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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_(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).

When the gas bubble is at a height y from the bottom , its temperature is :-

A

`T_(0)((P_(0) + rho_(l)gH)/(P_(0) + rho_(l)gy))^(2//5)`

B

`T_(0)((P_(0) + rho_(l)g(H-y))/(P_(0) + rho_(l)gH))^(2//5)`

C

`T_(0)((P_(0) + rho_(l)gH)/(P_(0) + rho_(l)gy))^(3//5)`

D

`T_(0)((P_(0) + rho_(l)g(H-y))/(P_(0) + rho_(l)gH))^(3//5)`

Text Solution

Verified by Experts

The correct Answer is:
B
`T^(gamma)P^(gamma-1)` = constant `implies T_(2) = T_(1)((P_(2))/(P_(1))^((gamma-1)/(gamma))`
=`T_(0)((P_(0) + rho_(l)g(H-y))/(P_(0) + rho_(l)gH))^(1-(3)/(5)) = T_(0) ((P_(0) + rho_(l)g(H-y))/(P_(0) + rho_(l)gH))^(2//5)`
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