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A small spherical monoatomic ideal gas b...

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

A

`rho_(1)nRgT_(0)((P_(0)+rho_(l)gH)^(2/5))/((P_(0)+rho_(1)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_(1)nRgT_(0)((P_(0)+rho_(1)gH)^(3/5))/((P_(0)+rho_(l)gy)^(8/5))`

D

`(rho_(1)nRgT_(0))/((P_(0)+rho_(l)gH)^(3/5)[P_(0)+rho_(l)g(H-y)]^(2/5))`

Text Solution

Verified by Experts

The correct Answer is:
B
Buoyancy force `F=` (volume of bubble)` (rho_(1))g`
`=((nRT_(2))/(p_(2)))rho_(1)g`
Here `T_(2)=[(p_(0)+p_(e)g(H-y))/(p_(0)+rho_(1)gh)\^(2/5)]`
and `p_(2)=p_(0)+rho_(1)g(H-y)`
Substituting the values we get
`F=(rho_(l)nR_(g)T_(0))/((p_(0)+p_(l)gH)^(2/5)[p_(0)+rho_(l)g(H-y)]^(3/5)`
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