JEE Advanced · Physics · 11. Properties of Fluids
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)
- A \(\rho_l n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}}{\left(\rho_0+\rho_l g y\right)^{\frac{7}{5}}}\)
- B \(\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}\)
- C \(\rho n R g T_0 \frac{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}}{\left(\rho_0+\rho_l g y\right)^{\frac{8}{5}}}\)
- D \(\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{3}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{2}{5}}}\)
Answer & Solution
Correct Answer
(B) \(\frac{\rho_l n R g T_0}{\left(\rho_0+\rho_l g H\right)^{\frac{2}{5}}\left[\rho_0+\rho_l g(H-y)\right]^{\frac{3}{5}}}\)
Step-by-step Solution
Detailed explanation
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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