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

`T^(gamma)P^(gamma-1)` = constant `implies T_(2) = T_(1)((P_(2))/(P_(1))^((gamma-1)/(gamma))`
=`T_(0)((P_(0) + rho_(l)g(H-y))/(P_(0) + rho_(l)gH))^(1-(3)/(5)) = T_(0) ((P_(0) + rho_(l)g(H-y))/(P_(0) + rho_(l)gH))^(2//5)`