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).

The buoyancy force acting on the gas bubble is (Assume R is the universal gas constant):

Options

A. `rho_(l)nRgT_(0) (P_(0) + rho_(l)gH)^(2//5)/(P_(0) + rho_(l)gy)^(7//5)`
B. `(rho_(l)nRgT_(0))/((P_(0)+rho_(l)gH)^(2//5)[P_(0)+rho_(l)g(H-y)]^(3//5))`
C. `rho_(l)nRgT_(0) (P_(0) + rho_(l)gH)^(3//5)/(P_(0) + rho_(l)gy)^(6//5)`
D. `(rho_(l)nRgT_(0))/((P_(0)+rho_(l)gH)^(3//5)[P_(0)+rho_(l)g(H-y)]^(2//5))`

Answer

B

Explanation

`F = V_(f)rho_(l).g = (nRT_(f))/(P_(f)).rho_(l)g`
nR. `T_(0) [(P_(0) + rho_(l)g(H - y))/(P_(0) + rho_(l)gH)]^((2)/(5)) xx (rho_(l).g)/(P_(0) + rho_(l)g(H - y)) = (rho_(l).n.RgT_(0))/((P_(0) + rho_(l)gH)^(2//5).[P_(0) + rho_(l)g(H - y)]^(3//5))`