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. `p_(1)n Rg T_(0) ((P_(0)+p_(1)gH))/((P_(0)+p_(1)gy)^(7/2))`
B. `(p_(1)nRgT_(0))/((P_(0)+p_(1)gH)^(2/5) (P_(0)+p_(1)g(H-y)^(3/5))`
C. `p_(1)nRg T_(0) ((P_(0)+p_(1)gH)^(3-5))/((P_(0)+p_(1)gy)^(8/5))`
D. `(p_(1)nRgT_(0))/((P_(0)+p_(1)p_(1)gH)^(2/5)[P_(0)+p_(1)g(H-y)^(3/5))`

Answer

B

Explanation

For ideal gas we can write
PV=nRT
Substituting appropriate values for the point at a height y from the bottom we can write the following :
`V=(nRT)/P= ((nRT_(0)[P_(0)+p_(1)g(H-y)]^(2/6))/([P_(0)+p_(1)gH]^(2/5)))/([P_(0)+p_(1)g(H-y)])`
`V=(nRT_(0))/([P_(0)+p_(1)gH]^(2/5)[P_(0)+p_(1)g(H-y)]^(3/5))`