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. `p_(l)nRgT_(0)(P_(0)+ rho_(l)gH)^(2//5)/((P_(0) + rho_(l)gy)^(7//5))`
B. `(p_(l)nRgT_(0))/((P_(0)+ rho_(l)gH)^(2//5)[P_(0) + rho_(l)g(H-y)]^(3//5))`
C. `p_(l)nRgT_(0)(P_(0)+ rho_(l)gH)^(3//5)/((P_(0) + rho_(l)gy)^(8//5))`
D. `(p_(l)nRgT_(0))/((P_(0)+ rho_(l)gH)^(3//5)[P_(0) + rho_(l)g(H-y)]^(2//5))`

Answer

B

Explanation

Buoyancy force =`Vrho_(l)g = ((nRT_(2))/(P_(2)))rho_(l)g`
=` (nRrho_(l)gT_(0))/(P_(0) + rho_(l)g(H-y))[(P_(0) + rho_(l)g(H-y))/(P_(0) + rho_(l)gH)]^(2)/(5)`
=`(rho_(l)nRgT_(0))/((P_(0) + rho_(l)gH)^(2)/(5)(P_(0) + rho_(l)g(H-y)^(3)/(5))`