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)+p_(1)gH)/(P_(0)+p_(1)gy))^(2/5)`
B. `T_(0) ((P_(0)+p_(1)g(H-y))/(P_(0)+p_(1)gH))^(2/5)`
C. `((P_(0)+p_(1)gH)/(P_(0)+p_(1)gy))^(3/5)`
D. `(P_(0)+p_(10g(H-y))/(P_(0)+p_(1)gyH))^(3/5)`

Answer

B

Explanation

It is given that bubble does not exchange heat with the surroundings hence we can assume adiabatic process as the bubbles moves up, As the bubble moves up pressure changes. Pressure inside bubble is taken to be equal to pressure outside.
For adiabatic prorcess we can write the following `(T^gamma)/(P^(gamma-1))="constant"`
`rArr (T_(0)^(gamma))/((P_(0)+p_(1)gH)^(gamma-1))=(T^(gamma))/(|P_(0)+p_(1)g(H-y)|^(gamma-1))`
`rArr T=T_(0) [(P_(0)+p_(1)g(H-y)]^((gamma-1)/(gamma))/((P_(0)+p_(1)gH)^((gamma-1)/(gamma)))]`
Substituting value of `gamma` we can write the following:
`rArr T=T_(0) ([P_(0)+p_(1)g(H-y)]^(2/5))/((P_(0)+p_(1)gH)^(2/5))`
`=T_(0) [(P_(0)+p_(q)g(H-y))/((P_(0)+p_(1)gH))]^(2/5)`