A small spherical monoatomic ideal gas bubble `(gamma=5//3)` is trapped inside a liquid of density `rho` (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 `P_0` (Neglect surface tension).

When the gas bubble is at a height y from the bottom, its temperature is-

Options

A. `T_0((P_0+rho_lgH)/(P_0+rho_lgy))^(2//5)`
B. `T_0((P_0+rho_lg(H-y))/(P_0+rho_lgH))^(2//5)`
C. `T_0((P_0+rho_lgH)/(P_0+rho_lgy))^(3//5)`
D. `T_0((P_0+rho_lg(H-y))/(P_0+rho_lgH))^(3//5)`

Answer

B

Explanation

(b) It is given that the bubble does not exchange any heat
with the liquid. This means that while the bubble moves
up and expand, the process is adiabatic.
For adiabatic expansion the pressure -temperature
relationship is
`T_2=T_1[(P_1)/(P_2)]^((1-gamma)/gamma)`
Here `T_1=T_0, P-1=P_0+Hrho_lg,`
`P_2=P_0+(H-y)rho_lg, gamma=5/3`
`:. T_2=T_0[(P_0+Hrho_lg)/(P_0+(H-y)rho_lg)]^(1-5/3//5//3)`
`T_0[(P_0+Hrho_lg)/(P_0+(H-y)rho_lg)]^((-2)/3xx3/5)`
`T_2=T_0[(P_0+(H-y)rho_lg)/(P_0+Hrho_lg)]^(2/5)