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_(l)gH)/(P_(0)+rho_(l)gy))^(2/5)`
B. `T_(0)((P_(0)+rho_(l)g(H-y))/(P_(0)+rho_(l)gy))^(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)gy))^(3/5)`

Answer

B

Explanation

As there is no exchange of heat, process is adiabatic.
Applying
`Tp^((1-gamma)/gamma)="constant"`
`T_(2)p_(2)^((1-gamma)/(gamma))=T_(1)p_(1)^((1-gamma)/(gamma))`
`T_(2)=T_(1)((p_(1))/(p_(2)))^((1-gamma)/(gamma))=T_(1)((p_(2))/(p_(1)))^((1-gamma)/(gamma))`
substituting the values, we have
`T_(2)=T_(0)[(p_(0)+rho_(1)g(H-y))/(p_(0)+rho_(1)gH)]^((5/3-1)/(5//3))`
`T_(0)=[(p_(0)+rh0_(l)g(H-y))/(p_(0)+rho_(l)gH)](-2//5)`