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: When the gas bubble is at height $y$ from the bottom, its temperature is
  1. $T_0\left(\frac{p_0+\rho \lg H}{p_0+\rho \lg y}\right)^{\frac{2}{5}}$
  2. $T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{2}{5}}$
  3. $T_0\left(\frac{p_0+\rho l g H}{p_0+\rho \lg y}\right)^{\frac{3}{5}}$
  4. $T_0\left(\frac{p_0+\rho l g(H-y)}{p_0+\rho l g H}\right)^{\frac{3}{5}}$

Solution

As there is no exchange of heat. Therefore, process is adiabatic. Applying, $ \begin{array}{rlrl} & T p^{\frac{1-\gamma}{\gamma}}=\text { constant } \\ \therefore \quad & T_2 p_2^{\frac{1-\gamma}{\gamma}}=T_1 p_1^{\frac{1-\gamma}{\gamma}} \\ \text { or } & T_2 & =T_1\left(\frac{p_1}{p_2}\right)^{\frac{1-\gamma}{\gamma}}=T_1\left(\frac{p_2}{p_1}\right)^{\frac{\gamma-1}{\gamma}} \end{array} $ Substituting the values we have, $ \begin{aligned} T_2 & =T_0\left[\frac{p_0+\rho \lg (H-y)^T}{p_0+\rho \lg H}\right]^{\frac{5 / 3-1}{5 / 3}} \\ & =T_0\left[\frac{p_0+\rho \lg (H-y)}{p_0+\rho \lg H}\right]^{2 / 5} \end{aligned} $ $\therefore$ correct option is (b)

Asked in: JEE Advanced 2008 (Paper 1)

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