A small spherical monoatomic ideal gas bubble `(gamma= (5)/(3))` is trapped inside a liquid of density `rho_(l)` (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).

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

Options

A. `T_0((P_0 + rho_l gH)/(P_0 + rho_l " gy"))^(2//5)`
B. `T_0((P_0 + rho_l g(H- Y))/(P_0 + rho_l gH))^(2//5)`
C. `T_0 ((P_0 + rho_l gH)/(P_0+rho_l gY))^(3//5)`
D. `T_0((P_0 + rho_l g (H-Y))/(P_0 + rho_l gH))^(3//5)`

Answer

B

Explanation

As the bubble goes up, gas inside it undergoes adiabatic expansion.
for adiabatic process `p prop T^(gamma//gamma-1)`
i.e., `P prop T^(5//2)`
So, `((P_0 + rho_1 gH))/((P_0 + rho_1 g(H-Y))) = (T_0/T)^(5//2)" So, "TT_0 = ([P_0+rho_1 g(H-y)])/([P_0 + rho_1 g H]^(2//5))`