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).

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

Options

A. `rho_l nRgT_0((P_0+rho_lgH)^(2//5))/([P_0+rho_lgy]^(7//5)`
B. `(rho_l nRgT_0)/((P_0+rho_lgH)^(2//5)[P_0rho_lg(H-y)]^(3//5)`
C. `rho_l nRgT_0((P_0+rho_lgH)^(3//5))/([P_0+rho_lgy]^(8//5)`
D. `(rho_l nRgT_0)/((P_0+rho_lgH)^(3//5)[P_0rho_lg(H-y)]^(2//5)`

Answer

B

Explanation

(b) Buoyency force=weight of fluid displaced
=(mass of flud displaced)g
`=Vrho_lg…(ii)`
Where V=Volume of fluid displaced
=Volume of the bubble.
Now, `PV=nRT`
`rArrV=(nRT)/P=(nRT)/(P_0+(h-y)rho_lg`
Where P is pressure of the bubble at an arbitrary
location distant y from the bottom.
Substituting the value of temperature from equation (i)
we get
`V=(nR)/([P_0+(H-y)rho_lg])xx(T_0[P_0+(H-y)rho_lg]^(2/5))/[P_0+Hrho_lg]^(2//5)`
`=(nRT_0)/([P_0+(H-y)rho_lg]^(3/5)[P_0+Hrho_lg]^(2/5)).....(iii)`
From (ii) and (iii)
`Bouyancy force=(nRT_0rho_lg)/([P_0+(H-y)rho_lg]^(3/5)[P_0+Hrho_lg]^(2/5))`