraph: A small spherical monoatomic ideal gas bubble $\left(\gamma=\frac{5}{3}\right)$ is trapped inside…

Paragraph: 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)
  1. $\rho_l n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}}{\left(\rho_0+\rho_l g y\right)^{\frac{7}{5}}}$
  2. $\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}$
  3. $\rho n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}}{\left(\rho_0+\rho_l g y\right)^{\frac{8}{5}}}$
  4. $\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{2}{5}}}$

Solution

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)

Asked in: JEE Advanced 2008 (Paper 1)

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